Finite Group Databasebeta

← #496393

Subgroups — #496393

Order 512.

1376 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1376 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×#2542C211yesmaximal, characteristic
2#20722C212yesmaximal
3#13298C218yesmaximal, complemented
4#12856C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic, complemented
6C22×C22⋊Q8—22no—
7C22×C4×D4C2211yescharacteristic
8#2542—22no—
9#2542—22no—
10#2542C414yescomplemented
11#2542C2214yes—
12C2×#399—22no—
13C2×#390C2212yes—
14C2×#384C412yescomplemented
15#2280—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×C22⋊Q8—24no—
33C2×C22⋊Q8—24no—
34C2×C22⋊Q8—24no—
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—416no—
37C2×C4×D4—41no—
38C2×C4×D4—42no—
39C2×C4×D4—41no—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117—42no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—24no—
51C22×C22⋊C4D412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C412yes—
54C2×#99—24no—
55C2×#99C2318yes—
56#399—24no—
57#399—24no—
58#399—24no—
59#399—24no—
60#390—24no—
61#390C2×C414yes—
62#390C2314yes—
63#384—22no—
64#384—22no—
65#384C2×C418yes—
66#381—42no—
67#380—24no—
68#380—42no—
69#379—24no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2×C414yes—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—42no—
74C2×#95—21no—
75#342—24no—
76#342—24no—
77C25C2×D412yes—
78C22×Q8—24no—
79C22×Q8—48no—
80C22×Q8—42no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×D4—24no—
84C22×D4—42no—
85C22×D4—41no—
86C22×D4—41no—
87C22×D4C22×C411yescharacteristic
88C22×D4C2×D412yes—
89C22×D4C2411yescharacteristic
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—48no—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C4⋊Q8—44no—
106C4⋊Q8—44no—
107C4⋊Q8—44no—
108C4⋊Q8—44no—
109C22⋊Q8—416no—
110C22⋊Q8—416no—
111C22⋊Q8—416no—
112C22⋊Q8—416no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118C4×D4—42no—
119C4×D4—42no—
120C4×D4—42no—
121#117—88no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—416no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C4⋊C4C22×C414yes—
139C2×C4⋊C4C22×C412yes—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—416no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4C22⋊C412yes—
149C2×C22⋊C4C22⋊C412yes—
150C2×C22⋊C4C2×D412yes—
151C2×C22⋊C4C2×D412yes—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42C22⋊C412yes—
159C2×C42C22×C412yes—
160C2×C42C22×C414yes—
161#99—216no—
162#99—48no—
163#95—24no—
164#95—44no—
165#95—44no—
166C24—24no—
167C24—22no—
168C24—22no—
169C24—42no—
170C24—44no—
171C24—44no—
172C24C2×C22⋊C411yescharacteristic
173C24C2×C22⋊C412yes—
174C24C22×D412yes—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—84no—
180C2×Q8—88no—
181C2×Q8—84no—
182C2×D4—24no—
183C2×D4—22no—
184C2×D4—24no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—48no—
191C2×D4—48no—
192C2×D42−1+414yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—48no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—48no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—48no—
226C22×C4—84no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4#9911yescharacteristic
230C22×C4#9912yes—
231C22×C4#9911yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C23×C411yescharacteristic
238C4⋊C4—216no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—416no—
244C4⋊C4—416no—
245C4⋊C4—44no—
246C4⋊C4—48no—
247C4⋊C4—88no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—416no—
253C22⋊C4—416no—
254C22⋊C4—48no—
255C22⋊C4—48no—
256C22⋊C4—88no—
257C22⋊C4—88no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—44no—
263C42—44no—
264C42—88no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—44no—
275C23—44no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—416no—
281C23—44no—
282C23—44no—
283C23—82no—
284C23—82no—
285C23—88no—
286C23C2×#9911yescharacteristic
287C23C2×#9911yescharacteristic
288C23C22×C22⋊C411yescharacteristic
289C23#53614yes—
290Q8—88no—
291Q8—88no—
292D4—44no—
293D4—48no—
294D4—44no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—416no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—88no—
331C2×C4—88no—
332C2×C4—88no—
333C2×C4#35314yes—
334C2×C4C2×#9912yes—
335C2×C4C2×#9912yes—
336C2×C4#41012yes—
337C2×C4#51912yes—
338C22—24no—
339C22—24no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—24no—
347C22—44no—
348C22—44no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—82no—
355C22—88no—
356C22—88no—
357C22—82no—
358C22C2×#35312yes—
359C22#212912yes—
360C22#231011yescharacteristic
361C22C22×#9911yescharacteristic
362C22C2×#41011yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—48no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—84no—
371C4—168no—
372C4C2×#35314yes—
373C2—21no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—44no—
378C2—82no—
379C2#2182511yescharacteristic
380C2C22×#35311yescharacteristic
381C2C2×#212911yescharacteristic
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. #13298C2semidirect
5. C22×C4⋊Q8C4semidirect
10. #2542C4semidirect
14. C2×#384C4semidirect
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×#25422
#207222
#132982
#128562