Finite Group Databasebeta

← #375175

Subgroups — #375175

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C2×#1899C214yesmaximal, complemented
3#13795C211yesmaximal, characteristic, complemented
4#13776C211yesmaximal, characteristic, complemented
5#11353C212yesmaximal, complemented
6#10980C212yesmaximal
7#10960C214yesmaximal, complemented
8C22×#123C2212yes—
9C22×#121C2211yescharacteristic
10C22×#117C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2218yes—
13#2618—24no—
14#2618—24no—
15C2×#398C412yescomplemented
16C2×#396C414yescomplemented
17C2×#394C412yescomplemented
18C2×#386C2212yes—
19C2×#381C2212yes—
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2212yes—
22#1899—216no—
23C2×#342C2218yes—
24#1436—24no—
25C4⋊#95—24no—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C22⋊#95C2212yes—
29C23×D4C2×C411yescharacteristic
30C23×D4C2312yes—
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C2×#123—24no—
35C2×#123—24no—
36C2×#123—28no—
37C2×#123—28no—
38C2×#121—22no—
39C2×#121—22no—
40C2×#121—44no—
41C2×#117—28no—
42C2×#117—28no—
43C2×#117—44no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—44no—
46C22×C4⋊C4C2×C414yes—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4D414yes—
52C22×C4⋊C4D414yes—
53C22×C22⋊C4C2×C414yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C414yes—
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2314yes—
58C22×C22⋊C4C2314yes—
59C22×C42—22no—
60C22×C42—22no—
61C2×#99—216no—
62C2×#99—216no—
63C2×#99—48no—
64#398—28no—
65#396—216no—
66#394—28no—
67#386—24no—
68#386—24no—
69#381—24no—
70#381D418yescomplemented
71#380—22no—
72#380—22no—
73C4×C22⋊C4—44no—
74C2×#95—24no—
75C2×#95—22no—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95C2×C412yes—
80C2×#95C2×C414yes—
81#342—216no—
82C25C22×C412yes—
83C25C2411yescharacteristic
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—28no—
89C22×D4—28no—
90C22×D4—22no—
91C22×D4—42no—
92C22×D4—82no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—82no—
100C23×C4—82no—
101C23×C4C22×C412yes—
102C23×C4C22×C412yes—
103C23×C4C22×C414yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C412yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D414yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4#4114yes—
115#123—416no—
116#123—48no—
117#123—48no—
118#121—48no—
119#121—48no—
120#117—432no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—216no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—82no—
136C2×C4⋊C4C22⋊C418yes—
137C2×C4⋊C4C22⋊C418yes—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—216no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—84no—
154C2×C22⋊C4C2×D418yes—
155C2×C22⋊C4#4118yes—
156C2×C42—28no—
157C2×C42—44no—
158C2×C42—44no—
159C2×C42—44no—
160C2×C42—84no—
161#99—432no—
162#95—24no—
163#95—28no—
164#95—216no—
165#95—44no—
166#95—84no—
167#95—84no—
168#95—84no—
169C24—22no—
170C24—24no—
171C24—24no—
172C24—24no—
173C24—42no—
174C24—42no—
175C24—44no—
176C24—44no—
177C24C23×C411yescharacteristic
178C24C22×D411yescharacteristic
179C24C22×D411yescharacteristic
180C24C22×D412yes—
181C24C2×#4112yes—
182C24C2×#4114yes—
183C24H(5,𝔽2)14yes—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—48no—
193C2×D4—48no—
194C2×D4—416no—
195C2×D4—84no—
196C2×D4—88no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—216no—
206C22×C4—216no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—88no—
241C22×C4—88no—
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C414yes—
246C22×C4C4×D418yes—
247C22×C4#12118yes—
248C22×C4#12414yes—
249C22×C4#12714yes—
250C4⋊C4—48no—
251C4⋊C4—48no—
252C4⋊C4—416no—
253C4⋊C4—44no—
254C4⋊C4—416no—
255C4⋊C4—84no—
256C22⋊C4—216no—
257C22⋊C4—216no—
258C22⋊C4—48no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—48no—
263C22⋊C4—48no—
264C22⋊C4—416no—
265C22⋊C4—88no—
266C42—416no—
267C42—88no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—44no—
275C23—44no—
276C23—48no—
277C23—44no—
278C23—44no—
279C23—42no—
280C23—44no—
281C23—44no—
282C23—42no—
283C23—42no—
284C23—48no—
285C23—84no—
286C23—88no—
287C23—88no—
288C23—84no—
289C23C22×C22⋊C411yescharacteristic
290C23C2×C4×D412yes—
291C23C4×#4112yes—
292C23#51812yes—
293C23C2×#12112yes—
294C23#52512yes—
295C23C2×#12411yescharacteristic
296C23C2×#12711yescharacteristic
297C23#53212yes—
298C23#53412yes—
299C23#53412yes—
300C23#54618yes—
301C23#55318yes—
302D4—88no—
303D4—84no—
304D4—84no—
305D4—88no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—216no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—216no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—416no—
325C2×C4—416no—
326C2×C4—42no—
327C2×C4—48no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—88no—
341C2×C4—88no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—88no—
346C2×C4#39018yes—
347C22—24no—
348C22—24no—
349C22—24no—
350C22—24no—
351C22—22no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—48no—
361C22—48no—
362C22—82no—
363C22—84no—
364C22—84no—
365C22—82no—
366C22—82no—
367C22—168no—
368C22C2×#39011yescharacteristic
369C22#229711yescharacteristic
370C22#230811yescharacteristic
371C22#236414yes—
372C22#253912yes—
373C22#254314yes—
374C22#259112yes—
375C22#288312yes—
376C22#288412yes—
377C4—28no—
378C4—42no—
379C4—44no—
380C4—42no—
381C4—88no—
382C4—88no—
383C4—82no—
384C4—84no—
385C2—22no—
386C2—24no—
387C2—24no—
388C2—22no—
389C2—162no—
390C2#1415211yescharacteristic
391C2#2121412yes—
392C2#2142312yes—
393C2#2174911yescharacteristic
394C2#2175111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1899C2semidirect
3. #13795C2semidirect
4. #13776C2semidirect
5. #11353C2semidirect
7. #10960C2semidirect
15. C2×#398C4semidirect
16. C2×#396C4semidirect
17. C2×#394C4semidirect
70. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C2×#18992
#137952
#137762
#113532
#109802
#109602