Finite Group Databasebeta

← #30571161

Subgroups — #30571161

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#21459686C211yesmaximal, characteristic
2C2×#16840487C211yesmaximal, characteristic, complemented
3#21458876C214yesmaximal, complemented
4#21458855C214yesmaximal, complemented
5#21458626C214yesmaximal, complemented
6#21458605C214yesmaximal, complemented
7#21456849C214yesmaximal, complemented
8#21456744C211yesmaximal, characteristic
9#21455991C214yesmaximal, complemented
10#21455854C211yesmaximal, characteristic, complemented
11#21455851C211yesmaximal, characteristic
12#21454902C211yesmaximal, characteristic, complemented
13C61×#3209C211yesmaximal, characteristic, complemented
14#16840497C2212yes—
15#16840491C2212yes—
16#16840487—22no—
17#16840487—22no—
18#16840487C2214yes—
19#16840487C2214yes—
20#16840486C2212yes—
21#16840485—22no—
22#16840473C2212yes—
23#16840450—22no—
24#16840361—22no—
25#16840338—21no—
26#16840312C2211yescharacteristic
27C2×#14519775C2212yes—
28#16840298C2214yes—
29C2×#14519768—21no—
30C2×#14519767—22no—
31#16840242—21no—
32C2×Q8×Dic61C2212yes—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×Dic122C2211yescharacteristic
35C2×#14519742—21no—
36#16840155C2218yes—
37#16840136C2212yes—
38#16840129C2218yes—
39#16840112C2212yes—
40#16840095C2212yes—
41#16840081C2212yes—
42#16840067C2218yes—
43#16840054C2212yes—
44#16840048C2212yes—
45#16839986C2218yes—
46#16839970C2218yes—
47#16839969C2212yes—
48#16839966C2218yes—
49#16839964C2214yes—
50#16839963C2212yes—
51#16839917C2212yes—
52#16839903C2212yes—
53#16839901C2214yes—
54#16839892C2214yes—
55#16839889C2214yes—
56#16839869C2218yes—
57#16839865C2218yes—
58C4×#11758202C2214yes—
59C4×#11758190C2214yes—
60#16839816C2214yes—
61#16839815C2211yescharacteristic
62C61×#533C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#494C2212yes—
65C61×#487C2212yes—
66C61×#462C2214yes—
67C61×#451C2214yes—
68C61×#431C2211yescharacteristic
69C2×#11758222—21no—
70C22×Dic122D411yescharacteristic
71C22×Dic122C2311yescharacteristic
72C2×#11758202C2314yes—
73C2×#11758190C2314yes—
74C2×#11758201C2318yes—
75#14519777—22no—
76#14519777C2311yescharacteristic
77#14519775—24no—
78#14519775C2318yes—
79#14519774C2311yescharacteristic
80#14519772—24no—
81#14519772—24no—
82#14519768—24no—
83#14519768—24no—
84#14519768—24no—
85#14519768C2314yes—
86#14519767—28no—
87#14519767—28no—
88#14519766C2314yes—
89#14519765—22no—
90#14519765—21no—
91#14519762—24no—
92#14519761—24no—
93#14519757—22no—
94#14519757—22no—
95#14519753—24no—
96#14519751—21no—
97#14519751C2312yes—
98Q8×Dic61—24no—
99Q8×Dic61C2314yes—
100Q8×Dic61C2318yes—
101C4×Dic122—21no—
102C4×Dic122—21no—
103C4×Dic122—22no—
104C4×Dic122D412yes—
105C4×Dic122D412yes—
106C4×Dic122C2312yes—
107C4×Dic122C2312yes—
108C4×Dic122C2312yes—
109C4×Dic122C2312yes—
110#14519742—24no—
111#14519742—24no—
112C4×D244—21no—
113C4×D244—22no—
114C4×D244C2312yes—
115#14519739C2314yes—
116#14519737—24no—
117#14519735—22no—
118#14519735—24no—
119C2×C4⋊Dic61—22no—
120C2×C4⋊Dic61D411yescharacteristic
121C2×C4⋊Dic61C2311yescharacteristic
122C2×#11758196—21no—
123C2×#11758196—22no—
124C2×#11758196—22no—
125C2×#11758196D412yes—
126C4⋊C4×D61—22no—
127C2×C4×Dic61—21no—
128C2×C4×Dic61—22no—
129C2×C4×Dic61C2312yes—
130#14519710C2314yes—
131#14519709C2314yes—
132C4⋊#92418C2312yes—
133#14519706C2318yes—
134#14519702C2314yes—
135#14519701C2318yes—
136#14519701C2314yes—
137#14519700C2314yes—
138#14519699C2314yes—
139C22⋊#92418C2312yes—
140#14519687C2318yes—
141C4×#92418C2312yes—
142C22×Q8×C61C2312yes—
143C2×Q16×C61C2318yes—
144C4⋊Q8×C61—21no—
145C4⋊Q8×C61—21no—
146C4⋊Q8×C61C2312yes—
147C4⋊Q8×C61C2312yes—
148C4⋊Q8×C61C2311yescharacteristic
149C4⋊Q8×C61C2311yescharacteristic
150#125×C61—21no—
151C22⋊Q8×C61—21no—
152C22⋊Q8×C61C2312yes—
153C4×Q8×C61—21no—
154C4×Q8×C61C2312yes—
155#117×C61—21no—
156C2×C4⋊C4×C61—21no—
157C2×C4⋊C4×C61—21no—
158C2×C42×C61C2311yescharacteristic
159C4⋊C8×C61C2312yes—
160#103×C61C2318yes—
161C22⋊C8×C61C2312yes—
162C4×C8×C61C2312yes—
163C2×Q8×C61—22no—
164C2×Q8×C61—21no—
165C2×Q8×C61—44no—
166C2×Q8×C61C2414yes—
167C2×Q8×C61C2412yes—
168C22×C4×C61—22no—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2411yescharacteristic
172C2×C22⋊D61—22no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61C2×D412yes—
176C22×Dic61C2×D412yes—
177#11758222—44no—
178C2×D244—21no—
179C2×D244—21no—
180C2×D244C2412yes—
181C2×C4×D61—22no—
182C2×C4×D61—24no—
183C2×Dic122—24no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—42no—
187C2×Dic122C2×D412yes—
188C2×Dic122C2×D412yes—
189C2×Dic122C2412yes—
190Q16×C61—48no—
191C2×C8×C61C2414yes—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61C2412yes—
202C22⋊C4×C61—22no—
203C42×C61—22no—
204C42×C61C2×D411yescharacteristic
205C42×C61C2×D411yescharacteristic
206C42×C61C2411yescharacteristic
207C42×C61C2411yescharacteristic
208#11758202—44no—
209#11758201—48no—
210#11758198—22no—
211#11758198—22no—
212#11758198—24no—
213#11758198—24no—
214#11758198—24no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—28no—
218C4⋊Dic61—24no—
219C4⋊Dic61—24no—
220C4⋊Dic61C2×D412yes—
221C4⋊Dic61C2×D412yes—
222C4⋊Dic61C2412yes—
223C4⋊Dic61C2412yes—
224#11758196—22no—
225#11758196—22no—
226#11758196—24no—
227#11758196—28no—
228#11758196—28no—
229#11758196C2×D414yes—
230#11758196C2×D414yes—
231C4×Dic61—22no—
232C4×Dic61—22no—
233C4×Dic61—28no—
234C4×Dic61—24no—
235C4×Dic61C2×D414yes—
236C4×Dic61C2414yes—
237C2×#92418C2414yes—
238#11758190—44no—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241Q8×C61—28no—
242Q8×C61—24no—
243Q8×C61—44no—
244Q8×C61—81no—
245C2×C4×C61—22no—
246C2×C4×C61—24no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—44no—
251C2×C4×C61#13714yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2511yescharacteristic
258C22⋊D61—44no—
259C2×Dic61—24no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61C22×D414yes—
267C2×Dic61C22×D414yes—
268D244—24no—
269D244—42no—
270D244—81no—
271C4×D61—44no—
272C4×D61—84no—
273Dic122—24no—
274Dic122—24no—
275Dic122—22no—
276Dic122—22no—
277Dic122—42no—
278Dic122—42no—
279C8×C61—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—21no—
291C4×C61—21no—
292C4×C61—42no—
293C4×C61—44no—
294C4×C61—42no—
295C4×C61C2×#13712yes—
296C4×C61C2×#13712yes—
297C4×C61#58512yes—
298Dic61—24no—
299Dic61—24no—
300Dic61—42no—
301Dic61—42no—
302Dic61—42no—
303#3507—611no—
304C2×#557—611no—
305#3400—614no—
306#3398—614no—
307#3374—614no—
308#3372—614no—
309#3218—614no—
310#3209D6111yescharacteristic, complemented
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—614no—
323#557—614no—
324#557—612no—
325#557—1222no—
326#557—1222no—
327#556—612no—
328#556—1222no—
329#554—612no—
330#552—1222no—
331#539—1222no—
332#536—1221no—
333#533C2×D6111yescharacteristic
334#532—611no—
335C2×C4⋊Q8—612no—
336C2×C4⋊Q8C2×D6111yescharacteristic
337#529—614no—
338C2×#125—1222no—
339C2×#125—1221no—
340#520—1221no—
341C2×C4×Q8—611no—
342C2×C4×Q8—612no—
343C2×C4×Q8—611no—
344C2×C4×Q8—1221no—
345#497—618no—
346#494C2×D6112yes—
347#492—612no—
348#491—618no—
349#487—612no—
350#487C2×D6112yes—
351#484—612no—
352#481—612no—
353#479—618no—
354#477—612no—
355#475—612no—
356#464—618no—
357#462—618no—
358#462—612no—
359#462C2×D6114yes—
360#461—614no—
361#461—618no—
362#460—612no—
363#451—612no—
364#451C2×D6114yes—
365#448—614no—
366#448—612no—
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#431C2×D6111yescharacteristic
376C61#6189311yescharacteristic, complemented, Sylow
377C2×#41—1221no—
378C22×Q8—611no—
379C22×Q8—611no—
380C22×Q8C22×D6112yes—
381C2×Q16—614no—
382C2×Q16C22×D6118yes—
383C2×SD16—618no—
384C2×SD16—614no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—618no—
388C4⋊Q8—611no—
389C4⋊Q8—1222no—
390C4⋊Q8—1224no—
391C4⋊Q8C22×D6112yes—
392C4⋊Q8C22×D6112yes—
393C4⋊Q8C22×D6111yescharacteristic
394C4⋊Q8C22×D6111yescharacteristic
395#127—611no—
396#126—1224no—
397#126—1224no—
398#125—21no—
399#125—614no—
400#125—1228no—
401#125—1228no—
402#125—1224no—
403#125—1224no—
404#125—1224no—
405#124—614no—
406#124—1222no—
407#124—1221no—
408#123—1224no—
409#123—1224no—
410C22⋊Q8—21no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1224no—
414C22⋊Q8C22×D6112yes—
415#121—612no—
416#121—1221no—
417C4×Q8—21no—
418C4×Q8—612no—
419C4×Q8—612no—
420C4×Q8—612no—
421C4×Q8—612no—
422C4×Q8—618no—
423C4×Q8—614no—
424C4×Q8—612no—
425C4×Q8—612no—
426C4×Q8—1222no—
427C4×Q8—1224no—
428C4×Q8—1224no—
429C4×Q8—1224no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×Q8C22×D6112yes—
433C4×D4—614no—
434C4×D4—612no—
435C4×D4—1222no—
436C4×D4—1224no—
437C4×D4—1221no—
438#117—21no—
439#117—1224no—
440#117—1222no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4—1221no—
451C2×C42—612no—
452C2×C42—1222no—
453C2×C42—1221no—
454C2×C42C22×D6111yescharacteristic
455#107—614no—
456#106—614no—
457C4⋊C8—618no—
458C4⋊C8—612no—
459C4⋊C8C22×D6112yes—
460#103—614no—
461#103—618no—
462#103—614no—
463#103—614no—
464#103C22×D6118yes—
465#102—614no—
466C22⋊C8—612no—
467C22⋊C8C22×D6112yes—
468#97—618no—
469C4×C8—612no—
470C4×C8C22×D6112yes—
471#41—2444no—
472C2×Q8—22no—
473C2×Q8—21no—
474C2×Q8—44no—
475C2×Q8—612no—
476C2×Q8—612no—
477C2×Q8—612no—
478C2×Q8—1224no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—2442no—
482C2×Q8C23×D6114yes—
483C2×Q8C23×D6112yes—
484C2×D4—612no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4—1221no—
488C22×C4—22no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—1224no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498Q16—48no—
499Q16—2444no—
500SD16—2444no—
501SD16—2448no—
502C2×C8—614no—
503C2×C8C23×D6114yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—612no—
514C4⋊C4—614no—
515C4⋊C4—614no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—1224no—
520C4⋊C4—1228no—
521C4⋊C4—1228no—
522C4⋊C4—1228no—
523C4⋊C4—1224no—
524C4⋊C4—1224no—
525C4⋊C4—1221no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1221no—
529C4⋊C4C23×D6112yes—
530C22⋊C4—22no—
531C22⋊C4—1224no—
532C22⋊C4—1224no—
533C22⋊C4—1224no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C42—22no—
537C42—614no—
538C42—614no—
539C42—1228no—
540C42—1224no—
541C42—1222no—
542C42—1222no—
543C42D4×D6111yescharacteristic
544C42D4×D6111yescharacteristic
545C42C23×D6111yescharacteristic
546C42C23×D6111yescharacteristic
547C23—1222no—
548C23C2×D4×D6111yescharacteristic
549Q8—28no—
550Q8—24no—
551Q8—44no—
552Q8—81no—
553Q8—1224no—
554Q8—1224no—
555Q8—1222no—
556Q8—1222no—
557Q8—2442no—
558Q8—2442no—
559D4—1224no—
560D4—4881no—
561D4—2444no—
562D4—2442no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—614no—
570C2×C4—614no—
571C2×C4—1224no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—4884no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2444no—
580C2×C4#1451980914yes—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4C2×D4×D6111yescharacteristic
586C2×C4C24×D6111yescharacteristic
587C8—24no—
588C8—1224no—
589C22—21no—
590C22—21no—
591C22—2442no—
592C22—2442no—
593C22—2442no—
594C22C2×#1451980912yes—
595C22C22×D4×D6111yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—1224no—
603C4—1224no—
604C4—2442no—
605C4—2442no—
606C4—2442no—
607C4C2×#1451980912yes—
608C4C2×#1451980912yes—
609C4#1684059712yes—
610C2—22no—
611C2—4882no—
612C2#2145947511yescharacteristic
613C2C22×#1451980911yescharacteristic
614C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3209C2
socleC22×C61C23×D4
prime radical#16840312C22
hypercenter#3209D61
hypocenterC61#61893
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
2. C2×#16840487C2semidirect
3. #21458876C2semidirect
4. #21458855C2semidirect
5. #21458626C2semidirect
6. #21458605C2semidirect
7. #21456849C2semidirect
9. #21455991C2semidirect
10. #21455854C2semidirect
12. #21454902C2semidirect
13. C61×#3209C2semidirect
310. #3209D61semidirect
376. C61#61893semidirect

Maximal subgroups

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

subgroupindex
#214596862
C2×#168404872
#214588762
#214588552
#214586262
#214586052
#214568492
#214567442
#214559912
#214558542
#214558512
#214549022
C61×#32092
#6189361