Finite Group Databasebeta

← #181128

Subgroups — #181128

Order 512.

1357 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#34192C211yesmaximal, characteristic
3#34128C211yesmaximal, characteristic
4#7945C213yesmaximal
5#7944C213yesmaximal
6#7941C213yesmaximal
7#7940C213yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2984—21no—
10#2984—23no—
11#2983C416yescomplemented
12#2982C412yescomplemented
13Q8×M16—21no—
14Q8×M16—23no—
15#2920C2213yes—
16#2920C2213yes—
17#2356—23no—
18#2356—23no—
19#2351—21no—
20#2351—21no—
21#2345—23no—
22#2344—23no—
23Q8×C22⋊C4—21no—
24Q8×C22⋊C4—21no—
25#2320C416yescomplemented
26#2295C416yescomplemented
27#2283C2213yes—
28#2283C2213yes—
29#2283C2213yes—
30#2276C412yescomplemented
31C42×Q8C412yescomplemented
32C4×#117C2213yes—
33#1284C2211yescharacteristic
34#1284C2213yes—
35#1284C2213yes—
36#1284C2213yes—
37#1284C2216yes—
38C23×Q8D411yescharacteristic
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8C2×C413yes—
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×Q8C2×C411yescharacteristic
44C2×C4×Q8C2×C413yes—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8C2×C413yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D413yes—
50C2×C4×Q8D413yes—
51C2×C4×Q8Q811yescharacteristic
52C2×C4×Q8Q813yes—
53C2×#117—23no—
54C22×C4⋊C4D413yes—
55C22×C42C2313yes—
56#446—26no—
57#446—26no—
58#446—212no—
59C8×Q8—26no—
60C8×Q8—22no—
61#432—23no—
62#432—23no—
63#432C2×C416yes—
64#432C2×C416yes—
65#431—23no—
66#431—23no—
67#431C2×C416yes—
68#431C2×C416yes—
69#423—23no—
70#423—23no—
71#407C2311yescharacteristic
72#407C2313yes—
73C4×M16—23no—
74C4×M16—23no—
75#391—26no—
76#391—26no—
77#389—26no—
78#389—26no—
79#387—212no—
80#387—212no—
81#383—23no—
82#383—23no—
83#383C2×C416yes—
84#382—23no—
85#382—23no—
86#382C2×C416yes—
87#381—23no—
88#381C2313yes—
89#381C2311yescharacteristic
90#380—23no—
91#380—23no—
92#380—23no—
93C4×C4⋊C4—26no—
94C4×C4⋊C4—26no—
95C4×C4⋊C4—23no—
96C4×C4⋊C4—23no—
97C4×C4⋊C4C2×C416yes—
98C4×C4⋊C4C2×C416yes—
99C4×C4⋊C4C2×C416yes—
100C4×C22⋊C4—23no—
101C4×C22⋊C4—23no—
102C4×C22⋊C4—23no—
103C4×C22⋊C4C2311yescharacteristic
104C4×C22⋊C4C2313yes—
105#376—23no—
106#376—23no—
107#376C2×C416yes—
108#376C2×C416yes—
109C43C2×C416yes—
110C22×Q8—44no—
111C22×Q8C22⋊C411yescharacteristic
112C22×Q8C22⋊C411yescharacteristic
113C22×Q8C22⋊C413yes—
114C22×Q8C22⋊C413yes—
115C22×Q8C4⋊C411yescharacteristic
116C22×Q8C4⋊C413yes—
117C23×C4C2×D413yes—
118C23×C4C2×D413yes—
119C23×C4C2411yescharacteristic
120C2×M16—23no—
121C2×M16—21no—
122C4×Q8—48no—
123C4×Q8—48no—
124C4×Q8C4213yes—
125C4×Q8C4213yes—
126C4×Q8C4213yes—
127C4×Q8C4216yes—
128C4×Q8C4211yescharacteristic
129C4×Q8C22⋊C413yes—
130C4×Q8C22⋊C411yescharacteristic
131C4×Q8C22⋊C413yes—
132C4×Q8C22⋊C413yes—
133C4×Q8C22⋊C416yes—
134C4×Q8C4⋊C413yes—
135C4×Q8C4⋊C413yes—
136C4×Q8C4⋊C413yes—
137C4×Q8C4⋊C413yes—
138C4×Q8C4⋊C413yes—
139C4×Q8C4⋊C413yes—
140C4×Q8C4⋊C416yes—
141C4×Q8C4⋊C416yes—
142C4×Q8C4⋊C411yescharacteristic
143C4×Q8C4⋊C411yescharacteristic
144#117—412no—
145C2×C4⋊C4—26no—
146C2×C4⋊C4—26no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4C22⋊C413yes—
150C2×C4⋊C4C22⋊C413yes—
151C2×C4⋊C4C22⋊C416yes—
152C2×C4⋊C4C22⋊C416yes—
153C2×C4⋊C4C22⋊C413yes—
154C2×C4⋊C4C22⋊C413yes—
155C2×C4⋊C4C4⋊C413yes—
156C2×C4⋊C4C4⋊C413yes—
157C2×C4⋊C4C4⋊C416yes—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—23no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—23no—
162C2×C42—26no—
163C2×C42—26no—
164C2×C42—26no—
165C2×C42—26no—
166C2×C42C22×C412yes—
167C2×C42C22×C416yes—
168C2×C42C22×C413yes—
169C2×C42C22×C413yes—
170C2×C42C22×C413yes—
171C2×C42C22×C413yes—
172C2×C42C22×C413yes—
173C2×C42C22×C413yes—
174C2×C42C2×D413yes—
175C2×C42C2×D413yes—
176C2×C42C2×D413yes—
177C2×C42C2×D413yes—
178C2×C42C2×Q813yes—
179C2×C42C2×Q813yes—
180C4⋊C8—26no—
181C4⋊C8—26no—
182C4⋊C8—26no—
183C4⋊C8—26no—
184C22⋊C8—21no—
185C22⋊C8—23no—
186C22⋊C8C22×C412yes—
187C22⋊C8C22×C416yes—
188#97—26no—
189#97—26no—
190C4×C8—26no—
191C4×C8—26no—
192#95—26no—
193#95—26no—
194#95—26no—
195#95—26no—
196#95—26no—
197#95—26no—
198#95—26no—
199#95—26no—
200#95—21no—
201#95—23no—
202#95—23no—
203#95—21no—
204#95C22×C412yes—
205#95C22×C416yes—
206C24C22×D411yescharacteristic
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—22no—
219C22×C4—26no—
220C22×C4—22no—
221C22×C4—26no—
222C22×C4—46no—
223C22×C4—46no—
224C22×C4—42no—
225C22×C4C2×C22⋊C413yes—
226C22×C4C2×C22⋊C413yes—
227C22×C4C2×C22⋊C413yes—
228C22×C4C2×C22⋊C413yes—
229C22×C4C2×C22⋊C413yes—
230C22×C4C2×C22⋊C413yes—
231C22×C4C2×C22⋊C413yes—
232C22×C4C2×C22⋊C413yes—
233C22×C4C2×C4⋊C413yes—
234C22×C4C2×C4⋊C413yes—
235C22×C4C2×C4⋊C413yes—
236C22×C4C2×C4⋊C413yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C23×C411yescharacteristic
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×Q811yescharacteristic
243M16—46no—
244M16—42no—
245C2×C8—26no—
246C2×C8—22no—
247C4⋊C4—412no—
248C4⋊C4—412no—
249C4⋊C4—412no—
250C4⋊C4—86no—
251C4⋊C4#9516yes—
252C4⋊C4#9516yes—
253C4⋊C4#9516yes—
254C4⋊C4#9516yes—
255C4⋊C4#9516yes—
256C4⋊C4#9516yes—
257C4⋊C4#9513yes—
258C4⋊C4#9513yes—
259C4⋊C4#9513yes—
260C4⋊C4#9513yes—
261C22⋊C4—24no—
262C22⋊C4—42no—
263C22⋊C4—46no—
264C22⋊C4—46no—
265C42—42no—
266C42—46no—
267C42—412no—
268C42—412no—
269C42—412no—
270C42—86no—
271C42C2×C4213yes—
272C42C2×C4213yes—
273C42C2×C4213yes—
274C42C2×C4213yes—
275C42C2×C22⋊C413yes—
276C42C2×C22⋊C413yes—
277C42C2×C22⋊C413yes—
278C42C2×C22⋊C413yes—
279C42C2×C4⋊C413yes—
280C42C2×C4⋊C413yes—
281C42C2×C4⋊C413yes—
282C42C2×C4⋊C413yes—
283C42C2×C4⋊C413yes—
284C42C2×C4⋊C413yes—
285C42C2×C4⋊C413yes—
286C42C2×C4⋊C413yes—
287C23—22no—
288C23—22no—
289C23—41no—
290C23C22×C22⋊C411yescharacteristic
291C23C22×C22⋊C411yescharacteristic
292C23C22×C4⋊C411yescharacteristic
293Q8—26no—
294Q8—26no—
295Q8—26no—
296Q8—46no—
297Q8H(3,ℤ/4)14yes—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—26no—
305C2×C4—24no—
306C2×C4—43no—
307C2×C4—43no—
308C2×C4—43no—
309C2×C4—43no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—43no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—46no—
321C2×C4—46no—
322C2×C4—46no—
323C2×C4—42no—
324C2×C4—46no—
325C2×C4—46no—
326C2×C4—46no—
327C2×C4—42no—
328C2×C4C2×#9513yes—
329C2×C4C2×#9513yes—
330C2×C4C2×#9513yes—
331C2×C4C2×#9513yes—
332C2×C4C2×#9513yes—
333C2×C4C2×#9513yes—
334C2×C4C2×#9513yes—
335C2×C4C2×#9513yes—
336C2×C4C22×C4211yescharacteristic
337C2×C4C22×C22⋊C411yescharacteristic
338C2×C4C22×C4⋊C411yescharacteristic
339C2×C4C22×C4⋊C411yescharacteristic
340C8—46no—
341C8—42no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—44no—
350C22C22×#9511yescharacteristic
351C22#288412yes—
352C22#290112yes—
353C4—23no—
354C4—23no—
355C4—23no—
356C4—23no—
357C4—23no—
358C4—23no—
359C4—23no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—44no—
365C4—86no—
366C4—82no—
367C4—86no—
368C4C2×H(3,ℤ/4)16yes—
369C2—22no—
370C2—22no—
371C2—41no—
372C2C22×H(3,ℤ/4)11yescharacteristic
373C2#2034311yescharacteristic
374C2#2035511yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ 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
11. #2983C4semidirect
12. #2982C4semidirect
25. #2320C4semidirect
26. #2295C4semidirect
30. #2276C4semidirect
31. C42×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#602482
#341922
#341282
#79452
#79442
#79412
#79402