Finite Group Databasebeta

← #29654861

Subgroups — #29654861

Order 15616.

998 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 191 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#21459629C211yesmaximal, characteristic, complemented
2#21456760C212yesmaximal
3#21444758C218yesmaximal, complemented
4#21444317C212yesmaximal, complemented
5#21444143C211yesmaximal, characteristic, complemented
6C61×#2548C211yesmaximal, characteristic
7Q8×Dic122—22no—
8#16840491—22no—
9#16840488—24no—
10#16840393—21no—
11#16840393C2214yes—
12#16840319—22no—
13#16840319C2212yes—
14#16840240—21no—
15#16840134—22no—
16#16840108—22no—
17#16839959—24no—
18#16839913C2218yes—
19#16839815C2212yes—
20#16839498C2218yes—
21#16839488C2214yes—
22C2×#127×C61C2211yescharacteristic
23C2×#124×C61—21no—
24C2×C4×Q8×C61C2211yescharacteristic
25C61×#394C2214yes—
26C61×#390—21no—
27C61×#390—22no—
28C4×C4⋊C4×C61C2211yescharacteristic
29C2×#11758200—28no—
30#14519777—22no—
31#14519777—22no—
32#14519777D412yes—
33#14519777D412yes—
34#14519776—24no—
35#14519776—28no—
36#14519769—24no—
37#14519769—22no—
38#14519768—24no—
39#14519768—28no—
40#14519756—22no—
41#14519756—24no—
42#14519756C2314yes—
43Q8×Dic61—24no—
44Q8×Dic61—24no—
45C4×Dic122—24no—
46C4×Dic122—24no—
47C4×Dic122—22no—
48C4×Dic122D414yes—
49#14519736—24no—
50#14519736—22no—
51C4⋊#92418—22no—
52#14519698—28no—
53C22⋊#92418C2314yes—
54C4×#92418—22no—
55C22×Q8×C61D411yescharacteristic
56C22×D4×C61—21no—
57C22×D4×C61—21no—
58C22×D4×C61C2312yes—
59#127×C61—22no—
60#127×C61—22no—
61#124×C61—44no—
62C4×Q8×C61—22no—
63C4×Q8×C61—22no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61C2312yes—
67C2×C22⋊C4×C61—24no—
68C2×C22⋊C4×C61—24no—
69C2×C42×C61—21no—
70C2×C42×C61D412yes—
71C2×C42×C61C2311yescharacteristic
72#95×C61C2312yes—
73C24×C61—22no—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61SD1614yescomplemented
77C2×D4×C61—24no—
78C2×D4×C61—44no—
79C2×D4×C61—41no—
80C2×D4×C61—42no—
81C2×D4×C61—41no—
82C2×D4×C61—42no—
83C2×D4×C61—42no—
84C2×D4×C61—82no—
85C22×C4×C61—22no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2411yescharacteristic
91C2×Dic122—22no—
92C2×Dic122—24no—
93C2×Dic122C2×D414yes—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—42no—
97C22⋊C4×C61—48no—
98C22⋊C4×C61—84no—
99C42×C61—22no—
100C42×C61—22no—
101C42×C61—42no—
102C42×C61C2×D412yes—
103C42×C61C2×D412yes—
104C22⋊Dic61—24no—
105#11758200—88no—
106C4⋊Dic61—24no—
107C4⋊Dic61—24no—
108C4⋊Dic61—24no—
109C4⋊Dic61—22no—
110C4⋊Dic61—22no—
111C4⋊Dic61—22no—
112C4⋊Dic61C2×D414yes—
113#11758196—28no—
114#11758196—24no—
115#11758196—24no—
116C4×Dic61—28no—
117C4×Dic61—24no—
118C2×#92418—24no—
119C23×C61—44no—
120C23×C61—44no—
121C23×C61—42no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61C22×D411yescharacteristic
125Q8×C61—28no—
126D4×C61—48no—
127D4×C61—82no—
128D4×C61—84no—
129D4×C61—82no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61#12011yescharacteristic
140C2×C4×C61#12012yes—
141C2×C4×C61#12011yescharacteristic
142C2×C4×C61C2×SD1614yes—
143C2×C4×C61C2×SD1612yes—
144C2×C4×C61#13712yes—
145C2×C4×C61C22×D412yes—
146C2×Dic61—24no—
147C2×Dic61—24no—
148Dic122—44no—
149Dic122—44no—
150Dic122—84no—
151#92418—44no—
152#18403—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—84no—
156C22×C61—84no—
157C22×C61—82no—
158C22×C61—82no—
159C22×C61—82no—
160C22×C61C2×#12011yescharacteristic
161C22×C61C22×SD1611yescharacteristic
162C22×C61C2×#13711yescharacteristic
163C4×C61—24no—
164C4×C61—21no—
165C4×C61—22no—
166C4×C61—42no—
167C4×C61—82no—
168C4×C61#44814yes—
169C4×C61#45412yes—
170Dic61—44no—
171Dic61—44no—
172#3502—611no—
173#3208—612no—
174#2548D6111yescharacteristic
175#1599—618no—
176#1503—612no—
177#1461—611no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61—162no—
181C2×C61C2×#44811yescharacteristic
182C2×C61C2×#45411yescharacteristic
183C2×C61#302711yescharacteristic
184Q82—1222no—
185#557—1224no—
186#557—1222no—
187#544—614no—
188#544—1221no—
189#533—612no—
190#533—1222no—
191C2×#127C2×D6111yescharacteristic
192C2×#124—21no—
193#519—1221no—
194C2×C4×Q8C2×D6111yescharacteristic
195#492—1222no—
196#486—1222no—
197#460—1224no—
198#450—618no—
199#431—612no—
200#394C2×D6114yes—
201#390—21no—
202#390—22no—
203C4×C4⋊C4C2×D6111yescharacteristic
204#328—618no—
205#324—614no—
206C61#1840311yescharacteristic, complemented, Sylow
207C22×Q8C22⋊D6111yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C22×D6112yes—
211C2×SD16—1228no—
212C4⋊Q8—612no—
213C4⋊Q8—612no—
214C4⋊Q8—1222no—
215C4⋊Q8—1222no—
216C4⋊Q8—1228no—
217C4⋊Q8—1224no—
218#127—22no—
219#127—22no—
220#125—1224no—
221#125—1228no—
222#125—1224no—
223#125—1222no—
224#124—44no—
225C22⋊Q8—614no—
226C22⋊Q8—1222no—
227C22⋊Q8—1224no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—614no—
231C4×Q8—1224no—
232C4×Q8—1224no—
233C4×Q8—1224no—
234C4×Q8—1224no—
235C4×Q8—1222no—
236#117—1222no—
237#117—1224no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D6111yescharacteristic
240C2×C4⋊C4C22×D6112yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C42—21no—
244C2×C42C22⋊D6112yes—
245C2×C42C22×D6111yescharacteristic
246C4⋊C8—1222no—
247#102—1228no—
248C22⋊C8—614no—
249C4×C8—1222no—
250#95C22×D6112yes—
251C24—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—614no—
255C2×Q8—1224no—
256C2×Q8—1222no—
257C2×Q8#1175820014yescomplemented
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—82no—
266C22×C4—22no—
267C22×C4C2×C22⋊D6111yescharacteristic
268C22×C4C2×C22⋊D6112yes—
269C22×C4C2×C22⋊D6112yes—
270C22×C4C2×C22⋊D6111yescharacteristic
271C22×C4C23×D6111yescharacteristic
272SD16—4888no—
273C2×C8—1224no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—614no—
278C4⋊C4—1228no—
279C4⋊C4—1224no—
280C4⋊C4—1224no—
281C4⋊C4—1224no—
282C4⋊C4—1224no—
283C4⋊C4—1224no—
284C4⋊C4—1222no—
285C4⋊C4—1222no—
286C4⋊C4—1222no—
287C22⋊C4—48no—
288C22⋊C4—84no—
289C22⋊C4—1224no—
290C42—22no—
291C42—22no—
292C42—42no—
293C42—1228no—
294C42—1224no—
295C42D4×D6112yes—
296C42D4×D6112yes—
297C23—44no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23C22×C22⋊D6111yescharacteristic
303Q8—28no—
304Q8—4884no—
305Q8—2444no—
306Q8—2444no—
307D4—48no—
308D4—82no—
309D4—84no—
310D4—82no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—1224no—
321C2×C4—1224no—
322C2×C4#1451974511yescharacteristic
323C2×C4#1451974512yes—
324C2×C4#1451974511yescharacteristic
325C2×C4C2×#1175820014yes—
326C2×C4C2×#1175820012yes—
327C2×C4#1451980712yes—
328C2×C4C2×D4×D6112yes—
329C8—2444no—
330C22—21no—
331C22—21no—
332C22—84no—
333C22—84no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22C2×#1451974511yescharacteristic
338C22C22×#1175820011yescharacteristic
339C22C2×#1451980711yescharacteristic
340C4—21no—
341C4—22no—
342C4—24no—
343C4—42no—
344C4—82no—
345C4—2444no—
346C4—2444no—
347C4#1683990014yes—
348C4#1683993412yes—
349C2—21no—
350C2—21no—
351C2—162no—
352C2C2×#1683990011yescharacteristic
353C2C2×#1683993411yescharacteristic
354C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2548C2
socleC22×C61C2×#120
prime radicalC2×#127×C61C22
hypercenter#2548D61
hypocenterC61#18403
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #21459629C2semidirect
3. #21444758C2semidirect
4. #21444317C2semidirect
5. #21444143C2semidirect
76. C2×Q8×C61SD16semidirect
206. C61#18403semidirect
257. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214567602
#214447582
#214443172
#214441432
C61×#25482
#1840361