Finite Group Databasebeta

← #29679423

Subgroups — #29679423

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839736C212yesmaximal
2#21448437C212yesmaximal, complemented
3#21445807C218yesmaximal, complemented
4#21445552C212yesmaximal, complemented
5C2×C61×#520C211yesmaximal, characteristic
6C2×#14519736C2214yes—
7#16839736—28no—
8C2×#14519692C2218yes—
9#16839661—28no—
10#16839622C2218yes—
11#16839529C2218yes—
12C22×#41×C61C2211yescharacteristic
13C61×#520—28no—
14C2×C4×Q8×C61C412yescomplemented
15C2×C4×D4×C61C412yescomplemented
16C2×C4×D4×C61C2214yes—
17C2×#117×C61C412yescomplemented
18C2×#117×C61C2211yescharacteristic
19C22×C4⋊C4×C61C412yescomplemented
20C22×C4⋊C4×C61C2211yescharacteristic
21#14519736—28no—
22#14519736—28no—
23C2×C4⋊Dic61—24no—
24C2×C4⋊Dic61—28no—
25C2×C22⋊Dic61C2318yes—
26C2×C4×Dic61—24no—
27#14519692—216no—
28#14519684—28no—
29C2×#41×C61—24no—
30C2×#41×C61—24no—
31C2×#41×C61D414yes—
32C2×#41×C61Q814yes—
33C22×Q8×C61C2×C411yescharacteristic
34C22×D4×C61C2×C411yescharacteristic
35C22×D4×C61C2312yes—
36C23×C4×C61C2×C412yes—
37C23×C4×C61C2×C414yes—
38C23×C4×C61C2312yes—
39C23×C4×C61C2311yescharacteristic
40C4×Q8×C61—28no—
41C4×D4×C61—216no—
42C4×D4×C61—28no—
43#117×C61—24no—
44#117×C61—28no—
45C2×C4⋊C4×C61—28no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C4⋊C4×C61C2×C414yes—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C412yes—
50C2×C4⋊C4×C61C2×C412yes—
51C2×C4⋊C4×C61D418yes—
52C2×C22⋊C4×C61C2×C414yes—
53C2×C22⋊C4×C61C2312yes—
54C2×C42×C61C2×C414yes—
55C2×C42×C61C2×C412yes—
56C24×C61C22×C412yes—
57C24×C61C2411yescharacteristic
58#41×C61—216no—
59#41×C61—216no—
60C2×Q8×C61—22no—
61C2×Q8×C61—22no—
62C2×Q8×C61C4⋊C414yescomplemented
63C2×D4×C61—24no—
64C2×D4×C61—24no—
65C2×D4×C61—22no—
66C2×D4×C61—22no—
67C2×D4×C61C4⋊C414yescomplemented
68C2×D4×C61C2×D414yes—
69C2×D4×C61C2×Q814yes—
70C22×C4×C61—24no—
71C22×C4×C61—24no—
72C22×C4×C61—28no—
73C22×C4×C61—82no—
74C22×C4×C61—84no—
75C22×C4×C61C4⋊C418yescomplemented
76C22×C4×C61C22×C412yes—
77C22×C4×C61C22×C412yes—
78C22×C4×C61C22×C412yes—
79C22×C4×C61C22×C414yes—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D418yes—
83C22×C4×C61C2×D414yes—
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×Q812yes—
86C22×C4×C61#4114yes—
87C22×Dic61—28no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—28no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61C22⋊C418yes—
92C4⋊C4×C61C22⋊C418yes—
93C22⋊C4×C61—24no—
94C22⋊C4×C61—28no—
95C42×C61—28no—
96C42×C61—24no—
97C22⋊Dic61—28no—
98C22⋊Dic61—28no—
99C4⋊Dic61—48no—
100C4⋊Dic61—88no—
101C4×Dic61—48no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61C2×C4⋊C414yes—
109C23×C61C23×C411yescharacteristic
110C23×C61C22×D412yes—
111C23×C61C22×D411yescharacteristic
112C23×C61C22×Q811yescharacteristic
113C23×C61C2×#4112yes—
114Q8×C61—28no—
115D4×C61—28no—
116D4×C61—28no—
117D4×C61—28no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—216no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—48no—
124C2×C4×C61—44no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—88no—
128C2×C4×C61C2×C22⋊C418yes—
129C2×C4×C61C2×C22⋊C412yes—
130C2×C4×C61C2×C22⋊C412yes—
131C2×C4×C61C2×C4⋊C414yes—
132C2×C4×C61C2×C4⋊C412yes—
133C2×C4×C61C2×C4⋊C412yes—
134C2×C4×C61#11712yes—
135C2×C4×C61#11712yes—
136C2×C4×C61#12118yes—
137C2×C4×C61C22⋊Q818yes—
138C2×Dic61—48no—
139C2×Dic61—48no—
140C2×Dic61—48no—
141#20634—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—21no—
146C22×C61—24no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—42no—
150C22×C61—44no—
151C22×C61—44no—
152C22×C61—42no—
153C22×C61—42no—
154C22×C61C22×C22⋊C411yescharacteristic
155C22×C61C22×C4⋊C411yescharacteristic
156C22×C61C2×#11711yescharacteristic
157C22×C61C2×#12112yes—
158C22×C61C2×C22⋊Q812yes—
159C4×C61—28no—
160C4×C61—24no—
161C4×C61—42no—
162C4×C61—42no—
163C4×C61#38014yes—
164C4×C61#38014yes—
165Dic61—88no—
166C2×#520D6111yescharacteristic
167C2×#410—612no—
168#2316—612no—
169#1799—618no—
170#1740—612no—
171C2×C61—22no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—22no—
175C2×C61—44no—
176C2×C61C2×#38011yescharacteristic
177C2×C61#288311yescharacteristic
178C2×C61#288411yescharacteristic
179C22×#41C2×D6111yescharacteristic
180#520—28no—
181C2×C4×Q8Dic6112yescomplemented
182C2×C4×D4Dic6112yescomplemented
183C2×C4×D4C2×D6114yes—
184C2×#117—614no—
185C2×#117Dic6112yescomplemented
186C2×#117C2×D6111yescharacteristic
187C22×C4⋊C4Dic6112yescomplemented
188C22×C4⋊C4C2×D6111yescharacteristic
189#410—1228no—
190C2×#99—618no—
191#389—1228no—
192#380—618no—
193#342—618no—
194C61#2063411yescharacteristic, complemented, Sylow
195C2×#41—24no—
196C2×#41—24no—
197C2×#41Dic12214yes—
198C2×#41D24414yes—
199C22×Q8C2×Dic6111yescharacteristic
200C22×D4C2×Dic6111yescharacteristic
201C22×D4C22×D6112yes—
202C23×C4C2×Dic6112yes—
203C23×C4C2×Dic6114yes—
204C23×C4C22×D6112yes—
205C23×C4C22×D6111yescharacteristic
206C4×Q8—28no—
207C4×D4—216no—
208C4×D4—28no—
209#117—24no—
210#117—28no—
211#117—1228no—
212#117—1228no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—1228no—
215C2×C4⋊C4—1224no—
216C2×C4⋊C4C2×Dic6111yescharacteristic
217C2×C4⋊C4C2×Dic6114yes—
218C2×C4⋊C4C2×Dic6111yescharacteristic
219C2×C4⋊C4C2×Dic6112yes—
220C2×C4⋊C4C2×Dic6112yes—
221C2×C4⋊C4C22⋊D6118yes—
222C2×C22⋊C4—618no—
223C2×C22⋊C4C2×Dic6114yes—
224C2×C22⋊C4C22×D6112yes—
225C2×C42—1224no—
226C2×C42C2×Dic6114yes—
227C2×C42C2×Dic6112yes—
228#99—12216no—
229#95—1228no—
230C24C22×Dic6112yes—
231C24C23×D6111yescharacteristic
232#41—216no—
233#41—216no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8C4⋊Dic6114yescomplemented
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4C4⋊Dic6114yescomplemented
242C2×D4C2×Dic12214yes—
243C2×D4C2×D24414yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—1228no—
250C22×C4C4⋊Dic6118yescomplemented
251C22×C4C2×Dic12212yes—
252C22×C4C2×D24412yes—
253C22×C4#1175822214yes—
254C22×C4C22×Dic6112yes—
255C22×C4C22×Dic6112yes—
256C22×C4C22×Dic6112yes—
257C22×C4C22×Dic6114yes—
258C22×C4C22×Dic6111yescharacteristic
259C22×C4C22×Dic6111yescharacteristic
260C22×C4C2×C22⋊D6118yes—
261C22×C4C2×C22⋊D6114yes—
262C4⋊C4—22no—
263C4⋊C4—28no—
264C4⋊C4—22no—
265C4⋊C4—4888no—
266C4⋊C4—2448no—
267C4⋊C4C22⋊Dic6118yes—
268C4⋊C4C22⋊Dic6118yes—
269C22⋊C4—24no—
270C22⋊C4—28no—
271C22⋊C4—1228no—
272C22⋊C4—1228no—
273C42—28no—
274C42—24no—
275C42—2448no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23C2×C4⋊Dic6114yes—
283C23C23×Dic6111yescharacteristic
284C23C22×C22⋊D6112yes—
285C23C22×D24411yescharacteristic
286C23C22×Dic12211yescharacteristic
287C23C2×#1175822212yes—
288Q8—28no—
289D4—28no—
290D4—28no—
291D4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—216no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—48no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—88no—
302C2×C4—2448no—
303C2×C4—2448no—
304C2×C4—2448no—
305C2×C4C2×C22⋊Dic6118yes—
306C2×C4C2×C22⋊Dic6112yes—
307C2×C4C2×C22⋊Dic6112yes—
308C2×C4C2×C4⋊Dic6114yes—
309C2×C4C2×C4⋊Dic6112yes—
310C2×C4C2×C4⋊Dic6112yes—
311C2×C4#1451973612yes—
312C2×C4#1451973612yes—
313C2×C4#1451975118yes—
314C2×C4#1451975618yes—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—24no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22C22×C22⋊Dic6111yescharacteristic
328C22C22×C4⋊Dic6111yescharacteristic
329C22C2×#1451973611yescharacteristic
330C22C2×#1451975112yes—
331C22C2×#1451975612yes—
332C4—24no—
333C4—28no—
334C4—42no—
335C4—42no—
336C4—4888no—
337C4#1683962214yes—
338C4#1683962214yes—
339C2—22no—
340C2—21no—
341C2—21no—
342C2—22no—
343C2—44no—
344C2C2×#1683962211yescharacteristic
345C2#2145428011yescharacteristic
346C2#2145428311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC2×C61×#520C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#41×C61C22
cosocleC24×C61C24
hypercenterC2×#520D61
hypocenterC61#20634
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
2. #21448437C2semidirect
3. #21445807C2semidirect
4. #21445552C2semidirect
14. C2×C4×Q8×C61C4semidirect
15. C2×C4×D4×C61C4semidirect
17. C2×#117×C61C4semidirect
19. C22×C4⋊C4×C61C4semidirect
62. C2×Q8×C61C4⋊C4semidirect
67. C2×D4×C61C4⋊C4semidirect
75. C22×C4×C61C4⋊C4semidirect
181. C2×C4×Q8Dic61semidirect
182. C2×C4×D4Dic61semidirect
185. C2×#117Dic61semidirect
187. C22×C4⋊C4Dic61semidirect
194. C61#20634semidirect
236. C2×Q8C4⋊Dic61semidirect
241. C2×D4C4⋊Dic61semidirect
250. C22×C4C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168397362
#214484372
#214458072
#214455522
C2×C61×#5202
#2063461