Finite Group Databasebeta

← #391825

Subgroups — #391825

Order 512.

1674 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1674 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×#2626C211yesmaximal, characteristic
2#14341C211yesmaximal, characteristic
3#14338C211yesmaximal, characteristic
4#11381C212yesmaximal, complemented
5#11353C212yesmaximal, complemented
6#11203C214yesmaximal, complemented
7#11196C214yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10#2626—24no—
11#2626—24no—
12C2×#399C412yescomplemented
13C2×#396C414yescomplemented
14C2×#393C412yescomplemented
15C2×#385C2212yes—
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×#379C2212yes—
19C2×#379C2212yes—
20C2×#379C2212yes—
21C2×#342C2218yes—
22C2×#342C2218yes—
23#1440—24no—
24#1439—24no—
25C22⋊#95C2212yes—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C26C2311yescharacteristic
29C23×Q8C2×C411yescharacteristic
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—44no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C414yes—
40C22×C22⋊C4—24no—
41C22×C22⋊C4—22no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—44no—
44C22×C22⋊C4C2×C414yes—
45C22×C22⋊C4C2×C414yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C412yes—
48C22×C22⋊C4D414yes—
49C22×C22⋊C4D414yes—
50C22×C22⋊C4C2314yes—
51C22×C22⋊C4C2314yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2312yes—
54#399—28no—
55#396—216no—
56#393—28no—
57#385—24no—
58#385—24no—
59#380—22no—
60#380—22no—
61#380—44no—
62#379—24no—
63#379—24no—
64#379—44no—
65#379—44no—
66C2×#95—22no—
67C2×#95—24no—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95C2×C414yes—
71C2×#95C2×C412yes—
72#342—48no—
73#342—48no—
74C25—24no—
75C25—22no—
76C25—24no—
77C25—22no—
78C25—82no—
79C25—82no—
80C25C22×C412yes—
81C25C2×D412yes—
82C25C2×D412yes—
83C25C2411yescharacteristic
84C22×Q8—22no—
85C22×Q8—42no—
86C22×Q8—82no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—82no—
92C23×C4—82no—
93C23×C4C22×C412yes—
94C23×C4C22×C414yes—
95C23×C4C22×C412yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C22⋊Q8—48no—
104C22⋊Q8—48no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—84no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4—84no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—44no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—48no—
140C2×C22⋊C4—84no—
141C2×C22⋊C4—84no—
142C2×C22⋊C4—84no—
143C2×C22⋊C4—84no—
144C2×C22⋊C4—84no—
145C2×C22⋊C4—84no—
146C2×C22⋊C4C22⋊C418yes—
147C2×C22⋊C4C22⋊C418yes—
148#95—24no—
149#95—28no—
150#95—44no—
151#95—84no—
152#95—84no—
153#95—84no—
154C24—24no—
155C24—22no—
156C24—22no—
157C24—22no—
158C24—22no—
159C24—24no—
160C24—24no—
161C24—24no—
162C24—24no—
163C24—24no—
164C24—24no—
165C24—42no—
166C24—42no—
167C24—44no—
168C24—44no—
169C24—44no—
170C24—44no—
171C24—44no—
172C24—44no—
173C24—48no—
174C24—48no—
175C24—88no—
176C24—88no—
177C24—88no—
178C24—816no—
179C24—88no—
180C24C2×C22⋊C414yes—
181C24C23×C411yescharacteristic
182C24C22×D411yescharacteristic
183C24C22×D411yescharacteristic
184C24H(5,𝔽2)14yes—
185C24H(5,𝔽2)12yes—
186C242−1+414yes—
187C242−1+412yes—
188C2×Q8—44no—
189C2×Q8—84no—
190C2×Q8—88no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—88no—
227C22×C4—88no—
228C22×C4C2×C22⋊C414yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C4⋊C4—48no—
232C4⋊C4—44no—
233C4⋊C4—84no—
234C4⋊C4—88no—
235C4⋊C4—88no—
236C22⋊C4—48no—
237C22⋊C4—48no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—416no—
244C22⋊C4—48no—
245C22⋊C4—48no—
246C22⋊C4—88no—
247C22⋊C4—88no—
248C22⋊C4—816no—
249C22⋊C4—88no—
250C22⋊C4—88no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—24no—
258C23—24no—
259C23—24no—
260C23—28no—
261C23—28no—
262C23—44no—
263C23—44no—
264C23—44no—
265C23—48no—
266C23—44no—
267C23—48no—
268C23—44no—
269C23—44no—
270C23—416no—
271C23—48no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23—44no—
276C23—48no—
277C23—44no—
278C23—44no—
279C23—88no—
280C23—88no—
281C23—88no—
282C23—88no—
283C23—84no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—832no—
288C23—832no—
289C23C22×C22⋊C411yescharacteristic
290C23#51812yes—
291C23#51912yes—
292C23#52012yes—
293C23#53412yes—
294C23#53412yes—
295C23#53511yescharacteristic
296C23#53612yes—
297C23#53611yescharacteristic
298C23#53612yes—
299C23#53712yes—
300C23#56218yes—
301C23#56318yes—
302Q8—88no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—24no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—88no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—88no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—88no—
332C2×C4—88no—
333C2×C4—88no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C22—24no—
338C22—24no—
339C22—24no—
340C22—22no—
341C22—24no—
342C22—24no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—48no—
351C22—48no—
352C22—48no—
353C22—44no—
354C22—48no—
355C22—44no—
356C22—44no—
357C22—88no—
358C22—88no—
359C22—88no—
360C22—88no—
361C22—816no—
362C22#230811yescharacteristic
363C22#231011yescharacteristic
364C22#231111yescharacteristic
365C22#237814yes—
366C22#281414yes—
367C22#281612yes—
368C22#281712yes—
369C22#288312yes—
370C22#288312yes—
371C4—42no—
372C4—42no—
373C4—44no—
374C4—84no—
375C4—82no—
376C4—82no—
377C4—164no—
378C4—164no—
379C2—22no—
380C2—24no—
381C2—22no—
382C2—24no—
383C2—84no—
384C2#1466011yescharacteristic
385C2#2174911yescharacteristic
386C2#2175011yescharacteristic
387C2#2182712yes—
388C2#2187612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11381C2semidirect
5. #11353C2semidirect
6. #11203C2semidirect
12. C2×#399C4semidirect
13. C2×#396C4semidirect
14. C2×#393C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
#143412
#143382
#113812
#113532
#112032
#111962