Finite Group Databasebeta

← #134829

Subgroups — #134829

Order 512.

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

normal subgroups only

This group has 381 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#22262C211yesmaximal, characteristic
2#11529C212yesmaximal
3#8612C211yesmaximal, characteristic, complemented
4#8611C211yesmaximal, characteristic, complemented
5#7249C212yesmaximal, complemented
6C2×#436C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8C4×#127C2211yescharacteristic
9#1954C411yescharacteristic
10#1954C2211yescharacteristic
11#1870—21no—
12#1870—21no—
13#1869—21no—
14#1869—21no—
15#1831C411yescharacteristic
16#1810—21no—
17#1810—21no—
18#1744C411yescharacteristic
19#1465C2212yes—
20#1465C2212yes—
21#1325—22no—
22#1318—22no—
23#1293—21no—
24#1293—21no—
25C22×M16—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#127Q811yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4—21no—
31C2×C4×D4C2×C411yescharacteristic
32C2×C4×D4C2311yescharacteristic
33#436—41no—
34#436—41no—
35C8×D4—41no—
36C8×D4—41no—
37C2×#108—41no—
38C2×#108—41no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41#407—22no—
42#407—22no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8D411yescharacteristic
49C2×C22⋊C8D411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8C2×C411yescharacteristic
52#390—21no—
53#390—21no—
54#390D411yescharacteristic
55#390Q811yescharacteristic
56#384D411yescharacteristic
57C43C2×C411yescharacteristic
58#341—41no—
59#341—41no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#324—22no—
65#324—22no—
66#323—22no—
67#323—22no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4C4⋊C411yescharacteristic
71C22×D4C2×Q811yescharacteristic
72C23×C4—21no—
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—81no—
79C2×M16—81no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—81no—
87C22×C8—81no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92#127—42no—
93C4×D4—41no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—41no—
97C4×D4—81no—
98C4×D4—81no—
99C4×D4—81no—
100C4×D4—81no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C4⋊C411yescharacteristic
104C2×C4⋊C4C2×D411yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4#4111yescharacteristic
111C2×C22⋊C4#4111yescharacteristic
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C4⋊C411yescharacteristic
115C2×C42C4⋊C411yescharacteristic
116C2×C42C22×C411yescharacteristic
117#108—82no—
118#108—82no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—81no—
130C22⋊C8—81no—
131C22⋊C8—81no—
132C22⋊C8—81no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—41no—
136C24—21no—
137C24C22⋊Q811yescharacteristic
138C24C22⋊Q811yescharacteristic
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—81no—
150C2×D4—81no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4C2×C4⋊C411yescharacteristic
170C22×C4#11711yescharacteristic
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173M16—84no—
174M16—82no—
175M16—82no—
176M16—82no—
177M16—82no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—41no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8#9911yescharacteristic
198C2×C8#9911yescharacteristic
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—42no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C42—21no—
211C42—21no—
212C42—22no—
213C42—41no—
214C42—42no—
215C42—41no—
216C42—41no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23#38011yescharacteristic
235D4—44no—
236D4—82no—
237D4—84no—
238D4—84no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
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—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4C2×#9911yescharacteristic
269C2×C4#41011yescharacteristic
270C2×C4#43012yes—
271C8—21no—
272C8—21no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—81no—
277C8—81no—
278C8—162no—
279C8—162no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22#179911yescharacteristic
300C22#181111yescharacteristic
301C22#181311yescharacteristic
302C4—22no—
303C4—21no—
304C4—42no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4#212711yescharacteristic
311C4#212811yescharacteristic
312C2—21no—
313C2—21no—
314C2—81no—
315C2—81no—
316C2—161no—
317C2#861311yescharacteristic
318C2#1279911yescharacteristic
319C2#1280911yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#127Q8
LCSG ▷ C2×C42 ▷ 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. #8612C2semidirect
4. #8611C2semidirect
5. #7249C2semidirect

Maximal subgroups

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

subgroupindex
#222622
#115292
#86122
#86112
#72492