Finite Group Databasebeta

← #29736802

Subgroups — #29736802

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456195C211yesmaximal, characteristic, complemented
2#21455299C211yesmaximal, characteristic
3#21454177C211yesmaximal, characteristic, complemented
4#21446972C214yesmaximal, complemented
5#21446809C211yesmaximal, characteristic, complemented
6#21446797C211yesmaximal, characteristic, complemented
7#21446195C214yesmaximal, complemented
8C4×#14519704C211yesmaximal, characteristic, complemented
9C61×#3503C211yesmaximal, characteristic, complemented
10C2×#14519802C2212yes—
11C2×#14519774C2211yescharacteristic
12C2×#14519766—22no—
13#16840128—22no—
14#16840110—21no—
15#16840110—21no—
16#16839962—21no—
17#16839962—21no—
18#16839957—21no—
19#16839957—21no—
20#16839786C2211yescharacteristic
21C2×#14519704C2212yes—
22#16839766C2214yes—
23#16839766C2212yes—
24C4×#11758194C2211yescharacteristic
25#16839663—22no—
26#16839663C2214yes—
27C42×Dic61C2211yescharacteristic
28#16839531C2214yes—
29#16839521C2214yes—
30#16839521C2212yes—
31Q82×C61—21no—
32C61×#557—21no—
33C61×#554—21no—
34C61×#554—22no—
35D4×Q8×C61—22no—
36D4×Q8×C61—21no—
37C61×#544—22no—
38C61×#544C2212yes—
39C61×#533—21no—
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61—21no—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520C2212yes—
44C4×#41×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48#14519802—28no—
49C2×#11758201—22no—
50C2×#11758199—22no—
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53#14519774—21no—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519766—48no—
58C2×#11758198—28no—
59C2×C4×Dic61—21no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61C2312yes—
62C4⋊#92418—21no—
63C4⋊#92418—21no—
64#14519704—24no—
65#14519697—22no—
66#14519697—22no—
67#14519697—24no—
68#14519697—24no—
69#14519688—21no—
70C4×#92418—21no—
71C2×#41×C61C2311yescharacteristic
72C22×Q8×C61—22no—
73C22×Q8×C61D412yes—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61—24no—
77C4⋊Q8×C61—22no—
78C4⋊Q8×C61—22no—
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61D411yescharacteristic
84C4⋊Q8×C61D411yescharacteristic
85#125×C61—24no—
86#125×C61—22no—
87#125×C61—22no—
88#125×C61D412yes—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—24no—
92C22⋊Q8×C61—24no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—24no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61—22no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101C4×Q8×C61D411yescharacteristic
102C4×Q8×C61D411yescharacteristic
103C4×D4×C61—22no—
104C4×D4×C61—21no—
105C4×D4×C61—24no—
106C4×D4×C61—21no—
107C4×D4×C61D411yescharacteristic
108C4×D4×C61D411yescharacteristic
109#117×C61—22no—
110#117×C61—21no—
111#117×C61C2312yes—
112#117×C61C2311yescharacteristic
113C2×C4⋊C4×C61—22no—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61D412yes—
116C2×C4⋊C4×C61C2312yes—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C23×D61—22no—
120#41×C61—22no—
121#41×C61#4114yes—
122C2×Q8×C61—24no—
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C2×Q8×C61—84no—
129C2×Q8×C61C2×D411yescharacteristic
130C2×D4×C61—21no—
131C2×D4×C61C2×D411yescharacteristic
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D414yes—
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—24no—
144C2×D244—24no—
145C2×D244—44no—
146C2×D244—44no—
147C2×D244—41no—
148C2×D244—41no—
149C2×D244—41no—
150C2×D244—41no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C22⋊C4×C61—24no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C42×C61—24no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—22no—
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183#11758201—44no—
184#11758199—44no—
185#11758198—416no—
186C4×Dic61—24no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61—42no—
190#11758194—22no—
191#11758194—24no—
192#11758194—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C23×C61—21no—
196C23×C61C22×D411yescharacteristic
197C22×D61—44no—
198C22×D61—44no—
199C22×D61—42no—
200C22×D61—44no—
201Q8×C61—21no—
202Q8×C61—44no—
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—82no—
206Q8×C61—84no—
207Q8×C61#12412yescomplemented
208D4×C61—24no—
209D4×C61—21no—
210D4×C61#12412yescomplemented
211C2×C4×C61—21no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61—84no—
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4112yes—
231C2×C4×C612−1+412yes—
232C2×Dic61—24no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235D244—42no—
236D244—42no—
237D244—44no—
238D244—82no—
239D244—82no—
240D244—82no—
241D244—82no—
242#92418—42no—
243#92418—42no—
244#25924—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61C2×#12011yescharacteristic
251C22×C61#54812yes—
252C2×D61—88no—
253C2×D61—82no—
254C2×D61—82no—
255C2×D61—82no—
256C4×C61—22no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—41no—
260C4×C61—44no—
261C4×C61—42no—
262C4×C61—41no—
263C4×C61—42no—
264C4×C61#45312yes—
265C4×C61#45412yes—
266C4×C61C2×#12411yescharacteristic
267C4×C61#53611yescharacteristic
268C4×C61#54812yes—
269Dic61—44no—
270#3503D6111yescharacteristic, complemented
271#3138—611no—
272#3038—611no—
273#2858—611no—
274#2027—614no—
275#1997—611no—
276#1994—611no—
277#1883—614no—
278C4×#104—611no—
279C2×C61—22no—
280C2×C61—42no—
281C2×C61#243611yescharacteristic
282C2×C61C2×#45311yescharacteristic
283C2×C61C2×#45411yescharacteristic
284D61—162no—
285C2×AGL(1,ℤ/8)—612no—
286Q82—21no—
287#557—21no—
288#554—21no—
289#554—22no—
290D4×Q8—22no—
291D4×Q8—21no—
292#544—22no—
293#544C2×D6112yes—
294#533—21no—
295#533C2×D6111yescharacteristic
296C2×C4⋊Q8—21no—
297C2×C4⋊Q8C2×D6111yescharacteristic
298C2×#127—611no—
299C2×#124—1222no—
300#520C2×D6112yes—
301C4×#41C2×D6111yescharacteristic
302#490—1222no—
303#486—1221no—
304#486—1221no—
305#460—1221no—
306#460—1221no—
307#459—1221no—
308#459—1221no—
309#423—611no—
310C2×#104—612no—
311#418—614no—
312#418—612no—
313C4×M16—611no—
314#390—614no—
315#390—1222no—
316C43—611no—
317#343—614no—
318#339—614no—
319#339—612no—
320C61#2592411yescharacteristic, complemented, Sylow
321C2×#41C22×D6111yescharacteristic
322C22×Q8—22no—
323C22×Q8C22⋊D6112yes—
324C22×D4—612no—
325C22×D4—1221no—
326C22×D4—1221no—
327AGL(1,ℤ/8)—1228no—
328C2×SD16—1222no—
329C2×D8—1222no—
330C2×M16—612no—
331C2×M16—612no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8—24no—
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8C22⋊D6111yescharacteristic
342C4⋊Q8C22⋊D6111yescharacteristic
343#127—1221no—
344#127—1221no—
345#127—1221no—
346#127—1221no—
347#125—24no—
348#125—22no—
349#125—22no—
350#125C22⋊D6112yes—
351#124—2448no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—24no—
355C22⋊Q8—24no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—24no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8C22⋊D6111yescharacteristic
365C4×Q8C22⋊D6111yescharacteristic
366C4×D4—22no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—24no—
370C4×D4C22⋊D6111yescharacteristic
371C4×D4C22⋊D6111yescharacteristic
372#117—22no—
373#117—21no—
374#117C22×D6111yescharacteristic
375#117C22×D6112yes—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4C22⋊D6112yes—
379C2×C4⋊C4C22×D6112yes—
380C2×C22⋊C4—1228no—
381C2×C42—21no—
382C2×C42—612no—
383C2×C42—1221no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386C4⋊C8—1221no—
387C4⋊C8—1221no—
388#104—1224no—
389#102—1222no—
390#102—1222no—
391#102—1224no—
392#102—1224no—
393#97—1221no—
394C4×C8—1221no—
395C24—1222no—
396#41—22no—
397#41#1175822614yes—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—84no—
405C2×Q8C2×C22⋊D6111yescharacteristic
406C2×D4—21no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1224no—
410C2×D4—1224no—
411C2×D4—2444no—
412C2×D4—2444no—
413C2×D4—2441no—
414C2×D4—2441no—
415C2×D4—2441no—
416C2×D4—2441no—
417C2×D4C2×C22⋊D6111yescharacteristic
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—1222no—
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C2×C22⋊D6114yes—
426C22×C4C23×D6111yescharacteristic
427SD16—2444no—
428D8—2444no—
429M16—1222no—
430M16—1224no—
431M16—1222no—
432C2×C8—1222no—
433C2×C8—1222no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4C2×C22⋊D6112yes—
449C4⋊C4C2×C22⋊D6112yes—
450C4⋊C4C2×C22⋊D6111yescharacteristic
451C4⋊C4C2×C22⋊D6111yescharacteristic
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—24416no—
458C42—24no—
459C42—21no—
460C42—22no—
461C42—21no—
462C42—21no—
463C42—1224no—
464C42—1222no—
465C42—1222no—
466C42—2442no—
467C42C2×C22⋊D6111yescharacteristic
468C42C2×C22⋊D6111yescharacteristic
469C42C2×C22⋊D6111yescharacteristic
470C42C2×C22⋊D6111yescharacteristic
471C23—21no—
472C23—2444no—
473C23—2444no—
474C23—2442no—
475C23—2444no—
476C23C22×C22⋊D6111yescharacteristic
477Q8—21no—
478Q8—44no—
479Q8—42no—
480Q8—42no—
481Q8—82no—
482Q8—84no—
483Q8#1451976612yescomplemented
484D4—24no—
485D4—21no—
486D4—4882no—
487D4—4882no—
488D4—4882no—
489D4—4882no—
490D4—2442no—
491D4—2442no—
492D4—2444no—
493D4#1451976612yescomplemented
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—84no—
507C2×C4—1224no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4#1451974611yescharacteristic
511C2×C4#1451974611yescharacteristic
512C2×C4#1451974611yescharacteristic
513C2×C4#1451974611yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4C2×#1175822612yes—
517C2×C4#1451982512yes—
518C8—2442no—
519C8—2442no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—4888no—
526C22—4882no—
527C22—4882no—
528C22—4882no—
529C22C2×#1451974611yescharacteristic
530C22#1684042312yes—
531C4—22no—
532C4—21no—
533C4—21no—
534C4—41no—
535C4—44no—
536C4—42no—
537C4—41no—
538C4—42no—
539C4—2444no—
540C4#1683993012yes—
541C4#1683993512yes—
542C4C2×#1451976611yescharacteristic
543C4#1684033611yescharacteristic
544C4#1684042312yes—
545C2—22no—
546C2—42no—
547C2—9762no—
548C2#2144947311yescharacteristic
549C2C2×#1683993011yescharacteristic
550C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3503C2
socleC22×C61C2×#120
prime radical#21455299C2
hypercenter#3503D61
hypocenterC61#25924
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #21456195C2semidirect
3. #21454177C2semidirect
4. #21446972C2semidirect
5. #21446809C2semidirect
6. #21446797C2semidirect
7. #21446195C2semidirect
8. C4×#14519704C2semidirect
9. C61×#3503C2semidirect
207. Q8×C61#124semidirect
210. D4×C61#124semidirect
270. #3503D61semidirect
320. C61#25924semidirect
483. Q8#14519766semidirect
493. D4#14519766semidirect

Maximal subgroups

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

subgroupindex
#214561952
#214552992
#214541772
#214469722
#214468092
#214467972
#214461952
C4×#145197042
C61×#35032
#2592461