Finite Group Databasebeta

← #29570557

Subgroups — #29570557

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#92418C211yesmaximal, characteristic
2C4×#14519692C211yesmaximal, characteristic
3#21445593C212yesmaximal, complemented
4#21443665C212yesmaximal, complemented
5C61×#1459C211yesmaximal, characteristic
6C23×#92418C411yescharacteristic, complemented
7C23×#92418C2211yescharacteristic
8D4×#92418—22no—
9D4×#92418—22no—
10C2×C4⋊#92418C411yescharacteristic, complemented
11C2×#14519692C412yes—
12C2×C22⋊#92418C411yescharacteristic, complemented
13C2×C22⋊#92418C2211yescharacteristic
14C2×C4×#92418C411yescharacteristic, complemented
15C2×C8×Dic61—22no—
16C4×C22⋊Dic61C2212yes—
17#16839529C412yes—
18#16839516—22no—
19C2×C4×D4×C61C2211yescharacteristic
20C2×C22⋊C8×C61C2212yes—
21C61×#324C412yescomplemented
22C61×#323C412yescomplemented
23C22×#92418—21no—
24C22×#92418—21no—
25C22×#92418—42no—
26C22×#92418—42no—
27C22×#92418C2×C411yescharacteristic
28C22×#92418C2×C411yescharacteristic
29C22×#92418C2×C411yescharacteristic
30C22×#92418C2×C411yescharacteristic
31C22×#92418D412yes—
32C22×#92418D412yes—
33C2×C22⋊Dic61C2×C414yes—
34C2×C4×Dic61—22no—
35C4⋊#92418—22no—
36#14519692C818yescomplemented
37C22⋊#92418—22no—
38C22⋊#92418—22no—
39C4×#92418—21no—
40C4×#92418—21no—
41C8×Dic61—44no—
42#14519684—22no—
43C22×D4×C61C2×C411yescharacteristic
44C23×C4×C61C2×C411yescharacteristic
45C23×C4×C61C2311yescharacteristic
46C22×C8×C61—22no—
47C4×D4×C61—22no—
48C4×D4×C61—22no—
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61C2×C411yescharacteristic
52C2×C42×C61C2×C411yescharacteristic
53C22⋊C8×C61—22no—
54C22⋊C8×C61C2×C414yes—
55C24×C61C4211yescharacteristic
56C24×C61C22×C411yescharacteristic
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61C2×C814yes—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61—41no—
66C22×C4×C61C4211yescharacteristic
67C22×C4×C61C4211yescharacteristic
68C22×C4×C61C4211yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C2×D412yes—
72C22×C4×C61#4112yes—
73C22×Dic61—24no—
74C2×C8×C61—24no—
75C2×C8×C61—42no—
76C2×C8×C61—42no—
77C4⋊C4×C61—22no—
78C22⋊C4×C61—22no—
79C22⋊C4×C61—22no—
80C42×C61—21no—
81C42×C61—21no—
82C22⋊Dic61—24no—
83C22⋊Dic61C2×C818yescomplemented
84C4×Dic61—42no—
85C4×Dic61—42no—
86C2×#92418—24no—
87C2×#92418—22no—
88C2×#92418—21no—
89C2×#92418—21no—
90C2×#92418—41no—
91C2×#92418—42no—
92C2×#92418—42no—
93C2×#92418—42no—
94C2×#92418—41no—
95C2×#92418—41no—
96C2×#92418C22⋊C411yescharacteristic
97C2×#92418C22⋊C412yes—
98C2×#92418C22⋊C411yescharacteristic
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61C4×C812yescomplemented
106C23×C61C2×C4211yescharacteristic
107C23×C61C4×D412yes—
108C23×C61C22×C812yes—
109C23×C61#13112yes—
110D4×C61—28no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—21no—
116C2×C4×C61—22no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—41no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61C4×C812yescomplemented
127C2×C4×C61C2×C22⋊C411yescharacteristic
128C2×C4×C61#11711yescharacteristic
129C2×C4×C61C4×D412yes—
130C2×Dic61—28no—
131C2×Dic61—44no—
132C2×Dic61—44no—
133C8×C61—44no—
134#92418—21no—
135#92418—22no—
136#92418—41no—
137#92418—41no—
138#92418#9912yescomplemented
139C8×#99—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61C4×C22⋊C411yescharacteristic
152C22×C61C2×C4×C811yescharacteristic
153C22×C61#40511yescharacteristic
154C22×C61C8×D414yes—
155C4×C61—22no—
156C4×C61—22no—
157C4×C61—21no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61C2×#9911yescharacteristic
161C4×C61#41011yescharacteristic
162Dic61—48no—
163C2×C8×D4—611no—
164C4×#99—611no—
165C8×C22⋊C4—612no—
166#1459D6111yescharacteristic
167#1282—612no—
168C2×C61—24no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—42no—
172C2×C61C8×C22⋊C411yescharacteristic
173C2×C61C4×#9911yescharacteristic
174C2×C61#175711yescharacteristic
175C23×C8—611no—
176C23×C8—611no—
177C2×C4×D4C2×D6111yescharacteristic
178C8×D4—1222no—
179C8×D4—1222no—
180C2×C4⋊C8—611no—
181C2×#99—612no—
182C2×C22⋊C8—611no—
183C2×C22⋊C8—611no—
184C2×C22⋊C8C2×D6112yes—
185C2×C4×C8—611no—
186C2×C4×C8—1222no—
187C4×C22⋊C4—612no—
188#342—612no—
189#336—1222no—
190#324Dic6112yescomplemented
191#323Dic6112yescomplemented
192C61C8×#9911yescharacteristic, complemented, Sylow
193C22×D4C4×D6111yescharacteristic
194C23×C4C4×D6111yescharacteristic
195C23×C4C22×D6111yescharacteristic
196C22×C8—22no—
197C22×C8—611no—
198C22×C8—612no—
199C22×C8—611no—
200C22×C8—612no—
201C22×C8—611no—
202C22×C8—611no—
203C22×C8—1221no—
204C22×C8—1221no—
205C22×C8—2442no—
206C22×C8—2442no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4C2×Dic6111yescharacteristic
210C2×C22⋊C4—614no—
211C2×C22⋊C4C4×D6111yescharacteristic
212C2×C22⋊C4C2×Dic6111yescharacteristic
213C2×C42—1222no—
214C2×C42C4×D6111yescharacteristic
215C4⋊C8—1222no—
216#99—618no—
217C22⋊C8—22no—
218C22⋊C8—1222no—
219C22⋊C8—1222no—
220C22⋊C8C2×Dic6114yes—
221C4×C8—1221no—
222C4×C8—1221no—
223C4×C8—2444no—
224#95—1222no—
225C24C4×Dic6111yescharacteristic
226C24C2×C4×D6111yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4C8×D6114yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—1224no—
237C22×C4C4×Dic6111yescharacteristic
238C22×C4C4×Dic6111yescharacteristic
239C22×C4C4×Dic6111yescharacteristic
240C22×C4C2×C4×D6111yescharacteristic
241C22×C4D4×D6112yes—
242C22×C4#1175822412yes—
243C22×C4C22×Dic6111yescharacteristic
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—611no—
248C2×C8—611no—
249C2×C8—612no—
250C2×C8—1224no—
251C2×C8—1222no—
252C2×C8—1221no—
253C2×C8—1221no—
254C2×C8—2442no—
255C2×C8—2441no—
256C2×C8—2442no—
257C2×C8—2442no—
258C2×C8—2441no—
259C2×C8—2441no—
260C4⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—618no—
264C22⋊C4—1224no—
265C42—21no—
266C42—21no—
267C42—2442no—
268C42—2442no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C8×Dic6112yescomplemented
276C23C2×C4×Dic6111yescharacteristic
277C23#1451973712yes—
278C23C2×C8×D6112yes—
279C23#1451978512yes—
280D4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—1228no—
297C2×C4—2444no—
298C2×C4—2444no—
299C2×C4C8×Dic6112yescomplemented
300C2×C4C22⋊C4×D6111yescharacteristic
301C2×C4#1451973311yescharacteristic
302C2×C4D4×Dic6112yes—
303C8—44no—
304C8—612no—
305C8—1222no—
306C8—1221no—
307C8—2441no—
308C8—2441no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22C22⋊C4×Dic6111yescharacteristic
321C22C2×C8×Dic6111yescharacteristic
322C22#1683972011yescharacteristic
323C22#1683982514yes—
324C4—22no—
325C4—22no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—2448no—
330C4#99×D6111yescharacteristic
331C4#1683973411yescharacteristic
332C2—24no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2144559311yescharacteristic
337C2#99×Dic6111yescharacteristic
338C2#2144562211yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC22×C61C2×C4×C8
FrattiniC23×C4C22×D61
FittingC61×#1459C2
socleC22×C61C4×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#1459D61
hypocenterC61C8×#99
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21445593C2semidirect
4. #21443665C2semidirect
6. C23×#92418C4semidirect
10. C2×C4⋊#92418C4semidirect
12. C2×C22⋊#92418C4semidirect
14. C2×C4×#92418C4semidirect
21. C61×#324C4semidirect
22. C61×#323C4semidirect
36. #14519692C8semidirect
83. C22⋊Dic61C2×C8semidirect
105. C23×C61C4×C8semidirect
126. C2×C4×C61C4×C8semidirect
138. #92418#99semidirect
190. #324Dic61semidirect
191. #323Dic61semidirect
192. C61C8×#99semidirect
275. C23C8×Dic61semidirect
299. C2×C4C8×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#924182
C4×#145196922
#214455932
#214436652
C61×#14592
C8×#9961