Finite Group Databasebeta

← #29657452

Subgroups — #29657452

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456741C211yesmaximal, characteristic, complemented
2C2×#16840158C211yesmaximal, characteristic, complemented
3#21455634C211yesmaximal, characteristic, complemented
4#21445103C214yesmaximal, complemented
5#21445082C214yesmaximal, complemented
6#21444491C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3499C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840158—22no—
12#16840158C2214yes—
13#16840134—21no—
14#16840114—21no—
15#16840057—21no—
16#16840052—21no—
17#16840047C2212yes—
18#16840042C2212yes—
19#16839967C2214yes—
20#16839900—22no—
21#16839815C2211yescharacteristic
22#16839805—21no—
23C2×#14519708—22no—
24#16839786C2211yescharacteristic
25#16839763—21no—
26C2×C4×#92418C2211yescharacteristic
27#16839508C2214yes—
28Q8⋊#92418C2212yes—
29D4⋊#92418C2212yes—
30D4⋊#92418C2214yes—
31C2×2−1+4×C61—21no—
32D4×Q8×C61—21no—
33D4×Q8×C61C2212yes—
34C61×#547—22no—
35C61×#547—21no—
36C61×#547—22no—
37C61×#547C2212yes—
38C61×#536—22no—
39C61×#536—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61—21no—
43C2×C4⋊Q8×C61C2211yescharacteristic
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46C2×#11758200—24no—
47C2×#11758194—21no—
48C22×#92418—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777C2312yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55#14519710—22no—
56#14519708—22no—
57#14519708—28no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519700—24no—
63#14519698—22no—
64C22⋊#92418—21no—
65C4×#92418—21no—
66C4×#92418—21no—
67C4×#92418C2312yes—
682−1+4×C61—44no—
692−1+4×C61—44no—
70C2×#41×C61—22no—
71C2×#41×C61—24no—
72C2×#41×C61—22no—
73C2×#41×C61—21no—
74C2×#41×C61D411yescharacteristic
75C22×Q8×C61—22no—
76C22×Q8×C61—22no—
77C22×Q8×C61—21no—
78C4⋊Q8×C61—22no—
79C4⋊Q8×C61—22no—
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61D411yescharacteristic
84C4⋊Q8×C61D412yes—
85C4⋊Q8×C61C2312yes—
86#127×C61D411yescharacteristic
87#124×C61—22no—
88#124×C61—21no—
89#123×C61—24no—
90#123×C61—24no—
91#123×C61—24no—
92C22⋊Q8×C61—24no—
93C22⋊Q8×C61—22no—
94C22⋊Q8×C61—24no—
95C22⋊Q8×C61—22no—
96#121×C61—22no—
97#121×C61—22no—
98#121×C61—21no—
99#121×C61D412yes—
100C4×Q8×C61D411yescharacteristic
101C4×Q8×C61C2311yescharacteristic
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61D411yescharacteristic
108C4×D4×C61C2312yes—
109C4×D4×C61C2311yescharacteristic
110#117×C61—21no—
111#117×C61D411yescharacteristic
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61D412yes—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#41×C61—22no—
119#41×C61—48no—
120#41×C61—416no—
121#41×C61—44no—
122#41×C61—44no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61SD1614yescomplemented
126C2×Q8×C61—24no—
127C2×Q8×C61—22no—
128C2×Q8×C61—21no—
129C2×Q8×C61—22no—
130C2×Q8×C61—48no—
131C2×Q8×C61—44no—
132C2×Q8×C61—44no—
133C2×Q8×C61—42no—
134C2×Q8×C61—42no—
135C2×Q8×C61C2×D411yescharacteristic
136C2×D4×C61—21no—
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×D411yescharacteristic
143C22×C4×C61—21no—
144C22×C4×C61—24no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×Dic61—41no—
154C2×Dic122—22no—
155C2×Dic122—41no—
156C2×Dic122—82no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D414yes—
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—22no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183#11758200—84no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—41no—
187C4⋊Dic61—41no—
188#11758194—42no—
189C2×#92418—24no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—21no—
194C23×C61—22no—
195C23×C61#12011yescharacteristic
196Q8×C61—21no—
197Q8×C61—48no—
198Q8×C61—42no—
199Q8×C61—42no—
200Q8×C61—42no—
201Q8×C61—44no—
202Q8×C61C2×SD1612yes—
203D4×C61—24no—
204D4×C61—21no—
205D4×C61—24no—
206D4×C61—48no—
207D4×C61—44no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61C2×SD1612yes—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—24no—
218C2×C4×C61—22no—
219C2×C4×C61—48no—
220C2×C4×C61—48no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61C2×SD1612yes—
230C2×C4×C61AGL(1,ℤ/8)12yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×Dic61—42no—
235C2×Dic61—82no—
236Dic122—44no—
237Dic122—82no—
238#92418—24no—
239#92418—42no—
240#92418—42no—
241#18651—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—22no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61#45012yes—
251C22×C61C2×#12011yescharacteristic
252C4×C61—22no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—44no—
256C4×C61—42no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61#45012yes—
260C4×C61#45612yes—
261C4×C61C22×SD1611yescharacteristic
262C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C4×C61#57712yes—
264Dic61—82no—
265#3499D6111yescharacteristic
266#3206—611no—
267C2×#498—611no—
268#3079—611no—
269#1654—614no—
270#1651—614no—
271#1546—611no—
272#1492—611no—
273#1476—611no—
274C2×C61—22no—
275C2×C61—42no—
276C2×C61—82no—
277C2×C61C2×#45011yescharacteristic
278C2×C61#301011yescharacteristic
279C2×C61C2×#45611yescharacteristic
280C2×2−1+4—21no—
281D4×Q8—21no—
282D4×Q8C2×D6112yes—
283#547—22no—
284#547—21no—
285#547—22no—
286#547C2×D6112yes—
287#536—22no—
288#536—21no—
289#532—21no—
290#532C2×D6111yescharacteristic
291C2×C4⋊Q8—21no—
292C2×C4⋊Q8—611no—
293C2×C4⋊Q8C2×D6111yescharacteristic
294C4×#41C2×D6111yescharacteristic
295#498—614no—
296#498—1222no—
297#492—1221no—
298#487—1221no—
299#477—1221no—
300#476—1221no—
301#475—612no—
302#474—612no—
303#461—614no—
304#448—1222no—
305#431—611no—
306#428—1221no—
307C2×#106—1222no—
308#423—611no—
309#417—1221no—
310C2×C4×C8—611no—
311#332—614no—
312Q8⋊C8—612no—
313D4⋊C8—614no—
314D4⋊C8—612no—
315C61#1865111yescharacteristic, complemented, Sylow
3162−1+4—44no—
3172−1+4—44no—
318C2×#41—22no—
319C2×#41—24no—
320C2×#41—22no—
321C2×#41—21no—
322C2×#41C22⋊D6111yescharacteristic
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—21no—
326C22×Q8—1221no—
327C2×SD16—1224no—
328C2×M16—1221no—
329C22×C8—1221no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—612no—
336C4⋊Q8—1221no—
337C4⋊Q8—1221no—
338C4⋊Q8—1221no—
339C4⋊Q8C22⋊D6111yescharacteristic
340C4⋊Q8C22⋊D6112yes—
341C4⋊Q8C22×D6112yes—
342#127C22⋊D6111yescharacteristic
343#124—22no—
344#124—21no—
345#123—24no—
346#123—24no—
347#123—24no—
348C22⋊Q8—24no—
349C22⋊Q8—22no—
350C22⋊Q8—24no—
351C22⋊Q8—22no—
352#121—22no—
353#121—22no—
354#121—21no—
355#121C22⋊D6112yes—
356C4×Q8C22⋊D6111yescharacteristic
357C4×Q8C22×D6111yescharacteristic
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C22⋊D6111yescharacteristic
364C4×D4C22×D6111yescharacteristic
365C4×D4C22×D6112yes—
366#117—21no—
367#117C22⋊D6111yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22⋊D6112yes—
374C2×C42—21no—
375C2×C42C22×D6111yescharacteristic
376#107—1222no—
377#106—1228no—
378#106—1222no—
379C4⋊C8—612no—
380C4⋊C8—612no—
381C4⋊C8—1221no—
382#103—1222no—
383#103—1224no—
384#102—1222no—
385C22⋊C8—1221no—
386C4×C8—612no—
387C4×C8—1221no—
388C4×C8—1221no—
389#41—22no—
390#41—48no—
391#41—416no—
392#41—44no—
393#41—44no—
394#41—42no—
395#41—42no—
396#41#1175820014yescomplemented
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—1222no—
407C2×Q8—4882no—
408C2×Q8—2441no—
409C2×Q8C2×C22⋊D6111yescharacteristic
410C2×D4—21no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4C2×C22⋊D6112yes—
416C2×D4C2×C22⋊D6111yescharacteristic
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—2441no—
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C2×C22⋊D6111yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428SD16—4884no—
429M16—2442no—
430C2×C8—1224no—
431C2×C8—1222no—
432C2×C8—1222no—
433C2×C8—1222no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—2441no—
445C4⋊C4—2441no—
446C4⋊C4C2×C22⋊D6112yes—
447C4⋊C4C2×C22⋊D6114yes—
448C4⋊C4C2×C22⋊D6111yescharacteristic
449C4⋊C4C2×C22⋊D6111yescharacteristic
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—21no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C22⋊C4—21no—
456C22⋊C4—22no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42C2×C22⋊D6111yescharacteristic
461C42C2×C22⋊D6111yescharacteristic
462C42C2×C22⋊D6111yescharacteristic
463C42C23×D6111yescharacteristic
464C23—21no—
465C23—22no—
466C23#1451974611yescharacteristic
467Q8—21no—
468Q8—48no—
469Q8—42no—
470Q8—42no—
471Q8—42no—
472Q8—44no—
473Q8—4882no—
474Q8—2444no—
475Q8C2×#1175820012yes—
476D4—24no—
477D4—21no—
478D4—24no—
479D4—48no—
480D4—44no—
481D4—42no—
482D4—42no—
483D4C2×#1175820012yes—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—4882no—
500C2×C4—2442no—
501C2×C4#1451974611yescharacteristic
502C2×C4#1451974611yescharacteristic
503C2×C4#1451974611yescharacteristic
504C2×C4C2×#1175820012yes—
505C2×C4#1451980312yes—
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C22×C22⋊D6111yescharacteristic
508C2×C4C22×C22⋊D6111yescharacteristic
509C8—1224no—
510C8—2442no—
511C8—2442no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22#1683991312yes—
521C22C2×#1451974611yescharacteristic
522C4—21no—
523C4—21no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—44no—
528C4—42no—
529C4—4882no—
530C4#1683991312yes—
531C4#1683994512yes—
532C4C22×#1175820011yescharacteristic
533C4C2×#1451980311yescharacteristic
534C4#1684057012yes—
535C2—22no—
536C2—42no—
537C2—82no—
538C2C2×#1683991311yescharacteristic
539C2#2145510511yescharacteristic
540C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3499C2
socleC22×C61C2×#120
hypercenter#3499D61
hypocenterC61#18651
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #21456741C2semidirect
2. C2×#16840158C2semidirect
3. #21455634C2semidirect
4. #21445103C2semidirect
5. #21445082C2semidirect
6. #21444491C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
125. #41×C61SD16semidirect
315. C61#18651semidirect
396. #41#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567412
C2×#168401582
#214556342
#214451032
#214450822
#214444912
#214442702
#214442012
C61×#34992
#1865161