Finite Group Databasebeta

← #29689805

Subgroups — #29689805

Order 15616.

1022 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×#16840372C211yesmaximal, characteristic
2C2×#16839820C211yesmaximal, characteristic
3#21448439C211yesmaximal, characteristic
4#21446786C211yesmaximal, characteristic, complemented
5#21446782C211yesmaximal, characteristic, complemented
6#21446637C211yesmaximal, characteristic, complemented
7#21446619C211yesmaximal, characteristic, complemented
8#21446131C211yesmaximal, characteristic
9#21446127C211yesmaximal, characteristic
10#21446124C211yesmaximal, characteristic
11#21445833C211yesmaximal, characteristic, complemented
12#21445832C211yesmaximal, characteristic, complemented
13#21445823C211yesmaximal, characteristic, complemented
14#21445822C211yesmaximal, characteristic, complemented
15C61×#1873C211yesmaximal, characteristic
16#16840372—24no—
17#16840372C418yescomplemented
18C2×#14519775C2211yescharacteristic
19C2×#14519770—21no—
20C2×#14519770—21no—
21C2×#14519768—21no—
22C2×#14519768C2211yescharacteristic
23C2×#14519767—21no—
24C2×#14519756C2211yescharacteristic
25C2×Q8×Dic61C2211yescharacteristic
26C2×#14519742C2211yescharacteristic
27C2×#14519736C2211yescharacteristic
28C22×C4⋊Dic61C2211yescharacteristic
29#16839820—24no—
30C2×#14519706C2211yescharacteristic
31C2×#14519706C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×C22⋊#92418C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40C2×C8×Dic61C2211yescharacteristic
41#16839660—21no—
42#16839660—21no—
43#16839634—21no—
44#16839634—21no—
45#16839631C2211yescharacteristic
46#16839631C2211yescharacteristic
47#16839627C2211yescharacteristic
48#16839627C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C22⋊Q8×C61C2211yescharacteristic
56C2×#117×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519775—22no—
67#14519775C2×C414yescomplemented
68#14519770—28no—
69#14519770—28no—
70#14519768—28no—
71#14519768—22no—
72#14519768C2×C414yescomplemented
73#14519767—28no—
74#14519756—22no—
75#14519756C2×C414yes—
76Q8×Dic61—22no—
77Q8×Dic61C2×C414yescomplemented
78#14519742—22no—
79#14519742C2×C414yescomplemented
80#14519736C2×C414yes—
81#14519736D414yes—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—41no—
86C2×C4⋊Dic61—41no—
87C2×C4⋊Dic61D411yescharacteristic
88C2×C4⋊Dic61D411yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×C22⋊Dic61—21no—
102C2×C22⋊Dic61D411yescharacteristic
103C2×C22⋊Dic61D411yescharacteristic
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108#14519706—22no—
109#14519706—22no—
110#14519702—41no—
111#14519702—41no—
112#14519701—41no—
113#14519701—41no—
114#14519700—41no—
115#14519700—41no—
116C22⋊#92418—22no—
117#14519687—22no—
118C8×Dic61—22no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×Q8×C61C2311yescharacteristic
124C23×C4×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22⋊Q8×C61—22no—
128C22⋊Q8×C61C2×C414yes—
129#117×C61—41no—
130#117×C61—41no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C22⋊C4×C61—21no—
136C2×C22⋊C4×C61—21no—
137C2×C42×C61—21no—
138#103×C61—41no—
139#103×C61—41no—
140C22⋊C8×C61—22no—
141C24×C61C2×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×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C22×C412yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—42no—
165C22×Dic61—81no—
166C22×Dic61—81no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
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×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—42no—
192C42×C61—42no—
193C22⋊Dic61—24no—
194C22⋊Dic61C22⋊C414yes—
195C22⋊Dic61C22⋊C414yes—
196C4⋊Dic61—21no—
197C4⋊Dic61—24no—
198C4⋊Dic61—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204C4⋊Dic61C22⋊C414yes—
205C4⋊Dic61C22⋊C414yes—
206C4⋊Dic61C22×C412yes—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—81no—
213#11758196—81no—
214#11758196—81no—
215#11758196—81no—
216C4×Dic61—24no—
217C4×Dic61—24no—
218C4×Dic61C22×C412yes—
219C4×Dic61C22×C412yes—
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D412yes—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61C2×C22⋊C412yes—
235C23×C61C22×D411yescharacteristic
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
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×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C23×C411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—84no—
286C2×Dic61C2×C22⋊C412yes—
287C2×Dic61C2×C22⋊C412yes—
288C2×Dic61C2×C22⋊C412yes—
289C2×Dic61C2×C22⋊C412yes—
290Dic122—22no—
291Dic122—22no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—41no—
295C8×C61—41no—
296#92418—41no—
297#92418—41no—
298#21873—611nomaximal, Sylow
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61C22×C22⋊C411yescharacteristic
307C22×C61#53511yescharacteristic
308C22×C61#53611yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57711yescharacteristic
311C22×C61#57811yescharacteristic
312C22×C61#57811yescharacteristic
313C4×C61—22no—
314C4×C61—42no—
315C4×C61—42no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61#52011yescharacteristic
319C4×C61#52011yescharacteristic
320C4×C61#53711yescharacteristic
321C4×C61#53711yescharacteristic
322Dic61—22no—
323Dic61—22no—
324Dic61—22no—
325Dic61—22no—
326Dic61—42no—
327Dic61—42no—
328Dic61—81no—
329Dic61—81no—
330C2×#541—611no—
331C2×#433—611no—
332#2317—611no—
333#1992—611no—
334#1991—611no—
335#1960—611no—
336#1956—611no—
337#1873—611no—
338#1873D6111yescharacteristic
339#1872—611no—
340#1872—611no—
341#1805—611no—
342#1805—611no—
343#1803—611no—
344#1803—611no—
345C2×C61—42no—
346C2×C61#231111yescharacteristic
347C2×C61#289811yescharacteristic
348C2×C61#289811yescharacteristic
349C2×C61#322211yescharacteristic
350C2×C61#322311yescharacteristic
351C2×C61#324211yescharacteristic
352C2×C61#324211yescharacteristic
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⋊Q8C2×D6111yescharacteristic
363C2×C4×Q8—611no—
364C2×C4×Q8—611no—
365C2×#117—611no—
366C2×#117C2×D6111yescharacteristic
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⋊C8C2×D6111yescharacteristic
381C2×#97—611no—
382C2×C4×C8—611no—
383#389—1221no—
384#389—1221no—
385#385—1221no—
386#385—1221no—
387#384—611no—
388#384—611no—
389#382—611no—
390#382—611no—
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#2187311yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C22×Q8C22×D6111yescharacteristic
402C23×C4—1221no—
403C23×C4C22×D6111yescharacteristic
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—1222no—
414#125—1228no—
415#125—1228no—
416C22⋊Q8—22no—
417C22⋊Q8—614no—
418C22⋊Q8—1222no—
419C22⋊Q8C4×D6114yes—
420C4×Q8—614no—
421C4×Q8—614no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424#117—41no—
425#117—41no—
426#117—614no—
427#117—614no—
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—1221no—
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—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—2441no—
447C2×C4⋊C4—2441no—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—611no—
454C2×C22⋊C4—611no—
455C2×C22⋊C4—1221no—
456C2×C42—21no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—1221no—
460C2×C42—1221no—
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463#103—41no—
464#103—41no—
465#103—2441no—
466#103—2441no—
467#103—2441no—
468#103—2441no—
469#103—2441no—
470#103—2441no—
471C22⋊C8—22no—
472C22⋊C8—1222no—
473#97—1222no—
474C4×C8—1222no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24D4×D6111yescharacteristic
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—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—4881no—
509C22×C4—4881no—
510C22×C4—2442no—
511C22×C4C2×C4×D6112yes—
512C22×C4D4×D6112yes—
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—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—614no—
534C4⋊C4—614no—
535C4⋊C4—612no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1224no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—4881no—
549C4⋊C4—4881no—
550C4⋊C4—4881no—
551C4⋊C4—4881no—
552C4⋊C4—2442no—
553C4⋊C4C2×C4×D6112yes—
554C22⋊C4—24no—
555C22⋊C4—42no—
556C22⋊C4—614no—
557C22⋊C4—614no—
558C22⋊C4—1224no—
559C42—42no—
560C42—612no—
561C42—612no—
562C42—612no—
563C42—612no—
564C42—1224no—
565C42—1224no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23C22⋊C4×D6112yes—
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—22no—
585C2×C4—22no—
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—41no—
599C2×C4—41no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—4884no—
615C2×C4—2441no—
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×C4—2441no—
625C2×C4C22⋊C4×D6112yes—
626C2×C4C22×C4×D6111yescharacteristic
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4#1451982211yescharacteristic
629C2×C4#1451982211yescharacteristic
630C2×C4#1451982411yescharacteristic
631C2×C4#1451982411yescharacteristic
632C8—41no—
633C8—41no—
634C8—2441no—
635C8—2441no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22C2×C22⋊C4×D6111yescharacteristic
644C22#1684032711yescharacteristic
645C22#1684033411yescharacteristic
646C22#1684056911yescharacteristic
647C22#1684057111yescharacteristic
648C22#1684057611yescharacteristic
649C22#1684057811yescharacteristic
650C4—22no—
651C4—42no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—1222no—
656C4—1222no—
657C4—1222no—
658C4—1222no—
659C4—4881no—
660C4—4881no—
661C4—2442no—
662C4—2442no—
663C4#1684024111yescharacteristic
664C4#1684024111yescharacteristic
665C4#1684033911yescharacteristic
666C4#1684033911yescharacteristic
667C2—42no—
668C2#2144840611yescharacteristic
669C2#2145434811yescharacteristic
670C2#2145435111yescharacteristic
671C2#2145688911yescharacteristic
672C2#2145690011yescharacteristic
673C2#2145708511yescharacteristic
674C2#2145709311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1873C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1873D61
hypocenterC61#21873
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. #21446786C2semidirect
5. #21446782C2semidirect
6. #21446637C2semidirect
7. #21446619C2semidirect
11. #21445833C2semidirect
12. #21445832C2semidirect
13. #21445823C2semidirect
14. #21445822C2semidirect
17. #16840372C4semidirect
67. #14519775C2×C4semidirect
72. #14519768C2×C4semidirect
77. Q8×Dic61C2×C4semidirect
79. #14519742C2×C4semidirect
399. C61#21873semidirect

Maximal subgroups

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

subgroupindex
C2×#168403722
C2×#168398202
#214484392
#214467862
#214467822
#214466372
#214466192
#214461312
#214461272
#214461242
#214458332
#214458322
#214458232
#214458222
C61×#18732
#2187361