Finite Group Databasebeta

← #29696185

Subgroups — #29696185

Order 15616.

1106 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840467C211yesmaximal, characteristic
2#21454551C212yesmaximal
3#21446806C212yesmaximal, complemented
4#21446286C212yesmaximal, complemented
5#21446285C212yesmaximal, complemented
6#21445898C211yesmaximal, characteristic, complemented
7#21445814C211yesmaximal, characteristic, complemented
8#21445614C211yesmaximal, characteristic, complemented
9C2×#16839531C211yesmaximal, characteristic, complemented
10#343×D61C211yesmaximal, characteristic, complemented
11C61×#1821C211yesmaximal, characteristic, complemented
12C22×#11758194C2212yes—
13C2×M16×D61C2212yes—
14#16840467—22no—
15#16840467—22no—
16#16840467—22no—
17#16840467—22no—
18C2×#14519776C2211yescharacteristic
19C2×#14519767C412yes—
20C2×#14519757C2211yescharacteristic
21C2×#14519756C412yes—
22C2×#14519756C2211yescharacteristic
23C2×#14519755C412yes—
24C2×#14519742C2211yescharacteristic
25C2×C4×C22⋊D61C412yes—
26C2×C4×C22⋊D61C2211yescharacteristic
27C2×C4⋊C4×D61C2211yescharacteristic
28#16839837—22no—
29#16839836—22no—
30#16839830—22no—
31#16839825—22no—
32#16839785C2212yes—
33C2×#14519695C2212yes—
34C2×#14519695C2212yes—
35#16839725C2212yes—
36M16×Dic61C2212yes—
37#16839531C2211yescharacteristic
38#16839531C2212yes—
39#16839531C2211yescharacteristic
40#16839530C2212yes—
41#16839530C2212yes—
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#407C2212yes—
44C61×#343C2211yescharacteristic
45C61×#343C2212yes—
46C61×#343C2211yescharacteristic
47C22×Dic122C2×C412yes—
48C22×Dic122C2311yescharacteristic
49C22×C22⋊D61C2×C411yescharacteristic
50C22×C4×D61C2×C412yes—
51C22×C4×D61C2311yescharacteristic
52C2×#11758194—22no—
53C2×#11758194—24no—
54C2×#11758194—42no—
55C2×#11758194C2312yes—
56C2×#11758194C2312yes—
57C2×#11758189—22no—
58M16×D61—24no—
59M16×D61—42no—
60C22×#92418—22no—
61C2×C8×D61—22no—
62#14519776—22no—
63#14519776—22no—
64#14519767—24no—
65#14519767—24no—
66#14519757—22no—
67#14519757—22no—
68#14519756—22no—
69#14519756—22no—
70#14519756—24no—
71#14519756—24no—
72#14519755—24no—
73#14519755—24no—
74#14519742—22no—
75#14519742—22no—
76C4×C22⋊D61—22no—
77C4×C22⋊D61—22no—
78C4×C22⋊D61—22no—
79C4×C22⋊D61—22no—
80C4×C22⋊D61D412yes—
81C4×C22⋊D61D412yes—
82C4×C22⋊D61D412yes—
83C4×C22⋊D61D412yes—
84C2×C4⋊Dic61C2×C412yes—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196C2×C412yes—
87C2×#11758196C2×C412yes—
88C2×#11758196C2×C412yes—
89C2×#11758196C2×C412yes—
90C2×#11758196C2×C411yescharacteristic
91C4⋊C4×D61—22no—
92C4⋊C4×D61—22no—
93C2×C22⋊Dic61C2×C412yes—
94C2×C22⋊Dic61C2×C411yescharacteristic
95C2×#11758198C2×C412yes—
96C2×#11758198C2×C411yescharacteristic
97C2×C4×Dic61C2×C412yes—
98C2×C4×Dic61C2311yescharacteristic
99#14519706—22no—
100#14519706—22no—
101#14519695—24no—
102#14519695—42no—
103#14519689—22no—
104#14519689—22no—
105#14519687—22no—
106C8×Dic61—22no—
107C23×C4×C61C2×C412yes—
108C23×C4×C61C2311yescharacteristic
109C2×M16×C61C2312yes—
110C2×M16×C61C2312yes—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61C2×C412yes—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C22⋊C8×C61—22no—
117C22⋊C8×C61—22no—
118C24×C61C22×C411yescharacteristic
119C23×D61C22×C411yescharacteristic
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—82no—
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—24no—
133C2×C22⋊D61C22⋊C411yescharacteristic
134C2×C22⋊D61C22⋊C411yescharacteristic
135C2×C22⋊D61C22⋊C411yescharacteristic
136C2×C22⋊D61C22⋊C411yescharacteristic
137C22×Dic61C22×C412yes—
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C22×Dic61C22×C412yes—
141C22×Dic61C22×C411yescharacteristic
142C2×C4×D61—22no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—82no—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—24no—
155C2×Dic122—82no—
156M16×C61—22no—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C22⋊Dic61—22no—
166C22⋊Dic61—22no—
167C22⋊Dic61C22⋊C412yes—
168C22⋊Dic61C22⋊C412yes—
169#11758198—22no—
170#11758198—22no—
171#11758198C22⋊C412yes—
172#11758198C22⋊C412yes—
173C4⋊Dic61—22no—
174C4⋊Dic61—24no—
175#11758196—24no—
176#11758196—24no—
177#11758196—22no—
178#11758196—22no—
179#11758196—24no—
180#11758196C22⋊C412yes—
181#11758196C22⋊C412yes—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61C2×D412yes—
185C4×Dic61C2×D412yes—
186#11758194—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—44no—
190#11758194#4114yes—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—44no—
194#11758189—44no—
195C8×D61—44no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—41no—
200C23×C61C2×C22⋊C411yescharacteristic
201C23×C61C2×C22⋊C411yescharacteristic
202C23×C61C23×C411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—22no—
206C22×D61—41no—
207C22×D61C2×C22⋊C411yescharacteristic
208C22×D61C2×C22⋊C411yescharacteristic
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—82no—
220C2×C4×C61—82no—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C22×D412yes—
223C2×C4×C61C2×#4112yes—
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C22⋊D61—22no—
227C22⋊D61—22no—
228C22⋊D61—22no—
229C22⋊D61#10114yescomplemented
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C411yescharacteristic
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—82no—
248C4×D61—82no—
249Dic122—42no—
250Dic122—42no—
251Dic122—84no—
252C8×C61—42no—
253C8×C61—42no—
254#92418—24no—
255#92418—42no—
256#22481—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61C2×#10112yes—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61C4×#4111yescharacteristic
269C22×C61#52011yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61C2×#10112yes—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61#54612yes—
282C4×C61#54712yes—
283Dic61—22no—
284Dic61—21no—
285Dic61—22no—
286Dic61—42no—
287Dic61—42no—
288Dic61C2×#10112yes—
289C2×#554—611no—
290D4×M16—612no—
291#1997—612no—
292#1901—612no—
293#1901—612no—
294#1821—611no—
295#1821D6111yescharacteristic, complemented
296#1801—611no—
297C4×#101—611no—
298C2×#343—611no—
299C2×#343—611no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—22no—
303C2×C61—41no—
304C2×C61#233911yescharacteristic
305C2×C61C22×#10111yescharacteristic
306C2×C61#288811yescharacteristic
307D61—22no—
308D61—41no—
309C22×M16—612no—
310C22×M16—612no—
311#554—1222no—
312#554—1222no—
313#554—1222no—
314#554—1222no—
315C2×C4⋊Q8—611no—
316C2×#125—612no—
317C2×C22⋊Q8—612no—
318C2×C22⋊Q8—611no—
319C2×C22⋊Q8—612no—
320C2×C22⋊Q8—611no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×C4×D4—612no—
324C22×C4⋊C4—611no—
325C22×C4⋊C4C2×D6111yescharacteristic
326#436—1222no—
327#435—1222no—
328#435—1222no—
329C8×D4—1222no—
330#423—612no—
331C2×#101—612no—
332C2×#101—612no—
333#407—612no—
334#407C2×D6112yes—
335C4×M16—612no—
336#343—612no—
337#343—612no—
338#343—611no—
339#343—612no—
340#343—611no—
341#343C2×D6111yescharacteristic
342#343C2×D6111yescharacteristic
343#343C2×D6112yes—
344C61#2248111yescharacteristic, complemented, Sylow
345C22×Q8—612no—
346C22×Q8—611no—
347C22×D4—611no—
348C23×C4—611no—
349C23×C4—612no—
350C23×C4C4×D6112yes—
351C23×C4C22×D6111yescharacteristic
352C2×M16—612no—
353C2×M16—612no—
354C2×M16—1222no—
355C2×M16—1222no—
356C2×M16—1224no—
357C2×M16—1224no—
358C2×M16—2442no—
359C2×M16—2442no—
360C2×M16C22×D6112yes—
361C2×M16C22×D6112yes—
362C22×C8—1222no—
363C22×C8—1222no—
364C4⋊Q8—1222no—
365C4⋊Q8—1222no—
366#125—1224no—
367#125—1224no—
368C22⋊Q8—1224no—
369C22⋊Q8—1224no—
370C22⋊Q8—1224no—
371C22⋊Q8—1224no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375C22⋊Q8—1222no—
376C4×Q8—1222no—
377C4×Q8—1222no—
378C4×D4—612no—
379C4×D4—612no—
380C4×D4—612no—
381C4×D4—612no—
382C4×D4—1222no—
383C4×D4—1222no—
384C4×D4—1222no—
385C4×D4—1222no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—612no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4C4×D6112yes—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—611no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—612no—
403C2×C22⋊C4—612no—
404C2×C42—611no—
405C2×C42—612no—
406C4⋊C8—1222no—
407C4⋊C8—1222no—
408#101—1224no—
409#101—2442no—
410C22⋊C8—22no—
411C22⋊C8—22no—
412C22⋊C8—1222no—
413C22⋊C8—1222no—
414#97—1222no—
415C4×C8—1222no—
416C24—611no—
417C24C2×C4×D6111yescharacteristic
418C2×Q8—1224no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—4882no—
424C2×D4—611no—
425C2×D4—611no—
426C2×D4—611no—
427C2×D4—611no—
428C2×D4—1224no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—82no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—4882no—
448C22×C4C2×C4×D6111yescharacteristic
449C22×C4C2×C4×D6112yes—
450C22×C4C2×C4×D6112yes—
451C22×C4D4×D6112yes—
452C22×C4D4×D6112yes—
453C22×C4C23×D6111yescharacteristic
454M16—22no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—614no—
459M16—1222no—
460M16—1222no—
461M16—1222no—
462M16—2444no—
463M16—2444no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—2444no—
469C2×C8—2444no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—612no—
474C4⋊C4—612no—
475C4⋊C4—1224no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4—1222no—
481C4⋊C4—1224no—
482C22⋊C4—612no—
483C22⋊C4—612no—
484C22⋊C4—612no—
485C22⋊C4—612no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—612no—
491C42—612no—
492C42—1222no—
493C42—1222no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—41no—
498C23—611no—
499C23—611no—
500C23—1222no—
501C23—1221no—
502C23—1221no—
503C23—2441no—
504C23C22⋊C4×D6111yescharacteristic
505C23C22⋊C4×D6111yescharacteristic
506C23C22×C4×D6111yescharacteristic
507Q8—4884no—
508Q8—2442no—
509Q8—2442no—
510D4—614no—
511D4—1222no—
512D4—1222no—
513D4—1222no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—1222no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—4882no—
542C2×C4—4882no—
543C2×C4—2441no—
544C2×C4—2442no—
545C2×C4C22⋊C4×D6112yes—
546C2×C4C2×D4×D6112yes—
547C2×C4#41×D6112yes—
548C2×C4#1451982311yescharacteristic
549C2×C4#1451982411yescharacteristic
550C8—42no—
551C8—42no—
552C8—1224no—
553C8—2442no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—612no—
564C22—1221no—
565C22—1221no—
566C22—1221no—
567C22—1221no—
568C22—1221no—
569C22—2442no—
570C22—2442no—
571C22#101×D6112yes—
572C22C2×C22⋊C4×D6111yescharacteristic
573C22#1684023011yescharacteristic
574C22#1684024211yescharacteristic
575C4—22no—
576C4—42no—
577C4—42no—
578C4—612no—
579C4—1222no—
580C4—1221no—
581C4—1222no—
582C4—2442no—
583C4—2442no—
584C4#1684040412yes—
585C4#1684041012yes—
586C2—21no—
587C2—21no—
588C2—22no—
589C2—41no—
590C2—1222no—
591C2—2441no—
592C2#2144864311yescharacteristic
593C2C2×#101×D6111yescharacteristic
594C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1821C2
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1821D61
hypocenterC61#22481
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #21446806C2semidirect
4. #21446286C2semidirect
5. #21446285C2semidirect
6. #21445898C2semidirect
7. #21445814C2semidirect
8. #21445614C2semidirect
9. C2×#16839531C2semidirect
10. #343×D61C2semidirect
11. C61×#1821C2semidirect
229. C22⋊D61#101semidirect
295. #1821D61semidirect
344. C61#22481semidirect

Maximal subgroups

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

subgroupindex
C2×#168404672
#214545512
#214468062
#214462862
#214462852
#214458982
#214458142
#214456142
C2×#168395312
#343×D612
C61×#18212
#2248161