Finite Group Databasebeta

← #29661861

Subgroups — #29661861

Order 15616.

992 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456830C212yesmaximal, complemented
2#21456164C211yesmaximal, characteristic, complemented
3#21445125C218yesmaximal, complemented
4#21444454C211yesmaximal, characteristic, complemented
5#21444430C212yesmaximal, complemented
6C61×#3549C211yesmaximal, characteristic
7#16840319C2211yescharacteristic, complemented
8#16840157C2214yescomplemented
9#16840130—21no—
10#16840126—21no—
11#16840121C2214yescomplemented
12#16840121C2214yescomplemented
13#16839969—22no—
14#16839959—22no—
15#16839924—24no—
16#16839819C2212yescomplemented
17#16839758—22no—
18C2×#14519688C2211yescharacteristic, complemented
19#16839502C2218yescomplemented
20#16839502C2214yescomplemented
21C61×H(3,𝔽4)C2214yes—
22C61×#555—22no—
23C61×#555—21no—
24D42×C61—22no—
25D42×C61—21no—
26C61×#535—22no—
27C61×#535—24no—
28C61×#532C2211yescharacteristic
29C61×#532C2212yes—
30C2×#11758202—24no—
31C2×#11758200—24no—
32C22×#92418—21no—
33#14519777C2312yes—
34#14519769—21no—
35#14519756—22no—
36C4×Dic122—22no—
37#14519736—21no—
38C4⋊#92418—22no—
39#14519701—24no—
40#14519701—24no—
41#14519698—24no—
42#14519698—24no—
43C22⋊#92418—22no—
44#14519688—21no—
45#14519688C2312yes—
46#14519688C2314yes—
47C2×#41×C61D412yes—
48C2×#41×C61D414yes—
49C22×D4×C61—24no—
50C22×D4×C61—22no—
51C4⋊Q8×C61D412yes—
52C4⋊Q8×C61D411yescharacteristic
53#127×C61—24no—
54#127×C61D412yes—
55#127×C61D411yescharacteristic
56#124×C61—24no—
57#124×C61—28no—
58#124×C61C2312yes—
59#124×C61C2314yes—
60#121×C61—24no—
61#121×C61—24no—
62#121×C61—24no—
63#121×C61—22no—
64#121×C61—22no—
65#121×C61—22no—
66#120×C61—24no—
67#120×C61—24no—
68#120×C61—28no—
69C4×D4×C61—22no—
70C4×D4×C61—22no—
71C4×D4×C61—22no—
72#117×C61—24no—
73#117×C61—22no—
74C2×C42×C61C2311yescharacteristic
75C24×C61—24no—
76#41×C61—44no—
77#41×C61—84no—
78C2×Q8×C61C2×D412yes—
79C2×Q8×C61C2×D414yes—
80C2×D4×C61—24no—
81C2×D4×C61—28no—
82C2×D4×C61—22no—
83C2×D4×C61—24no—
84C2×D4×C61—22no—
85C2×D4×C61—22no—
86C2×D4×C61—22no—
87C2×D4×C61—88no—
88C2×D4×C61—84no—
89C2×D4×C61C2×D412yes—
90C2×D4×C61C2×D414yes—
91C22×C4×C61—24no—
92C22×C4×C61—22no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D412yes—
95C2×Dic122—22no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C22⋊C4×C61—28no—
100C22⋊C4×C61—24no—
101C22⋊C4×C61—24no—
102C22⋊C4×C61—24no—
103C22⋊C4×C61—24no—
104C42×C61—24no—
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D412yes—
107C42×C61C2411yescharacteristic
108C22⋊Dic61—41no—
109#11758202—84no—
110#11758200—84no—
111C4⋊Dic61—22no—
112C4⋊Dic61—22no—
113#11758196—42no—
114C4×Dic61—41no—
115C2×#92418—22no—
116C2×#92418—24no—
117C2×#92418—42no—
118C23×C61—22no—
119C23×C61—24no—
120C23×C61—84no—
121C23×C61—88no—
122C23×C61#12011yescharacteristic
123Q8×C61—44no—
124Q8×C61—42no—
125D4×C61—44no—
126D4×C61—42no—
127D4×C61—44no—
128D4×C61—44no—
129D4×C61—44no—
130D4×C61—82no—
131D4×C61—84no—
132D4×C61—84no—
133D4×C61—88no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—24no—
137C2×C4×C61—42no—
138C2×C4×C61—48no—
139C2×C4×C61—44no—
140C2×C4×C61#12012yes—
141C2×C4×C61#12011yescharacteristic
142C2×C4×C61C22×D411yescharacteristic
143C2×C4×C61C22×D412yes—
144C2×Dic61—42no—
145Dic122—42no—
146Dic122—42no—
147#92418—42no—
148#92418—44no—
149#19056—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—22no—
152C22×C61—42no—
153C22×C61—44no—
154C22×C61—44no—
155C22×C61—44no—
156C22×C61—44no—
157C22×C61—168no—
158C22×C61C2×#12011yescharacteristic
159C4×C61—21no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—42no—
163C4×C61—44no—
164C4×C61#45512yes—
165Dic61—82no—
166#3549D6111yescharacteristic
167#3216—612no—
168#3135—611no—
169#1659—618no—
170#1536—611no—
171#1530—612no—
172C2×C61—41no—
173C2×C61—82no—
174C2×C61—84no—
175C2×C61C2×#45511yescharacteristic
176C2×C61#301912yes—
177H(3,𝔽4)C2×D6114yes—
178#555—22no—
179#555—21no—
180D42—22no—
181D42—21no—
182#535—22no—
183#535—24no—
184#533—611no—
185#532C2×D6111yescharacteristic
186#532C2×D6112yes—
187#497—614no—
188#491—1221no—
189#490—1221no—
190#489—614no—
191#489—614no—
192#462—1222no—
193#460—1222no—
194#452—1224no—
195#432—612no—
196#416—1222no—
197C2×#97—611no—
198#329—614no—
199#329—618no—
200C61#1905611yescharacteristic, complemented, Sylow
201C2×#41C22⋊D6112yes—
202C2×#41C22⋊D6114yes—
203C22×D4—24no—
204C22×D4—22no—
205C2×Q16—1224no—
206C2×SD16—1224no—
207C22×C8—1221no—
208C4⋊Q8—612no—
209C4⋊Q8C22⋊D6112yes—
210C4⋊Q8C22⋊D6111yescharacteristic
211#127—24no—
212#127C22⋊D6112yes—
213#127C22⋊D6111yescharacteristic
214#125—1221no—
215#124—24no—
216#124—28no—
217#124C22×D6112yes—
218#124C22×D6114yes—
219C22⋊Q8—1222no—
220#121—24no—
221#121—24no—
222#121—24no—
223#121—22no—
224#121—22no—
225#121—22no—
226#120—24no—
227#120—24no—
228#120—28no—
229C4×Q8—1222no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233#117—24no—
234#117—22no—
235#117—1221no—
236C2×C42C22×D6111yescharacteristic
237C4⋊C8—1222no—
238#103—1224no—
239#103—1224no—
240#102—1224no—
241#102—1224no—
242C22⋊C8—1222no—
243#97—612no—
244#97—614no—
245#97—1221no—
246C24—24no—
247#41—44no—
248#41—84no—
249C2×Q8—1222no—
250C2×Q8C2×C22⋊D6112yes—
251C2×Q8C2×C22⋊D6114yes—
252C2×D4—24no—
253C2×D4—28no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—88no—
260C2×D4—84no—
261C2×D4C2×C22⋊D6112yes—
262C2×D4C2×C22⋊D6114yes—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4C2×C22⋊D6111yescharacteristic
266C22×C4C2×C22⋊D6112yes—
267Q16—4884no—
268SD16—4884no—
269C2×C8—1222no—
270C2×C8—1224no—
271C2×C8—2442no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—1222no—
276C4⋊C4—1222no—
277C4⋊C4—2442no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—2441no—
284C42—24no—
285C42—2441no—
286C42C2×C22⋊D6111yescharacteristic
287C42C2×C22⋊D6112yes—
288C42C23×D6111yescharacteristic
289C23—22no—
290C23—24no—
291C23—84no—
292C23—88no—
293C23#1451974611yescharacteristic
294Q8—44no—
295Q8—42no—
296Q8—2442no—
297Q8—2442no—
298D4—44no—
299D4—42no—
300D4—44no—
301D4—44no—
302D4—44no—
303D4—82no—
304D4—84no—
305D4—84no—
306D4—88no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—2442no—
314C2×C4#1451974612yes—
315C2×C4#1451974611yescharacteristic
316C2×C4C22×C22⋊D6111yescharacteristic
317C2×C4C22×C22⋊D6112yes—
318C8—2442no—
319C8—2444no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—168no—
328C22C2×#1451974611yescharacteristic
329C4—21no—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—44no—
334C4—4882no—
335C4#1683994012yes—
336C2—41no—
337C2—82no—
338C2—84no—
339C2C2×#1683994011yescharacteristic
340C2#2145517612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3549C2
socleC22×C61C2×#120
hypercenter#3549D61
hypocenterC61#19056
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3549

Complements

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

NH ≅ G/Ntype
1. #21456830C2semidirect
2. #21456164C2semidirect
3. #21445125C2semidirect
4. #21444454C2semidirect
5. #21444430C2semidirect
7. #16840319C22semidirect
8. #16840157C22semidirect
11. #16840121C22semidirect
12. #16840121C22semidirect
16. #16839819C22semidirect
18. C2×#14519688C22semidirect
19. #16839502C22semidirect
20. #16839502C22semidirect
200. C61#19056semidirect

Maximal subgroups

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

subgroupindex
#214568302
#214561642
#214451252
#214444542
#214444302
C61×#35492
#1905661