Finite Group Databasebeta

← #137907

Subgroups — #137907

Order 512.

403 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 90 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#33487C211yesmaximal, characteristic, complemented
2#22262C211yesmaximal, characteristic
3#11636C212yesmaximal, complemented
4#11488C211yesmaximal, characteristic, complemented
5#7143C212yesmaximal, complemented
6C2×#436C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8C4×#127C411yescharacteristic
9#2168—24no—
10#2163C2212yes—
11#2163C2212yes—
12#2112—21no—
13C2×C22⋊C16—21no—
14#1954C411yescharacteristic
15#1954C411yescharacteristic
16#1870—21no—
17#1870—21no—
18#1869—21no—
19#1869—21no—
20#1831C411yescharacteristic
21#1744C2211yescharacteristic
22#1330—24no—
23#1314—21no—
24#1314—21no—
25C22×M16—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#127C811yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4—21no—
31C2×C4×D4C2×C411yescharacteristic
32C2×C4×D4C2×C411yescharacteristic
33C2×M32—22no—
34C22×C16—22no—
35#436—22no—
36#436—22no—
37C8×D4—22no—
38C8×D4—22no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8D411yescharacteristic
47C2×C22⋊C8D411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#390—21no—
51#390—21no—
52#390C811yescharacteristic
53#390C811yescharacteristic
54#384C811yescharacteristic
55C43C2×C411yescharacteristic
56C4⋊C16—22no—
57C4⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61C22⋊C16—22no—
62#346—22no—
63#346D414yescomplemented
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4C2×C811yescharacteristic
71C22×D4C2×C811yescharacteristic
72C23×C4—21no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C2×M16—42no—
76C2×M16—81no—
77C2×M16—81no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—81no—
85C22×C8—81no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90#127—42no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—81no—
96C4×D4—81no—
97C4×D4—81no—
98C4×D4—81no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2×C811yescharacteristic
102C2×C4⋊C4C2×C811yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4M1611yescharacteristic
109C2×C22⋊C4M1611yescharacteristic
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C2×C811yescharacteristic
113C2×C42C2×C811yescharacteristic
114C2×C42C22×C411yescharacteristic
115M32—82no—
116C2×C16—24no—
117C2×C16—42no—
118C2×C16—82no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—81no—
126C22⋊C8—81no—
127C22⋊C8—81no—
128C22⋊C8—81no—
129C4×C8—22no—
130C4×C8C2×D412yes—
131C4×C8#4112yes—
132C24—21no—
133C24C22⋊C811yescharacteristic
134C24C22⋊C811yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—81no—
146C2×D4—81no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
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×C4C22⋊C811yescharacteristic
165C22×C4C22⋊C811yescharacteristic
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C22×C811yescharacteristic
168C22×C4C2×M1611yescharacteristic
169M16—84no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8#12011yescharacteristic
187C2×C8#12112yes—
188C2×C8#12311yescharacteristic
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—41no—
201C42—41no—
202C42—42no—
203C42—41no—
204C42—41no—
205C42C4×D412yes—
206C42#13112yes—
207C16—44no—
208C16—82no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23#34912yes—
226C23C2×C22⋊C811yescharacteristic
227D4—44no—
228D4—82no—
229D4—84no—
230D4—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4#38611yescharacteristic
260C2×C4#40811yescharacteristic
261C2×C4C8×D412yes—
262C2×C4#43512yes—
263C8—22no—
264C8—21no—
265C8—21no—
266C8—41no—
267C8—41no—
268C8—81no—
269C8—81no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22#186911yescharacteristic
290C22C2×#34911yescharacteristic
291C22#211511yescharacteristic
292C4—21no—
293C4—22no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4#191011yescharacteristic
301C4#191111yescharacteristic
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2—161no—
307C2#910111yescharacteristic
308C2#1298211yescharacteristic
309C2#1298511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC2×#127C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33487C2semidirect
3. #11636C2semidirect
4. #11488C2semidirect
5. #7143C2semidirect
63. #346D4semidirect

Maximal subgroups

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

subgroupindex
#334872
#222622
#116362
#114882
#71432