Finite Group Databasebeta

← #29657359

Subgroups — #29657359

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 532 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#21444922C214yesmaximal, complemented
5#21444915C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3503C211yesmaximal, 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
31Q82×C61—21no—
32C61×#557C2212yescomplemented
33C61×#554—21no—
34C61×#554—22no—
35D4×Q8×C61—22no—
36D4×Q8×C61C2212yescomplemented
37C61×#544—22no—
38C61×#544—21no—
39C61×#533—21no—
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61—21no—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520—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—22no—
71C22×Q8×C61—21no—
72C4⋊Q8×C61—22no—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61—24no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61D411yescharacteristic
80C4⋊Q8×C61D412yes—
81C4⋊Q8×C61D411yescharacteristic
82C4⋊Q8×C61C2312yes—
83#125×C61—22no—
84#125×C61—24no—
85#125×C61—22no—
86#125×C61—21no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—24no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—22no—
92C22⋊Q8×C61—24no—
93C22⋊Q8×C61D412yes—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61—21no—
98C4×Q8×C61D411yescharacteristic
99C4×Q8×C61C2312yes—
100C4×Q8×C61C2311yescharacteristic
101C4×D4×C61—24no—
102C4×D4×C61—22no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61D411yescharacteristic
106C4×D4×C61C2311yescharacteristic
107#117×C61—22no—
108#117×C61—21no—
109#117×C61—21no—
110#117×C61D411yescharacteristic
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
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×C61SD1614yescomplemented
120C2×Q8×C61—24no—
121C2×Q8×C61—21no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—82no—
125C2×Q8×C61—84no—
126C2×Q8×C61C2×D412yes—
127C2×Q8×C61C2×D411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61C2×D411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—44no—
142C2×D244—42no—
143C2×D244—42no—
144C2×D244—41no—
145C2×D244—41no—
146C2×D244—41no—
147C2×D244—82no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C2×D414yes—
163C4⋊C4×C61C2×D411yescharacteristic
164C4⋊C4×C61C2×D411yescharacteristic
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
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—24no—
196Q8×C61—44no—
197Q8×C61—42no—
198Q8×C61—82no—
199Q8×C61—84no—
200Q8×C61C2×SD1612yes—
201D4×C61—24no—
202D4×C61—21no—
203D4×C61C2×SD1612yes—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—82no—
216C2×C4×C61—84no—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61C2×SD1612yes—
221C2×C4×C61#13712yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225D244—44no—
226D244—42no—
227D244—42no—
228D244—44no—
229D244—82no—
230D244—82no—
231D244—82no—
232#92418—42no—
233#92418—42no—
234#92418—42no—
235#18642—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—22no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61#44812yes—
242C22×C61C2×#12011yescharacteristic
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C2×D61—164no—
247C4×C61—22no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—41no—
251C4×C61—42no—
252C4×C61—44no—
253C4×C61—41no—
254C4×C61—42no—
255C4×C61#44812yes—
256C4×C61#45412yes—
257C4×C61C22×SD1611yescharacteristic
258C4×C61C2×#13711yescharacteristic
259C4×C61#57612yes—
260#3503D6111yescharacteristic, complemented
261#3208—611no—
262C2×#486—611no—
263#3038—611no—
264#1627—614no—
265#1626—614no—
266#1549—611no—
267#1493—611no—
268#1476—611no—
269C2×C61—22no—
270C2×C61—42no—
271C2×C61C2×#44811yescharacteristic
272C2×C61#300811yescharacteristic
273C2×C61C2×#45411yescharacteristic
274D61—162no—
275C2×AGL(1,ℤ/8)—1221no—
276Q82—21no—
277#557C2×D6112yescomplemented
278#554—21no—
279#554—22no—
280D4×Q8—22no—
281D4×Q8C2×D6112yescomplemented
282#544—22no—
283#544—21no—
284#533—21no—
285#533C2×D6111yescharacteristic
286C2×C4⋊Q8—21no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×#127—611no—
289#520—21no—
290C4×#41C2×D6111yescharacteristic
291#492—1221no—
292#486—614no—
293#486—1222no—
294#486—1221no—
295#460—612no—
296#460—614no—
297#460—1221no—
298#459—612no—
299#459—1221no—
300#450—1222no—
301#431—611no—
302#423—611no—
303#418—1221no—
304C2×#102—1222no—
305C2×C4×C8—611no—
306#332—614no—
307Q8⋊C8—612no—
308Q8⋊C8—614no—
309D4⋊C8—612no—
310C61#1864211yescharacteristic, complemented, Sylow
311C2×#41C22⋊D6111yescharacteristic
312C22×Q8—22no—
313C22×Q8—21no—
314C22×D4—1221no—
315C22×D4—1221no—
316C22×D4—1221no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—1224no—
319C2×SD16—1222no—
320C2×D8—1222no—
321C2×M16—1221no—
322C22×C8—1221no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—24no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8C22⋊D6111yescharacteristic
331C4⋊Q8C22⋊D6112yes—
332C4⋊Q8C22⋊D6111yescharacteristic
333C4⋊Q8C22×D6112yes—
334#127—612no—
335#127—1221no—
336#127—1221no—
337#127—1221no—
338#125—22no—
339#125—24no—
340#125—22no—
341#125—21no—
342C22⋊Q8—22no—
343C22⋊Q8—24no—
344C22⋊Q8—24no—
345C22⋊Q8—21no—
346C22⋊Q8—22no—
347C22⋊Q8—24no—
348C22⋊Q8C22⋊D6112yes—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—21no—
353C4×Q8C22⋊D6111yescharacteristic
354C4×Q8C22×D6111yescharacteristic
355C4×Q8C22×D6112yes—
356C4×D4—24no—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4C22⋊D6111yescharacteristic
361C4×D4C22×D6111yescharacteristic
362#117—22no—
363#117—21no—
364#117—21no—
365#117C22⋊D6111yescharacteristic
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C22⋊D6112yes—
370C2×C42—21no—
371C2×C42C22×D6111yescharacteristic
372C4⋊C8—612no—
373C4⋊C8—612no—
374C4⋊C8—1221no—
375#102—1228no—
376#102—1222no—
377#102—1222no—
378#102—1224no—
379C22⋊C8—1221no—
380C4×C8—612no—
381C4×C8—1221no—
382C4×C8—1221no—
383C24—2441no—
384#41—22no—
385#41#1175820114yescomplemented
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—84no—
392C2×Q8C2×C22⋊D6112yes—
393C2×Q8C2×C22⋊D6111yescharacteristic
394C2×D4—21no—
395C2×D4—1222no—
396C2×D4—1222no—
397C2×D4—1222no—
398C2×D4—4882no—
399C2×D4—2444no—
400C2×D4—2442no—
401C2×D4—2441no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2442no—
405C2×D4C2×C22⋊D6111yescharacteristic
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414SD16—4884no—
415SD16—2444no—
416D8—2444no—
417M16—2442no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—2442no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4C2×C22⋊D6114yes—
437C4⋊C4C2×C22⋊D6111yescharacteristic
438C4⋊C4C2×C22⋊D6111yescharacteristic
439C4⋊C4C2×C22⋊D6112yes—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C42—24no—
446C42—21no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42C2×C22⋊D6111yescharacteristic
451C42C2×C22⋊D6111yescharacteristic
452C42C2×C22⋊D6111yescharacteristic
453C42C23×D6111yescharacteristic
454C23—21no—
455C23—4882no—
456C23—4882no—
457C23—4882no—
458C23—2442no—
459C23#1451974611yescharacteristic
460Q8—21no—
461Q8—24no—
462Q8—44no—
463Q8—42no—
464Q8—82no—
465Q8—84no—
466Q8C2×#1175820112yes—
467D4—24no—
468D4—21no—
469D4—4882no—
470D4—4882no—
471D4—4882no—
472D4—2444no—
473D4—2442no—
474D4—2442no—
475D4—2444no—
476D4C2×#1175820112yes—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4#1451974611yescharacteristic
491C2×C4#1451974611yescharacteristic
492C2×C4#1451974611yescharacteristic
493C2×C4C2×#1175820112yes—
494C2×C4#1451980812yes—
495C2×C4C22×C22⋊D6111yescharacteristic
496C2×C4C22×C22⋊D6111yescharacteristic
497C2×C4C22×C22⋊D6111yescharacteristic
498C8—2442no—
499C8—2442no—
500C8—2442no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—41no—
505C22—41no—
506C22—9764no—
507C22—4882no—
508C22—4882no—
509C22—4882no—
510C22#1683990112yes—
511C22C2×#1451974611yescharacteristic
512C4—21no—
513C4—22no—
514C4—21no—
515C4—41no—
516C4—42no—
517C4—41no—
518C4—42no—
519C4—44no—
520C4#1683990112yes—
521C4#1683993512yes—
522C4C22×#1175820111yescharacteristic
523C4C2×#1451980811yescharacteristic
524C4#1684056512yes—
525C2—22no—
526C2—42no—
527C2—9762no—
528C2C2×#1683990111yescharacteristic
529C2#2145508711yescharacteristic
530C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3503C2
socleC22×C61C2×#120
prime radical#21455299C2
hypercenter#3503D61
hypocenterC61#18642
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. C2×#16840110C2semidirect
4. #21444922C2semidirect
5. #21444915C2semidirect
6. #21444504C2semidirect
7. #21444274C2semidirect
8. #21444201C2semidirect
9. C61×#3503C2semidirect
27. #16839508C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
32. C61×#557C22semidirect
36. D4×Q8×C61C22semidirect
119. #41×C61SD16semidirect
260. #3503D61semidirect
277. #557C2×D61semidirect
281. D4×Q8C2×D61semidirect
310. C61#18642semidirect
385. #41#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168401102
#214552992
#214449222
#214449152
#214445042
#214442742
#214442012
C61×#35032
#1864261