Finite Group Databasebeta

← #30610785

Subgroups — #30610785

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462624C211yesmaximal, characteristic
2#21462464C211yesmaximal, characteristic
3#21461169C211yesmaximal, characteristic, complemented
4#21461167C212yesmaximal, complemented
5#21461149C212yesmaximal, complemented
6#21461148C211yesmaximal, characteristic, complemented
7#21461129C211yesmaximal, characteristic, complemented
8#21461127C212yesmaximal, complemented
9C2×#16840612C212yesmaximal, complemented
10D61⋊(Q8⋊C8)C211yesmaximal, characteristic, complemented
11#21459398C211yesmaximal, characteristic
12C2×#14519864—21no—
13#16840811—21no—
14#16840789—22no—
15D4×#92420—21no—
16#16840781C2211yescharacteristic
17#16840779C2212yes—
18#16840769—21no—
19#16840768—22no—
20#16840766C2211yescharacteristic
21#16840765C2212yes—
22D61⋊(C4⋊C8)C2211yescharacteristic
23C2×C4⋊#92420C2212yes—
24C4×D61⋊C8C2211yescharacteristic
25C2×C4×#92420C2212yes—
26#16840612C2211yescharacteristic
27#16840612C2211yescharacteristic
28#16840612C2212yes—
29#16840612C2212yes—
30#16840612C2212yes—
31#16840612C2212yes—
32#16840612C2212yes—
33Q8⋊#92420C2211yescharacteristic
34Q8⋊#92420C2211yescharacteristic
35Q8⋊#92420C2212yes—
36#16840492C411yescharacteristic
37#16840490C411yescharacteristic
38#16840489—21no—
39#16840471—21no—
40#16840467C412yes—
41#16840466—22no—
42D4×Dic122C411yescharacteristic
43#16840425C412yes—
44#16840318—21no—
45#16840317C2211yescharacteristic
46#16840314—22no—
47#16840314C2212yes—
48C4×#11758224—21no—
49C4×#11758224C2211yescharacteristic
50C2×#14519742C2212yes—
51C4×Q8×D61C2211yescharacteristic
52C61×#554C411yescharacteristic
53#14519864—44no—
54C2×#11758213—21no—
55C2×#11758217—22no—
56C2×D61⋊C8—21no—
57C2×D61⋊C8—21no—
58C22×#92420—22no—
59C22×#92420—22no—
60#14519844—21no—
61#14519844—22no—
62#14519844—22no—
63C4⋊#92420—22no—
64C4⋊#92420—21no—
65C4⋊#92420—21no—
66C4⋊#92420C2311yescharacteristic
67C4⋊#92420C2312yes—
68C4⋊#92420C2311yescharacteristic
69#14519834—21no—
70C22⋊#92420—22no—
71C4×#92420—22no—
72C4×#92420—21no—
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C4×#92420C2311yescharacteristic
76C2×#11758224D411yescharacteristic
77C22×Dic122C2×C412yes—
78C2×Q8×D61C2×C411yescharacteristic
79#14519777C2×C411yescharacteristic
80#14519776—21no—
81#14519776—21no—
82#14519776—22no—
83#14519776C2×C412yes—
84#14519775—22no—
85#14519775C2×C412yes—
86#14519769—21no—
87#14519768—21no—
88#14519767—22no—
89#14519767—22no—
90#14519767—22no—
91#14519757C2×C412yes—
92#14519756—22no—
93#14519756C2×C412yes—
94#14519755—22no—
95#14519755—22no—
96#14519754—22no—
97#14519754—22no—
98#14519754—22no—
99#14519753—21no—
100Q8×Dic61—21no—
101Q8×Dic61C2311yescharacteristic
102C4×Dic122—21no—
103C4×Dic122—21no—
104C4×Dic122C2×C411yescharacteristic
105C4×Dic122C2×C411yescharacteristic
106#14519742—21no—
107#14519742—22no—
108#14519742—21no—
109#14519742—22no—
110#14519742—22no—
111#14519742C2×C412yes—
112#14519742C2×C412yes—
113#14519742D411yescharacteristic
114#14519742C2311yescharacteristic
115#14519742C2312yes—
116D4×Dic61—21no—
117D4×Dic61D411yescharacteristic
118C4×C22⋊D61—22no—
119C4×C22⋊D61C2×C412yes—
120#14519737—22no—
121#14519737D412yes—
122#14519735—21no—
123#14519735D411yescharacteristic
124#14519733—22no—
125#14519733D412yes—
126C2×#11758196—22no—
127C4⋊C4×D61—21no—
128C4⋊C4×D61C2×C411yescharacteristic
129C2×C4×Dic61—22no—
130C2×C4×Dic61C2312yes—
131C42×D61—21no—
132C42×D61C2311yescharacteristic
133C4⋊Q8×C61C2×C411yescharacteristic
134#125×C61—21no—
135C22⋊Q8×C61—22no—
136C22⋊Q8×C61C2×C412yes—
137C4×Q8×C61C2×C411yescharacteristic
138C4×D4×C61—21no—
139C4×D4×C61C2×C411yescharacteristic
140C2×C4⋊C4×C61C2×C412yes—
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61C22⋊C411yescharacteristic
144C22×C4×C61—22no—
145C22×C4×C61C22×C412yes—
146C2×C22⋊D61C22⋊C412yes—
147C22×Dic61—22no—
148C22×Dic61C2×D412yes—
149C22×Dic61C2×D412yes—
150Q8×D61—22no—
151Q8×D61—41no—
152#11758224SD1614yes—
153#11758224Q1614yes—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—24no—
160C2×Dic122—22no—
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—42no—
164C2×Dic122C22⋊C411yescharacteristic
165C2×Dic122C22×C412yes—
166C2×Dic122C22×C411yescharacteristic
167#11758217—44no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—22no—
171C2×#92420—21no—
172C2×#92420—22no—
173C2×#92420—21no—
174C2×#92420—44no—
175C2×#92420—44no—
176#11758213—42no—
177D61⋊C8—42no—
178D61⋊C8—42no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C22⋊C4×C61C22⋊C412yes—
188C42×C61—21no—
189C42×C61C22×C411yescharacteristic
190C22⋊Dic61—22no—
191C22⋊Dic61C22⋊C412yes—
192#11758198—22no—
193#11758198C22⋊C412yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61C22⋊C411yescharacteristic
199C4⋊Dic61C22×C411yescharacteristic
200#11758196—21no—
201#11758196—22no—
202#11758196—21no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—21no—
209#11758196—22no—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212C4×Dic61—21no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—22no—
218C4×Dic61C22×C412yes—
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D412yes—
223C4×Dic61C2411yescharacteristic
224C23×C61C2×C22⋊C412yes—
225C22×D61C2×C22⋊C411yescharacteristic
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—81no—
229D4×C61#10314yes—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—82no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C22⋊D61#10318yes—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—42no—
245C2×Dic61—44no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61C2×SD1614yes—
250C2×Dic61C2×Q1614yes—
251C2×Dic61C22×D411yescharacteristic
252C2×Dic61C22×D411yescharacteristic
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—41no—
256C4×D61—42no—
257C4×D61—81no—
258C4×D61C2×SD1612yes—
259C4×D61C2×Q1612yes—
260Dic122—21no—
261Dic122—22no—
262Dic122—22no—
263Dic122—42no—
264Dic122—44no—
265Dic122—82no—
266Dic122#10314yes—
267Dic122#13112yescomplemented
268#92420—22no—
269#92420—41no—
270#92420—42no—
271#92420—41no—
272#92420—41no—
273#92420—42no—
274#17509—611nomaximal, Sylow
275C22×C61—22no—
276C22×C61—22no—
277C22×C61C2×#10314yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C2×D61—21no—
280C2×D61—21no—
281C2×D61C2×#10312yes—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—41no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61C2×#10312yes—
288C4×C61C2×#13111yescharacteristic
289C4×C61#56811yescharacteristic
290Dic61—22no—
291Dic61—21no—
292Dic61—21no—
293Dic61—22no—
294Dic61—42no—
295Dic61—41no—
296Dic61C2×#10314yes—
297Dic61C22×SD1611yescharacteristic
298Dic61C22×Q1611yescharacteristic
299Q8×#41—611no—
300C8×#41—611no—
301#2906—611no—
302#1493—612no—
303#1493—611no—
304#1487—611no—
305#1487—612no—
306#1481—612no—
307#1481—611no—
308C2×Q8⋊C8—611no—
309C2×Q8⋊C8—612no—
310C2×C61—24no—
311C2×C61#288011yescharacteristic
312C2×C61C22×#10311yescharacteristic
313C2×C61#290111yescharacteristic
314D61—22no—
315C2×#131—1221no—
316Q82—611no—
317#557—611no—
318#557—1221no—
319#554—612no—
320#554—1221no—
321#554—1222no—
322#554#692311yescharacteristic
323D4×Q8—612no—
324D4×Q8—611no—
325#533—611no—
326#533—612no—
327#533—1222no—
328#533—1221no—
329C4×#41—611no—
330C4×#41—1221no—
331C2×C4×Q8—611no—
332C2×C4×Q8—612no—
333C8×Q8—1221no—
334C8×D4—1221no—
335C8×D4—1222no—
336#431—612no—
337#431—611no—
338#424—1222no—
339#424—1221no—
340#423—612no—
341#423—611no—
342C2×C4⋊C8—612no—
343C2×C4⋊C8—611no—
344C2×C4×C8—612no—
345C2×C4×C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8—612no—
349Q8⋊C8—611no—
350Q8⋊C8—612no—
351Q8⋊C8—611no—
352Q8⋊C8—612no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355Q8⋊C8—612no—
356C61#1750911yescharacteristic, complemented, Sylow
357C2×#41—611no—
358C22×Q8—611no—
359C22×Q8—612no—
360#131—2444no—
361C2×M16—1222no—
362C2×M16—1221no—
363C22×C8—1222no—
364C22×C8—1222no—
365C22×C8—1221no—
366C22×C8—1221no—
367C4⋊Q8—612no—
368C4⋊Q8—612no—
369C4⋊Q8—611no—
370C4⋊Q8—1222no—
371C4⋊Q8—1221no—
372C4⋊Q8—1222no—
373C4⋊Q8—1221no—
374C4⋊Q8C2×#692311yescharacteristic
375#125—21no—
376#125—1221no—
377#125—1222no—
378#125—1221no—
379#125—1222no—
380#125—1222no—
381C22⋊Q8—22no—
382C22⋊Q8—612no—
383C22⋊Q8—612no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1221no—
390C22⋊Q8—1222no—
391C22⋊Q8C2×#692312yes—
392C4×Q8—611no—
393C4×Q8—611no—
394C4×Q8—611no—
395C4×Q8—611no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—612no—
399C4×Q8—611no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×Q8C2×#692311yescharacteristic
409C4×D4—21no—
410C4×D4—611no—
411C4×D4—612no—
412C4×D4—612no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1221no—
416C4×D4C2×#692311yescharacteristic
417#117—611no—
418#117—612no—
419#117—1222no—
420#117—1221no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4C2×#692312yes—
425C2×C42—612no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42—1222no—
429C4⋊C8—611no—
430C4⋊C8—612no—
431C4⋊C8—611no—
432C4⋊C8—1222no—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435C4⋊C8—1221no—
436C4⋊C8—1222no—
437C4⋊C8—1222no—
438C22⋊C8—1222no—
439C22⋊C8—1221no—
440C4×C8—611no—
441C4×C8—612no—
442C4×C8—611no—
443C4×C8—1222no—
444C4×C8—1221no—
445#41—614no—
446#41—614no—
447C2×Q8—21no—
448C2×Q8—611no—
449C2×Q8—611no—
450C2×Q8—612no—
451C2×Q8—1222no—
452C2×Q8—1221no—
453C2×Q8—1224no—
454C2×Q8—1222no—
455C2×Q8—1222no—
456C2×Q8—2442no—
457C2×Q8—2441no—
458C2×Q8C22×#692311yescharacteristic
459C2×D4—612no—
460C2×D4C22⋊#692311yescharacteristic
461C22×C4—22no—
462C22×C4—612no—
463C22×C4—611no—
464C22×C4—612no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4C22×#692312yes—
471M16—2442no—
472M16—2444no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1222no—
476C2×C8—1221no—
477C2×C8—1222no—
478C2×C8—1221no—
479C2×C8—2442no—
480C2×C8—2444no—
481C2×C8—2442no—
482C2×C8—2444no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—612no—
488C4⋊C4—612no—
489C4⋊C4—611no—
490C4⋊C4—611no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1221no—
502C4⋊C4—1222no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4C22⋊#692311yescharacteristic
506C4⋊C4C22×#692311yescharacteristic
507C4⋊C4C22×#692312yes—
508C22⋊C4—22no—
509C22⋊C4—612no—
510C22⋊C4—612no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4C22⋊#692312yes—
514C42—21no—
515C42—612no—
516C42—611no—
517C42—612no—
518C42—611no—
519C42—611no—
520C42—612no—
521C42—1221no—
522C42—1221no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42—1221no—
527C42C22×#692311yescharacteristic
528C23—611no—
529C23C2×C22⋊#692312yes—
530Q8—21no—
531Q8—21no—
532Q8—81no—
533Q8—614no—
534Q8—612no—
535Q8—1221no—
536Q8—1222no—
537Q8—1222no—
538Q8—4882no—
539Q8—2444no—
540Q8—2442no—
541D4—618no—
542D4#1451984014yes—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—82no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—612no—
552C2×C4—611no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—611no—
556C2×C4—612no—
557C2×C4—611no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—4881no—
564C2×C4—2441no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2444no—
568C2×C4C2×C22⋊#692312yes—
569C2×C4C2×C22⋊#692311yescharacteristic
570C2×C4C2×C22⋊#692311yescharacteristic
571C2×C4C23×#692311yescharacteristic
572C8—1222no—
573C8—2441no—
574C8—2441no—
575C8—2442no—
576C8—2441no—
577C8—2442no—
578C22—22no—
579C22—22no—
580C22—612no—
581C22—1221no—
582C22—1221no—
583C22C2×#1451984014yes—
584C22C22×C22⋊#692311yescharacteristic
585C4—22no—
586C4—21no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—611no—
591C4—611no—
592C4—614no—
593C4—1221no—
594C4—1222no—
595C4—1221no—
596C4—1222no—
597C4—2442no—
598C4—2441no—
599C4C2×#1451984012yes—
600C4D61⋊#13111yescharacteristic
601C4#1684084111yescharacteristic
602C2—24no—
603C2—1222no—
604C2#2146240911yescharacteristic
605C2C22×#1451984011yescharacteristic
606C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#554C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758224D4
cosocleC4×Dic61C24
hypercenter#554#6923
hypocenterC61#17509
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
3. #21461169C2semidirect
4. #21461167C2semidirect
5. #21461149C2semidirect
6. #21461148C2semidirect
7. #21461129C2semidirect
8. #21461127C2semidirect
9. C2×#16840612C2semidirect
10. D61⋊(Q8⋊C8)C2semidirect
267. Dic122#131semidirect
356. C61#17509semidirect

Maximal subgroups

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

subgroupindex
#214626242
#214624642
#214611692
#214611672
#214611492
#214611482
#214611292
#214611272
C2×#168406122
D61⋊(Q8⋊C8)2
#214593982
#1750961