Finite Group Databasebeta

← #30571157

Subgroups — #30571157

Order 15616.

1698 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic
2#21458868C214yesmaximal, complemented
3#21458846C214yesmaximal, complemented
4#21458618C214yesmaximal, complemented
5#21458596C214yesmaximal, complemented
6#21456846C214yesmaximal, complemented
7#21456767C211yesmaximal, characteristic, complemented
8#21456744C211yesmaximal, characteristic
9#21455990C214yesmaximal, complemented
10#21455858C211yesmaximal, characteristic, complemented
11#21455850C211yesmaximal, characteristic
12#21454904C211yesmaximal, characteristic, complemented
13C2×C61×#557C211yesmaximal, characteristic, complemented
14#16840497C2212yes—
15#16840490C2212yes—
16#16840485C2212yes—
17#16840484—22no—
18#16840472C2212yes—
19#16840453—22no—
20#16840359—22no—
21#16840336—21no—
22#16840319C2211yescharacteristic
23#16840312C2211yescharacteristic
24#16840296C2214yes—
25#16840243—21no—
26#16840154C2218yes—
27#16840141C2212yes—
28#16840136C2212yes—
29#16840130C2218yes—
30#16840112C2212yes—
31#16840112C2212yes—
32#16840097C2212yes—
33#16840080C2212yes—
34#16840069C2218yes—
35#16840057C2212yes—
36#16840047C2212yes—
37#16839984C2218yes—
38#16839972C2212yes—
39#16839969C2214yes—
40#16839969C2218yes—
41#16839964C2214yes—
42#16839964C2218yes—
43#16839962C2212yes—
44#16839918C2214yes—
45#16839918C2212yes—
46#16839901C2214yes—
47#16839901C2212yes—
48#16839893C2214yes—
49#16839893C2214yes—
50#16839867C2218yes—
51#16839860C2218yes—
52C4×#11758202C2214yes—
53C4×#11758201C2214yes—
54#16839819C2214yes—
55#16839815C2211yescharacteristic
56#16839815C2211yescharacteristic
57C61×#557—22no—
58C61×#557—22no—
59C61×#557C2214yes—
60C61×#557C2214yes—
61C2×C4⋊Q8×C61C2211yescharacteristic
62C2×C4⋊Q8×C61C2212yes—
63C2×#125×C61—21no—
64C2×#125×C61—22no—
65C2×C4×Q8×C61—21no—
66C2×C4×Q8×C61C2212yes—
67C2×C4×Q8×C61C2211yescharacteristic
68C2×C4×Q8×C61C2211yescharacteristic
69C2×#11758222—21no—
70C2×#11758202C2314yes—
71C2×#11758202C2318yes—
72C2×#11758201C2314yes—
73C2×#11758201C2318yes—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519775—22no—
78#14519774C2311yescharacteristic
79#14519772—24no—
80#14519771—24no—
81#14519769—21no—
82#14519769C2314yes—
83#14519767—24no—
84#14519766—22no—
85#14519765—21no—
86#14519765C2314yes—
87#14519762—24no—
88#14519760—24no—
89#14519756—21no—
90#14519756C2312yes—
91#14519755—24no—
92#14519753—22no—
93#14519753—22no—
94#14519751—21no—
95#14519751C2312yes—
96C4×Dic122—21no—
97C4×Dic122C2314yes—
98C4×Dic122C2312yes—
99#14519742—22no—
100C4×D244—21no—
101C4×D244C2312yes—
102C4×D244C2314yes—
103#14519739—22no—
104C4×C22⋊D61—24no—
105#14519736—21no—
106#14519735—22no—
107#14519734—24no—
108C4⋊C4×D61—22no—
109#14519710C2314yes—
110#14519708C2314yes—
111C4⋊#92418C2312yes—
112C4⋊#92418C2312yes—
113C4⋊#92418C2318yes—
114#14519701C2314yes—
115#14519701C2318yes—
116#14519701C2318yes—
117#14519701C2314yes—
118#14519700C2314yes—
119#14519697C2314yes—
120C22⋊#92418C2312yes—
121C22⋊#92418C2312yes—
122#14519688C2318yes—
123C4×#92418C2312yes—
124C4×#92418C2312yes—
125C22×Q8×C61D411yescharacteristic
126C22×Q8×C61C2312yes—
127C22×Q8×C61C2311yescharacteristic
128C4⋊Q8×C61—24no—
129C4⋊Q8×C61—21no—
130C4⋊Q8×C61—21no—
131C4⋊Q8×C61C2312yes—
132C4⋊Q8×C61C2318yes—
133C4⋊Q8×C61C2312yes—
134#125×C61—24no—
135#125×C61—24no—
136#125×C61—28no—
137#125×C61—28no—
138C4×Q8×C61—24no—
139C4×Q8×C61—24no—
140C4×Q8×C61—24no—
141C4×Q8×C61—21no—
142C4×Q8×C61—21no—
143C4×Q8×C61D412yes—
144C4×Q8×C61D412yes—
145C4×Q8×C61C2312yes—
146C4×Q8×C61C2312yes—
147C4×Q8×C61C2318yes—
148C4×Q8×C61C2312yes—
149C4×Q8×C61C2312yes—
150C2×C4⋊C4×C61—22no—
151C2×C4⋊C4×C61—21no—
152C2×C4⋊C4×C61—22no—
153C2×C4⋊C4×C61—21no—
154C2×C4⋊C4×C61—22no—
155C2×C4⋊C4×C61—21no—
156C2×C4⋊C4×C61D411yescharacteristic
157C2×C4⋊C4×C61D412yes—
158C2×C4⋊C4×C61C2311yescharacteristic
159C2×C42×C61—22no—
160C2×C42×C61—21no—
161C2×C42×C61C2312yes—
162C2×C42×C61C2311yescharacteristic
163C2×Q8×C61—24no—
164C2×Q8×C61—22no—
165C2×Q8×C61—21no—
166C2×Q8×C61—42no—
167C2×Q8×C61—44no—
168C2×Q8×C61C2×D412yes—
169C2×Q8×C61C2×D412yes—
170C2×Q8×C61C2412yes—
171C2×Q8×C61C2414yes—
172C22×C4×C61—21no—
173C22×C4×C61—22no—
174C22×C4×C61—21no—
175C22×C4×C61C2×D412yes—
176C22×C4×C61C2×D412yes—
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D411yescharacteristic
179C22×C4×C61C2411yescharacteristic
180C2×C22⋊D61—22no—
181#11758222—44no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244C2412yes—
185C2×C4×D61—22no—
186C2×C4×D61—24no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122C2412yes—
190C4⋊C4×C61—28no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—28no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—28no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61C2×D412yes—
201C4⋊C4×C61C2×D412yes—
202C4⋊C4×C61C2×D414yes—
203C4⋊C4×C61C2×D414yes—
204C4⋊C4×C61C2412yes—
205C42×C61—28no—
206C42×C61—22no—
207C42×C61—22no—
208C42×C61C2×D414yes—
209C42×C61C2×D411yescharacteristic
210C42×C61C2×D411yescharacteristic
211C42×C61C2411yescharacteristic
212C42×C61C2411yescharacteristic
213C42×C61C2414yes—
214C22⋊Dic61—22no—
215#11758202—44no—
216#11758202—48no—
217#11758201—48no—
218#11758201—44no—
219#11758198—22no—
220#11758198—24no—
221#11758198—24no—
222#11758198—22no—
223#11758198—24no—
224C4⋊Dic61—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61—21no—
227C4⋊Dic61—21no—
228C4⋊Dic61C2412yes—
229C4⋊Dic61C2412yes—
230#11758196—24no—
231#11758196—22no—
232#11758196—24no—
233#11758196—24no—
234#11758196—22no—
235C4×Dic61—24no—
236C4×Dic61—22no—
237C2×#92418C2414yes—
238C2×#92418C2414yes—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241Q8×C61—24no—
242Q8×C61—24no—
243Q8×C61—22no—
244Q8×C61—28no—
245Q8×C61—22no—
246Q8×C61—42no—
247Q8×C61—44no—
248C2×C4×C61—24no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—24no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—44no—
258C2×C4×C61#13714yes—
259C2×C4×C61C22×D414yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D414yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2511yescharacteristic
267C22⋊D61—44no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271D244—24no—
272D244—42no—
273D244—81no—
274C4×D61—44no—
275C4×D61—84no—
276Dic122—24no—
277Dic122—42no—
278Dic122—81no—
279#92418—24no—
280#92418—24no—
281#61893—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61C2×#13712yes—
285C22×C61C23×D411yescharacteristic
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C4×C61—21no—
290C4×C61—24no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—24no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61—44no—
297C4×C61C2×#13712yes—
298C4×C61C2×#13712yes—
299C4×C61#58512yes—
300Dic61—42no—
301Dic61—42no—
302Dic61—42no—
303#3507—611no—
304C2×#557D6111yescharacteristic, complemented
305#3400—614no—
306#3398—614no—
307#3374—614no—
308#3372—614no—
309#3218—614no—
310#3209—611no—
311#3206—611no—
312#3117—614no—
313#3104—611no—
314#3103—611no—
315#2983—611no—
316C2×C61—22no—
317C2×C61#348711yescharacteristic
318C2×C61C22×#13711yescharacteristic
319C2×C61#358611yescharacteristic
320D61—82no—
321#559—612no—
322#557—22no—
323#557—22no—
324#557—612no—
325#557C2×D6114yes—
326#557C2×D6114yes—
327#556—612no—
328#556—1222no—
329#554—612no—
330#552—1222no—
331#539—1222no—
332#536—1221no—
333#533—611no—
334#532—611no—
335C2×C4⋊Q8C2×D6111yescharacteristic
336C2×C4⋊Q8C2×D6112yes—
337#529—614no—
338C2×#125—21no—
339C2×#125—22no—
340#520—1221no—
341C2×C4×Q8—21no—
342C2×C4×Q8C2×D6111yescharacteristic
343C2×C4×Q8C2×D6112yes—
344C2×C4×Q8C2×D6111yescharacteristic
345#497—618no—
346#494—612no—
347#492—612no—
348#491—618no—
349#487—612no—
350#487—612no—
351#484—612no—
352#481—612no—
353#479—618no—
354#477—612no—
355#475—612no—
356#464—618no—
357#462—618no—
358#462—614no—
359#462—612no—
360#461—614no—
361#461—618no—
362#460—612no—
363#451—612no—
364#451—614no—
365#448—612no—
366#448—614no—
367#446—614no—
368#446—614no—
369#441—618no—
370#440—618no—
371C4×Q16—614no—
372C4×SD16—614no—
373#432—614no—
374#431—611no—
375#431—611no—
376C61#6189311yescharacteristic, complemented, Sylow
377C2×#41—1221no—
378C22×Q8C22⋊D6111yescharacteristic
379C22×Q8C22×D6111yescharacteristic
380C22×Q8C22×D6112yes—
381C2×Q16—614no—
382C2×Q16—618no—
383C2×SD16—614no—
384C2×SD16—618no—
385C4⋊Q8—24no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—611no—
389C4⋊Q8—611no—
390C4⋊Q8—611no—
391C4⋊Q8—1222no—
392C4⋊Q8C22×D6112yes—
393C4⋊Q8C22×D6118yes—
394C4⋊Q8C22×D6112yes—
395#127—611no—
396#126—1224no—
397#126—1224no—
398#125—24no—
399#125—24no—
400#125—28no—
401#125—28no—
402#125—614no—
403#125—1224no—
404#125—1221no—
405#124—614no—
406#124—1222no—
407#124—1221no—
408#123—1224no—
409#123—1224no—
410C22⋊Q8—612no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1221no—
414C22⋊Q8—1224no—
415#121—612no—
416#121—1221no—
417C4×Q8—24no—
418C4×Q8—24no—
419C4×Q8—24no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—614no—
423C4×Q8—612no—
424C4×Q8—1222no—
425C4×Q8—1221no—
426C4×Q8C22⋊D6112yes—
427C4×Q8C22⋊D6112yes—
428C4×Q8C22×D6112yes—
429C4×Q8C22×D6112yes—
430C4×Q8C22×D6118yes—
431C4×Q8C22×D6112yes—
432C4×Q8C22×D6112yes—
433C4×D4—614no—
434C4×D4—612no—
435C4×D4—1224no—
436C4×D4—1222no—
437C4×D4—1221no—
438#117—1221no—
439#117—1222no—
440#117—1224no—
441C2×C4⋊C4—22no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—22no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4C22⋊D6111yescharacteristic
449C2×C4⋊C4C22⋊D6112yes—
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C42—22no—
452C2×C42—21no—
453C2×C42C22×D6112yes—
454C2×C42C22×D6111yescharacteristic
455#107—614no—
456#106—614no—
457C4⋊C8—612no—
458C4⋊C8—612no—
459C4⋊C8—618no—
460#103—614no—
461#103—614no—
462#103—618no—
463#103—618no—
464#103—614no—
465#102—614no—
466C22⋊C8—612no—
467C22⋊C8—612no—
468#97—618no—
469C4×C8—612no—
470C4×C8—612no—
471#41—2444no—
472C2×Q8—24no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—42no—
476C2×Q8—44no—
477C2×Q8—612no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8C2×C22⋊D6112yes—
481C2×Q8C2×C22⋊D6112yes—
482C2×Q8C23×D6112yes—
483C2×Q8C23×D6114yes—
484C2×D4—612no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4—1221no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—1224no—
492C22×C4—1222no—
493C22×C4C2×C22⋊D6112yes—
494C22×C4C2×C22⋊D6112yes—
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498Q16—2444no—
499Q16—2448no—
500SD16—2448no—
501SD16—2444no—
502C2×C8—614no—
503C2×C8—614no—
504C4⋊C4—28no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—28no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—28no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—1221no—
517C4⋊C4—1224no—
518C4⋊C4—1222no—
519C4⋊C4—1224no—
520C4⋊C4—1222no—
521C4⋊C4—1224no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4C2×C22⋊D6112yes—
526C4⋊C4C2×C22⋊D6112yes—
527C4⋊C4C2×C22⋊D6114yes—
528C4⋊C4C2×C22⋊D6114yes—
529C4⋊C4C23×D6112yes—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1224no—
533C22⋊C4—1224no—
534C22⋊C4—1222no—
535C22⋊C4—1224no—
536C42—28no—
537C42—22no—
538C42—22no—
539C42—1224no—
540C42—1222no—
541C42C2×C22⋊D6114yes—
542C42C2×C22⋊D6111yescharacteristic
543C42C2×C22⋊D6111yescharacteristic
544C42C23×D6111yescharacteristic
545C42C23×D6111yescharacteristic
546C42C23×D6114yes—
547C23—1222no—
548C23C22×C22⋊D6111yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—22no—
552Q8—28no—
553Q8—22no—
554Q8—42no—
555Q8—44no—
556Q8—1224no—
557Q8—4881no—
558Q8—2442no—
559D4—1224no—
560D4—4881no—
561D4—2444no—
562D4—2442no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—4884no—
577C2×C4—2444no—
578C2×C4#1451980814yes—
579C2×C4C22×C22⋊D6114yes—
580C2×C4C22×C22⋊D6114yes—
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C22×C22⋊D6111yescharacteristic
585C2×C4C22×C22⋊D6111yescharacteristic
586C2×C4C24×D6111yescharacteristic
587C8—1224no—
588C8—1224no—
589C22—21no—
590C22—21no—
591C22—2442no—
592C22—2442no—
593C22—2442no—
594C22C2×#1451980812yes—
595C22C23×C22⋊D6111yescharacteristic
596C4—21no—
597C4—24no—
598C4—24no—
599C4—21no—
600C4—21no—
601C4—42no—
602C4—44no—
603C4—42no—
604C4—2442no—
605C4—2442no—
606C4—2442no—
607C4C2×#1451980812yes—
608C4C2×#1451980812yes—
609C4#1684059812yes—
610C2—22no—
611C2—4882no—
612C2#2145947711yescharacteristic
613C2C22×#1451980811yescharacteristic
614C2#2146060611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#557C2
socleC22×C61C23×D4
prime radical#16840312C22
hypercenterC2×#557D61
hypocenterC61#61893
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #21458868C2semidirect
3. #21458846C2semidirect
4. #21458618C2semidirect
5. #21458596C2semidirect
6. #21456846C2semidirect
7. #21456767C2semidirect
9. #21455990C2semidirect
10. #21455858C2semidirect
12. #21454904C2semidirect
13. C2×C61×#557C2semidirect
304. C2×#557D61semidirect
376. C61#61893semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214588682
#214588462
#214586182
#214585962
#214568462
#214567672
#214567442
#214559902
#214558582
#214558502
#214549042
C2×C61×#5572
#6189361