Finite Group Databasebeta

← #181371

Subgroups — #181371

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic
3#34196C211yesmaximal, characteristic
4#8110C213yesmaximal
5#8109C213yesmaximal
6#8106C213yesmaximal
7#8105C213yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2984—23no—
10#2984—21no—
11#2984—23no—
12#2984—21no—
13#2983—23no—
14#2983C416yescomplemented
15#2982—21no—
16#2982C412yescomplemented
17Q8×M16—21no—
18Q8×M16—23no—
19Q8×M16C412yescomplemented
20Q8×M16C416yescomplemented
21#2920C2213yes—
22#2920C2213yes—
23#2920C2213yes—
24#2920C2213yes—
25#2355—23no—
26Q8×C4⋊C4—21no—
27#2344C416yes—
28Q8×C22⋊C4C412yes—
29#2320—23no—
30#2285C2213yes—
31#2283C2213yes—
32#2276—21no—
33#1305C2213yes—
34#1305C2216yes—
35#1305C2211yescharacteristic
36#1305C2213yes—
37#1305C2213yes—
38C23×Q8C2×C411yescharacteristic
39C2×C4×Q8—22no—
40C2×C4×Q8C2×C411yescharacteristic
41C2×C4×Q8C2×C413yes—
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8C2×C413yes—
44C2×C4×Q8D413yes—
45C2×C4×Q8D411yescharacteristic
46C2×C4×Q8D413yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D413yes—
50C2×C4×Q8Q811yescharacteristic
51C2×C4×Q8Q813yes—
52C2×#117C2×C416yes—
53C22×C4⋊C4C2×C413yes—
54C22×C42C2313yes—
55#446—26no—
56#446—26no—
57#446—212no—
58#446—26no—
59#446—26no—
60#446—212no—
61C8×Q8—26no—
62C8×Q8—22no—
63C8×Q8—26no—
64C8×Q8—22no—
65#432—23no—
66#432—23no—
67#432—23no—
68#432—23no—
69#432—23no—
70#432—23no—
71#432C2×C416yes—
72#432C2×C416yes—
73#431—23no—
74#431—23no—
75#431—23no—
76#431—23no—
77#431—23no—
78#431—23no—
79#431C2×C416yes—
80#431C2×C416yes—
81#423—23no—
82#423—23no—
83#423C2×C416yes—
84#423C2×C416yes—
85#407C2313yes—
86#407C2311yescharacteristic
87#407C2311yescharacteristic
88#407C2313yes—
89C4×M16—23no—
90C4×M16—23no—
91C4×M16C2×C416yes—
92C4×M16C2×C416yes—
93#391—26no—
94#389—26no—
95#387—212no—
96#384—23no—
97#383—23no—
98#382—23no—
99#382—23no—
100#381C2×C416yes—
101#380C2×C416yes—
102#380C2311yescharacteristic
103#380C2313yes—
104C4×C4⋊C4—26no—
105C4×C4⋊C4—23no—
106C4×C4⋊C4—23no—
107C4×C4⋊C4—23no—
108C4×C22⋊C4C2×C416yes—
109#376—23no—
110C22×Q8—44no—
111C22×Q8C4211yescharacteristic
112C22×Q8C4213yes—
113C22×Q8C22⋊C411yescharacteristic
114C22×Q8C22⋊C413yes—
115C22×Q8C4⋊C411yescharacteristic
116C22×Q8C4⋊C413yes—
117C23×C4C22×C413yes—
118C23×C4C22×C413yes—
119C23×C4C2411yescharacteristic
120C2×M16—21no—
121C2×M16—23no—
122C2×M16C22×C416yes—
123C2×M16C22×C412yes—
124C4×Q8—48no—
125C4×Q8C22⋊C413yes—
126C4×Q8C22⋊C413yes—
127C4×Q8C22⋊C411yescharacteristic
128C4×Q8C22⋊C411yescharacteristic
129C4×Q8C22⋊C413yes—
130C4×Q8C22⋊C413yes—
131C4×Q8C22⋊C413yes—
132C4×Q8C22⋊C413yes—
133C4×Q8C22⋊C416yes—
134C4×Q8C22⋊C416yes—
135C4×Q8C4⋊C413yes—
136C4×Q8C4⋊C413yes—
137C4×Q8C4⋊C413yes—
138C4×Q8C4⋊C413yes—
139C4×Q8C4⋊C413yes—
140C4×Q8C4⋊C413yes—
141C4×Q8C4⋊C416yes—
142C4×Q8C4⋊C416yes—
143C4×Q8C4⋊C411yescharacteristic
144C4×Q8C4⋊C411yescharacteristic
145#117—86no—
146C2×C4⋊C4—26no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—23no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4C4213yes—
151C2×C4⋊C4C4213yes—
152C2×C4⋊C4C4216yes—
153C2×C4⋊C4C22⋊C413yes—
154C2×C4⋊C4C22⋊C416yes—
155C2×C4⋊C4C22⋊C413yes—
156C2×C4⋊C4C4⋊C413yes—
157C2×C4⋊C4C4⋊C413yes—
158C2×C4⋊C4C4⋊C416yes—
159C2×C22⋊C4C22×C416yes—
160C2×C22⋊C4C22×C412yes—
161C2×C42—26no—
162C2×C42—26no—
163C2×C42C22×C413yes—
164C2×C42C22×C413yes—
165C2×C42C22×C413yes—
166C2×C42C22×C413yes—
167C2×C42C2×D413yes—
168C2×C42C2×D413yes—
169C2×C42C2×D413yes—
170C2×C42C2×D413yes—
171C2×C42C2×D413yes—
172C2×C42C2×D413yes—
173C2×C42C2×Q813yes—
174C2×C42C2×Q813yes—
175C4⋊C8—26no—
176C4⋊C8—26no—
177C4⋊C8—26no—
178C4⋊C8—26no—
179C4⋊C8—26no—
180C4⋊C8—26no—
181C4⋊C8—26no—
182C4⋊C8—26no—
183C22⋊C8—23no—
184C22⋊C8—21no—
185C22⋊C8—23no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—23no—
189C22⋊C8C22×C412yes—
190C22⋊C8C22×C416yes—
191#97—26no—
192#97—26no—
193#97—26no—
194#97—26no—
195C4×C8—26no—
196C4×C8—26no—
197C4×C8—26no—
198C4×C8—26no—
199#95—26no—
200#95—26no—
201#95—26no—
202#95—26no—
203#95—21no—
204#95—23no—
205C24C23×C411yescharacteristic
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—46no—
210C2×Q8—46no—
211C2×Q8—46no—
212C2×Q8#9513yes—
213C2×Q8#9513yes—
214C2×Q8#9513yes—
215C2×Q8#9516yes—
216C2×Q8#9511yescharacteristic
217C22×C4—22no—
218C22×C4—26no—
219C22×C4—46no—
220C22×C4—42no—
221C22×C4—43no—
222C22×C4—43no—
223C22×C4C2×C4213yes—
224C22×C4C2×C4213yes—
225C22×C4C2×C4213yes—
226C22×C4C2×C4213yes—
227C22×C4C2×C22⋊C413yes—
228C22×C4C2×C22⋊C413yes—
229C22×C4C2×C22⋊C413yes—
230C22×C4C2×C22⋊C413yes—
231C22×C4C2×C4⋊C413yes—
232C22×C4C2×C4⋊C413yes—
233C22×C4C2×C4⋊C413yes—
234C22×C4C2×C4⋊C413yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×Q811yescharacteristic
241M16—42no—
242M16—46no—
243M16—46no—
244M16—42no—
245C2×C8—26no—
246C2×C8—22no—
247C2×C8—26no—
248C2×C8—22no—
249C4⋊C4—42no—
250C4⋊C4—46no—
251C4⋊C4—412no—
252C4⋊C4—86no—
253C4⋊C4#9516yes—
254C4⋊C4#9516yes—
255C4⋊C4#9516yes—
256C4⋊C4#9516yes—
257C4⋊C4#9516yes—
258C4⋊C4#9516yes—
259C4⋊C4#9513yes—
260C4⋊C4#9513yes—
261C4⋊C4#9513yes—
262C4⋊C4#9513yes—
263C22⋊C4—42no—
264C22⋊C4—46no—
265C42—412no—
266C42—86no—
267C42C2×C22⋊C413yes—
268C42C2×C22⋊C413yes—
269C42C2×C22⋊C413yes—
270C42C2×C22⋊C413yes—
271C42C2×C22⋊C413yes—
272C42C2×C22⋊C413yes—
273C42C2×C22⋊C413yes—
274C42C2×C22⋊C413yes—
275C42C2×C4⋊C413yes—
276C42C2×C4⋊C413yes—
277C42C2×C4⋊C413yes—
278C42C2×C4⋊C413yes—
279C42C2×C4⋊C413yes—
280C42C2×C4⋊C413yes—
281C42C2×C4⋊C413yes—
282C42C2×C4⋊C413yes—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23C22×C4211yescharacteristic
288C23C22×C22⋊C411yescharacteristic
289C23C22×C4⋊C411yescharacteristic
290Q8—26no—
291Q8—26no—
292Q8—26no—
293Q8—46no—
294Q8#34414yes—
295C2×C4—23no—
296C2×C4—23no—
297C2×C4—23no—
298C2×C4—23no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—26no—
305C2×C4—43no—
306C2×C4—43no—
307C2×C4—43no—
308C2×C4—43no—
309C2×C4—43no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—46no—
317C2×C4—42no—
318C2×C4—46no—
319C2×C4—46no—
320C2×C4—42no—
321C2×C4C2×#9513yes—
322C2×C4C2×#9513yes—
323C2×C4C2×#9513yes—
324C2×C4C2×#9513yes—
325C2×C4C2×#9513yes—
326C2×C4C2×#9513yes—
327C2×C4C2×#9513yes—
328C2×C4C2×#9513yes—
329C2×C4C22×C22⋊C411yescharacteristic
330C2×C4C22×C22⋊C411yescharacteristic
331C2×C4C22×C4⋊C411yescharacteristic
332C2×C4C22×C4⋊C411yescharacteristic
333C8—46no—
334C8—42no—
335C8—46no—
336C8—42no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—42no—
346C22—42no—
347C22C22×#9511yescharacteristic
348C22#288811yescharacteristic
349C22#288911yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—23no—
354C4—23no—
355C4—23no—
356C4—23no—
357C4—23no—
358C4—23no—
359C4—23no—
360C4—86no—
361C4—82no—
362C4C2×#34416yes—
363C4#288412yes—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2#2035511yescharacteristic
369C2#2035811yescharacteristic
370C2C22×#34411yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2983C4semidirect
16. #2982C4semidirect
19. Q8×M16C4semidirect
20. Q8×M16C4semidirect

Maximal subgroups

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

subgroupindex
#602482
#341962
#81102
#81092
#81062
#81052