Finite Group Databasebeta

← #134334

Subgroups — #134334

Order 512.

421 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 86 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#21050C211yesmaximal, characteristic
2#11425C212yesmaximal
3#8602C211yesmaximal, characteristic, complemented
4#8601C211yesmaximal, characteristic, complemented
5#7190C212yesmaximal, complemented
6C2×#424C411yescharacteristic
7C2×#407C411yescharacteristic
8C22×C22⋊C8C411yescharacteristic
9C2×#405C2211yescharacteristic
10#2280C2211yescharacteristic
11#1924—21no—
12#1924—21no—
13#1923—21no—
14#1923—21no—
15#1729C411yescharacteristic
16#1729C2211yescharacteristic
17#1564—21no—
18#1564—21no—
19#1458C2212yes—
20#1458C2212yes—
21#1317—22no—
22#1316—22no—
23#1275—21no—
24#1275—21no—
25C24×C4C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×#117C2311yescharacteristic
32#431—22no—
33#431—22no—
34#424—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37#407—42no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—41no—
43C2×C22⋊C8—41no—
44#405—41no—
45#405—41no—
46C2×#97D411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#381D411yescharacteristic
51#381Q811yescharacteristic
52#380—21no—
53#380—21no—
54#379D411yescharacteristic
55#379Q811yescharacteristic
56C4×C22⋊C4—21no—
57C4×C22⋊C4—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#335—41no—
63#335—41no—
64#334—41no—
65#334—41no—
66#324—22no—
67#324—22no—
68#323—22no—
69#323—22no—
70C25C4⋊C411yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4C4⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100#117—41no—
101#117—41no—
102#117—41no—
103#117—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4#4111yescharacteristic
106C2×C4⋊C4#4111yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4C4⋊C411yescharacteristic
112C2×C22⋊C4C4⋊C411yescharacteristic
113C2×C22⋊C4C2×D411yescharacteristic
114C2×C22⋊C4C2×Q811yescharacteristic
115C2×C42—21no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#97—41no—
130#97—41no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—42no—
136C4×C8—42no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24—22no—
143C24—41no—
144C24—42no—
145C24—41no—
146C24—81no—
147C24C2×C4⋊C411yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4#11711yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4#13112yes—
178C22×C4#13112yes—
179M16—44no—
180M16—82no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8#9911yescharacteristic
203C2×C8#9911yescharacteristic
204C2×C8#10012yes—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—41no—
221C42—41no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—42no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—82no—
236C23—84no—
237C23—84no—
238C23—82no—
239C23#38011yescharacteristic
240C23#42412yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—84no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4#40811yescharacteristic
271C2×C4#40811yescharacteristic
272C2×C4C2×#9911yescharacteristic
273C2×C4#41011yescharacteristic
274C2×C4C2×#10011yescharacteristic
275C2×C4#41311yescharacteristic
276C2×C4#43312yes—
277C8—21no—
278C8—21no—
279C8—22no—
280C8—41no—
281C8—41no—
282C8—81no—
283C8—81no—
284C8—162no—
285C8—162no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—81no—
297C22—81no—
298C22—84no—
299C22—84no—
300C22—81no—
301C22—81no—
302C22#179811yescharacteristic
303C22#179911yescharacteristic
304C22#180011yescharacteristic
305C4—22no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4#146811yescharacteristic
314C4#146911yescharacteristic
315C4#147111yescharacteristic
316C4#147311yescharacteristic
317C2—21no—
318C2—21no—
319C2—42no—
320C2—82no—
321C2#856111yescharacteristic
322C2#1280511yescharacteristic
323C2#1280611yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC25C4⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8602C2semidirect
4. #8601C2semidirect
5. #7190C2semidirect

Maximal subgroups

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

subgroupindex
#210502
#114252
#86022
#86012
#71902