Finite Group Databasebeta

← #29685294

Subgroups — #29685294

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840372C211yesmaximal, characteristic
2C2×#16839720C211yesmaximal, characteristic
3#21448426C211yesmaximal, characteristic
4#21446786C211yesmaximal, characteristic, complemented
5#21446782C211yesmaximal, characteristic, complemented
6#21446637C211yesmaximal, characteristic, complemented
7#21446619C211yesmaximal, characteristic, complemented
8#21446252C211yesmaximal, characteristic
9#21446243C211yesmaximal, characteristic
10#21446240C211yesmaximal, characteristic
11#21445651C211yesmaximal, characteristic, complemented
12#21445648C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1894C211yesmaximal, characteristic
16C23×#92418C2211yescharacteristic
17#16840372—24no—
18#16840372C418yescomplemented
19C2×#14519775C2211yescharacteristic
20C2×#14519770—21no—
21C2×#14519770—21no—
22C2×#14519768—21no—
23C2×#14519768C2211yescharacteristic
24C2×#14519767—21no—
25C2×#14519756C2211yescharacteristic
26C2×Q8×Dic61C2211yescharacteristic
27C2×#14519742C2211yescharacteristic
28C2×#14519736C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35#16839720—24no—
36C2×#14519687C2211yescharacteristic
37C2×#14519687C2211yescharacteristic
38C2×C8×Dic61C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40#16839656—21no—
41#16839650—21no—
42#16839647—21no—
43#16839645—21no—
44#16839631C2211yescharacteristic
45#16839627C2211yescharacteristic
46#16839622C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C22×M16×C61C2211yescharacteristic
56C2×C22⋊Q8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×#92418—22no—
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519775—22no—
69#14519775C2×C414yes—
70#14519770—28no—
71#14519770—28no—
72#14519768—28no—
73#14519768—22no—
74#14519768C2×C414yes—
75#14519767—28no—
76#14519756—22no—
77#14519756C2×C414yes—
78Q8×Dic61—22no—
79Q8×Dic61C2×C414yes—
80#14519742—22no—
81#14519742C2×C414yes—
82#14519736C2×C414yes—
83#14519736D414yes—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61D411yescharacteristic
87C2×C4⋊Dic61D411yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61D411yescharacteristic
101C2×C22⋊Dic61D411yescharacteristic
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108#14519702—22no—
109#14519702—22no—
110#14519701—22no—
111#14519701—22no—
112#14519700—22no—
113#14519700—22no—
114#14519687—22no—
115#14519687—22no—
116C8×Dic61—22no—
117C8×Dic61—22no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×Q8×C61C2311yescharacteristic
124C23×C4×C61C2311yescharacteristic
125C2×M16×C61—22no—
126C2×M16×C61—22no—
127C2×M16×C61—22no—
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130C22⋊Q8×C61—22no—
131C22⋊Q8×C61C2×C414yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C22⋊C4×C61—21no—
137C2×C22⋊C4×C61—21no—
138#103×C61—22no—
139#103×C61—22no—
140#95×C61—21no—
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×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174M16×C61—28no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61#4112yes—
180C2×C8×C61#4112yes—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C22×C412yes—
188C22⋊C4×C61—24no—
189C22⋊C4×C61—81no—
190C22⋊Dic61—24no—
191C22⋊Dic61—81no—
192C22⋊Dic61C22⋊C414yes—
193C22⋊Dic61C22⋊C414yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—24no—
196C4⋊Dic61—24no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C22⋊C414yes—
202C4⋊Dic61C22⋊C414yes—
203C4⋊Dic61C22×C412yes—
204#11758196—24no—
205#11758196—24no—
206#11758196—24no—
207#11758196—24no—
208#11758196—24no—
209#11758196—81no—
210#11758196—81no—
211C4×Dic61—24no—
212C4×Dic61—24no—
213C4×Dic61—81no—
214C4×Dic61—81no—
215C4×Dic61C22×C412yes—
216C4×Dic61C22×C412yes—
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D412yes—
219C2×#92418—28no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418#4112yes—
225C2×#92418#4112yes—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61C2×C22⋊C412yes—
233C23×C61C22×D411yescharacteristic
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
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×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
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×Dic61—41no—
283C2×Dic61C2×C22⋊C412yes—
284C2×Dic61C2×C22⋊C412yes—
285C2×Dic61C2×C22⋊C412yes—
286C2×Dic61C2×C22⋊C412yes—
287Dic122—22no—
288Dic122—22no—
289Dic122—41no—
290Dic122—41no—
291C8×C61—22no—
292C8×C61—22no—
293#92418—22no—
294#92418—22no—
295#21413—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61C22×C22⋊C411yescharacteristic
304C22×C61#53211yescharacteristic
305C22×C61#53211yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57711yescharacteristic
308C22×C61#57811yescharacteristic
309C22×C61#57811yescharacteristic
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61C4×#4111yescharacteristic
316C4×C61C4×#4111yescharacteristic
317C4×C61#52511yescharacteristic
318C4×C61#52511yescharacteristic
319Dic61—22no—
320Dic61—22no—
321Dic61—22no—
322Dic61—22no—
323Dic61—42no—
324Dic61—42no—
325Dic61—81no—
326Dic61—81no—
327C2×#541—611no—
328C2×#405—611no—
329#2314—611no—
330#1992—611no—
331#1991—611no—
332#1960—611no—
333#1956—611no—
334#1894—611no—
335#1894D6111yescharacteristic
336#1892—611no—
337#1892—611no—
338#1763—611no—
339#1763—611no—
340C4×#103—611no—
341C4×#103—611no—
342C2×C61—42no—
343C2×C61#229711yescharacteristic
344C2×C61#289811yescharacteristic
345C2×C61#289811yescharacteristic
346C2×C61#317411yescharacteristic
347C2×C61#317711yescharacteristic
348C2×C61#317811yescharacteristic
349C2×C61#317811yescharacteristic
350C22×M16C2×D6111yescharacteristic
351C23×C8—611no—
352#541—618no—
353#541—1224no—
354C2×C4⋊Q8—611no—
355C2×#126—1221no—
356C2×#126—1221no—
357C2×#125—611no—
358C2×#125—1221no—
359C2×#125—1221no—
360C2×C22⋊Q8—611no—
361C2×C22⋊Q8C2×D6111yescharacteristic
362C2×C4×Q8—611no—
363C2×C4×Q8—611no—
364C2×#117—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103C2×D6111yescharacteristic
372C2×#103C2×D6111yescharacteristic
373#405—1224no—
374C2×#97—611no—
375C2×#97—611no—
376C2×C4×C8—611no—
377C2×C4×C8—611no—
378#388—1221no—
379#388—1221no—
380#387—1221no—
381#387—1221no—
382#384—611no—
383#382—611no—
384#380—611no—
385#380C2×D6111yescharacteristic
386C4×C4⋊C4—611no—
387C4×C4⋊C4—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340C2×D6111yescharacteristic
395#340C2×D6111yescharacteristic
396C61#2141311yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8C22×D6111yescharacteristic
399C23×C4C22×D6111yescharacteristic
400C2×M16—22no—
401C2×M16—22no—
402C2×M16—22no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—1222no—
406C22×C8—1222no—
407C22×C8—1222no—
408C22×C8C22×D6111yescharacteristic
409C22×C8C22×D6111yescharacteristic
410C4⋊Q8—614no—
411C4⋊Q8—1222no—
412#126—1228no—
413#126—1228no—
414#125—614no—
415#125—1222no—
416#125—1228no—
417#125—1228no—
418C22⋊Q8—22no—
419C22⋊Q8—614no—
420C22⋊Q8—1222no—
421C22⋊Q8C4×D6114yes—
422C4×Q8—614no—
423C4×Q8—614no—
424C4×Q8—1222no—
425C4×Q8—1222no—
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⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—611no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42—1221no—
458#103—22no—
459#103—22no—
460#103—1222no—
461#103—1222no—
462#103—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#97—1222no—
467#97—1222no—
468C4×C8—1222no—
469C4×C8—1222no—
470#95—21no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24D4×D6111yescharacteristic
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8C2×C4×D6112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4C2×C4×D6112yes—
506C22×C4D4×D6112yes—
507C22×C4D4×D6111yescharacteristic
508C22×C4C23×D6111yescharacteristic
509M16—28no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—612no—
515C2×C8—612no—
516C2×C8—1228no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8#1175822412yes—
522C2×C8#1175822412yes—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—614no—
530C4⋊C4—614no—
531C4⋊C4—612no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1221no—
539C4⋊C4—1224no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—4881no—
545C4⋊C4—4881no—
546C4⋊C4C2×C4×D6112yes—
547C22⋊C4—24no—
548C22⋊C4—81no—
549C22⋊C4—614no—
550C22⋊C4—614no—
551C22⋊C4—1224no—
552C22⋊C4—4881no—
553C42—612no—
554C42—612no—
555C42—612no—
556C42—612no—
557C42—1224no—
558C42—1224no—
559C42—4881no—
560C42—4881no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22⋊C4×D6112yes—
568C23C2×D4×D6111yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—1222no—
574Q8—1222no—
575Q8—2441no—
576Q8—2441no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—612no—
596C2×C4—612no—
597C2×C4—612no—
598C2×C4—612no—
599C2×C4—1221no—
600C2×C4—1221no—
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—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4C22⋊C4×D6112yes—
620C2×C4C22×C4×D6111yescharacteristic
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4#41×D6111yescharacteristic
624C2×C4C2×#1175822411yescharacteristic
625C2×C4C2×#1175822411yescharacteristic
626C8—22no—
627C8—22no—
628C8—1222no—
629C8—1222no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×C22⋊C4×D6111yescharacteristic
638C22#1684030711yescharacteristic
639C22#1684030711yescharacteristic
640C22#1684057111yescharacteristic
641C22#1684057211yescharacteristic
642C22#1684057811yescharacteristic
643C22#1684057811yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—1222no—
650C4—1222no—
651C4—1222no—
652C4—1222no—
653C4—4881no—
654C4—4881no—
655C4—2442no—
656C4—2442no—
657C4C4×#1175822411yescharacteristic
658C4C4×#1175822411yescharacteristic
659C4#1684026811yescharacteristic
660C4#1684027011yescharacteristic
661C2—42no—
662C2#2144830411yescharacteristic
663C2#2145435011yescharacteristic
664C2#2145435111yescharacteristic
665C2#2145644211yescharacteristic
666C2#2145647311yescharacteristic
667C2#2145648011yescharacteristic
668C2#2145648811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1894C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1894D61
hypocenterC61#21413
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
4. #21446786C2semidirect
5. #21446782C2semidirect
6. #21446637C2semidirect
7. #21446619C2semidirect
11. #21445651C2semidirect
12. #21445648C2semidirect
13. #103×Dic61C2semidirect
14. #21445638C2semidirect
18. #16840372C4semidirect
396. C61#21413semidirect

Maximal subgroups

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

subgroupindex
C2×#168403722
C2×#168397202
#214484262
#214467862
#214467822
#214466372
#214466192
#214462522
#214462432
#214462402
#214456512
#214456482
#103×Dic612
#214456382
C61×#18942
#2141361