Finite Group Databasebeta

← #29779565

Subgroups — #29779565

Order 15616.

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

normal subgroups only

This group has 862 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#21454129C211yesmaximal, characteristic
2C2×#16839668C211yesmaximal, characteristic
3#21447239C212yesmaximal, complemented
4#21446971C212yesmaximal, complemented
5#21446509C214yesmaximal, complemented
6#21445910C212yesmaximal
7#21445755C212yesmaximal
8C61×#1749C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C2×#14519769C2211yescharacteristic
11C22×C4⋊Dic61C2211yescharacteristic
12C2×#14519702C2214yes—
13C2×#14519702C2214yes—
14#16839668—22no—
15#16839668C2212yes—
16#16839668C2212yes—
17#16839660C2212yes—
18#16839646—22no—
19#16839635—22no—
20#16839628C2211yescharacteristic
21#16839622C2212yes—
22C2×#14519684—22no—
23#16839525C2214yes—
24#16839525C2214yes—
25C22×C42×C61C2211yescharacteristic
26C2×C22⋊C8×C61C2212yes—
27C2×#97×C61C2212yes—
28C61×#336C2212yes—
29C22×Dic122—21no—
30C22×Dic122—24no—
31C22×Dic122—24no—
32C22×Dic122—42no—
33C22×Dic122C2312yes—
34C23×Dic61—21no—
35C23×Dic61—21no—
36#14519769—42no—
37C2×C4⋊Dic61—24no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2312yes—
40C2×C4⋊Dic61C2312yes—
41C2×C4⋊Dic61C2312yes—
42C2×#11758196—22no—
43C2×#11758196—22no—
44C2×C22⋊Dic61—22no—
45#14519702—28no—
46#14519702—44no—
47#14519684—24no—
48#14519684—24no—
49#14519684—22no—
50#14519684—22no—
51#14519684—22no—
52C23×C4×C61D412yes—
53C23×C4×C61C2311yescharacteristic
54C22×C8×C61C2314yes—
55C2×C42×C61—42no—
56C2×C42×C61D412yes—
57C2×C42×C61C2311yescharacteristic
58C2×C42×C61C2311yescharacteristic
59C22⋊C8×C61—24no—
60#97×C61—42no—
61C24×C61C2×D411yescharacteristic
62C22×C4×C61—22no—
63C22×C4×C61—22no—
64C22×C4×C61—24no—
65C22×C4×C61—42no—
66C22×C4×C61—42no—
67C22×C4×C61C2×D412yes—
68C22×C4×C61C2×D412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4114yes—
71C22×C4×C61C2411yescharacteristic
72C22×Dic61—24no—
73C22×Dic61—22no—
74C22×Dic61—22no—
75C22×Dic61—22no—
76C22×Dic61—22no—
77C22×Dic61—42no—
78C22×Dic61—42no—
79C22×Dic61—44no—
80C22×Dic61—42no—
81C2×Dic122—24no—
82C2×Dic122—24no—
83C2×Dic122—24no—
84C2×Dic122—22no—
85C2×Dic122—416no—
86C2×Dic122—42no—
87C2×Dic122—42no—
88C2×Dic122#4114yes—
89C2×C8×C61—24no—
90C2×C8×C61—24no—
91C2×C8×C61—24no—
92C42×C61—22no—
93C42×C61—22no—
94C42×C61—42no—
95C22⋊Dic61—44no—
96C4⋊Dic61—24no—
97C4⋊Dic61—24no—
98C4⋊Dic61—24no—
99C4⋊Dic61—42no—
100#11758196—44no—
101#11758196—44no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—22no—
105C23×C61#12412yes—
106C23×C61#13714yes—
107C23×C61C22×D411yescharacteristic
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—28no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—48no—
118C2×C4×C61#12412yes—
119C2×C4×C61C2×#4112yes—
120C2×C4×C61C2×#4112yes—
121C2×C4×C61H(5,𝔽2)12yes—
122C2×Dic61—24no—
123C2×Dic61—48no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—44no—
134C2×Dic61—84no—
135Dic122—28no—
136Dic122—48no—
137Dic122—44no—
138C8×C61—44no—
139#29112—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—24no—
142C22×C61—22no—
143C22×C61—24no—
144C22×C61#49114yes—
145C22×C61#52511yescharacteristic
146C22×C61C2×#12411yescharacteristic
147C22×C61#53411yescharacteristic
148C22×C61C2×#13712yes—
149C22×C61#57512yes—
150C4×C61—22no—
151C4×C61—42no—
152C4×C61—42no—
153C4×C61#55114yes—
154Dic61—44no—
155Dic61—82no—
156Dic61—82no—
157Dic61—82no—
158#2851—611no—
159C2×#391—611no—
160#2074—612no—
161#2026—612no—
162#1935—614no—
163#1824—612no—
164#1787—612no—
165#1749D6111yescharacteristic
166C2×C61—24no—
167C2×C61#256111yescharacteristic
168C2×C61#311712yes—
169C2×C61C2×#49111yescharacteristic
170C2×C61#314011yescharacteristic
171C2×C61#319612yes—
172C23×Q8—611no—
173C2×#125—611no—
174C22×C4⋊C4—611no—
175C22×C42C2×D6111yescharacteristic
176C2×#103—614no—
177C2×#103—614no—
178C2×C22⋊C8C2×D6112yes—
179C2×#97C2×D6112yes—
180#391—612no—
181#391—612no—
182#391—1222no—
183#389—612no—
184#387—1222no—
185#385—1222no—
186#382—611no—
187#380—612no—
188C2×#95—1222no—
189#340—614no—
190#340—614no—
191#336C2×D6112yes—
192C61#2911211yescharacteristic, complemented, Sylow
193C22×Q8—612no—
194C22×Q8—1224no—
195C22×Q8—1221no—
196C22×Q8—1224no—
197C22×Q8—2442no—
198C23×C4—1221no—
199C23×C4—1221no—
200C23×C4D24412yes—
201C23×C4C22×D6111yescharacteristic
202C22×C8C22×D6114yes—
203#125—2442no—
204C2×C4⋊C4—612no—
205C2×C4⋊C4—612no—
206C2×C4⋊C4—612no—
207C2×C4⋊C4—1224no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1222no—
210C2×C4⋊C4—1222no—
211C2×C22⋊C4—1222no—
212C2×C42—42no—
213C2×C42D24412yes—
214C2×C42C22×D6111yescharacteristic
215C2×C42C22×D6111yescharacteristic
216#103—1228no—
217#103—2444no—
218C22⋊C8—24no—
219#97—42no—
220#95—1224no—
221#95—1224no—
222#95—1222no—
223#95—1222no—
224#95—1222no—
225C24C2×D24411yescharacteristic
226C2×Q8—614no—
227C2×Q8—1224no—
228C2×Q8—1222no—
229C2×Q8—1224no—
230C2×Q8—1224no—
231C2×Q8—24416no—
232C2×Q8—2442no—
233C2×Q8—2442no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—1224no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—2444no—
245C22×C4—2442no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4C2×D24412yes—
249C22×C4C2×D24412yes—
250C22×C4C2×D24411yescharacteristic
251C22×C4#1175822214yes—
252C22×C4C23×D6111yescharacteristic
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C4⋊C4—1224no—
257C4⋊C4—1224no—
258C4⋊C4—1224no—
259C4⋊C4—2444no—
260C4⋊C4—2444no—
261C4⋊C4—2442no—
262C22⋊C4—2444no—
263C42—22no—
264C42—22no—
265C42—42no—
266C23—21no—
267C23—22no—
268C23—21no—
269C23#1451976512yes—
270C23#1451981014yes—
271C23C22×D24411yescharacteristic
272Q8—1228no—
273Q8—2448no—
274Q8—2444no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—28no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—48no—
285C2×C4—1224no—
286C2×C4—4884no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2444no—
294C2×C4—2442no—
295C2×C4—2448no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4#1451976512yes—
299C2×C4#41×D6112yes—
300C2×C4C2×#1175822212yes—
301C2×C4#1451982212yes—
302C8—44no—
303C22—24no—
304C22—22no—
305C22—24no—
306C22—22no—
307C22#1684013114yes—
308C22#1684027411yescharacteristic
309C22C2×#1451976511yescharacteristic
310C22#1684032411yescharacteristic
311C22C2×#1451981012yes—
312C22#1684056212yes—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—4882no—
317C4—4882no—
318C4—4882no—
319C4—2444no—
320C4#1684044114yes—
321C2—24no—
322C2#2145097211yescharacteristic
323C2#2145599212yes—
324C2C2×#1684013111yescharacteristic
325C2#2145620611yescharacteristic
326C2#2145665112yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1749C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1749D61
hypocenterC61#29112
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #21447239C2semidirect
4. #21446971C2semidirect
5. #21446509C2semidirect
192. C61#29112semidirect

Maximal subgroups

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

subgroupindex
#214541292
C2×#168396682
#214472392
#214469712
#214465092
#214459102
#214457552
C61×#17492
#2911261