Finite Group Databasebeta

← #30606313

Subgroups — #30606313

Order 15616.

822 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#413C211yesmaximal, characteristic
2#21462280C211yesmaximal, characteristic
3#21460915C214yesmaximal, complemented
4#413×D61C211yesmaximal, characteristic
5C2×D61⋊M16—21no—
6C2×#14519856C411yescharacteristic
7C22×C4×#6923C411yescharacteristic
8#16840749—21no—
9#16840749C412yescomplemented
10#16840748—21no—
11#16840748C412yescomplemented
12D61⋊#101C2212yes—
13D61⋊#100—21no—
14(C2×#6923)⋊Q8C2212yes—
15#16840714C412yes—
16#16840710—21no—
17#16840710—21no—
18C4×C22⋊#6923—21no—
19C4×C22⋊#6923—21no—
20C2×#41×D61C2211yescharacteristic
21C2×M16×D61—21no—
22#16839746C411yescharacteristic, complemented
23#16839745C412yescomplemented
24#16839744—21no—
25#16839743—21no—
26#101×D61C2212yes—
27#100×D61—21no—
28C61×#413C411yescharacteristic, complemented
29C23×#6923—21no—
30C2×#11758213—41no—
31D61⋊M16—22no—
32D61⋊M16—22no—
33D61⋊M16—41no—
34D61⋊M16—41no—
35C2×#11758217—22no—
36C2×D61⋊C8—41no—
37#14519856—41no—
38#14519856—41no—
39C2×C4⋊#6923C2×C412yes—
40C2×C22⋊#6923—21no—
41C2×C22⋊#6923—22no—
42C2×C22⋊#6923—22no—
43C2×C4×#6923—21no—
44C2×C4×#6923—21no—
45C2×C4×#6923—41no—
46C2×C4×#6923—41no—
47C2×C4×#6923C2×C412yes—
48#14519838—22no—
49#14519838C2×C414yescomplemented
50#14519837—22no—
51#14519837—22no—
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—22no—
54C2×#11758226C2×C411yescharacteristic
55C2×#11758222C2×C411yescharacteristic
56C2×#11758222D411yescharacteristic
57C2×#11758222Q811yescharacteristic
58C2×#11758224C2×C411yescharacteristic
59C2×#11758224D411yescharacteristic
60C2×#11758224Q811yescharacteristic
61#41×D61—41no—
62#41×D61—41no—
63#41×D61—41no—
64#41×D61—41no—
65C2×Q8×D61C2311yescharacteristic
66C2×D4×D61C2×C411yescharacteristic
67C2×D4×D61D411yescharacteristic
68C2×D4×D61D411yescharacteristic
69C22×C4×D61C2×C411yescharacteristic
70C22×C4×D61D411yescharacteristic
71C22×C4×D61D411yescharacteristic
72C2×#11758194—21no—
73C2×#11758189—41no—
74M16×D61—22no—
75M16×D61—22no—
76M16×D61—41no—
77M16×D61—41no—
78C2×C8×D61—41no—
79#14519696C2×C412yescomplemented
80#14519695—22no—
81#14519694—21no—
82#14519693—22no—
83C2×#41×C61C2×C411yescharacteristic
84C2×M16×C61—21no—
85#101×C61C2×C412yescomplemented
86#100×C61—21no—
87C23×D61C22⋊C411yescharacteristic
88C23×D61C22⋊C411yescharacteristic
89C23×D61C22⋊C411yescharacteristic
90C22×#6923—22no—
91C22×#6923—22no—
92C22×#6923—44no—
93C22×#6923—42no—
94#41×C61—41no—
95#41×C61—41no—
96C2×Q8×C61C22×C411yescharacteristic
97C2×D4×C61C4211yescharacteristic
98C2×D4×C61C22⋊C411yescharacteristic
99C2×D4×C61C4⋊C411yescharacteristic
100C22×C4×C61C4211yescharacteristic
101C22×C4×C61C22⋊C411yescharacteristic
102C22×C4×C61C4⋊C411yescharacteristic
103C2×C22⋊D61C22⋊C411yescharacteristic
104C2×C22⋊D61C22⋊C411yescharacteristic
105C2×C22⋊D61C4⋊C411yescharacteristic
106C2×C22⋊D61C4⋊C411yescharacteristic
107C2×C22⋊D61C4⋊C411yescharacteristic
108C2×C22⋊D61C4⋊C411yescharacteristic
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61C4⋊C411yescharacteristic
111C22×Dic61C4⋊C411yescharacteristic
112#11758226—41no—
113#11758226—41no—
114Q8×D61—41no—
115Q8×D61—41no—
116#11758224—41no—
117#11758224—41no—
118#11758224—41no—
119#11758224—41no—
120#11758224—41no—
121#11758224—41no—
122D4×D61—41no—
123D4×D61—41no—
124D4×D61—41no—
125D4×D61—41no—
126D4×D61—41no—
127D4×D61—41no—
128#11758222—41no—
129#11758222—41no—
130#11758222—41no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134C2×D244C4211yescharacteristic
135C2×D244C22⋊C411yescharacteristic
136C2×D244C4⋊C411yescharacteristic
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61C4211yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C22⋊C411yescharacteristic
153C2×C4×D61C4⋊C411yescharacteristic
154C2×C4×D61C4⋊C411yescharacteristic
155C2×C4×D61C4⋊C411yescharacteristic
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122C22×C411yescharacteristic
160C2×Dic122C2×D411yescharacteristic
161C2×Dic122C2×Q811yescharacteristic
162C22⋊#6923—44no—
163C22⋊#6923—44no—
164C22⋊#6923—42no—
165#11758217—24no—
166#11758217—42no—
167C2×#92420—42no—
168C4⋊#6923—42no—
169C4×#6923—42no—
170C4×#6923—41no—
171C4×#6923—41no—
172C4×#6923—41no—
173C4×#6923—41no—
174#11758213—42no—
175#11758213—42no—
176D61⋊C8—42no—
177D61⋊C8—42no—
178M16×C61—22no—
179M16×C61—41no—
180C2×C8×C61—41no—
181#11758194—22no—
182#11758194—41no—
183C2×#92418—41no—
184#11758189—42no—
185#11758189—42no—
186C8×D61—42no—
187C8×D61—42no—
188C23×C61#9511yescharacteristic
189C23×C61#9511yescharacteristic
190C23×C61#9511yescharacteristic
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61#9511yescharacteristic
201C22×D61#9511yescharacteristic
202C22×D61#9511yescharacteristic
203C22×D61C2×C22⋊C411yescharacteristic
204C2×#6923—24no—
205C2×#6923—42no—
206C2×#6923—48no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923—42no—
211Q8×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61#9511yescharacteristic
222C2×C4×C61C2×C4211yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C2×C4⋊C411yescharacteristic
225C22⋊D61—41no—
226C22⋊D61—41no—
227C22⋊D61—41no—
228C22⋊D61—41no—
229C22⋊D61—41no—
230C22⋊D61—41no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61#9511yescharacteristic
238C2×Dic61C2×C22⋊C411yescharacteristic
239C2×Dic61C2×C4⋊C411yescharacteristic
240C2×Dic61C2×C4⋊C411yescharacteristic
241D244—41no—
242D244—41no—
243D244—41no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61—21no—
262C4×D61#9911yescharacteristic
263C4×D61#9911yescharacteristic
264C4×D61#9911yescharacteristic
265C4×D61#9911yescharacteristic
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269#92420—44no—
270C8×C61—42no—
271#92418—42no—
272#12585—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61C2×#9511yescharacteristic
280C2×D61—22no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61C2×#9911yescharacteristic
294C2×D61C2×#9911yescharacteristic
295#6923—44no—
296#6923—82no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61#34211yescharacteristic
301C4×C61#34211yescharacteristic
302Dic61—21no—
303Dic61—21no—
304Dic61—21no—
305Dic61#34211yescharacteristic
306Dic61#34211yescharacteristic
307C2×#413—611no—
308C2×#413—611no—
309#2297—611no—
310#1394—614no—
311C2×C61—21no—
312C2×C61—41no—
313C2×C61—41no—
314C2×C61—41no—
315C2×C61C2×#34211yescharacteristic
316D61—21no—
317D61—41no—
318D61—41no—
319D61—41no—
320D61#213212yes—
321C22×#41—611no—
322C22×M16—1221no—
323C22×M16—1221no—
324C2×#117—611no—
325C22×C42—611no—
326#413—612no—
327#413—612no—
328#413—611no—
329#413—612no—
330#413—1221no—
331#413—1221no—
332#413—1221no—
333#413—1221no—
334#413#692311yescharacteristic, complemented
335C2×#101—612no—
336C2×#101—612no—
337C2×#100—1221no—
338C2×#100—1221no—
339#391—612no—
340#390—612no—
341#386—1221no—
342#386—1221no—
343C4×C22⋊C4—1221no—
344C4×C22⋊C4—1221no—
345C61#1258511yescharacteristic, complemented, Sylow
346C2×#41—611no—
347C2×#41—611no—
348C2×#41—611no—
349C2×#41—611no—
350C2×#41—611no—
351C2×#41—611no—
352C2×#41—611no—
353C2×#41—2441no—
354C2×#41—2441no—
355C2×#41—2441no—
356C2×#41—2441no—
357C2×#41C2×#692311yescharacteristic
358C22×Q8—611no—
359C22×D4—611no—
360C22×D4—611no—
361C22×D4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—1221no—
366C2×M16—21no—
367C2×M16—1222no—
368C2×M16—1222no—
369C2×M16—1222no—
370C2×M16—1222no—
371C2×M16—1222no—
372C2×M16—1221no—
373C2×M16—2441no—
374C2×M16—2441no—
375C2×M16—2441no—
376C2×M16—2441no—
377C2×M16—2441no—
378C2×M16—2441no—
379C22×C8—2441no—
380C22×C8—2441no—
381#117—2441no—
382#117—2441no—
383C2×C4⋊C4—612no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C42—612no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42—2441no—
391C2×C42—2441no—
392#101—614no—
393#101—612no—
394#101—1222no—
395#101—1222no—
396#101C2×#692312yescomplemented
397#100—21no—
398#100—1222no—
399#100—1222no—
400#100—1222no—
401#100—1221no—
402#95—1222no—
403#95—1222no—
404C24—611no—
405C24—611no—
406C24—611no—
407#41—41no—
408#41—41no—
409#41—2441no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41—2441no—
415#41—2441no—
416#41—2441no—
417#41—2441no—
418#41—2441no—
419#41—2441no—
420#41—2441no—
421#41—2441no—
422#41—2441no—
423C2×Q8—611no—
424C2×Q8—611no—
425C2×Q8—611no—
426C2×Q8—2441no—
427C2×Q8—2441no—
428C2×Q8C22×#692311yescharacteristic
429C2×D4—611no—
430C2×D4—611no—
431C2×D4—611no—
432C2×D4—611no—
433C2×D4—611no—
434C2×D4—611no—
435C2×D4—611no—
436C2×D4—611no—
437C2×D4—611no—
438C2×D4—2441no—
439C2×D4—2441no—
440C2×D4—2441no—
441C2×D4—2441no—
442C2×D4—2441no—
443C2×D4—2441no—
444C2×D4C4×#692311yescharacteristic
445C2×D4C4⋊#692311yescharacteristic
446C2×D4C22⋊#692311yescharacteristic
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—2444no—
475C22×C4—2442no—
476C22×C4C4×#692311yescharacteristic
477C22×C4C4⋊#692311yescharacteristic
478C22×C4C22⋊#692311yescharacteristic
479M16—22no—
480M16—41no—
481M16—1224no—
482M16—1222no—
483M16—2442no—
484M16—2442no—
485M16—2441no—
486M16—2442no—
487M16—2442no—
488M16—2442no—
489C2×C8—41no—
490C2×C8—2442no—
491C2×C8—2442no—
492C2×C8—2441no—
493C2×C8—2442no—
494C2×C8—2442no—
495C2×C8—2442no—
496C4⋊C4—2442no—
497C22⋊C4—2444no—
498C22⋊C4—2444no—
499C22⋊C4—2442no—
500C42—2441no—
501C42—2441no—
502C42—2441no—
503C42—2441no—
504C42—2442no—
505C23—611no—
506C23—611no—
507C23—611no—
508C23—611no—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23(C2×#6923)⋊C411yescharacteristic
519C23(C2×#6923)⋊C411yescharacteristic
520C23(C2×#6923)⋊C411yescharacteristic
521Q8—41no—
522Q8—2441no—
523Q8—2441no—
524Q8—2441no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—2441no—
529D4—2441no—
530D4—2441no—
531D4—2441no—
532D4—2441no—
533D4—2441no—
534D4—2441no—
535D4—2441no—
536D4—2441no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1224no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2448no—
582C2×C4(C2×#6923)⋊C411yescharacteristic
583C2×C4C2×C4×#692311yescharacteristic
584C2×C4C2×C22⋊#692311yescharacteristic
585C2×C4C2×C4⋊#692311yescharacteristic
586C8—42no—
587C8—2442no—
588C8—2444no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—611no—
596C22—611no—
597C22—1222no—
598C22—1221no—
599C22—1221no—
600C22—1221no—
601C22—4881no—
602C22—4881no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22—2442no—
607C22—2442no—
608C22—2442no—
609C22—2442no—
610C22C2×(C2×#6923)⋊C411yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—611no—
615C4—611no—
616C4—1221no—
617C4—1221no—
618C4—1221no—
619C4—4882no—
620C4—2444no—
621C4#1684063911yescharacteristic
622C4#1684063911yescharacteristic
623C2—21no—
624C2—41no—
625C2—41no—
626C2—41no—
627C2—612no—
628C2—1221no—
629C2—2441no—
630C2—2441no—
631C2—2441no—
632C2C2×#1684063911yescharacteristic
subgroupG/H
centerC4#16840639
commutatorC2×C4×C61C2×C42
FrattiniC2×Q8C22×#6923
FittingC61×#413C4
socleC2×C61C2×#342
prime radicalC2×#41×D61C22
cosocleC2×Q8×D61C23
hypercenter#413#6923
hypocenterC61#12585
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #21460915C2semidirect
9. #16840749C4semidirect
11. #16840748C4semidirect
22. #16839746C4semidirect
23. #16839745C4semidirect
28. C61×#413C4semidirect
49. #14519838C2×C4semidirect
79. #14519696C2×C4semidirect
85. #101×C61C2×C4semidirect
334. #413#6923semidirect
345. C61#12585semidirect
396. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4132
#214622802
#214609152
#413×D612
#1258561