Finite Group Databasebeta

← #134825

Subgroups — #134825

Order 512.

410 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 72 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#22030C211yesmaximal, characteristic
2#11525C212yesmaximal
3#8620C211yesmaximal, characteristic, complemented
4#8612C211yesmaximal, characteristic, complemented
5#7247C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C22⋊C8C411yescharacteristic
8C4×#120C2211yescharacteristic
9#1955—21no—
10#1924—21no—
11#1923—21no—
12#1869—21no—
13#1869C2211yescharacteristic
14#1850C411yescharacteristic
15#1810—21no—
16#1779C411yescharacteristic
17#1752C411yescharacteristic
18#1568—21no—
19#1466C2212yes—
20#1465C2212yes—
21#1326—22no—
22#1316—22no—
23#1293—21no—
24#1290—21no—
25C24×C4C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×#120Q811yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2311yescharacteristic
32C8×D4—41no—
33C8×D4—41no—
34#432—22no—
35C2×#108—41no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8D411yescharacteristic
38#407—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—41no—
45C2×C22⋊C8—41no—
46C2×C22⋊C8C2×C411yescharacteristic
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8D411yescharacteristic
49C2×#97—21no—
50C2×C4×C8D411yescharacteristic
51#386—21no—
52#386Q811yescharacteristic
53#385—21no—
54#385D411yescharacteristic
55#379D411yescharacteristic
56C4×C22⋊C4—21no—
57C4×C22⋊C4C2×C411yescharacteristic
58#341—41no—
59#336—21no—
60#336—21no—
61#336—21no—
62#335—41no—
63#334—41no—
64#324—22no—
65#323—22no—
66#323—22no—
67#323—22no—
68C25C4⋊C411yescharacteristic
69C22×D4—21no—
70C22×D4C2×Q811yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4C4⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C2×M16—42no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8—81no—
89C22×C8—81no—
90C22×C8—81no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95#120—42no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—81no—
99C4×D4—81no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4#4111yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4C4⋊C411yescharacteristic
107C2×C22⋊C4C4⋊C411yescharacteristic
108C2×C22⋊C4C2×D411yescharacteristic
109C2×C22⋊C4#4111yescharacteristic
110C2×C42—21no—
111C2×C42C2×D411yescharacteristic
112#108—82no—
113C4⋊C8—41no—
114C4⋊C8—42no—
115C4⋊C8—81no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118C22⋊C8—42no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—81no—
126C22⋊C8—81no—
127C22⋊C8—81no—
128#97—42no—
129#97—81no—
130C4×C8—21no—
131C4×C8—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135C24—21no—
136C24—22no—
137C24—41no—
138C24—42no—
139C24—41no—
140C24—81no—
141C24C2×C4⋊C411yescharacteristic
142C24C22⋊Q811yescharacteristic
143C2×D4—22no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—81no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—41no—
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—41no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—84no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4#11711yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179M16—82no—
180M16—82no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8#9911yescharacteristic
202C2×C8#9911yescharacteristic
203C4⋊C4—41no—
204C4⋊C4—81no—
205C22⋊C4—22no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—41no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
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#43012yes—
241D4—44no—
242D4—84no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
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—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—84no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4C2×#9911yescharacteristic
276C2×C4#41011yescharacteristic
277C8—21no—
278C8—21no—
279C8—41no—
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—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—84no—
304C22—84no—
305C22—81no—
306C22—81no—
307C22#179911yescharacteristic
308C22#181111yescharacteristic
309C22#181311yescharacteristic
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4#212111yescharacteristic
319C4#212211yescharacteristic
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2—82no—
325C2#861311yescharacteristic
326C2#1279511yescharacteristic
327C2#1280511yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#120Q8
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8620C2semidirect
4. #8612C2semidirect
5. #7247C2semidirect

Maximal subgroups

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

subgroupindex
#220302
#115252
#86202
#86122
#72472