Finite Group Databasebeta

← #29669765

Subgroups — #29669765

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758199C211yesmaximal, characteristic
2#21458468C211yesmaximal, characteristic
3#21458104C211yesmaximal, characteristic, complemented
4#21445462C211yesmaximal, characteristic, complemented
5#21445391C211yesmaximal, characteristic, complemented
6#21445279C211yesmaximal, characteristic, complemented
7#21445242C211yesmaximal, characteristic, complemented
8#21445211C211yesmaximal, characteristic, complemented
9#21445087C211yesmaximal, characteristic, complemented
10#21445082C211yesmaximal, characteristic, complemented
11#21445033C211yesmaximal, characteristic, complemented
12#21444982C211yesmaximal, characteristic, complemented
13#21444648C211yesmaximal, characteristic
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3299C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17Q8×D244C2211yescharacteristic
18#16840165—21no—
19#16840165C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840119—21no—
22#16840114C2211yescharacteristic
23#16840113C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840105—21no—
26#16840066—21no—
27#16840049C2211yescharacteristic
28#16840047C2211yescharacteristic
29#16840042—21no—
30#16840042C2211yescharacteristic
31#16840039—21no—
32#16840039C2211yescharacteristic
33#16840019—21no—
34#16839977—21no—
35#16839972C2211yescharacteristic
36#16839967C2211yescharacteristic
37#16839916—21no—
38Q8×#92418C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C4×#11758200—21no—
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43#16839841C2211yescharacteristic
44#16839769—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#484—21no—
60C61×#477C2211yescharacteristic
61C61×#467—21no—
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C2×Q8×D61—21no—
68C2×#11758202—21no—
69C2×#11758200—21no—
70C2×#11758199D411yescharacteristic
71C2×#11758189—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519772—41no—
76#14519765—21no—
77#14519757—21no—
78#14519757—41no—
79#14519753—21no—
80C4×Dic122—21no—
81C4×D244—21no—
82C4×D244C2311yescharacteristic
83#14519735—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710D411yescharacteristic
89#14519708—21no—
90#14519708—21no—
91#14519708—41no—
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519701—41no—
101#14519700—21no—
102#14519700—21no—
103#14519700—21no—
104#14519700—21no—
105#14519698—21no—
106#14519698D411yescharacteristic
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×Q16×C61—21no—
117C2×Q16×C61D411yescharacteristic
118C22×C8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C22⋊Q8×C61—41no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—42no—
148C2×Q8×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155Q8×D61—42no—
156Q8×D61—81no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—41no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—41no—
163Q16×C61—22no—
164Q16×C61—81no—
165C2×C8×C61—21no—
166C2×C8×C61—81no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61C2411yescharacteristic
187#11758202—81no—
188#11758200—42no—
189#11758199—22no—
190#11758198—21no—
191#11758198—41no—
192#11758198—41no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—41no—
201C4×Dic61—41no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208#11758189—81no—
209C23×C61—21no—
210C22×D61—21no—
211Q8×C61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—82no—
217Q8×C61—81no—
218Q8×C61#13512yes—
219D4×C61—21no—
220D4×C61#13712yes—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61—161no—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238D244—21no—
239D244—21no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—82no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—82no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#19782—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61D4211yescharacteristic
262C2×D61—41no—
263C2×D61—41no—
264C2×D61—41no—
265C4×C61—22no—
266C4×C61—21no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61C2×#13511yescharacteristic
274C4×C61C2×#13711yescharacteristic
275C4×C61#57611yescharacteristic
276C4×C61#57811yescharacteristic
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280Q8×D8—611no—
281#3360—611no—
282#3328—611no—
283#3299D6111yescharacteristic, complemented
284#1718—611no—
285#1703—611no—
286#1684—611no—
287#1679—611no—
288#1674—611no—
289#1652—611no—
290#1651—611no—
291#1644—611no—
292#1637—611no—
293#1581—611no—
294#1579—611no—
295C2×C61—42no—
296C2×C61#327011yescharacteristic
297C2×C61#328011yescharacteristic
298C2×C61#328911yescharacteristic
299D61—81no—
300#556—611no—
301#554C2×D6111yescharacteristic
302D4×Q8—611no—
303D4×Q8C2×D6111yescharacteristic
304#500—611no—
305#500—1221no—
306#498—611no—
307#488—1221no—
308#487—611no—
309#487—611no—
310#487—611no—
311#487C2×D6111yescharacteristic
312#485—1221no—
313#484—21no—
314#479—1221no—
315#477C2×D6111yescharacteristic
316#476—611no—
317#475—611no—
318#474—611no—
319#474—611no—
320#474—1221no—
321#474—1221no—
322#470—1221no—
323#467—21no—
324#463—1221no—
325#462—611no—
326#462C2×D6111yescharacteristic
327#461—611no—
328#451—21no—
329#451—1221no—
330C8×Q8—611no—
331C8×Q8—611no—
332C4×Q16C2×D6111yescharacteristic
333C4×SD16—1221no—
334C4×D8—611no—
335C4×D8—1221no—
336#436—611no—
337C8×D4C2×D6111yescharacteristic
338#419—1221no—
339C2×#103—21no—
340#335—611no—
341#334—611no—
342#333—611no—
343#332—611no—
344#332—611no—
345#332—611no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348D4⋊C8—611no—
349D4⋊C8—611no—
350#322—611no—
351C61#1978211yescharacteristic, complemented, Sylow
352C22×Q8—21no—
353C22×Q8—1221no—
354C2×Q16—21no—
355C2×Q16—1221no—
356C2×Q16C22⋊D6111yescharacteristic
357C2×SD16—1221no—
358C2×D8—611no—
359C2×M16—1221no—
360C22×C8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—611no—
363C4⋊Q8—611no—
364C4⋊Q8—1221no—
365C4⋊Q8C22×D6111yescharacteristic
366C4⋊Q8C22×D6111yescharacteristic
367#126—2441no—
368#125—21no—
369#124—1221no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373C22⋊Q8—41no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—2441no—
377C4×Q8—1221no—
378C4×Q8C22×D6111yescharacteristic
379C4×Q8C22×D6111yescharacteristic
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—611no—
383C4×D4—1221no—
384C4×D4C22×D6111yescharacteristic
385#117—1221no—
386C2×C4⋊C4—21no—
387#107—21no—
388#107—611no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107C22⋊D6111yescharacteristic
394#106—1221no—
395#106—1221no—
396#106—2441no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—1221no—
401C4⋊C8—1221no—
402C4⋊C8C22×D6111yescharacteristic
403#103—21no—
404#103—21no—
405#103—21no—
406#103—21no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—2441no—
415#103C22⋊D6111yescharacteristic
416#103C22⋊D6111yescharacteristic
417#102—611no—
418#102—611no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—21no—
423C22⋊C8—1221no—
424C4×C8—611no—
425C4×C8—611no—
426C4×C8—1221no—
427C4×C8—1221no—
428C4×C8C22×D6111yescharacteristic
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—41no—
432C2×Q8—42no—
433C2×Q8—81no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—4881no—
437C2×Q8—2441no—
438C2×Q8—2442no—
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4—611no—
442C2×D4D4×D6111yescharacteristic
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—1221no—
447C22×C4—2441no—
448Q16—22no—
449Q16—81no—
450Q16—4881no—
451SD16—2442no—
452D8—1222no—
453M16—4881no—
454C2×C8—21no—
455C2×C8—81no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8C2×C22⋊D6111yescharacteristic
463C2×C8C2×C22⋊D6111yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—2441no—
479C4⋊C4—2441no—
480C4⋊C4D4×D6111yescharacteristic
481C4⋊C4C2×C22⋊D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—1221no—
487C22⋊C4—2441no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C42—21no—
491C42—21no—
492C42—2441no—
493C42C23×D6111yescharacteristic
494C23—21no—
495C23—1221no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—81no—
503Q8—4882no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508Q8#1451979712yes—
509D4—21no—
510D4—1221no—
511D4—1221no—
512D4#1451980912yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—82no—
523C2×C4—161no—
524C2×C4—1221no—
525C2×C4—4882no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#1451982211yescharacteristic
533C8—21no—
534C8—21no—
535C8—41no—
536C8—1221no—
537C8—1221no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C8—2441no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—2441no—
546C22—2441no—
547C22—2441no—
548C22D4×C22⋊D6111yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—81no—
557C4—4881no—
558C4—4881no—
559C4—2441no—
560C4C2×#1451979711yescharacteristic
561C4C2×#1451980911yescharacteristic
562C4#1684056511yescharacteristic
563C4#1684057511yescharacteristic
564C2—42no—
565C2—4881no—
566C2#2145738911yescharacteristic
567C2#2145750011yescharacteristic
568C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3299C2
socleC22×C61D42
prime radicalC2×#11758199D4
hypercenter#3299D61
hypocenterC61#19782
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3299

Complements

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

NH ≅ G/Ntype
3. #21458104C2semidirect
4. #21445462C2semidirect
5. #21445391C2semidirect
6. #21445279C2semidirect
7. #21445242C2semidirect
8. #21445211C2semidirect
9. #21445087C2semidirect
10. #21445082C2semidirect
11. #21445033C2semidirect
12. #21444982C2semidirect
14. #21444638C2semidirect
15. C61×#3299C2semidirect
283. #3299D61semidirect
351. C61#19782semidirect

Maximal subgroups

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

subgroupindex
Q8×#117581992
#214584682
#214581042
#214454622
#214453912
#214452792
#214452422
#214452112
#214450872
#214450822
#214450332
#214449822
#214446482
#214446382
C61×#32992
#1978261