Finite Group Databasebeta

← #496325

Subgroups — #496325

Order 512.

1048 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 202 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
1#35316C211yesmaximal, characteristic
2#20728C212yesmaximal
3#13227C218yesmaximal, complemented
4#12704C214yesmaximal
5C2×#536C2212yes—
6C22×C4⋊Q8C411yescharacteristic
7C2×C4×#41C2211yescharacteristic
8#2920—22no—
9#2857—22no—
10#2621—22no—
11#2547C414yes—
12#2546C2214yes—
13#2486—22no—
14#2486—22no—
15#2310C412yes—
16#2285C411yescharacteristic
17C2×#356C22116yes—
18#1822—22no—
19#1801C2214yes—
20#1800—22no—
21#1785C2214yes—
22C4×#101C2214yes—
23C4×#100—22no—
24C22×#41C2311yescharacteristic
25C23×Q8C2×C412yes—
26#536—48no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—41no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—41no—
32C2×C4⋊Q8C2×C414yes—
33C2×C4⋊Q8C2314yes—
34C2×#124C2×C414yes—
35C2×#123—24no—
36C2×C22⋊Q8—24no—
37C4×#41—41no—
38C4×#41—42no—
39C4×#41—41no—
40C2×C4×Q8C2312yes—
41C2×C4×D4C2×C412yes—
42C2×C4×D4D412yes—
43C2×C4×D4D412yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C412yes—
46C2×#117D412yes—
47C22×C4⋊C4—22no—
48C22×C42C2×C411yescharacteristic
49C22×C42D412yes—
50#432—42no—
51#432—42no—
52#431—42no—
53#431—42no—
54#423—24no—
55C2×#101C2318yes—
56C2×#100—24no—
57#407—24no—
58C4×M16—24no—
59#398—24no—
60#398—24no—
61#397—24no—
62#397—24no—
63#391—24no—
64#391C2×C418yes—
65#389—24no—
66#389—24no—
67#384—21no—
68#382—21no—
69#381—22no—
70#381—22no—
71#380—24no—
72#380—24no—
73#380—22no—
74C4×C4⋊C4C2×C414yes—
75#356—416no—
76#343—24no—
77#343—24no—
78C2×#41—24no—
79C2×#41—41no—
80C2×#41—41no—
81C2×#41—42no—
82C22×Q8—22no—
83C22×Q8—44no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—44no—
87C22×Q8C22×C414yes—
88C22×Q8C2411yescharacteristic
89C22×D4C22×C411yescharacteristic
90C22×D4C2×D412yes—
91C23×C4—22no—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C412yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D412yes—
97C2×M16—28no—
98C4⋊Q8—216no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—44no—
102C4⋊Q8—44no—
103C4⋊Q8—41no—
104C4⋊Q8—42no—
105C4⋊Q8—41no—
106#124—48no—
107#123—416no—
108C22⋊Q8—416no—
109C4×Q8—42no—
110C4×Q8—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117#117—41no—
118#117—41no—
119#117—44no—
120#117—42no—
121#117—42no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4C22×C412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4C22⋊C411yescharacteristic
135C2×C22⋊C4C22⋊C411yescharacteristic
136C2×C22⋊C4C22⋊C412yes—
137C2×C22⋊C4C22⋊C412yes—
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C411yescharacteristic
147C2×C42C22⋊C411yescharacteristic
148C2×C42C22×C414yes—
149C2×C42C22×C412yes—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C4⋊C8—44no—
153C4⋊C8—44no—
154#101—216no—
155#100—48no—
156C22⋊C8—44no—
157C22⋊C8—44no—
158#97—44no—
159C4×C8—44no—
160#95—24no—
161#95—28no—
162#95—28no—
163C24C2×C22⋊C412yes—
164C24C2×C22⋊C411yescharacteristic
165#41—48no—
166#41—48no—
167C2×Q8—28no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—88no—
177C2×Q8—88no—
178C2×Q8H(5,𝔽2)14yes—
179C2×D4—24no—
180C2×D4—22no—
181C2×D4—24no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—41no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C23×C411yescharacteristic
213C22×C4C22×D412yes—
214M16—48no—
215M16—48no—
216C2×C8—48no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—48no—
221C4⋊C4—48no—
222C4⋊C4—44no—
223C4⋊C4—416no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—44no—
231C22⋊C4—416no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—21no—
239C42—21no—
240C42—22no—
241C42—22no—
242C42—28no—
243C42#9911yescharacteristic
244C42#9912yes—
245C42#9911yescharacteristic
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23#41012yes—
255C23C22×C22⋊C411yescharacteristic
256Q8—44no—
257Q8—88no—
258Q8—84no—
259Q8—84no—
260D4—48no—
261D4—44no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—88no—
291C2×C4C2×#9912yes—
292C2×C4C2×#9911yescharacteristic
293C2×C4C2×#9911yescharacteristic
294C2×C4C2×#9912yes—
295C2×C4#51812yes—
296C2×C4#53514yes—
297C8—88no—
298C22—22no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22#230911yescharacteristic
309C22C22×#9911yescharacteristic
310C22C2×#41011yescharacteristic
311C4—24no—
312C4—21no—
313C4—21no—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—42no—
319C4—84no—
320C4—84no—
321C4#213514yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—82no—
326C2#2179311yescharacteristic
327C2C2×#213512yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13227C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#207282
#132272
#127042