Finite Group Databasebeta

← #134465

Subgroups — #134465

Order 512.

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

normal subgroups only

This group has 414 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#21492C211yesmaximal, characteristic
2#11697C212yesmaximal
3#10815C211yesmaximal, characteristic, complemented
4#8601C211yesmaximal, characteristic, complemented
5#7253C212yesmaximal, complemented
6C2×#435—21no—
7C2×C8×D4C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2392C2211yescharacteristic
10#1955C411yescharacteristic
11#1940C411yescharacteristic
12#1923—21no—
13#1869—21no—
14#1798C2211yescharacteristic
15#1770C411yescharacteristic
16#1752—21no—
17#1742C2212yes—
18C2×#341—21no—
19C2×#338—21no—
20#1564—21no—
21#1458C2212yes—
22#1346—22no—
23#1345—22no—
24#1275—21no—
25C22×M16—22no—
26C22×M16D411yescharacteristic
27C23×C8—21no—
28C23×C8D411yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2×C411yescharacteristic
31C2×#117C2311yescharacteristic
32#435—44no—
33C8×D4—41no—
34C8×D4—41no—
35#431—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43#405—41no—
44#405—41no—
45C2×#97—21no—
46C2×#97C2×C411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#398D411yescharacteristic
51#397Q811yescharacteristic
52#396—21no—
53#394—21no—
54#394D411yescharacteristic
55#392Q811yescharacteristic
56C4×C4⋊C4C2×C411yescharacteristic
57C4×C22⋊C4—21no—
58#343—22no—
59#343—22no—
60#341—41no—
61#341—41no—
62#338—24no—
63#336—21no—
64#336—21no—
65#336—21no—
66#335—41no—
67#334—41no—
68#324—22no—
69#323—22no—
70C22×D4—21no—
71C22×D4C4⋊C411yescharacteristic
72C23×C4—21no—
73C23×C4C2×D411yescharacteristic
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—24no—
77C2×M16—41no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—42no—
100C4×D4—42no—
101#117—41no—
102#117—41no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4C4⋊C411yescharacteristic
106C2×C4⋊C4C2×D411yescharacteristic
107C2×C4⋊C4#4111yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C2×Q811yescharacteristic
114C2×C22⋊C4#4111yescharacteristic
115C2×C42—21no—
116C2×C42C22×C411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—42no—
128#97—41no—
129#97—41no—
130#97—42no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—42no—
136#95—21no—
137#95—21no—
138#95C4⋊C411yescharacteristic
139#95C4⋊C411yescharacteristic
140C24—21no—
141C24C22⋊Q811yescharacteristic
142C2×D4—41no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
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×C4C2×C22⋊C411yescharacteristic
161C22×C4C2×C4⋊C411yescharacteristic
162C22×C4#11711yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4C22⋊Q811yescharacteristic
166C22×C4#13512yes—
167C22×C4AGL(1,ℤ/8)12yes—
168M16—44no—
169M16—44no—
170M16—44no—
171M16—82no—
172M16—82no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—44no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—84no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8#9911yescharacteristic
197C2×C8#9911yescharacteristic
198C2×C8#10412yes—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—81no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23#38011yescharacteristic
229C23#47812yes—
230D4—42no—
231D4—42no—
232D4—84no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4C2×#9911yescharacteristic
261C2×C4#41011yescharacteristic
262C2×C4#41611yescharacteristic
263C2×C4#41811yescharacteristic
264C2×C4C2×#10411yescharacteristic
265C2×C4#42111yescharacteristic
266C2×C4#46312yes—
267C8—22no—
268C8—21no—
269C8—21no—
270C8—41no—
271C8—41no—
272C8—84no—
273C8—81no—
274C8—81no—
275C8—162no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22#179911yescharacteristic
291C22#180411yescharacteristic
292C22#180611yescharacteristic
293C4—22no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4#150011yescharacteristic
302C4#150711yescharacteristic
303C4#151411yescharacteristic
304C4#151611yescharacteristic
305C2—21no—
306C2—21no—
307C2—42no—
308C2—82no—
309C2#857311yescharacteristic
310C2#1280511yescharacteristic
311C2#1281011yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radical#392Q8
LCSG ▷ C2×C42 ▷ C23 ▷ 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. #10815C2semidirect
4. #8601C2semidirect
5. #7253C2semidirect

Maximal subgroups

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

subgroupindex
#214922
#116972
#108152
#86012
#72532