Finite Group Databasebeta

← #29657443

Subgroups — #29657443

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456937C211yesmaximal, characteristic, complemented
2C2×#16840165C211yesmaximal, characteristic, complemented
3#21455306C211yesmaximal, characteristic, complemented
4#21445068C214yesmaximal, complemented
5#21445054C214yesmaximal, complemented
6#21444491C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3499C211yesmaximal, characteristic
10C2×#14519807—21no—
11C2×#14519777C2211yescharacteristic
12#16840165—21no—
13#16840165—22no—
14#16840165C2214yes—
15#16840108—21no—
16#16840073—22no—
17#16840042C2214yes—
18#16839972—21no—
19#16839972C2212yes—
20#16839967—21no—
21#16839967C2212yes—
22#16839815C2211yescharacteristic
23C2×#14519710—22no—
24#16839786C2211yescharacteristic
25#16839770—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—
46#14519807—44no—
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758194—21no—
50C22×#92418—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2312yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57#14519710—24no—
58#14519710—28no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519700—22no—
63#14519700—22no—
64#14519698—24no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68C4×#92418C2312yes—
692−1+4×C61—44no—
702−1+4×C61—44no—
71C2×#41×C61—22no—
72C2×#41×C61—24no—
73C2×#41×C61—22no—
74C2×#41×C61—21no—
75C2×#41×C61D411yescharacteristic
76C22×Q8×C61—22no—
77C22×Q8×C61—22no—
78C22×Q8×C61—21no—
79C4⋊Q8×C61—22no—
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61D411yescharacteristic
85C4⋊Q8×C61D412yes—
86C4⋊Q8×C61C2312yes—
87#127×C61D411yescharacteristic
88#124×C61—22no—
89#124×C61—21no—
90#123×C61—24no—
91#123×C61—24no—
92#123×C61—24no—
93C22⋊Q8×C61—24no—
94C22⋊Q8×C61—22no—
95C22⋊Q8×C61—24no—
96C22⋊Q8×C61—22no—
97#121×C61—22no—
98#121×C61—22no—
99#121×C61—21no—
100#121×C61D412yes—
101C4×Q8×C61D411yescharacteristic
102C4×Q8×C61C2311yescharacteristic
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61D411yescharacteristic
109C4×D4×C61C2312yes—
110C4×D4×C61C2311yescharacteristic
111#117×C61—21no—
112#117×C61D411yescharacteristic
113C2×C4⋊C4×C61—22no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—22no—
116C2×C4⋊C4×C61D412yes—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#41×C61—22no—
120#41×C61—48no—
121#41×C61—416no—
122#41×C61—44no—
123#41×C61—44no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61D814yes—
127C2×Q8×C61—24no—
128C2×Q8×C61—22no—
129C2×Q8×C61—21no—
130C2×Q8×C61—22no—
131C2×Q8×C61—48no—
132C2×Q8×C61—44no—
133C2×Q8×C61—44no—
134C2×Q8×C61—42no—
135C2×Q8×C61—42no—
136C2×Q8×C61C2×D411yescharacteristic
137C2×D4×C61—21no—
138C2×D4×C61—24no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×D412yes—
143C2×D4×C61C2×D411yescharacteristic
144C22×C4×C61—21no—
145C22×C4×C61—24no—
146C22×C4×C61—24no—
147C22×C4×C61—22no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—22no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×Dic61—41no—
155C2×Dic122—22no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158C2×Dic122—41no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D414yes—
169C4⋊C4×C61C2×D411yescharacteristic
170C4⋊C4×C61C2×D411yescharacteristic
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—22no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2411yescharacteristic
185#11758202—44no—
186#11758200—44no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—41no—
190C4⋊Dic61—41no—
191#11758194—42no—
192C2×#92418—24no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—21no—
197C23×C61—22no—
198C23×C61#12011yescharacteristic
199Q8×C61—21no—
200Q8×C61—48no—
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—42no—
204Q8×C61—44no—
205Q8×C61C2×D812yes—
206D4×C61—24no—
207D4×C61—21no—
208D4×C61—24no—
209D4×C61—48no—
210D4×C61—44no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61C2×D812yes—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—48no—
223C2×C4×C61—48no—
224C2×C4×C61—44no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61C2×D812yes—
233C2×C4×C61AGL(1,ℤ/8)12yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×Dic61—42no—
238C2×Dic61—82no—
239Dic122—42no—
240Dic122—42no—
241Dic122—82no—
242#92418—24no—
243#92418—42no—
244#92418—42no—
245#18650—611nomaximal, Sylow
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—22no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61#44712yes—
255C22×C61C2×#12011yescharacteristic
256C4×C61—22no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—44no—
260C4×C61—42no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61#44712yes—
264C4×C61#45612yes—
265C4×C61C22×D811yescharacteristic
266C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C4×C61#57812yes—
268Dic61—82no—
269#3499D6111yescharacteristic
270#3226—611no—
271C2×#500—611no—
272#3039—611no—
273#1649—614no—
274#1647—614no—
275#1546—611no—
276#1492—611no—
277#1476—611no—
278C2×C61—22no—
279C2×C61—42no—
280C2×C61—82no—
281C2×C61C2×#44711yescharacteristic
282C2×C61#301111yescharacteristic
283C2×C61C2×#45611yescharacteristic
284C2×2−1+4—21no—
285C2×#137—1221no—
286D4×Q8—21no—
287D4×Q8C2×D6112yes—
288#547—22no—
289#547—21no—
290#547—22no—
291#547C2×D6112yes—
292#536—22no—
293#536—21no—
294#532—21no—
295#532C2×D6111yescharacteristic
296C2×C4⋊Q8—21no—
297C2×C4⋊Q8—611no—
298C2×C4⋊Q8C2×D6111yescharacteristic
299C4×#41C2×D6111yescharacteristic
300#500—614no—
301#500—1222no—
302#500—1221no—
303#486—1221no—
304#480—1222no—
305#474—614no—
306#462—612no—
307#462—1221no—
308#461—612no—
309#461—1221no—
310#431—611no—
311C2×#107—1222no—
312#423—611no—
313#419—1221no—
314C2×C4×C8—611no—
315#332—614no—
316Q8⋊C8—612no—
317D4⋊C8—612no—
318D4⋊C8—614no—
319C61#1865011yescharacteristic, complemented, Sylow
3202−1+4—44no—
3212−1+4—44no—
322C2×#41—22no—
323C2×#41—24no—
324C2×#41—22no—
325C2×#41—21no—
326C2×#41C22⋊D6111yescharacteristic
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—21no—
330C22×Q8—1221no—
331#137—2444no—
332C2×Q16—1222no—
333C2×SD16—1222no—
334C2×M16—1221no—
335C22×C8—1221no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—612no—
342C4⋊Q8—1221no—
343C4⋊Q8—1221no—
344C4⋊Q8—1221no—
345C4⋊Q8C22⋊D6111yescharacteristic
346C4⋊Q8C22⋊D6112yes—
347C4⋊Q8C22×D6112yes—
348#127C22⋊D6111yescharacteristic
349#124—22no—
350#124—21no—
351#123—24no—
352#123—24no—
353#123—24no—
354C22⋊Q8—24no—
355C22⋊Q8—22no—
356C22⋊Q8—24no—
357C22⋊Q8—22no—
358#121—22no—
359#121—22no—
360#121—21no—
361#121C22⋊D6112yes—
362C4×Q8C22⋊D6111yescharacteristic
363C4×Q8C22×D6111yescharacteristic
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4C22⋊D6111yescharacteristic
370C4×D4C22×D6111yescharacteristic
371C4×D4C22×D6112yes—
372#117—21no—
373#117C22⋊D6111yescharacteristic
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C22⋊D6112yes—
380C2×C42—21no—
381C2×C42C22×D6111yescharacteristic
382#107—1228no—
383#107—1224no—
384C4⋊C8—612no—
385C4⋊C8—612no—
386C4⋊C8—1221no—
387#103—1222no—
388#103—1222no—
389#102—1224no—
390C22⋊C8—1221no—
391C4×C8—612no—
392C4×C8—1221no—
393C4×C8—1221no—
394#41—22no—
395#41—48no—
396#41—416no—
397#41—44no—
398#41—44no—
399#41—42no—
400#41—42no—
401#41#1175819914yes—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—48no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—1222no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×Q8—2441no—
415C2×Q8C2×C22⋊D6111yescharacteristic
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×C22⋊D6112yes—
422C2×D4C2×C22⋊D6111yescharacteristic
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—2441no—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434Q16—2444no—
435SD16—2444no—
436M16—2442no—
437C2×C8—1224no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—1222no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—2441no—
452C4⋊C4—2441no—
453C4⋊C4C2×C22⋊D6111yescharacteristic
454C4⋊C4C2×C22⋊D6111yescharacteristic
455C4⋊C4C2×C22⋊D6112yes—
456C4⋊C4C2×C22⋊D6114yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—21no—
463C22⋊C4—22no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42C2×C22⋊D6111yescharacteristic
468C42C2×C22⋊D6111yescharacteristic
469C42C2×C22⋊D6111yescharacteristic
470C42C23×D6111yescharacteristic
471C23—21no—
472C23—22no—
473C23#1451974611yescharacteristic
474Q8—21no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—44no—
480Q8—4882no—
481Q8—2442no—
482Q8—2442no—
483Q8C2×#1175819912yes—
484D4—24no—
485D4—21no—
486D4—24no—
487D4—48no—
488D4—44no—
489D4—42no—
490D4—42no—
491D4C2×#1175819912yes—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—4882no—
508C2×C4—2442no—
509C2×C4#1451974611yescharacteristic
510C2×C4#1451974611yescharacteristic
511C2×C4#1451974611yescharacteristic
512C2×C4C2×#1175819912yes—
513C2×C4#1451980312yes—
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4C22×C22⋊D6111yescharacteristic
517C8—1224no—
518C8—2442no—
519C8—2442no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22#1683989612yes—
529C22C2×#1451974611yescharacteristic
530C4—21no—
531C4—21no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—44no—
536C4—42no—
537C4—4882no—
538C4#1683989612yes—
539C4#1683994512yes—
540C4C22×#1175819911yescharacteristic
541C4C2×#1451980311yescharacteristic
542C4#1684057712yes—
543C2—22no—
544C2—42no—
545C2—82no—
546C2C2×#1683989611yescharacteristic
547C2#2145511411yescharacteristic
548C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3499C2
socleC22×C61C2×#120
hypercenter#3499D61
hypocenterC61#18650
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. #21456937C2semidirect
2. C2×#16840165C2semidirect
3. #21455306C2semidirect
4. #21445068C2semidirect
5. #21445054C2semidirect
6. #21444491C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
319. C61#18650semidirect

Maximal subgroups

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

subgroupindex
#214569372
C2×#168401652
#214553062
#214450682
#214450542
#214444912
#214442702
#214442012
C61×#34992
#1865061