Finite Group Databasebeta

← #181092

Subgroups — #181092

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic
3#34128C211yesmaximal, characteristic
4#7930C213yesmaximal
5#7928C213yesmaximal
6#7926C213yesmaximal
7#7924C213yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2984—21no—
10#2984—23no—
11#2984—21no—
12#2984—23no—
13#2984—21no—
14#2984—23no—
15#2984C412yescomplemented
16#2984C416yescomplemented
17C2×#446—23no—
18C2×#446C416yescomplemented
19C2×C8×Q8—21no—
20C2×C8×Q8C412yescomplemented
21C2×#432C2213yes—
22C2×#432C2213yes—
23C2×#431C2213yes—
24C2×#431C2213yes—
25#2356—23no—
26#2351—21no—
27#2344C416yes—
28Q8×C22⋊C4C412yes—
29#2295—23no—
30#2283C2213yes—
31C42×Q8—21no—
32C4×#117C2213yes—
33#1282C2211yescharacteristic
34#1282C2213yes—
35#1282C2213yes—
36#1282C2213yes—
37#1282C2216yes—
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×Q8D411yescharacteristic
45C2×C4×Q8D413yes—
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8D413yes—
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—
81C2×C4⋊C8—23no—
82C2×C4⋊C8—23no—
83C2×C4⋊C8C2×C416yes—
84C2×C4⋊C8C2×C416yes—
85C2×C22⋊C8C2311yescharacteristic
86C2×C22⋊C8C2313yes—
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2313yes—
89C2×#97—23no—
90C2×#97C2×C416yes—
91C2×C4×C8—23no—
92C2×C4×C8C2×C416yes—
93#391—26no—
94#389—26no—
95#387—212no—
96#383—23no—
97#382—23no—
98#381C2×C416yes—
99#380C2×C416yes—
100C4×C4⋊C4—26no—
101C4×C4⋊C4—23no—
102C4×C4⋊C4—23no—
103C4×C4⋊C4—23no—
104C4×C22⋊C4C2×C416yes—
105C4×C22⋊C4C2311yescharacteristic
106C4×C22⋊C4C2313yes—
107#376—23no—
108#376—23no—
109C43—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
120C22×C8—21no—
121C22×C8—23no—
122C22×C8C22×C412yes—
123C22×C8C22×C416yes—
124C4×Q8—48no—
125C4×Q8C22⋊C413yes—
126C4×Q8C22⋊C411yescharacteristic
127C4×Q8C22⋊C413yes—
128C4×Q8C22⋊C411yescharacteristic
129C4×Q8C22⋊C413yes—
130C4×Q8C22⋊C413yes—
131C4×Q8C22⋊C416yes—
132C4×Q8C22⋊C413yes—
133C4×Q8C22⋊C413yes—
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—212no—
146#117—46no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4C4213yes—
150C2×C4⋊C4C4213yes—
151C2×C4⋊C4C4216yes—
152C2×C4⋊C4C22⋊C413yes—
153C2×C4⋊C4C22⋊C416yes—
154C2×C4⋊C4C22⋊C413yes—
155C2×C4⋊C4C4⋊C413yes—
156C2×C4⋊C4C4⋊C413yes—
157C2×C4⋊C4C4⋊C416yes—
158C2×C22⋊C4C22×C416yes—
159C2×C22⋊C4C22×C412yes—
160C2×C42—26no—
161C2×C42—26no—
162C2×C42—21no—
163C2×C42—23no—
164C2×C42C22×C413yes—
165C2×C42C22×C413yes—
166C2×C42C22×C413yes—
167C2×C42C22×C413yes—
168C2×C42C2×D413yes—
169C2×C42C2×D413yes—
170C2×C42C2×D413yes—
171C2×C42C2×D413yes—
172C2×C42C2×D413yes—
173C2×C42C2×D413yes—
174C2×C42C2×Q813yes—
175C2×C42C2×Q813yes—
176C4⋊C8—26no—
177C4⋊C8—26no—
178C4⋊C8—26no—
179C4⋊C8—26no—
180C4⋊C8—26no—
181C4⋊C8—26no—
182C4⋊C8—26no—
183C4⋊C8—26no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—23no—
188C22⋊C8—23no—
189C22⋊C8—23no—
190C22⋊C8C22×C412yes—
191C22⋊C8C22×C416yes—
192#97—26no—
193#97—26no—
194#97—26no—
195#97—26no—
196C4×C8—26no—
197C4×C8—26no—
198C4×C8—26no—
199C4×C8—26no—
200#95—26no—
201#95—26no—
202#95—26no—
203#95—26no—
204#95—21no—
205#95—23no—
206C24C23×C411yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—46no—
211C2×Q8—46no—
212C2×Q8—46no—
213C2×Q8#9513yes—
214C2×Q8#9513yes—
215C2×Q8#9513yes—
216C2×Q8#9516yes—
217C2×Q8#9511yescharacteristic
218C22×C4—26no—
219C22×C4—26no—
220C22×C4—22no—
221C22×C4—26no—
222C22×C4—46no—
223C22×C4—42no—
224C22×C4C2×C4213yes—
225C22×C4C2×C4213yes—
226C22×C4C2×C4213yes—
227C22×C4C2×C4213yes—
228C22×C4C2×C22⋊C413yes—
229C22×C4C2×C22⋊C413yes—
230C22×C4C2×C22⋊C413yes—
231C22×C4C2×C22⋊C413yes—
232C22×C4C2×C4⋊C413yes—
233C22×C4C2×C4⋊C413yes—
234C22×C4C2×C4⋊C413yes—
235C22×C4C2×C4⋊C413yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×Q811yescharacteristic
242C2×C8—22no—
243C2×C8—26no—
244C2×C8—22no—
245C2×C8—26no—
246C2×C8—42no—
247C2×C8—46no—
248C2×C8—42no—
249C2×C8—46no—
250C4⋊C4—412no—
251C4⋊C4—412no—
252C4⋊C4#9516yes—
253C4⋊C4#9516yes—
254C4⋊C4#9516yes—
255C4⋊C4#9516yes—
256C4⋊C4#9516yes—
257C4⋊C4#9516yes—
258C4⋊C4#9513yes—
259C4⋊C4#9513yes—
260C4⋊C4#9513yes—
261C4⋊C4#9513yes—
262C22⋊C4—22no—
263C22⋊C4—26no—
264C22⋊C4—26no—
265C22⋊C4#13114yes—
266C42—412no—
267C42—42no—
268C42—46no—
269C42—412no—
270C42C2×C22⋊C413yes—
271C42C2×C22⋊C413yes—
272C42C2×C22⋊C413yes—
273C42C2×C22⋊C413yes—
274C42C2×C22⋊C413yes—
275C42C2×C22⋊C413yes—
276C42C2×C22⋊C413yes—
277C42C2×C22⋊C413yes—
278C42C2×C4⋊C413yes—
279C42C2×C4⋊C413yes—
280C42C2×C4⋊C413yes—
281C42C2×C4⋊C413yes—
282C42C2×C4⋊C413yes—
283C42C2×C4⋊C413yes—
284C42C2×C4⋊C413yes—
285C42C2×C4⋊C413yes—
286C23—41no—
287C23—41no—
288C23C22×C4211yescharacteristic
289C23C22×C22⋊C411yescharacteristic
290C23C22×C4⋊C411yescharacteristic
291C23C2×#13112yes—
292C23#56812yes—
293Q8—26no—
294Q8—26no—
295Q8—26no—
296Q8—46no—
297Q8#34414yes—
298C2×C4—23no—
299C2×C4—23no—
300C2×C4—23no—
301C2×C4—23no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—26no—
305C2×C4—26no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—26no—
312C2×C4—24no—
313C2×C4—43no—
314C2×C4—43no—
315C2×C4—43no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—46no—
322C2×C4—46no—
323C2×C4—46no—
324C2×C4C2×#9513yes—
325C2×C4C2×#9513yes—
326C2×C4C2×#9513yes—
327C2×C4C2×#9513yes—
328C2×C4C2×#9513yes—
329C2×C4C2×#9513yes—
330C2×C4C2×#9513yes—
331C2×C4C2×#9513yes—
332C2×C4C22×C22⋊C411yescharacteristic
333C2×C4C22×C22⋊C411yescharacteristic
334C2×C4C22×C4⋊C411yescharacteristic
335C2×C4C22×C4⋊C411yescharacteristic
336C8—42no—
337C8—46no—
338C8—42no—
339C8—46no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—42no—
347C22—42no—
348C22C22×#9511yescharacteristic
349C22C4×#13111yescharacteristic
350C22#287511yescharacteristic
351C22#288012yes—
352C22#290612yes—
353C4—23no—
354C4—23no—
355C4—23no—
356C4—23no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—23no—
361C4—23no—
362C4—23no—
363C4—44no—
364C4—86no—
365C4C2×#34416yes—
366C4#288412yes—
367C2—22no—
368C2—21no—
369C2—21no—
370C2—22no—
371C2#2033611yescharacteristic
372C2#2035511yescharacteristic
373C2C22×#34411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC22C22×#95
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2984C4semidirect
16. #2984C4semidirect
18. C2×#446C4semidirect
20. C2×C8×Q8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#341282
#79302
#79282
#79262
#79242