Finite Group Databasebeta

← #362809

Subgroups — #362809

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic
3C4×#381C211yesmaximal, characteristic
4#11203C214yesmaximal
5#10912C214yesmaximal
6#10726C212yesmaximal
7#10726C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2296—22no—
12#2296—22no—
13#2296—22no—
14#2296—22no—
15C2×#389C412yescomplemented
16C2×#389C412yescomplemented
17C2×#385C2212yes—
18C2×#385C2212yes—
19C2×#381C2212yes—
20C2×#381C2212yes—
21C2×#380C2212yes—
22C2×C4×C4⋊C4C414yescomplemented
23C2×C4×C22⋊C4C2212yes—
24C2×#342C2218yes—
25C2×#342C2218yes—
26C4⋊#95—22no—
27#1434—22no—
28#1434—22no—
29C22⋊#95C2212yes—
30C22⋊#95C2212yes—
31C4×#95—22no—
32C24×C4C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—28no—
40C2×#117—44no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C414yes—
46C22×C4⋊C4C2×C414yes—
47C22×C22⋊C4C2314yes—
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2314yes—
50C22×C22⋊C4C2312yes—
51C22×C42—22no—
52C22×C42—44no—
53C22×C42C2×C412yes—
54C22×C42C2×C412yes—
55C22×C42C2×C412yes—
56C22×C42C2×C412yes—
57C22×C42D412yes—
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42D412yes—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#385—22no—
70#385—22no—
71#385—24no—
72#385—24no—
73#385—22no—
74#385—22no—
75#381—24no—
76#381—28no—
77#381Q818yes—
78#380—44no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C22⋊C4—44no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—24no—
90C2×#95C2×C414yes—
91C2×#95C2×C414yes—
92#342—216no—
93#342—48no—
94C25C2411yescharacteristic
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—44no—
102C23×C4—44no—
103C23×C4C22×C412yes—
104C23×C4C22×C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C22×C414yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×Q814yes—
119C23×C4#4114yes—
120#117—216no—
121#117—216no—
122#117—48no—
123#117—48no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—48no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—44no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4C2×Q818yes—
152C2×C22⋊C4#4118yes—
153C2×C42—24no—
154C2×C42—28no—
155C2×C42—24no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42—44no—
159C2×C42—44no—
160C2×C42—44no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42—48no—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C412yes—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—216no—
178#95—48no—
179#95—48no—
180#95—44no—
181#95—44no—
182#95—48no—
183C24—42no—
184C24—42no—
185C24C23×C411yescharacteristic
186C24C22×D411yescharacteristic
187C24C22×D411yescharacteristic
188C24C22×Q812yes—
189C24C2×#4114yes—
190C24C2×#4112yes—
191C24H(5,𝔽2)14yes—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—216no—
203C22×C4—216no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—28no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—416no—
230C22×C4—416no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4#11718yes—
251C22×C4C4×Q814yes—
252C22×C4C4×Q814yes—
253C22×C4C22⋊Q818yes—
254C22×C4#12414yes—
255C22×C4C4⋊Q814yes—
256C4⋊C4—28no—
257C4⋊C4—28no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—88no—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—216no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—48no—
271C22⋊C4—416no—
272C42—216no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42—48no—
282C42—88no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—48no—
294C23C22×C22⋊C411yescharacteristic
295C23C2×#11712yes—
296C23C2×C4×Q812yes—
297C23#51812yes—
298C23C2×C22⋊Q812yes—
299C23#52512yes—
300C23#52512yes—
301C23C2×#12411yescharacteristic
302C23C2×C4⋊Q811yescharacteristic
303C23#53412yes—
304C23#53412yes—
305C23#55318yes—
306C23#55418yes—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—216no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—416no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—84no—
346C2×C4—88no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—88no—
352C2×C4—88no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4#39114yes—
356C2×C4#39114yes—
357C2×C4#41014yes—
358C2×C4#41014yes—
359C22—24no—
360C22—22no—
361C22—24no—
362C22—22no—
363C22—24no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—48no—
373C22—48no—
374C22#228011yescharacteristic
375C22C2×#39111yescharacteristic
376C22#230811yescharacteristic
377C22#234514yes—
378C22#257914yes—
379C22#258312yes—
380C22#259112yes—
381C22C2×#41012yes—
382C22C2×#41012yes—
383C4—24no—
384C4—24no—
385C4—24no—
386C4—24no—
387C4—48no—
388C4—42no—
389C4—42no—
390C4—88no—
391C4—82no—
392C4—82no—
393C4—164no—
394C2—24no—
395C2—22no—
396C2—22no—
397C2—24no—
398C2#1392311yescharacteristic
399C2#2072312yes—
400C2#2133212yes—
401C2#2174811yescharacteristic
402C2#2174811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#389C4semidirect
16. C2×#389C4semidirect
22. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#146852
C4×#3812
#112032
#109122
#107262