Finite Group Databasebeta

← #496320

Subgroups — #496320

Order 512.

1094 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1094 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×#2546C211yesmaximal, characteristic
2#20727C212yesmaximal
3#13227C218yesmaximal, complemented
4#12700C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4⋊Q8C2212yes—
7C22×C4×Q8C2211yescharacteristic
8#2920—22no—
9#2546—22no—
10#2546—22no—
11#2546C414yes—
12#2546C2214yes—
13C2×#398—22no—
14C2×#391C412yes—
15C2×#389—22no—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×#356C22116yes—
18#1821—22no—
19#1801—22no—
20#1801C2214yes—
21#1784C2214yes—
22C4×#101—22no—
23C4×#101C2214yes—
24C23×Q8C2×C412yes—
25C23×Q8C2311yescharacteristic
26C2×C4⋊Q8—24no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—41no—
29C2×C4⋊Q8—42no—
30C2×C4⋊Q8—41no—
31C2×C4⋊Q8—48no—
32C2×C4⋊Q8C2×C414yes—
33C2×C4⋊Q8C2×C414yes—
34C2×C4⋊Q8C2314yes—
35C2×C4×Q8—41no—
36C2×C4×Q8—42no—
37C2×C4×Q8—41no—
38C2×C4×Q8C2×C412yes—
39C2×C4×Q8D412yes—
40C2×C4×Q8D412yes—
41C2×C4×Q8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4D412yes—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C412yes—
50C22×C42D412yes—
51#432—42no—
52#432—42no—
53#431—42no—
54#431—42no—
55#423—24no—
56C2×#101—24no—
57C2×#101C2318yes—
58#407—24no—
59C4×M16—24no—
60#398—24no—
61#398—24no—
62#398—24no—
63#398—24no—
64#391—22no—
65#391—22no—
66#391C2×C418yes—
67#389—24no—
68#389—24no—
69#389—24no—
70#389—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2×C414yes—
73C2×#95—21no—
74C2×#95—24no—
75#356—416no—
76#343—24no—
77#343—24no—
78C22×Q8—24no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—42no—
83C22×Q8—44no—
84C22×Q8—44no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8C22×C414yes—
88C22×Q8C22×C411yescharacteristic
89C22×Q8C2×D412yes—
90C22×Q8C2411yescharacteristic
91C23×C4—22no—
92C23×C4—22no—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D412yes—
98C2×M16—28no—
99C4⋊Q8—216no—
100C4⋊Q8—48no—
101C4⋊Q8—44no—
102C4⋊Q8—44no—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C4⋊Q8—416no—
106C4×Q8—42no—
107C4×Q8—42no—
108C4×Q8—42no—
109C4×Q8—42no—
110C4×Q8—42no—
111C4×Q8—42no—
112C4×Q8—42no—
113C4×Q8—42no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—416no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C22×C412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—44no—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C412yes—
149C2×C42C22×C414yes—
150C2×C42C22×C412yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C4⋊C8—44no—
154C4⋊C8—44no—
155#101—216no—
156#101—48no—
157C22⋊C8—44no—
158C22⋊C8—44no—
159#97—44no—
160C4×C8—44no—
161#95—24no—
162#95—28no—
163#95—28no—
164C24C2×C22⋊C411yescharacteristic
165C2×Q8—22no—
166C2×Q8—24no—
167C2×Q8—24no—
168C2×Q8—28no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—48no—
175C2×Q8—48no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×Q8—88no—
182C2×Q8—88no—
183C2×Q8H(5,𝔽2)14yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4#9911yescharacteristic
209C22×C4#9912yes—
210C22×C4#9911yescharacteristic
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C23×C411yescharacteristic
218C22×C4C22×D412yes—
219M16—48no—
220M16—48no—
221C2×C8—48no—
222C4⋊C4—216no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—416no—
231C4⋊C4—48no—
232C4⋊C4—416no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—28no—
248C42—44no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23C2×#9911yescharacteristic
254C23C2×#9911yescharacteristic
255C23C22×C22⋊C411yescharacteristic
256Q8—44no—
257Q8—48no—
258Q8—44no—
259Q8—88no—
260Q8—88no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—28no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—416no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—82no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4#35614yescomplemented
297C2×C4C2×#9912yes—
298C2×C4C2×#9912yes—
299C2×C4#41012yes—
300C2×C4#51812yes—
301C2×C4#53514yes—
302C8—88no—
303C22—24no—
304C22—24no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22C2×#35612yes—
312C22#213512yes—
313C22#230911yescharacteristic
314C22C22×#9911yescharacteristic
315C22C2×#41011yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—48no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—84no—
326C4C2×#35614yes—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—22no—
331C2#2179311yescharacteristic
332C2C22×#35611yescharacteristic
333C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13227C2semidirect
296. C2×C4#356semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
#207272
#132272
#127002