Finite Group Databasebeta

← #21449266

Subgroups — #21449266

Order 7808.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14519753C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D61C211yesmaximal, characteristic
3C2×C4⋊C4×D61C211yesmaximal, characteristic, complemented
4#16839693C212yesmaximal, complemented
5#16839685C212yesmaximal, complemented
6#16839669C211yesmaximal, characteristic, complemented
7#16839661C211yesmaximal, characteristic, complemented
8#16839638C212yesmaximal
9#16839622C211yesmaximal, characteristic, complemented
10#16839619C212yesmaximal, complemented
11C2×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
12C22×C22⋊D61C2211yescharacteristic
13C22×C4×D61C2212yes—
14C22×C4×D61C2211yescharacteristic
15#14519753—24no—
16C4×C22⋊D61—24no—
17C2×C4⋊Dic61C2211yescharacteristic
18C2×C4⋊Dic61C2211yescharacteristic
19C2×C4⋊Dic61C2212yes—
20C2×#11758196C2212yes—
21C2×#11758196C2212yes—
22C2×#11758196C2211yescharacteristic
23C4⋊C4×D61—24no—
24C2×C22⋊Dic61C2211yescharacteristic
25C2×C22⋊Dic61C2212yes—
26C2×#11758198C2211yescharacteristic
27C2×#11758198C2212yes—
28C2×C4×Dic61C2211yescharacteristic
29#14519684C2212yes—
30#14519684C2212yes—
31#14519684C2212yes—
32#14519684C2212yes—
33C22×Q8×C61C2211yescharacteristic
34C23×C4×C61C2211yescharacteristic
35C22⋊Q8×C61—24no—
36C2×C4⋊C4×C61C2212yes—
37C2×C4⋊C4×C61C2211yescharacteristic
38C2×C22⋊C4×C61C2212yes—
39C24×C61C2311yescharacteristic
40C23×D61C2311yescharacteristic
41C2×Q8×C61—22no—
42C2×Q8×C61—22no—
43C2×Q8×C61—22no—
44C22×C4×C61—22no—
45C22×C4×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61C2312yes—
48C22×C4×C61C2311yescharacteristic
49C22×C4×C61C2312yes—
50C22×C4×C61C2311yescharacteristic
51C2×C22⋊D61—22no—
52C2×C22⋊D61—22no—
53C2×C22⋊D61Q814yescomplemented
54C22×Dic61C2312yes—
55C22×Dic61C2311yescharacteristic
56C22×Dic61C2312yes—
57C22×Dic61C2311yescharacteristic
58C22×Dic61C2311yescharacteristic
59C2×C4×D61—22no—
60C2×C4×D61—22no—
61C2×C4×D61—24no—
62C2×C4×D61—24no—
63C2×C4×D61—42no—
64C2×C4×D61D414yes—
65C4⋊C4×C61—24no—
66C4⋊C4×C61D414yescomplemented
67C22⋊C4×C61—24no—
68C22⋊Dic61—22no—
69C22⋊Dic61—42no—
70#11758198—22no—
71#11758198—24no—
72C4⋊Dic61—22no—
73C4⋊Dic61—22no—
74C4⋊Dic61—42no—
75#11758196—24no—
76#11758196—22no—
77#11758196—24no—
78C4×Dic61—22no—
79C23×C61—21no—
80C23×C61—21no—
81C23×C61—21no—
82C23×C61—21no—
83C23×C61—21no—
84C23×C61—21no—
85C23×C61C2×Q812yes—
86C23×C61C2411yescharacteristic
87C22×D61—21no—
88C22×D61—21no—
89C22×D61—21no—
90C22×D61—21no—
91C22×D61C2×D412yes—
92C22×D61C2×Q812yes—
93C22×D61#4112yes—
94Q8×C61—44no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—21no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—44no—
104C2×C4×C61C2×D414yes—
105C2×C4×C61C2×D412yes—
106C2×C4×C61C2×D412yes—
107C2×C4×C61#4112yes—
108C22⋊D61—28no—
109C2×Dic61—21no—
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—21no—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—21no—
121C2×Dic61—22no—
122C2×Dic61C2×D412yes—
123C2×Dic61C2×Q812yes—
124C4×D61—28no—
125C4×D61—48no—
126C22×C61—22no—
127C22×C61—22no—
128C22×C61—22no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61—41no—
132C22×C61—41no—
133C22×C61C22×D411yescharacteristic
134C22×C61C22×D411yescharacteristic
135C22×C61C22×Q811yescharacteristic
136C22×C61C2×#4111yescharacteristic
137C22×C61C2×#4111yescharacteristic
138C22×C61H(5,𝔽2)11yescharacteristic
139C22×C612−1+411yescharacteristic
140C2×D61—22no—
141C2×D61—22no—
142C2×D61—22no—
143C2×D61—22no—
144C2×D61—22no—
145C2×D61—22no—
146C2×D61C22⋊Q814yescomplemented
147C4×C61—24no—
148C4×C61—41no—
149C4×C61—42no—
150C4×C61#12114yes—
151Dic61—22no—
152Dic61—22no—
153Dic61—41no—
154Dic61—42no—
155Dic61—42no—
156#2417—611nomaximal, Sylow
157C2×C61—42no—
158C2×C61C2×#12111yescharacteristic
159C2×C61C2×C22⋊Q811yescharacteristic
160C2×C61#53711yescharacteristic
161C2×C61#54611yescharacteristic
162C2×C61D4×Q811yescharacteristic
163C2×C61#55211yescharacteristic
164C2×C61#55411yescharacteristic
165D61—24no—
166C2×C22⋊Q8—611no—
167C2×C22⋊Q8D6111yescharacteristic, complemented
168C2×C4×D4—611no—
169C22×C4⋊C4—611no—
170#398—612no—
171#396—612no—
172#391—611no—
173#389—611no—
174#386—612no—
175#380—611no—
176#380—612no—
177C61#241711yescharacteristic, complemented, Sylow
178C22×Q8C2×D6111yescharacteristic
179C22×D4—611no—
180C23×C4—611no—
181C23×C4—612no—
182C23×C4C2×D6111yescharacteristic
183C22⋊Q8—24no—
184C22⋊Q8—1224no—
185C4×D4—1224no—
186C2×C4⋊C4—611no—
187C2×C4⋊C4—612no—
188C2×C4⋊C4—611no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—612no—
191C2×C4⋊C4—612no—
192C2×C4⋊C4—1224no—
193C2×C4⋊C4C2×D6112yes—
194C2×C4⋊C4C2×D6111yescharacteristic
195C2×C22⋊C4—611no—
196C2×C22⋊C4—611no—
197C2×C22⋊C4—612no—
198C2×C22⋊C4—612no—
199C2×C22⋊C4C2×D6112yes—
200C2×C42—611no—
201#95—612no—
202#95—612no—
203#95—612no—
204#95—612no—
205C24—611no—
206C24C22×D6111yescharacteristic
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×D4—614no—
211C2×D4—1222no—
212C2×D4—1222no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—611no—
217C22×C4—611no—
218C22×C4—614no—
219C22×C4—612no—
220C22×C4—611no—
221C22×C4—612no—
222C22×C4—1224no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—1224no—
226C22×C4—2442no—
227C22×C4C22×D6112yes—
228C22×C4C22×D6111yescharacteristic
229C22×C4C22×D6112yes—
230C22×C4C22×D6111yescharacteristic
231C4⋊C4—24no—
232C4⋊C4—1222no—
233C4⋊C4—1224no—
234C4⋊C4—1222no—
235C4⋊C4—1224no—
236C4⋊C4—1222no—
237C4⋊C4—2442no—
238C4⋊C4C22⋊D6114yescomplemented
239C22⋊C4—24no—
240C22⋊C4—1222no—
241C22⋊C4—1224no—
242C22⋊C4—1222no—
243C22⋊C4—2442no—
244C42—1222no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—612no—
252C23—612no—
253C23—612no—
254C23—1221no—
255C23—1221no—
256C23—1221no—
257C23—1221no—
258C23Q8×D6112yes—
259C23C23×D6111yescharacteristic
260Q8—44no—
261D4—1228no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—44no—
271C2×C4—612no—
272C2×C4—612no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1228no—
278C2×C4—1221no—
279C2×C4—1222no—
280C2×C4—1222no—
281C2×C4—1222no—
282C2×C4—1222no—
283C2×C4—1222no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1222no—
287C2×C4—2448no—
288C2×C4D4×D6112yes—
289C2×C4#1175822412yes—
290C2×C4C2×C22⋊D6114yes—
291C2×C4C2×C22⋊D6112yes—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—614no—
300C22—1222no—
301C22—1222no—
302C22—1222no—
303C22—1222no—
304C22—1222no—
305C22—1222no—
306C22C2×D4×D6111yescharacteristic
307C22C22×C22⋊D6111yescharacteristic
308C22C2×Q8×D6111yescharacteristic
309C22#41×D6111yescharacteristic
310C22C2×#1175822411yescharacteristic
311C22#1451982311yescharacteristic
312C22#1451982511yescharacteristic
313C4—24no—
314C4—41no—
315C4—42no—
316C4—1222no—
317C4—1222no—
318C4—2441no—
319C4—2442no—
320C4—2442no—
321C4#1451974814yes—
322C2—42no—
323C2—1224no—
324C2C2×#1451974811yescharacteristic
325C2C22⋊Q8×D6111yescharacteristic
326C2#1684034211yescharacteristic
327C2#1684040011yescharacteristic
328C2Q8×C22⋊D6111yescharacteristic
329C2#1684044611yescharacteristic
330C2#1684047011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×C22⋊Q8×C61C2
hypercenterC2×C22⋊Q8D61
hypocenterC61#2417
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14519753C2semidirect
3. C2×C4⋊C4×D61C2semidirect
4. #16839693C2semidirect
5. #16839685C2semidirect
6. #16839669C2semidirect
7. #16839661C2semidirect
9. #16839622C2semidirect
10. #16839619C2semidirect
11. C2×C22⋊Q8×C61C2semidirect
53. C2×C22⋊D61Q8semidirect
66. C4⋊C4×C61D4semidirect
146. C2×D61C22⋊Q8semidirect
167. C2×C22⋊Q8D61semidirect
177. C61#2417semidirect
238. C4⋊C4C22⋊D61semidirect