Finite Group Databasebeta

← #29657377

Subgroups — #29657377

Order 15616.

968 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic, complemented
2C2×#16840110C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic
4#21444929C214yesmaximal, complemented
5#21444915C214yesmaximal, complemented
6#21444491C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3509C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C2×#14519774C2211yescharacteristic
12#16840136—21no—
13#16840110—21no—
14#16840110—22no—
15#16840110C2214yes—
16#16839962—21no—
17#16839962C2212yes—
18#16839962C2214yes—
19#16839957—21no—
20#16839957C2212yes—
21#16839912—22no—
22#16839815C2211yescharacteristic
23#16839786C2211yescharacteristic
24#16839766—21no—
25C2×#14519697—22no—
26C2×C4×#92418C2211yescharacteristic
27#16839508C2214yescomplemented
28Q8⋊#92418C2212yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2212yescomplemented
31C61×#557—21no—
32C61×#557C2212yescomplemented
33C61×#556—21no—
34C61×#556C2212yescomplemented
35C61×#552—22no—
36C61×#552—22no—
37C61×#541—22no—
38C61×#536—21no—
39C61×#533—21no—
40C61×#533—21no—
41C61×#533C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#519—21no—
44C4×#41×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244—21no—
48#14519802—44no—
49C2×#11758201—22no—
50C2×#11758201—24no—
51C2×#11758199—22no—
52C2×#11758194—21no—
53C22×#92418—21no—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519697—22no—
62#14519697—22no—
63#14519697—24no—
64#14519697—28no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68C4×#92418C2312yes—
69C2×#41×C61D411yescharacteristic
70C22×Q8×C61—21no—
71C4⋊Q8×C61—22no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D412yes—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61C2312yes—
78#126×C61—24no—
79#126×C61—24no—
80#125×C61—22no—
81#125×C61—24no—
82#125×C61—22no—
83#125×C61—24no—
84#125×C61—21no—
85#124×C61—22no—
86#123×C61—24no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61D412yes—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61—22no—
99C4×Q8×C61—21no—
100C4×Q8×C61D411yescharacteristic
101C4×Q8×C61C2312yes—
102C4×Q8×C61C2311yescharacteristic
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61D411yescharacteristic
106C4×D4×C61C2311yescharacteristic
107#117×C61—22no—
108#117×C61—24no—
109#117×C61—21no—
110#117×C61—22no—
111#117×C61—21no—
112#117×C61D411yescharacteristic
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61D412yes—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C23×D61—41no—
118#41×C61—22no—
119#41×C61—22no—
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—42no—
124C2×Q8×C61—42no—
125C2×Q8×C61C2×D412yes—
126C2×Q8×C61C2×D411yescharacteristic
127C2×D4×C61—21no—
128C2×D4×C61C2×D411yescharacteristic
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—44no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—41no—
143C2×D244—41no—
144C2×D244—41no—
145C2×D244—82no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D414yes—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D411yescharacteristic
163C4⋊C4×C61C2×D411yescharacteristic
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C42×C61—24no—
171C42×C61—24no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—22no—
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2411yescharacteristic
180#11758201—44no—
181#11758201—84no—
182#11758199—44no—
183#11758194—42no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—42no—
188C23×C61—21no—
189C23×C61#12011yescharacteristic
190C22×D61—42no—
191C22×D61—82no—
192C22×D61—82no—
193C22×D61—82no—
194Q8×C61—21no—
195Q8×C61—21no—
196Q8×C61—24no—
197Q8×C61—42no—
198Q8×C61—42no—
199Q8×C61—44no—
200Q8×C61—42no—
201D4×C61—21no—
202D4×C61—21no—
203D4×C61—42no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—21no—
212C2×C4×C61—44no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#13512yes—
220C2×C4×C61AGL(1,ℤ/8)12yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224D244—44no—
225D244—42no—
226D244—42no—
227D244—44no—
228D244—82no—
229D244—82no—
230D244—82no—
231#92418—42no—
232#92418—42no—
233#92418—42no—
234#18644—611nomaximal, Sylow
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61#44912yes—
241C22×C61C2×#12011yescharacteristic
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—164no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—41no—
249C4×C61—44no—
250C4×C61—41no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61—42no—
254C4×C61#44912yes—
255C4×C61#45412yes—
256C4×C61C2×#13511yescharacteristic
257C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C4×C61#57712yes—
259#3509D6111yescharacteristic, complemented
260#3208—611no—
261C2×#486—611no—
262#3038—611no—
263#1628—614no—
264#1626—614no—
265#1546—611no—
266#1493—611no—
267#1476—611no—
268C2×C61—22no—
269C2×C61—81no—
270C2×C61C2×#44911yescharacteristic
271C2×C61#301011yescharacteristic
272C2×C61C2×#45411yescharacteristic
273D61—162no—
274C2×AGL(1,ℤ/8)—1221no—
275#557—21no—
276#557C2×D6112yescomplemented
277#556—21no—
278#556C2×D6112yescomplemented
279#552—22no—
280#552—22no—
281#541—22no—
282#536—21no—
283#533—21no—
284#533—21no—
285#533C2×D6111yescharacteristic
286C2×C4⋊Q8C2×D6111yescharacteristic
287C2×#127—611no—
288#519—21no—
289C4×#41C2×D6111yescharacteristic
290#492—1221no—
291#486—614no—
292#486—1222no—
293#486—1221no—
294#460—612no—
295#460—614no—
296#460—1221no—
297#459—612no—
298#459—1221no—
299#450—1222no—
300#431—611no—
301#423—611no—
302#418—1221no—
303C2×#102—1222no—
304C2×C4×C8—611no—
305#332—614no—
306Q8⋊C8—612no—
307Q8⋊C8—614no—
308D4⋊C8—612no—
309C61#1864411yescharacteristic, complemented, Sylow
310C2×#41C22⋊D6111yescharacteristic
311C22×Q8—21no—
312C22×D4—1221no—
313C22×D4—1221no—
314C22×D4—1221no—
315AGL(1,ℤ/8)—2444no—
316C2×SD16—1224no—
317C2×SD16—1222no—
318C2×D8—1222no—
319C2×M16—1221no—
320C22×C8—1221no—
321C4⋊Q8—22no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8C22⋊D6111yescharacteristic
325C4⋊Q8C22⋊D6112yes—
326C4⋊Q8C22⋊D6111yescharacteristic
327C4⋊Q8C22×D6112yes—
328#127—612no—
329#127—1221no—
330#127—1221no—
331#127—1221no—
332#126—24no—
333#126—24no—
334#125—22no—
335#125—24no—
336#125—22no—
337#125—24no—
338#125—21no—
339#124—22no—
340#123—24no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—21no—
346C22⋊Q8C22⋊D6112yes—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—21no—
354C4×Q8C22⋊D6111yescharacteristic
355C4×Q8C22×D6111yescharacteristic
356C4×Q8C22×D6112yes—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4C22⋊D6111yescharacteristic
360C4×D4C22×D6111yescharacteristic
361#117—24no—
362#117—22no—
363#117—21no—
364#117—22no—
365#117—21no—
366#117C22⋊D6111yescharacteristic
367C2×C4⋊C4—21no—
368C2×C4⋊C4C22⋊D6112yes—
369C2×C42—21no—
370C2×C42C22×D6111yescharacteristic
371C4⋊C8—612no—
372C4⋊C8—612no—
373C4⋊C8—1221no—
374#102—1228no—
375#102—1222no—
376#102—1222no—
377#102—1224no—
378C22⋊C8—1221no—
379C4×C8—612no—
380C4×C8—1221no—
381C4×C8—1221no—
382C24—2441no—
383#41—22no—
384#41—22no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8C2×C22⋊D6112yes—
391C2×Q8C2×C22⋊D6111yescharacteristic
392C2×D4—21no—
393C2×D4—1222no—
394C2×D4—1222no—
395C2×D4—1222no—
396C2×D4—4882no—
397C2×D4—2444no—
398C2×D4—2441no—
399C2×D4—2441no—
400C2×D4—2441no—
401C2×D4—2442no—
402C2×D4—2442no—
403C2×D4C2×C22⋊D6111yescharacteristic
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411SD16—4884no—
412SD16—2444no—
413D8—2444no—
414M16—2442no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—2442no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4C2×C22⋊D6114yes—
434C4⋊C4C2×C22⋊D6112yes—
435C4⋊C4C2×C22⋊D6111yescharacteristic
436C4⋊C4C2×C22⋊D6111yescharacteristic
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—21no—
440C22⋊C4—22no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C42—24no—
444C42—24no—
445C42—21no—
446C42—22no—
447C42—21no—
448C42—21no—
449C42C2×C22⋊D6111yescharacteristic
450C42C2×C22⋊D6111yescharacteristic
451C42C2×C22⋊D6111yescharacteristic
452C42C23×D6111yescharacteristic
453C23—21no—
454C23—4882no—
455C23—4882no—
456C23—4882no—
457C23—2442no—
458C23#1451974611yescharacteristic
459Q8—21no—
460Q8—21no—
461Q8—24no—
462Q8—42no—
463Q8—42no—
464Q8—44no—
465Q8—42no—
466D4—21no—
467D4—21no—
468D4—42no—
469D4—4882no—
470D4—4882no—
471D4—4882no—
472D4—2444no—
473D4—2442no—
474D4—2442no—
475D4—2444no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4#1451974611yescharacteristic
489C2×C4#1451974611yescharacteristic
490C2×C4#1451974611yescharacteristic
491C2×C4#1451979712yes—
492C2×C4#1451980212yes—
493C2×C4C22×C22⋊D6111yescharacteristic
494C2×C4C22×C22⋊D6111yescharacteristic
495C2×C4C22×C22⋊D6111yescharacteristic
496C8—2442no—
497C8—2442no—
498C8—2442no—
499C22—21no—
500C22—21no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—9764no—
505C22—4882no—
506C22—4882no—
507C22—4882no—
508C22#1683990712yes—
509C22C2×#1451974611yescharacteristic
510C4—21no—
511C4—21no—
512C4—41no—
513C4—42no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—44no—
518C4#1683990712yes—
519C4#1683993512yes—
520C4C2×#1451979711yescharacteristic
521C4C2×#1451980211yescharacteristic
522C4#1684057012yes—
523C2—22no—
524C2—81no—
525C2—9762no—
526C2C2×#1683990711yescharacteristic
527C2#2145510511yescharacteristic
528C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3509C2
socleC22×C61C2×#120
prime radical#21455299C2
hypercenter#3509D61
hypocenterC61#18644
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. C2×#16840110C2semidirect
4. #21444929C2semidirect
5. #21444915C2semidirect
6. #21444491C2semidirect
7. #21444274C2semidirect
8. #21444201C2semidirect
9. C61×#3509C2semidirect
27. #16839508C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
32. C61×#557C22semidirect
34. C61×#556C22semidirect
259. #3509D61semidirect
276. #557C2×D61semidirect
278. #556C2×D61semidirect
309. C61#18644semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168401102
#214552992
#214449292
#214449152
#214444912
#214442742
#214442012
C61×#35092
#1864461