Finite Group Databasebeta

← #29658268

Subgroups — #29658268

Order 15616.

996 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 179 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#21456734C212yesmaximal, complemented
2#21456195C211yesmaximal, characteristic, complemented
3#21444908C218yesmaximal, complemented
4#21444484C211yesmaximal, characteristic, complemented
5#21444236C212yesmaximal, complemented
6C61×#3548C211yesmaximal, characteristic, complemented
7C2×#14519774C2211yescharacteristic
8#16840138—22no—
9#16840128—22no—
10#16840110—22no—
11#16840110C2214yes—
12#16839957C2218yes—
13#16839895—24no—
14#16839815C2212yes—
15#16839766—22no—
16C4×#11758194C2211yescharacteristic
17#16839508C2214yescomplemented
18D4⋊#92418C2218yescomplemented
19C61×#562—22no—
20C61×#554—22no—
21C61×#550—22no—
22C61×#547C2214yescomplemented
23C61×#540—22no—
24C61×#537—24no—
25C61×#533C2211yescharacteristic
26C61×#532C2212yes—
27C22×D244—21no—
28C22×D244—22no—
29C2×#11758199—28no—
30C2×#11758194—21no—
31#14519774—22no—
32#14519774—21no—
33#14519774C2312yes—
34C4⋊#92418C2314yes—
35#14519697—28no—
36#14519697—28no—
37C22⋊#92418—22no—
38#14519688—21no—
39C4×#92418—22no—
40C4×#92418C2312yes—
41C2×#41×C61—24no—
42C2×#41×C61—22no—
43C4⋊Q8×C61D412yes—
44C4⋊Q8×C61C2312yes—
45#127×C61—22no—
46#127×C61D412yes—
47#125×C61—22no—
48#125×C61D412yes—
49#124×C61—22no—
50#123×C61—28no—
51C22⋊Q8×C61—28no—
52C22⋊Q8×C61—22no—
53C22⋊Q8×C61—24no—
54#121×C61—28no—
55#121×C61—28no—
56#121×C61—24no—
57#121×C61—22no—
58#121×C61D414yes—
59C4×Q8×C61—22no—
60C4×D4×C61—24no—
61C4×D4×C61—22no—
62C4×D4×C61C2314yes—
63#117×C61D412yes—
64C2×C4⋊C4×C61—24no—
65C2×C42×C61C2311yescharacteristic
66C23×D61—41no—
67#41×C61—84no—
68#41×C61—88no—
69C2×Q8×C61—22no—
70C2×Q8×C61—22no—
71C2×D4×C61—28no—
72C2×D4×C61—24no—
73C2×D4×C61—22no—
74C2×D4×C61—22no—
75C2×D4×C61C2×D414yes—
76C22×C4×C61—28no—
77C22×C4×C61—24no—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D412yes—
80C2×D244—22no—
81C2×D244—24no—
82C2×D244—44no—
83C2×D244—42no—
84C2×D244—41no—
85C2×D244—84no—
86C4⋊C4×C61—28no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61C2×D414yes—
91C4⋊C4×C61C2×D414yes—
92C22⋊C4×C61—28no—
93C22⋊C4×C61—22no—
94C22⋊C4×C61—24no—
95C22⋊C4×C61—24no—
96C42×C61—22no—
97C42×C61C2×D412yes—
98C42×C61C2×D411yescharacteristic
99C42×C61C2411yescharacteristic
100#11758199—88no—
101#11758194—42no—
102C2×#92418—24no—
103C2×#92418—22no—
104C23×C61—24no—
105C23×C61#12011yescharacteristic
106C22×D61—42no—
107C22×D61—82no—
108C22×D61—84no—
109Q8×C61—44no—
110Q8×C61—82no—
111D4×C61—28no—
112D4×C61—44no—
113D4×C61—82no—
114D4×C61—84no—
115D4×C61—88no—
116C2×C4×C61—24no—
117C2×C4×C61—22no—
118C2×C4×C61—24no—
119C2×C4×C61—24no—
120C2×C4×C61—48no—
121C2×C4×C61—88no—
122C2×C4×C61#12012yes—
123C2×C4×C61#12011yescharacteristic
124C2×C4×C61C22×D411yescharacteristic
125C2×C4×C61C22×D412yes—
126D244—44no—
127D244—48no—
128D244—84no—
129D244—82no—
130#92418—42no—
131#92418—44no—
132#18727—611nomaximal, Sylow
133C22×C61—22no—
134C22×C61—21no—
135C22×C61—44no—
136C22×C61—44no—
137C22×C61—44no—
138C22×C61C2×#12011yescharacteristic
139C2×D61—84no—
140C2×D61—82no—
141C2×D61—164no—
142C4×C61—22no—
143C4×C61—21no—
144C4×C61—44no—
145C4×C61—44no—
146C4×C61#45412yes—
147C4×C61#57614yes—
148#3548D6111yescharacteristic, complemented
149#3205—612no—
150#3138—611no—
151#1625—618no—
152#1544—611no—
153#1484—612no—
154C2×C61—41no—
155C2×C61—84no—
156C2×C61#300812yes—
157C2×C61C2×#45411yescharacteristic
158D61—162no—
159#562—22no—
160#554—22no—
161#550—22no—
162#547C2×D6114yescomplemented
163#540—22no—
164#537—24no—
165#533C2×D6111yescharacteristic
166#532C2×D6112yes—
167C2×#127—611no—
168#493—1222no—
169#490—1222no—
170#486—614no—
171#486—1222no—
172#459—618no—
173#447—1224no—
174#431—612no—
175#418—1222no—
176C4×M16—611no—
177#332—614no—
178D4⋊C8—618no—
179C61#1872711yescharacteristic, complemented, Sylow
180C2×#41—24no—
181C2×#41—22no—
182C22×D4—1222no—
183C22×D4—1221no—
184C2×D8—1228no—
185C2×M16—1221no—
186C4⋊Q8C22⋊D6112yes—
187C4⋊Q8C22×D6112yes—
188#127—22no—
189#127—612no—
190#127—1222no—
191#127—1221no—
192#127C22⋊D6112yes—
193#125—22no—
194#125C22⋊D6112yes—
195#124—22no—
196#123—28no—
197C22⋊Q8—28no—
198C22⋊Q8—22no—
199C22⋊Q8—24no—
200#121—28no—
201#121—28no—
202#121—24no—
203#121—22no—
204#121C22⋊D6114yes—
205C4×Q8—22no—
206C4×D4—24no—
207C4×D4—22no—
208C4×D4C22×D6114yes—
209#117C22⋊D6112yes—
210C2×C4⋊C4—24no—
211C2×C42C22×D6111yescharacteristic
212C4⋊C8—614no—
213#102—1228no—
214#102—1228no—
215C22⋊C8—1222no—
216#97—1221no—
217C4×C8—612no—
218C4×C8—1222no—
219C24—2441no—
220#41—84no—
221#41—88no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×D4—28no—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—1222no—
229C2×D4—1224no—
230C2×D4—4884no—
231C2×D4—2444no—
232C2×D4—2442no—
233C2×D4—2441no—
234C2×D4C2×C22⋊D6114yes—
235C22×C4—28no—
236C22×C4—24no—
237C22×C4C2×C22⋊D6112yes—
238C22×C4C2×C22⋊D6111yescharacteristic
239D8—4888no—
240M16—2442no—
241C2×C8—1222no—
242C2×C8—1224no—
243C4⋊C4—28no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4C2×C22⋊D6114yes—
248C4⋊C4C2×C22⋊D6114yes—
249C22⋊C4—28no—
250C22⋊C4—22no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C42—22no—
254C42C2×C22⋊D6112yes—
255C42C2×C22⋊D6111yescharacteristic
256C42C23×D6111yescharacteristic
257C23—24no—
258C23—4884no—
259C23—4882no—
260C23—2442no—
261C23#1451974611yescharacteristic
262Q8—44no—
263Q8—82no—
264D4—28no—
265D4—44no—
266D4—82no—
267D4—84no—
268D4—88no—
269D4—4884no—
270D4—4882no—
271D4—2444no—
272D4—2448no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—48no—
278C2×C4—88no—
279C2×C4#1451974612yes—
280C2×C4#1451974611yescharacteristic
281C2×C4C22×C22⋊D6111yescharacteristic
282C2×C4C22×C22⋊D6112yes—
283C8—2442no—
284C8—2444no—
285C22—21no—
286C22—22no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—9764no—
291C22—4882no—
292C22—4884no—
293C22C2×#1451974611yescharacteristic
294C4—21no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4#1683993512yes—
299C4#1684056514yes—
300C2—41no—
301C2—84no—
302C2—9762no—
303C2#2145508712yes—
304C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#18727
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. #21456195C2semidirect
3. #21444908C2semidirect
4. #21444484C2semidirect
5. #21444236C2semidirect
6. C61×#3548C2semidirect
17. #16839508C22semidirect
18. D4⋊#92418C22semidirect
22. C61×#547C22semidirect
148. #3548D61semidirect
162. #547C2×D61semidirect
179. C61#18727semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214561952
#214449082
#214444842
#214442362
C61×#35482
#1872761