Finite Group Databasebeta

← #213508

Subgroups — #213508

Order 512.

1190 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1190 conjugacy classes of subgroups — too many to draw. Only its 234 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×#2855C211yesmaximal, characteristic
2#20714C212yesmaximal
3#9082C218yesmaximal, complemented
4#8704C212yesmaximal
5#8308C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C23×C42C2211yescharacteristic
8C2×#431C2214yes—
9#2855—22no—
10#2855—22no—
11#2855—22no—
12#2855—22no—
13C2×#391C412yes—
14C2×#389C412yes—
15C2×#387C2214yes—
16C2×#385C412yes—
17C2×#381C412yes—
18C2×#380C2211yescharacteristic
19#1817C2214yes—
20#1797—22no—
21#1797—22no—
22#1779C2214yes—
23C4×C22⋊C8—22no—
24C4×C22⋊C8—22no—
25C2×#324C2218yes—
26#1280C2218yes—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4C2311yescharacteristic
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—44no—
34C22×C4⋊C4C2×C414yes—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2312yes—
38C22×C22⋊C4C2×C412yes—
39C22×C22⋊C4C2×C412yes—
40C22×C42—44no—
41C22×C42C2×C412yes—
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42D412yes—
45C22×C42D412yes—
46C22×C42C2312yes—
47#431—48no—
48C2×C4⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8C2318yes—
52C2×C4×C8—24no—
53#391—22no—
54#391—22no—
55#391—22no—
56#391—22no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#387—24no—
62#387—24no—
63#387—24no—
64#387—24no—
65#385—24no—
66#385—22no—
67#385—22no—
68#381—24no—
69#381—22no—
70#381—22no—
71#380—22no—
72#380—22no—
73C2×#95C2×C414yes—
74C2×#95C2×C414yes—
75C2×#95C2×C414yes—
76C2×#95C2312yes—
77#336—24no—
78#336—24no—
79#324—48no—
80C25C22×C411yescharacteristic
81C22×Q8—22no—
82C22×Q8—42no—
83C22×Q8—44no—
84C23×C4—24no—
85C23×C4—28no—
86C23×C4—24no—
87C23×C4—44no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C412yes—
100C23×C4C22×C412yes—
101C23×C4C22×C414yes—
102C23×C4C22×C412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2411yescharacteristic
106C22×C8—28no—
107C22⋊Q8—48no—
108C22⋊Q8—48no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—216no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C2×C42—28no—
126C2×C42—44no—
127C2×C42—44no—
128C2×C42—44no—
129C2×C42—44no—
130C2×C42—44no—
131C2×C42C22⋊C412yes—
132C2×C42C22⋊C412yes—
133C2×C42C22⋊C412yes—
134C2×C42C22⋊C412yes—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C4⋊C8—48no—
140C22⋊C8—216no—
141C22⋊C8—48no—
142C22⋊C8—48no—
143C4×C8—48no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24—24no—
155C24—42no—
156C24C2×C22⋊C411yescharacteristic
157C24C2×C22⋊C411yescharacteristic
158C24C23×C411yescharacteristic
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—88no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—28no—
185C22×C4—24no—
186C22×C4—28no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—416no—
200C22×C4—416no—
201C22×C4—416no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4#10114yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4#13118yes—
212C22×C4C22×D412yes—
213C22×C4H(5,𝔽2)14yes—
214C2×C8—48no—
215C2×C8—48no—
216C2×C8—48no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—48no—
220C4⋊C4—48no—
221C4⋊C4—48no—
222C4⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C42—24no—
228C42—24no—
229C42—24no—
230C42—24no—
231C42—42no—
232C42—42no—
233C42—42no—
234C42—42no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—28no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—21no—
247C23—48no—
248C23—48no—
249C23—44no—
250C23—44no—
251C23C2×#10112yes—
252C23#41312yes—
253C23C22×C22⋊C411yescharacteristic
254C23#51812yes—
255C23C2×#13114yes—
256Q8—88no—
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—216no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—44no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—88no—
291C2×C4—84no—
292C2×C4#40814yes—
293C2×C4#40814yes—
294C2×C4#41014yes—
295C2×C4C2×#10114yes—
296C2×C4#53414yes—
297C8—88no—
298C22—24no—
299C22—24no—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—416no—
307C22—42no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22#230811yescharacteristic
314C22C2×#40812yes—
315C22C2×#41012yes—
316C22C22×#10111yescharacteristic
317C22C2×#41311yescharacteristic
318C22#297714yes—
319C4—24no—
320C4—24no—
321C4—24no—
322C4—24no—
323C4—24no—
324C4—42no—
325C4—84no—
326C4—82no—
327C4—82no—
328C2—24no—
329C2—42no—
330C2—42no—
331C2#1738012yes—
332C2#1738512yes—
333C2#2173911yescharacteristic
334C2#2174711yescharacteristic
335C2#2175311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9082C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
#207142
#90822
#87042
#83082