Finite Group Databasebeta

← #29835349

Subgroups — #29835349

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456359C211yesmaximal, characteristic, complemented
2#21456306C211yesmaximal, characteristic
3#21448059C212yesmaximal, complemented
4#21448057C212yesmaximal, complemented
5#21448052C212yesmaximal, complemented
6#21448041C212yesmaximal, complemented
7#21447534C211yesmaximal, characteristic, complemented
8#21447531C211yesmaximal, characteristic, complemented
9C2×#16839564C211yesmaximal, characteristic, complemented
10#21447415C211yesmaximal, characteristic, complemented
11C61×#2138C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C22×D488C2211yescharacteristic
14#16840390—21no—
15#16840326—21no—
16C2×#14519717C2211yescharacteristic
17#16840169—22no—
18#16840166C2212yescomplemented
19#16840163C2212yescomplemented
20#16840148—22no—
21#16840147C2212yes—
22#16840137C2212yes—
23#16839805—21no—
24C2×#14519711C2211yescharacteristic
25#16839717C2211yescharacteristic
26#16839566C2212yescomplemented
27#16839564C2212yes—
28#16839564C2212yes—
29#16839563C2212yescomplemented
30#16839562C2212yes—
31#16839562C2212yes—
32#16839537C2212yescomplemented
33C16×Dic61C2212yescomplemented
34C2×#135×C61C2211yescharacteristic
35C22×C16×C61C2211yescharacteristic
36C61×#427C2211yescharacteristic
37C61×#358C2211yescharacteristic, complemented
38C61×#358C2211yescharacteristic, complemented
39C61×#357C2211yescharacteristic, complemented
40C61×#357C2211yescharacteristic, complemented
41C22×D244—21no—
42#14519802—44no—
43C2×#11758201—22no—
44C2×D488—22no—
45C2×D488—22no—
46C2×D488—21no—
47C2×D488C2312yes—
48C2×#11758199—22no—
49C2×#11758194D411yescharacteristic
50#14519777—41no—
51#14519776—22no—
52#14519773—22no—
53#14519769—41no—
54#14519768—22no—
55#14519766—22no—
56#14519754—42no—
57#14519751—41no—
58#14519751—41no—
59#14519745—42no—
60#14519736D411yescharacteristic
61C2×C4⋊Dic61—21no—
62#14519717—22no—
63C2×#11758185C2312yes—
64#14519711—21no—
65#14519711C2312yes—
66#14519710—22no—
67#14519708—22no—
68#14519687D412yes—
69C8×Dic61C2312yes—
70C2×#41×C61—21no—
71#135×C61—22no—
72#135×C61—22no—
73C2×Q16×C61C2311yescharacteristic
74C2×SD16×C61—21no—
75C2×D8×C61C2311yescharacteristic
76C22×C8×C61C2311yescharacteristic
77#117×C61—21no—
78C2×C16×C61—22no—
79C2×C16×C61C2311yescharacteristic
80C2×C16×C61C2311yescharacteristic
81#107×C61C2311yescharacteristic
82#107×C61C2311yescharacteristic
83#106×C61—21no—
84C23×D61—41no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61C2×D411yescharacteristic
92C2×C22⋊D61—42no—
93C2×C22⋊D61—42no—
94C22×Dic61—41no—
95C2×D244—22no—
96C2×D244—42no—
97C2×D244—42no—
98C2×D244—41no—
99C2×Dic122—42no—
100Q16×C61—21no—
101Q16×C61—21no—
102SD16×C61—42no—
103D8×C61—21no—
104D8×C61—21no—
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61#4112yes—
107C2×C8×C61#4112yes—
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C22⋊C4×C61—41no—
112C42×C61—41no—
113C22⋊Dic61D812yes—
114#11758201—44no—
115#11758199—44no—
116#11758198—42no—
117#11758198—42no—
118C4⋊Dic61—22no—
119C4⋊Dic61—41no—
120C4⋊Dic61—42no—
121C4⋊Dic61D811yescharacteristic
122C4⋊Dic61D811yescharacteristic
123#11758196—42no—
124#11758196—42no—
125C4×Dic61C2×D412yes—
126#11758194SD1614yes—
127C2×#92418C2×D412yes—
128D488—22no—
129D488—22no—
130D488—42no—
131C16×C61—21no—
132C16×C61—21no—
133#11758185—22no—
134C23×C61—41no—
135C23×C61C2×D811yescharacteristic
136C22×D61—42no—
137C22×D61—82no—
138C22×D61—161no—
139C22×D61—161no—
140Q8×C61—41no—
141Q8×C61—41no—
142D4×C61—41no—
143D4×C61—41no—
144D4×C61—82no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—82no—
149C2×C4×C61#12412yes—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×SD1612yes—
152C2×C4×C61C22×D411yescharacteristic
153C22⋊D61—162no—
154C22⋊D61—162no—
155C2×Dic61—42no—
156C2×Dic61—161no—
157C2×Dic61C2×D812yes—
158C2×Dic61C2×D812yes—
159D244—42no—
160D244—42no—
161D244—82no—
162Dic122—162no—
163C8×C61#12412yes—
164C8×C61C2×#4111yescharacteristic
165C8×C61C2×#4111yescharacteristic
166#92418C2×SD1614yes—
167#33702—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61#48612yes—
174C22×C61C22×D811yescharacteristic
175C2×D61—82no—
176C2×D61—82no—
177C2×D61—82no—
178C2×D61—164no—
179C4×C61—81no—
180C4×C61—81no—
181C4×C61—81no—
182C4×C61#48612yes—
183C4×C61C2×#12411yescharacteristic
184C4×C61C22×SD1611yescharacteristic
185Dic61—22no—
186Dic61—22no—
187Dic61—82no—
188#3164—611no—
189#3155—611no—
190#2244—612no—
191#2243—612no—
192#2242—612no—
193#2240—612no—
194#2140—611no—
195#2139—611no—
196#2138D6111yescharacteristic, complemented
197C2×#357—611no—
198#2108—611no—
199C2×C61—22no—
200C2×C61—161no—
201C2×C61C2×#48611yescharacteristic
202C2×C61#341511yescharacteristic
203C2×C61#341611yescharacteristic
204D61—162no—
205C2×AGL(1,ℤ/8)—1221no—
206C2×#135C2×D6111yescharacteristic
207C22×D8—611no—
208#544—1221no—
209#535—1221no—
210C2×M32—611no—
211C22×C16C2×D6111yescharacteristic
212#501—1222no—
213#500—612no—
214#499—612no—
215#496—1222no—
216#495—612no—
217#493—612no—
218#428—1221no—
219#427C2×D6111yescharacteristic
220C2×#107—611no—
221#405—611no—
222#358—612no—
223#358C2×D6111yescharacteristic, complemented
224#358C2×D6111yescharacteristic, complemented
225#357—612no—
226#357—612no—
227#357—612no—
228#357—612no—
229#357—612no—
230#357C2×D6111yescharacteristic, complemented
231#357C2×D6111yescharacteristic, complemented
232#346—612no—
233C4×C16—612no—
234C61#3370211yescharacteristic, complemented, Sylow
235C2×#41—21no—
236C22×D4—1221no—
237AGL(1,ℤ/8)—2444no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D6111yescharacteristic
241C2×SD16—21no—
242C2×SD16—1222no—
243C2×D8—612no—
244C2×D8—1222no—
245C2×D8—1222no—
246C2×D8—1221no—
247C2×D8—1222no—
248C2×D8C22×D6111yescharacteristic
249C2×M16—611no—
250C22×C8C22×D6111yescharacteristic
251C4⋊Q8—1222no—
252C4⋊Q8—2441no—
253#127—1222no—
254#125—1222no—
255#125—2441no—
256#124—1222no—
257C22⋊Q8—2442no—
258#121—2441no—
259#121—2441no—
260#120—2442no—
261#117—21no—
262#117—611no—
263C2×C4⋊C4—1221no—
264M32—1222no—
265C2×C16—22no—
266C2×C16—612no—
267C2×C16C22×D6111yescharacteristic
268C2×C16C22×D6111yescharacteristic
269#107—612no—
270#107—1222no—
271#107—1221no—
272#107C22×D6111yescharacteristic
273#107C22×D6111yescharacteristic
274#106—21no—
275#106—1222no—
276#97—612no—
277C4×C8—612no—
278C24—2441no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—2442no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—1222no—
286C2×D4—2442no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2441no—
290C2×D4—2442no—
291C22×C4—41no—
292C22×C4—2441no—
293C22×C4D4×D6111yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—2444no—
298D8—21no—
299D8—21no—
300D8—1222no—
301D8—1222no—
302D8—2444no—
303D8—2442no—
304M16—614no—
305C2×C8—612no—
306C2×C8#1175822212yes—
307C2×C8D4×D6111yescharacteristic
308C2×C8#1175822412yes—
309C2×C8C23×D6111yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—611no—
313C4⋊C4—611no—
314C4⋊C4—1222no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4—2442no—
318C4⋊C4—2441no—
319C22⋊C4—41no—
320C22⋊C4—612no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C42—41no—
324C42—612no—
325C16—21no—
326C16—21no—
327C16—1222no—
328C23—41no—
329C23—9761no—
330C23—9761no—
331C23—4882no—
332C23—2442no—
333C23D8×D6111yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—9762no—
337D4—41no—
338D4—41no—
339D4—82no—
340D4—9762no—
341D4—9762no—
342D4—4882no—
343D4—2442no—
344D4—2442no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—612no—
350C2×C4—612no—
351C2×C4—9761no—
352C2×C4—2442no—
353C2×C4#1451976312yes—
354C2×C4D8×D6111yescharacteristic
355C2×C4SD16×D6112yes—
356C2×C4C2×D4×D6111yescharacteristic
357C8—614no—
358C8#1451976312yes—
359C8C2×#1175822411yescharacteristic
360C8C2×#1175822211yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—9764no—
367C22—4882no—
368C22—4882no—
369C22—4882no—
370C22#1684010712yes—
371C22C2×D8×D6111yescharacteristic
372C4—81no—
373C4—81no—
374C4—81no—
375C4—1222no—
376C4—1222no—
377C4—4882no—
378C4#1684010712yes—
379C4C2×#1451976311yescharacteristic
380C4C2×SD16×D6111yescharacteristic
381C2—22no—
382C2—161no—
383C2—9762no—
384C2C2×#1684010711yescharacteristic
385C2#2145895711yescharacteristic
386C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radical#21456306C2
hypercenter#2138D61
hypocenterC61#33702
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #21456359C2semidirect
3. #21448059C2semidirect
4. #21448057C2semidirect
5. #21448052C2semidirect
6. #21448041C2semidirect
7. #21447534C2semidirect
8. #21447531C2semidirect
9. C2×#16839564C2semidirect
10. #21447415C2semidirect
11. C61×#2138C2semidirect
18. #16840166C22semidirect
19. #16840163C22semidirect
26. #16839566C22semidirect
29. #16839563C22semidirect
32. #16839537C22semidirect
33. C16×Dic61C22semidirect
37. C61×#358C22semidirect
38. C61×#358C22semidirect
39. C61×#357C22semidirect
40. C61×#357C22semidirect
196. #2138D61semidirect
223. #358C2×D61semidirect
224. #358C2×D61semidirect
230. #357C2×D61semidirect
231. #357C2×D61semidirect
234. C61#33702semidirect

Maximal subgroups

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

subgroupindex
#214563592
#214563062
#214480592
#214480572
#214480522
#214480412
#214475342
#214475312
C2×#168395642
#214474152
C61×#21382
#3370261