Finite Group Databasebeta

← #496395

Subgroups — #496395

Order 512.

1180 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2563C211yesmaximal, characteristic
2#20722C212yesmaximal
3#13299C218yesmaximal, complemented
4#12856C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7#2563—22no—
8#2563—22no—
9#2563C414yescomplemented
10#2563C2214yes—
11C2×#393—22no—
12C2×#390C2212yes—
13C2×#382C412yescomplemented
14#2280—22no—
15C2×#380—22no—
16C2×C4×C4⋊C4C411yescharacteristic, complemented
17#1819—22no—
18#1799—22no—
19#1799C2214yes—
20#1781C2214yes—
21C4×#99—22no—
22C4×#99C2214yes—
23#1393C22116yes—
24C23×Q8—21no—
25C23×D4C2311yescharacteristic
26C24×C4D412yes—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—41no—
29C2×C4⋊Q8—42no—
30C2×C4⋊Q8—41no—
31C2×C4⋊Q8C2×C414yes—
32C2×C4×D4—41no—
33C2×C4×D4—42no—
34C2×C4×D4—41no—
35C2×C4×D4C2×C412yes—
36C2×C4×D4D412yes—
37C2×C4×D4D412yes—
38C2×C4×D4C2312yes—
39C2×#117—42no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C22⋊C4—24no—
45C22×C22⋊C4D412yes—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C412yes—
48C2×#99—24no—
49C2×#99C2318yes—
50#393—24no—
51#393—24no—
52#393—24no—
53#393—24no—
54#390—24no—
55#390C2×C414yes—
56#390C2314yes—
57#382—22no—
58#382—22no—
59#382C2×C418yes—
60#381—42no—
61#380—24no—
62#380—24no—
63#380—24no—
64#380—24no—
65#380—24no—
66#380—42no—
67#379—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2×C414yes—
70C4×C22⋊C4—24no—
71C4×C22⋊C4—42no—
72C2×#95—21no—
73C2×#95—24no—
74#342—24no—
75#342—24no—
76C25C2×D412yes—
77C22×Q8—24no—
78C22×Q8—42no—
79C22×Q8—42no—
80C22×Q8—84no—
81C22×D4—24no—
82C22×D4—42no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4C22×C411yescharacteristic
86C22×D4C2×D412yes—
87C22×D4C2411yescharacteristic
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—48no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C4⋊Q8—44no—
106C4⋊Q8—44no—
107C4×D4—42no—
108C4×D4—42no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115#117—88no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22×C414yes—
131C2×C4⋊C4C22×C412yes—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—88no—
140C2×C22⋊C4C22⋊C412yes—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4C2×D412yes—
143C2×C22⋊C4C2×D412yes—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—22no—
148C2×C42—44no—
149C2×C42—44no—
150C2×C42C22⋊C412yes—
151C2×C42C22×C412yes—
152C2×C42C22×C414yes—
153#99—216no—
154#99—48no—
155#95—24no—
156#95—28no—
157#95—28no—
158#95—44no—
159#95—44no—
160C24—24no—
161C24—22no—
162C24—22no—
163C24—42no—
164C24—44no—
165C24—44no—
166C24C2×C22⋊C411yescharacteristic
167C24C2×C22⋊C412yes—
168C24C22×D412yes—
169C2×Q8—48no—
170C2×Q8—88no—
171C2×Q8—88no—
172C2×Q8—88no—
173C2×D4—24no—
174C2×D4—22no—
175C2×D4—24no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—48no—
182C2×D4—48no—
183C2×D42−1+414yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—48no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4#9911yescharacteristic
221C22×C4#9912yes—
222C22×C4#9911yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C23×C411yescharacteristic
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—48no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—88no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—88no—
245C22⋊C4—88no—
246C22⋊C4—88no—
247C22⋊C4—88no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—44no—
253C42—44no—
254C42—88no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—44no—
265C23—44no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—416no—
271C23—44no—
272C23—44no—
273C23—82no—
274C23—82no—
275C23—88no—
276C23C2×#9911yescharacteristic
277C23C2×#9911yescharacteristic
278C23C22×C22⋊C411yescharacteristic
279C23#53614yes—
280Q8—816no—
281D4—44no—
282D4—48no—
283D4—44no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—88no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—88no—
320C2×C4—88no—
321C2×C4—88no—
322C2×C4#35414yes—
323C2×C4C2×#9912yes—
324C2×C4C2×#9912yes—
325C2×C4#41012yes—
326C2×C4#51912yes—
327C22—24no—
328C22—24no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—44no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—82no—
344C22—88no—
345C22—88no—
346C22—82no—
347C22C2×#35412yes—
348C22#212812yes—
349C22#231011yescharacteristic
350C22C22×#9911yescharacteristic
351C22C2×#41011yescharacteristic
352C4—22no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—84no—
359C4—84no—
360C4—168no—
361C4C2×#35414yes—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—22no—
366C2—44no—
367C2—82no—
368C2#2182511yescharacteristic
369C2C22×#35411yescharacteristic
370C2C2×#212811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13299C2semidirect
5. C22×C4⋊Q8C4semidirect
9. #2563C4semidirect
13. C2×#382C4semidirect
16. 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×#25632
#207222
#132992
#128562