Finite Group Databasebeta

← #29657641

Subgroups — #29657641

Order 15616.

936 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic
2#21456379C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic, complemented
4#21445089C212yesmaximal, complemented
5#21445033C212yesmaximal, complemented
6#21444929C212yesmaximal, complemented
7#21444891C212yesmaximal, complemented
8#21444509C211yesmaximal, characteristic, complemented
9#21444270C211yesmaximal, characteristic, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3507C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13#16840312C2211yescharacteristic
14#16840158C2211yescharacteristic
15#16840136C2211yescharacteristic
16#16840112C2211yescharacteristic
17#16840110C2211yescharacteristic
18#16840042C2212yes—
19#16840039—21no—
20#16839996—22no—
21#16839993—22no—
22#16839984—22no—
23#16839972C2212yes—
24#16839962C2212yes—
25#16839957C2212yes—
26#16839954—21no—
27C4×#11758201—22no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839761—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#559—21no—
39C61×#557—21no—
40C61×#556—22no—
41C61×#556C2212yes—
42C61×#554C2212yes—
43C61×#552—22no—
44C61×#539—22no—
45C61×#536—21no—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#529—22no—
49C61×#520—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758222—21no—
52#14519808—44no—
53C2×#11758202—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C22×#92418—21no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67#14519710—22no—
68#14519708—22no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73#14519701—22no—
74#14519700—22no—
75#14519698—22no—
76#14519697—22no—
77#14519697—22no—
78#14519697—22no—
79C4×#92418—21no—
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
82C2×#41×C61—21no—
83C22×Q8×C61D411yescharacteristic
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61D411yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61D411yescharacteristic
89#126×C61—24no—
90#126×C61—24no—
91#125×C61—24no—
92#125×C61—22no—
93#125×C61—21no—
94#124×C61—22no—
95#124×C61—22no—
96#124×C61—21no—
97#123×C61—24no—
98#123×C61—24no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—24no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61D412yes—
104#121×C61—21no—
105#121×C61D412yes—
106C4×Q8×C61—22no—
107C4×Q8×C61—22no—
108C4×Q8×C61—21no—
109C4×Q8×C61—21no—
110C4×Q8×C61—21no—
111C4×Q8×C61D412yes—
112C4×Q8×C61D412yes—
113C4×Q8×C61C2312yes—
114C4×D4×C61—24no—
115C4×D4×C61—22no—
116C4×D4×C61—22no—
117C4×D4×C61—21no—
118C4×D4×C61C2312yes—
119#117×C61—24no—
120#117×C61—21no—
121#117×C61—22no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#41×C61—42no—
128#41×C61—42no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—22no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C2×D412yes—
140C22×C4×C61—21no—
141C22×C4×C61—24no—
142C22×C4×C61—21no—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—41no—
148#11758222—42no—
149#11758222—42no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×C4×D61—41no—
154C2×Dic122—21no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C42×C61—24no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2411yescharacteristic
184#11758202—44no—
185#11758201—44no—
186#11758201—44no—
187#11758199—82no—
188#11758198—41no—
189#11758198—41no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192#11758194—42no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—42no—
198C2×#92418—81no—
199C23×C61—22no—
200C23×C61#12011yescharacteristic
201C22×D61—41no—
202Q8×C61—22no—
203Q8×C61—22no—
204Q8×C61—42no—
205Q8×C61—41no—
206Q8×C61—41no—
207Q8×C61—42no—
208Q8×C61—42no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—44no—
212D4×C61—42no—
213D4×C61—81no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—81no—
226C2×C4×C61—84no—
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#13512yes—
231C2×C4×C61AGL(1,ℤ/8)12yes—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C22⋊D61—82no—
236C2×Dic61—41no—
237D244—42no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242C4×D61—82no—
243Dic122—41no—
244Dic122—41no—
245#92418—22no—
246#92418—41no—
247#92418—42no—
248#92418—42no—
249#18668—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61#44912yes—
256C22×C61C2×#12011yescharacteristic
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61#44912yes—
269C4×C61C2×#13511yescharacteristic
270C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C4×C61#57711yescharacteristic
272C4×C61#57811yescharacteristic
273Dic61—81no—
274#3507D6111yescharacteristic, complemented
275#3245—611no—
276#3167—611no—
277#3037—611no—
278#1652—612no—
279#1644—612no—
280#1628—612no—
281#1622—612no—
282#1550—611no—
283#1492—611no—
284C2×Q8⋊C8—611no—
285C2×C61—22no—
286C2×C61—82no—
287C2×C61C2×#44911yescharacteristic
288C2×C61#301311yescharacteristic
289C2×C61#301811yescharacteristic
290D61—161no—
291C2×#137—1221no—
292#559—21no—
293#557—21no—
294#556—22no—
295#556C2×D6112yes—
296#554C2×D6112yes—
297#552—22no—
298#539—22no—
299#536—21no—
300#533C2×D6111yescharacteristic
301#532—611no—
302#532C2×D6111yescharacteristic
303#529—22no—
304#520—21no—
305C2×C4×Q8C2×D6111yescharacteristic
306#498—611no—
307#492—611no—
308#487—611no—
309#486—611no—
310#474—612no—
311#474—1221no—
312#466—1222no—
313#465—1222no—
314#464—1222no—
315#462—612no—
316#460—612no—
317#459—612no—
318#459—1221no—
319C4×SD16—1222no—
320#423—611no—
321C2×C4⋊C8—611no—
322#417—1221no—
323C2×C4×C8—611no—
324#333—612no—
325#332—612no—
326Q8⋊C8—612no—
327Q8⋊C8—612no—
328D4⋊C8—612no—
329D4⋊C8—612no—
330C61#1866811yescharacteristic, complemented, Sylow
331C2×#41—21no—
332C2×#41—1221no—
333C22×Q8C22⋊D6111yescharacteristic
334#137—2444no—
335C2×Q16—1222no—
336C2×SD16—1222no—
337C2×SD16—1222no—
338C2×D8—1222no—
339C2×M16—1221no—
340C22×C8—1221no—
341C22×C8—1221no—
342C4⋊Q8—22no—
343C4⋊Q8—611no—
344C4⋊Q8C22⋊D6111yescharacteristic
345C4⋊Q8C22×D6111yescharacteristic
346C4⋊Q8C22×D6111yescharacteristic
347#127—611no—
348#127C22⋊D6111yescharacteristic
349#126—24no—
350#126—24no—
351#125—24no—
352#125—22no—
353#125—21no—
354#124—22no—
355#124—22no—
356#124—21no—
357#124—1221no—
358#123—24no—
359#123—24no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—24no—
363C22⋊Q8—21no—
364C22⋊Q8C22⋊D6112yes—
365#121—21no—
366#121—1221no—
367#121—1221no—
368#121C22⋊D6112yes—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8C22⋊D6112yes—
375C4×Q8C22⋊D6112yes—
376C4×Q8C22×D6112yes—
377C4×D4—24no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—1221no—
382C4×D4—1221no—
383C4×D4C22×D6112yes—
384#117—24no—
385#117—21no—
386#117—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4C22⋊D6111yescharacteristic
390C2×C42—21no—
391C2×C42C22×D6111yescharacteristic
392#107—1222no—
393#106—1222no—
394C4⋊C8—612no—
395C4⋊C8—612no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398#103—1222no—
399#103—1222no—
400#102—1222no—
401#102—1222no—
402#102—1222no—
403#102—1222no—
404C4×C8—611no—
405C4×C8—611no—
406C4×C8—1221no—
407#41—42no—
408#41—42no—
409#41—2442no—
410#41—2442no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—1221no—
417C2×Q8C2×C22⋊D6112yes—
418C2×Q8C2×C22⋊D6112yes—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—2441no—
426C2×D4C2×C22⋊D6112yes—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—2441no—
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435Q16—2444no—
436SD16—2444no—
437SD16—2444no—
438D8—4882no—
439M16—2442no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—4881no—
445C2×C8—2442no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4C2×C22⋊D6112yes—
461C4⋊C4C2×C22⋊D6112yes—
462C4⋊C4C2×C22⋊D6112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—2441no—
470C22⋊C4—2441no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42C2×C22⋊D6111yescharacteristic
476C42C2×C22⋊D6111yescharacteristic
477C42C2×C22⋊D6111yescharacteristic
478C42C23×D6111yescharacteristic
479C23—22no—
480C23—2441no—
481C23#1451974611yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485Q8—41no—
486Q8—41no—
487Q8—42no—
488Q8—42no—
489Q8—2441no—
490Q8—2441no—
491D4—22no—
492D4—22no—
493D4—44no—
494D4—42no—
495D4—81no—
496D4—4882no—
497D4—2442no—
498D4—2441no—
499D4—2441no—
500D4—2441no—
501D4—2441no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—81no—
514C2×C4—84no—
515C2×C4—4882no—
516C2×C4—2441no—
517C2×C4#1451974611yescharacteristic
518C2×C4#1451974611yescharacteristic
519C2×C4#1451974611yescharacteristic
520C2×C4#1451979712yes—
521C2×C4#1451980212yes—
522C2×C4C22×C22⋊D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C8—1222no—
526C8—2441no—
527C8—2442no—
528C8—2442no—
529C22—21no—
530C22—21no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22#1683990712yes—
538C22C2×#1451974611yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—4881no—
548C4#1683990712yes—
549C4C2×#1451979711yescharacteristic
550C4C2×#1451980211yescharacteristic
551C4#1684057011yescharacteristic
552C4#1684057711yescharacteristic
553C2—22no—
554C2—82no—
555C2—9761no—
556C2C2×#1683990711yescharacteristic
557C2#2145513411yescharacteristic
558C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radical#21457126C2
hypercenter#3507D61
hypocenterC61#18668
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
2. #21456379C2semidirect
3. #21455291C2semidirect
4. #21445089C2semidirect
5. #21445033C2semidirect
6. #21444929C2semidirect
7. #21444891C2semidirect
8. #21444509C2semidirect
9. #21444270C2semidirect
10. C2×Q8⋊#92418C2semidirect
11. C61×#3507C2semidirect
274. #3507D61semidirect
330. C61#18668semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214563792
#214552912
#214450892
#214450332
#214449292
#214448912
#214445092
#214442702
C2×Q8⋊#924182
C61×#35072
#1866861