Finite Group Databasebeta

← #29685166

Subgroups — #29685166

Order 15616.

1086 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840370C211yesmaximal, characteristic
2C2×#16839821C211yesmaximal, characteristic
3#21448173C211yesmaximal, characteristic
4#21446775C211yesmaximal, characteristic, complemented
5#21446773C211yesmaximal, characteristic, complemented
6#21446740C211yesmaximal, characteristic, complemented
7#21446740C211yesmaximal, characteristic, complemented
8#21446130C211yesmaximal, characteristic
9#21446128C211yesmaximal, characteristic
10#21446125C211yesmaximal, characteristic
11#21446123C211yesmaximal, characteristic
12#21445652C211yesmaximal, characteristic, complemented
13#21445649C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C4×#103×C61C211yesmaximal, characteristic, complemented
16#16840370—24no—
17#16840370C418yescomplemented
18C2×#14519775C2211yescharacteristic
19C2×#14519772—21no—
20C2×#14519771—21no—
21C2×#14519769—21no—
22C2×#14519768C2211yescharacteristic
23C2×#14519767—21no—
24C2×#14519757C2211yescharacteristic
25C2×#14519753C2211yescharacteristic
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519735C2211yescharacteristic
28C2×#14519734C2211yescharacteristic
29C2×C4⋊C4×D61C2211yescharacteristic
30#16839821—24no—
31C2×#14519706C2211yescharacteristic
32C2×#14519706C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39C2×#14519689C2211yescharacteristic
40C2×#14519689C2211yescharacteristic
41C2×#14519688C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46#16839613—21no—
47C4×#11758198—21no—
48#16839609—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519775—22no—
68#14519775C2×C414yescomplemented
69#14519772—28no—
70#14519771—28no—
71#14519769—28no—
72#14519768—22no—
73#14519768C2×C414yescomplemented
74#14519767—28no—
75#14519757—22no—
76#14519757C2×C414yes—
77#14519753—22no—
78#14519753C2×C414yes—
79C4×Dic122—22no—
80C4×Dic122C2×C414yescomplemented
81#14519735—22no—
82#14519735—22no—
83#14519734C2×C414yes—
84#14519734D414yes—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196D411yescharacteristic
97C2×#11758196D411yescharacteristic
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198D411yescharacteristic
104C2×#11758198D411yescharacteristic
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61C2311yescharacteristic
111#14519706—22no—
112#14519706—22no—
113#14519702—22no—
114#14519702—22no—
115#14519701—22no—
116#14519701—22no—
117#14519700—22no—
118#14519700—22no—
119#14519689—22no—
120#14519689—22no—
121#14519688—22no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125C22×Q8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4×Q8×C61—22no—
129C4×Q8×C61C2×C414yescomplemented
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137#103×C61—22no—
138#103×C61—22no—
139C4×C8×C61—22no—
140#95×C61—21no—
141C23×D61C2×D411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—24no—
167C2×C4×D61—81no—
168C2×C4×D61—81no—
169C2×C4×D61C22×C412yes—
170C2×C4×D61C2×D412yes—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122C22×C412yes—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C22×C412yes—
188C4⋊C4×C61#4112yes—
189C4⋊C4×C61#4112yes—
190C42×C61—24no—
191C42×C61—81no—
192C42×C61C22×C412yes—
193C42×C61C2×D412yes—
194#11758198—24no—
195#11758198—24no—
196#11758198—24no—
197#11758198C22⋊C414yes—
198#11758198C22⋊C414yes—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—24no—
203C4⋊Dic61—24no—
204C4⋊Dic61C22×C412yes—
205C4⋊Dic61#4112yes—
206C4⋊Dic61#4112yes—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196C22⋊C414yes—
214#11758196C22⋊C414yes—
215C4×Dic61—24no—
216C4×Dic61—24no—
217C4×Dic61—81no—
218C4×Dic61—81no—
219C4×Dic61—81no—
220C4×Dic61C22×C412yes—
221C4×Dic61C2×D412yes—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61C2×C22⋊C412yes—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61C2×C22⋊C412yes—
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61C2×C22⋊C412yes—
283C2×Dic61C2×C22⋊C412yes—
284C2×Dic61C2×C22⋊C412yes—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—84no—
290Dic122—22no—
291Dic122—22no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—41no—
295C8×C61—41no—
296#92418—41no—
297#92418—41no—
298#21398—611nomaximal, Sylow
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61#52511yescharacteristic
301C22×C61#52511yescharacteristic
302C22×C61#57711yescharacteristic
303C22×C61#57711yescharacteristic
304C22×C61#57811yescharacteristic
305C22×C61#57811yescharacteristic
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—42no—
316C4×C61—42no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61C4×#4111yescharacteristic
320C4×C61C4×#4111yescharacteristic
321C4×C61#53211yescharacteristic
322C4×C61#53211yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—22no—
326Dic61—42no—
327Dic61—42no—
328Dic61—42no—
329Dic61—42no—
330C2×#541—611no—
331C2×#433—611no—
332#2274—611no—
333#1989—611no—
334#1989—611no—
335#1981—611no—
336#1981—611no—
337#1873—611no—
338#1873—611no—
339#1872—611no—
340#1872—611no—
341#1763—611no—
342#1763—611no—
343C4×#103—611no—
344C4×#103D6111yescharacteristic, complemented
345C2×C61#229711yescharacteristic
346C2×C61#289811yescharacteristic
347C2×C61#289811yescharacteristic
348C2×C61#312011yescharacteristic
349C2×C61#312011yescharacteristic
350C2×C61#312311yescharacteristic
351C2×C61#312711yescharacteristic
352D61—42no—
353#541—618no—
354#541—1224no—
355C2×C4⋊Q8—611no—
356C2×#126—1221no—
357C2×#126—1221no—
358C2×#125—611no—
359C2×#125—1221no—
360C2×#125—1221no—
361C2×C22⋊Q8—611no—
362C2×C22⋊Q8—611no—
363C2×C4×Q8—611no—
364C2×C4×Q8C2×D6111yescharacteristic
365C2×#117—611no—
366C2×#117—611no—
367C22×C4⋊C4—611no—
368#433—1224no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103C2×D6111yescharacteristic
378C2×#103C2×D6111yescharacteristic
379C2×C22⋊C8—611no—
380C2×C22⋊C8—611no—
381C2×#97—611no—
382C2×C4×C8C2×D6111yescharacteristic
383C4×C4⋊C4—611no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4—611no—
386C4×C4⋊C4—1221no—
387C4×C4⋊C4—1221no—
388C4×C4⋊C4C2×D6111yescharacteristic
389C4×C22⋊C4—1221no—
390C4×C22⋊C4—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340C2×D6111yescharacteristic
398#340C2×D6111yescharacteristic
399C61#2139811yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C22×Q8C22×D6111yescharacteristic
402C23×C4—611no—
403C23×C4—1221no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4⋊Q8—614no—
409C4⋊Q8—1222no—
410#126—1228no—
411#126—1228no—
412#125—614no—
413#125—1228no—
414#125—1222no—
415#125—1228no—
416C22⋊Q8—614no—
417C22⋊Q8—614no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C4×Q8—22no—
421C4×Q8—614no—
422C4×Q8—1222no—
423C4×Q8C4×D6114yescomplemented
424#117—614no—
425#117—614no—
426#117—1222no—
427#117—1222no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—611no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—21no—
453C2×C42—21no—
454C2×C42—611no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42—1221no—
460C2×C42C22×D6111yescharacteristic
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463#103—22no—
464#103—22no—
465#103—1222no—
466#103—1222no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470#103—1222no—
471C22⋊C8—1222no—
472C22⋊C8—1222no—
473#97—1222no—
474C4×C8—22no—
475#95—21no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—611no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—612no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8C2×C4×D6112yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—1224no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—4881no—
511C22×C4—4881no—
512C22×C4D4×D6111yescharacteristic
513C22×C4D4×D6111yescharacteristic
514C22×C4C23×D6111yescharacteristic
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—24no—
531C4⋊C4—614no—
532C4⋊C4—614no—
533C4⋊C4—612no—
534C4⋊C4—612no—
535C4⋊C4—612no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1224no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1224no—
547C4⋊C4C2×C4×D6112yes—
548C4⋊C4#1175822212yes—
549C4⋊C4#1175822212yes—
550C22⋊C4—614no—
551C22⋊C4—614no—
552C22⋊C4—1224no—
553C22⋊C4—1224no—
554C22⋊C4—1224no—
555C42—24no—
556C42—81no—
557C42—612no—
558C42—612no—
559C42—1224no—
560C42—1224no—
561C42—4881no—
562C42—4881no—
563C42—4881no—
564C42C2×C4×D6112yes—
565C42D4×D6112yes—
566C23—612no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23C2×D4×D6111yescharacteristic
574Q8—22no—
575Q8—22no—
576Q8—41no—
577Q8—41no—
578Q8—1222no—
579Q8—1222no—
580Q8—2441no—
581Q8—2441no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—4884no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4C22⋊C4×D6112yes—
625C2×C4C22⋊C4×D6112yes—
626C2×C4C22×C4×D6111yescharacteristic
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4#41×D6111yescharacteristic
630C2×C4C2×#1175822211yescharacteristic
631C2×C4C2×#1175822211yescharacteristic
632C8—41no—
633C8—41no—
634C8—2441no—
635C8—2441no—
636C22—1222no—
637C22—1222no—
638C22—1222no—
639C22—2441no—
640C22—2441no—
641C22—2441no—
642C22—2441no—
643C22C2×C22⋊C4×D6111yescharacteristic
644C22#1684027111yescharacteristic
645C22#1684027111yescharacteristic
646C22#1684056811yescharacteristic
647C22#1684056911yescharacteristic
648C22#1684057511yescharacteristic
649C22#1684057611yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—1222no—
657C4—1222no—
658C4—1222no—
659C4—2442no—
660C4—2442no—
661C4—2442no—
662C4—2442no—
663C4#1684023011yescharacteristic
664C4C4×#1175822211yescharacteristic
665C4#1684030911yescharacteristic
666C4#1684030911yescharacteristic
667C2—2442no—
668C2#2144830511yescharacteristic
669C2#2145434711yescharacteristic
670C2#2145434811yescharacteristic
671C2#2145602911yescharacteristic
672C2#2145603911yescharacteristic
673C2#2145606811yescharacteristic
674C2#2145611611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenterC4×#103D61
hypocenterC61#21398
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21446775C2semidirect
5. #21446773C2semidirect
6. #21446740C2semidirect
7. #21446740C2semidirect
12. #21445652C2semidirect
13. #21445649C2semidirect
14. C4×#14519702C2semidirect
15. C4×#103×C61C2semidirect
17. #16840370C4semidirect
68. #14519775C2×C4semidirect
73. #14519768C2×C4semidirect
80. C4×Dic122C2×C4semidirect
129. C4×Q8×C61C2×C4semidirect
344. C4×#103D61semidirect
399. C61#21398semidirect
423. C4×Q8C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168403702
C2×#168398212
#214481732
#214467752
#214467732
#214467402
#214461302
#214461282
#214461252
#214461232
#214456522
#214456492
C4×#145197022
C4×#103×C612
#2139861