Finite Group Databasebeta

← #22602416

Subgroups — #22602416

Order 9088.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986731C213yesmaximal, complemented
2Q8×Dic142C211yesmaximal, characteristic, complemented
3#17986707C213yesmaximal, complemented
4#17986668C211yesmaximal, characteristic, complemented
5#17986660C211yesmaximal, characteristic, complemented
6#17986653C213yesmaximal, complemented
7#17986649C213yesmaximal, complemented
8#17986622C213yesmaximal, complemented
9#17986618C213yesmaximal, complemented
10#17986544C213yesmaximal, complemented
11#17986529C213yesmaximal, complemented
12#17986482C211yesmaximal, characteristic, complemented
13#17986481C211yesmaximal, characteristic, complemented
14C2×Q8×Dic71C211yesmaximal, characteristic, complemented
15#548×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765864C2216yes—
18#14556128C2213yes—
19#14556127C2213yes—
20#14556127C2213yes—
21#14556124C2213yes—
22#14556123C2213yes—
23#14556121C2213yes—
24#14556118C2213yes—
25#14556117C2213yes—
26#14556114C2216yes—
27#14556114C2213yes—
28#14556112C2216yes—
29#14556112C2213yes—
30#14556110C2216yes—
31#14556109C2213yes—
32#14556108C2216yes—
33#14556105C2216yes—
34#14556105C2211yescharacteristic
35#14556104C2213yes—
36#14556102C2213yes—
37#14556102C2213yes—
38#14556100C2213yes—
39#14556099C2213yes—
40Q8×Dic71C2213yes—
41Q8×Dic71C2211yescharacteristic
42Q8×Dic71C2211yescharacteristic
43Q8×Dic71C2213yes—
44Q8×Dic71C2211yescharacteristic
45C4×Dic142C2213yes—
46C4×D284C2213yes—
47D4×Dic71C2213yes—
48D4×Dic71C2213yes—
49#14556090C2213yes—
50C4×C22⋊D71C2213yes—
51C4×C22⋊D71C2213yes—
52(C22⋊D71)⋊C4C2216yes—
53#14556087C2213yes—
54#14556085C2213yes—
55C2×C4⋊Dic71C2213yes—
56C2×C4⋊Dic71C2213yes—
57C4⋊C4×D71C2213yes—
58C2×C4×Dic71C2213yes—
59C2×#41×C71C2211yescharacteristic
60C22×Q8×C71C2211yescharacteristic
61#124×C71C2213yes—
62C22⋊Q8×C71C2213yes—
63#121×C71C2213yes—
64C4×Q8×C71C2211yescharacteristic
65C4×D4×C71C2213yes—
66#11765866—24no—
67#11765864—412no—
68C2×Dic142C2311yescharacteristic
69C2×Dic142C2316yes—
70C2×D284C2313yes—
71C2×C22⋊D71C2313yes—
72C2×C22⋊D71C2313yes—
73C2×C22⋊D71C2316yes—
74C22×Dic71C2313yes—
75C22×Dic71C2311yescharacteristic
76C22×Dic71C2316yes—
77C2×C4×D71C2313yes—
78C2×C4×D71C2316yes—
79C4⋊Dic71C2313yes—
80C4⋊Dic71C2313yes—
81C4⋊Dic71C2313yes—
82C4⋊Dic71C2313yes—
83C4⋊Dic71C2316yes—
84C4⋊Dic71C2311yescharacteristic
85C4⋊Dic71C2313yes—
86Dic71⋊C4C2313yes—
87Dic71⋊C4C2311yescharacteristic
88Dic71⋊C4C2313yes—
89Dic71⋊C4C2311yescharacteristic
90Dic71⋊C4C2313yes—
91Dic71⋊C4C2313yes—
92C22⋊Dic71C2313yes—
93C22⋊Dic71C2313yes—
94C22⋊Dic71C2311yescharacteristic
95C22⋊Dic71C2313yes—
96(C2×D71)⋊C4C2313yes—
97(C2×D71)⋊C4C2313yes—
98(C2×D71)⋊C4C2316yes—
99(C2×D71)⋊C4C2313yes—
100(C2×D71)⋊C4C2313yes—
101C4×Dic71C2313yes—
102C4×Dic71C2313yes—
103C4×Dic71C2313yes—
104C4×Dic71C2313yes—
105C4×Dic71C2313yes—
106#41×C71—24no—
107C2×Q8×C71—24no—
108C2×Q8×C71C2313yes—
109C2×Q8×C71C2311yescharacteristic
110C2×D4×C71C2313yes—
111C2×D4×C71C2313yes—
112C22×C4×C71C2313yes—
113C22×C4×C71C2313yes—
114C4⋊C4×C71C2313yes—
115C4⋊C4×C71C2313yes—
116C22⋊C4×C71C2311yescharacteristic
117C22⋊C4×C71C2313yes—
118C22⋊C4×C71C2316yes—
119C42×C71C2313yes—
120C23×C71C2413yes—
121C23×C71C2411yescharacteristic
122C22×D71C2413yes—
123Q8×C71—26no—
124Q8×C71#4114yescomplemented
125D4×C71—26no—
126D4×C71—26no—
127C2×C4×C71—26no—
128C2×C4×C71—26no—
129C2×C4×C71C2411yescharacteristic
130C2×C4×C71C2413yes—
131C2×C4×C71C2413yes—
132C2×C4×C71C2411yescharacteristic
133C2×C4×C71C2413yes—
134C22⋊D71—26no—
135C22⋊D71—26no—
136C22⋊D71—46no—
137C2×Dic71—26no—
138C2×Dic71—22no—
139C2×Dic71—46no—
140C2×Dic71C2411yescharacteristic
141C2×Dic71C2411yescharacteristic
142C2×Dic71C2413yes—
143C2×Dic71C2413yes—
144C2×Dic71C2413yes—
145C2×Dic71C2411yescharacteristic
146C2×Dic71C2411yescharacteristic
147D284—26no—
148C4×D71—26no—
149C4×D71—46no—
150Dic142—22no—
151Dic142—46no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—23no—
155C22×C71—23no—
156C22×C71—23no—
157C22×C71H(5,𝔽2)12yes—
158C22×C71C2511yescharacteristic
159C2×D71—23no—
160C2×D71—23no—
161C2×D71—23no—
162C4×C71—21no—
163C4×C71—23no—
164C4×C71—23no—
165C4×C71—21no—
166C4×C71C2×#4116yes—
167Dic71—21no—
168Dic71—23no—
169Dic71—23no—
170Dic71—23no—
171Dic71—21no—
172Dic71—21no—
173Dic712−1+412yes—
174C2×C71—22no—
175C2×C71—43no—
176C2×C71C22×#4111yescharacteristic
177C2×C71C2×H(5,𝔽2)11yescharacteristic
178C2×C71C2×2−1+411yescharacteristic
179D71—43no—
180#3522—711nomaximal, Sylow
181C71#352211yescharacteristic, complemented, Sylow
182#559—713no—
183Q82—711no—
184#554—713no—
185#550—711no—
186#548—711no—
187#548D7111yescharacteristic, complemented
188#547—713no—
189#547—713no—
190#542—713no—
191#542—713no—
192#532—713no—
193#529—713no—
194#520—711no—
195#520—711no—
196C2×C4×Q8—711no—
197C2×#41—711no—
198C2×#41—716no—
199C2×#41C2×D7111yescharacteristic
200C22×Q8C2×D7111yescharacteristic
201C4⋊Q8—713no—
202C4⋊Q8—713no—
203C4⋊Q8—713no—
204#127—713no—
205#126—713no—
206#126—713no—
207#125—713no—
208#124—713no—
209#124—713no—
210#124—716no—
211#124C2×D7113yes—
212#123—713no—
213#123—713no—
214#123—716no—
215#123—716no—
216C22⋊Q8—713no—
217C22⋊Q8—716no—
218C22⋊Q8—711no—
219C22⋊Q8—716no—
220C22⋊Q8C2×D7113yes—
221#121—713no—
222#121—713no—
223#121—713no—
224#121—713no—
225#121C2×D7113yes—
226C4×Q8—711no—
227C4×Q8—713no—
228C4×Q8—711no—
229C4×Q8—713no—
230C4×Q8—711no—
231C4×Q8—713no—
232C4×Q8C2×D7111yescharacteristic
233C4×D4—713no—
234C4×D4—713no—
235C4×D4—713no—
236C4×D4—713no—
237C4×D4—713no—
238C4×D4—716no—
239C4×D4—713no—
240C4×D4C2×D7113yes—
241#117—713no—
242#117—713no—
243C2×C4⋊C4—713no—
244C2×C4⋊C4—713no—
245C2×C4⋊C4—713no—
246C2×C42—713no—
247#41—24no—
248#41—28412no—
249#41—1424no—
250C2×Q8—24no—
251C2×Q8—716no—
252C2×Q8—711no—
253C2×Q8C22×D7113yes—
254C2×Q8C22×D7111yescharacteristic
255C2×D4—716no—
256C2×D4—713no—
257C2×D4—713no—
258C2×D4—713no—
259C2×D4C22×D7113yes—
260C2×D4C22×D7113yes—
261C22×C4—716no—
262C22×C4—716no—
263C22×C4—711no—
264C22×C4—713no—
265C22×C4—713no—
266C22×C4C22×D7113yes—
267C22×C4C22×D7113yes—
268C4⋊C4—713no—
269C4⋊C4—711no—
270C4⋊C4—711no—
271C4⋊C4—713no—
272C4⋊C4—713no—
273C4⋊C4—713no—
274C4⋊C4—713no—
275C4⋊C4—713no—
276C4⋊C4—713no—
277C4⋊C4—713no—
278C4⋊C4—716no—
279C4⋊C4—711no—
280C4⋊C4—713no—
281C4⋊C4C22×D7113yes—
282C4⋊C4C22×D7113yes—
283C22⋊C4—713no—
284C22⋊C4—713no—
285C22⋊C4—711no—
286C22⋊C4—713no—
287C22⋊C4—713no—
288C22⋊C4—713no—
289C22⋊C4—713no—
290C22⋊C4—716no—
291C22⋊C4—713no—
292C22⋊C4C22×D7116yes—
293C22⋊C4C22×D7111yescharacteristic
294C22⋊C4C22×D7113yes—
295C42—713no—
296C42—713no—
297C42—713no—
298C42—713no—
299C42—713no—
300C42C22×D7113yes—
301C23—713no—
302C23C23×D7113yes—
303C23C23×D7111yescharacteristic
304Q8—26no—
305Q8—2846no—
306Q8—1422no—
307Q8#1176586414yescomplemented
308D4—26no—
309D4—26no—
310D4—2846no—
311D4—1426no—
312D4—1426no—
313D4—1426no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2846no—
317C2×C4—2846no—
318C2×C4—711no—
319C2×C4—711no—
320C2×C4—711no—
321C2×C4—713no—
322C2×C4—713no—
323C2×C4—713no—
324C2×C4—711no—
325C2×C4—1422no—
326C2×C4—1426no—
327C2×C4—1426no—
328C2×C4C23×D7113yes—
329C2×C4C23×D7111yescharacteristic
330C2×C4C23×D7113yes—
331C2×C4C23×D7111yescharacteristic
332C2×C4C23×D7113yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1423no—
339C22—1423no—
340C22—1423no—
341C22#1455617412yes—
342C22C24×D7111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—712no—
348C4—1421no—
349C4—1423no—
350C4—1423no—
351C4—1423no—
352C4—1421no—
353C4—1421no—
354C4C2×#1176586416yes—
355C2—22no—
356C2—43no—
357C2—2843no—
358C2C22×#1176586411yescharacteristic
359C2C2×#1455617411yescharacteristic
360C22−1+4×D7111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#548×C71C2
hypercenter#548D71
hypocenterC71#3522
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #17986731C2semidirect
2. Q8×Dic142C2semidirect
3. #17986707C2semidirect
4. #17986668C2semidirect
5. #17986660C2semidirect
6. #17986653C2semidirect
7. #17986649C2semidirect
8. #17986622C2semidirect
9. #17986618C2semidirect
10. #17986544C2semidirect
11. #17986529C2semidirect
12. #17986482C2semidirect
13. #17986481C2semidirect
14. C2×Q8×Dic71C2semidirect
15. #548×C71C2semidirect
124. Q8×C71#41semidirect
181. C71#3522semidirect
187. #548D71semidirect
307. Q8#11765864semidirect