Finite Group Databasebeta

← #391780

Subgroups — #391780

Order 512.

1568 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1568 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2#14342C211yesmaximal, characteristic, complemented
3#14339C211yesmaximal, characteristic, complemented
4#11353C212yesmaximal, complemented
5#11043C214yesmaximal, complemented
6#10980C212yesmaximal
7#10966C214yesmaximal, complemented
8C22×#123C2212yes—
9C22×#121C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×#99C2218yes—
12#2618—24no—
13#2618—24no—
14C2×#398C412yescomplemented
15C2×#396C414yescomplemented
16C2×#394C412yescomplemented
17C2×#386C2212yes—
18C2×#381C2212yes—
19C2×#381C2212yes—
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×#342C2218yes—
23#1445—24no—
24#1441—24no—
25C22⋊#95C2212yes—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C23×D4C2×C411yescharacteristic
29C23×D4C2312yes—
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C2×#123—24no—
34C2×#123—24no—
35C2×#123—48no—
36C2×#121—22no—
37C2×#121—22no—
38C2×#121—44no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—44no—
41C22×C4⋊C4C2×C414yes—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4D414yes—
47C22×C4⋊C4D414yes—
48C22×C22⋊C4C2×C414yes—
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C414yes—
51C22×C22⋊C4C2314yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2312yes—
54C2×#99—48no—
55C2×#99—48no—
56C2×#99—48no—
57#398—28no—
58#396—216no—
59#394—28no—
60#386—24no—
61#386—24no—
62#381—44no—
63#381—44no—
64#380—22no—
65#380—22no—
66#380—44no—
67C2×#95—22no—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95—24no—
71C2×#95—24no—
72C2×#95—24no—
73C2×#95C2×C412yes—
74C2×#95C2×C414yes—
75#342—48no—
76C25C22×C412yes—
77C25C2411yescharacteristic
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—28no—
81C22×D4—28no—
82C22×D4—24no—
83C22×D4—24no—
84C22×D4—22no—
85C22×D4—42no—
86C22×D4—82no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—82no—
96C23×C4—82no—
97C23×C4C22×C412yes—
98C23×C4C22×C412yes—
99C23×C4C22×C414yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109#123—416no—
110#123—416no—
111#121—48no—
112#121—48no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—48no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—84no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4C22⋊C418yes—
127C2×C4⋊C4C22⋊C418yes—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—84no—
145#99—432no—
146#95—24no—
147#95—28no—
148#95—44no—
149#95—48no—
150#95—48no—
151#95—84no—
152#95—84no—
153#95—84no—
154C24—22no—
155C24—24no—
156C24—24no—
157C24—24no—
158C24—42no—
159C24—42no—
160C24—44no—
161C24—44no—
162C24C23×C411yescharacteristic
163C24C22×D411yescharacteristic
164C24C22×D411yescharacteristic
165C24H(5,𝔽2)14yes—
166C24H(5,𝔽2)14yes—
167C24H(5,𝔽2)12yes—
168C242−1+412yes—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—44no—
173C2×D4—44no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—48no—
178C2×D4—48no—
179C2×D4—416no—
180C2×D4—84no—
181C2×D4—88no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—48no—
214C22×C4—48no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—48no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—88no—
228C22×C4—88no—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C414yes—
233C4⋊C4—48no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—416no—
237C4⋊C4—84no—
238C4⋊C4—88no—
239C22⋊C4—48no—
240C22⋊C4—416no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—48no—
246C22⋊C4—48no—
247C22⋊C4—88no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—44no—
255C23—48no—
256C23—44no—
257C23—44no—
258C23—44no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—44no—
263C23—44no—
264C23—48no—
265C23—84no—
266C23—88no—
267C23—88no—
268C23—84no—
269C23C22×C22⋊C411yescharacteristic
270C23#51812yes—
271C23#51812yes—
272C23#52012yes—
273C23#53412yes—
274C23#53412yes—
275C23#53512yes—
276C23#53512yes—
277C23#53511yescharacteristic
278C23#53611yescharacteristic
279C23#53712yes—
280C23#56018yes—
281C23#56218yes—
282D4—88no—
283D4—84no—
284D4—88no—
285D4—84no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—88no—
315C2×C4—84no—
316C2×C4—88no—
317C2×C4—88no—
318C2×C4—88no—
319C2×C4—88no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—22no—
329C22—24no—
330C22—24no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—48no—
339C22—48no—
340C22—84no—
341C22—82no—
342C22—84no—
343C22—82no—
344C22—82no—
345C22—168no—
346C22#230811yescharacteristic
347C22#230911yescharacteristic
348C22#231111yescharacteristic
349C22#237714yes—
350C22#281314yes—
351C22#281512yes—
352C22#281612yes—
353C22#288312yes—
354C22#288412yes—
355C4—44no—
356C4—42no—
357C4—44no—
358C4—42no—
359C4—84no—
360C4—82no—
361C4—164no—
362C4—164no—
363C2—22no—
364C2—24no—
365C2—24no—
366C2—22no—
367C2—162no—
368C2#1465911yescharacteristic
369C2#2174911yescharacteristic
370C2#2175111yescharacteristic
371C2#2179612yes—
372C2#2187712yes—
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. #14342C2semidirect
3. #14339C2semidirect
4. #11353C2semidirect
5. #11043C2semidirect
7. #10966C2semidirect
14. C2×#398C4semidirect
15. C2×#396C4semidirect
16. C2×#394C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
#143422
#143392
#113532
#110432
#109802
#109662