Finite Group Databasebeta

← #93341

Subgroups — #93341

Order 512.

454 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 76 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#10921C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4C22×C22⋊C8—21no—
5C22×C4×C8C411yescharacteristic
6C2×C4×C22⋊C4C2211yescharacteristic
7#1779—21no—
8#1749—21no—
9C2×#336—21no—
10C2×#336C411yescharacteristic
11C2×#323C412yes—
12#1280—21no—
13#1280—21no—
14#1280C412yes—
15C24×C4D411yescharacteristic
16C23×C8—21no—
17C23×C8—21no—
18C22×C22⋊C4C2×C411yescharacteristic
19C22×C22⋊C4Q811yescharacteristic
20C22×C42C2×C411yescharacteristic
21C22×C42D411yescharacteristic
22C2×C22⋊C8—22no—
23C2×C22⋊C8—24no—
24C2×C22⋊C8—22no—
25C2×C22⋊C8—42no—
26C2×C22⋊C8—42no—
27C2×C22⋊C8—41no—
28C2×C22⋊C8—41no—
29C2×#97—41no—
30C2×C4×C8—41no—
31C2×C4×C8—42no—
32C4×C22⋊C4—41no—
33C4×C22⋊C4—41no—
34C2×#95C2×C411yescharacteristic
35C2×#95D411yescharacteristic
36#336—41no—
37#336—41no—
38#336—42no—
39#323—22no—
40#323C814yescomplemented
41C25C4⋊C411yescharacteristic
42C23×C4—22no—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4—41no—
46C23×C4—41no—
47C23×C4—41no—
48C23×C4—41no—
49C23×C4—41no—
50C23×C4C4211yescharacteristic
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22⋊C411yescharacteristic
53C23×C4C22⋊C411yescharacteristic
54C23×C4C4⋊C411yescharacteristic
55C23×C4C4⋊C411yescharacteristic
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—42no—
70C22×C8—41no—
71C22×C8—42no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—41no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4C2×C812yes—
78C2×C22⋊C4M1612yes—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42—41no—
87C22⋊C8—24no—
88C22⋊C8—48no—
89C22⋊C8—42no—
90C22⋊C8—42no—
91C22⋊C8—42no—
92C22⋊C8—81no—
93C22⋊C8—81no—
94C22⋊C8—81no—
95C22⋊C8—81no—
96#97—81no—
97#97—81no—
98C4×C8—42no—
99C4×C8—42no—
100#95—41no—
101#95—41no—
102C24—21no—
103C24—21no—
104C24—21no—
105C24—21no—
106C24—21no—
107C24—21no—
108C24—41no—
109C24—41no—
110C24—41no—
111C24—41no—
112C24#9511yescharacteristic
113C24C4⋊C812yes—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4—81no—
157C22×C4C4×C811yescharacteristic
158C22×C4#9711yescharacteristic
159C22×C4C22⋊C812yes—
160C22×C4#9911yescharacteristic
161C22×C4#9911yescharacteristic
162C22×C4#10412yes—
163C22×C4#10812yes—
164C2×C8—24no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8—82no—
179C2×C8—82no—
180C2×C8—81no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—44no—
187C42—42no—
188C42—41no—
189C42—41no—
190C42—41no—
191C42—41no—
192C42—81no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—81no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23—81no—
220C23#33611yescharacteristic
221C23H(3,ℤ/4)11yescharacteristic
222C23#33811yescharacteristic
223C23#33911yescharacteristic
224C23#34111yescharacteristic
225C23#34211yescharacteristic
226C23#34411yescharacteristic
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
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—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4AΣL(1,𝔽16)11yescharacteristic
271C2×C4#35211yescharacteristic
272C2×C4#35311yescharacteristic
273C2×C4#35411yescharacteristic
274C8—44no—
275C8—82no—
276C8—84no—
277C8—162no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22#127511yescharacteristic
302C22#127811yescharacteristic
303C22#128211yescharacteristic
304C22#128411yescharacteristic
305C22#128711yescharacteristic
306C22#129111yescharacteristic
307C22#129311yescharacteristic
308C22#139011yescharacteristic
309C22#139211yescharacteristic
310C22#139311yescharacteristic
311C22#139511yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—81no—
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2—82no—
327C2#702111yescharacteristic
328C2#720311yescharacteristic
329C2#720511yescharacteristic
330C2#721311yescharacteristic
331C2#721611yescharacteristic
332C2#724711yescharacteristic
333C2#724911yescharacteristic
subgroupG/H
center, socleC23#342
commutatorC22×C4C4×C8
Frattini, cosocleC2×C4×C22⋊C4C22
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×C4×C22⋊C4 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
40. #323C8semidirect

Maximal subgroups

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

subgroupindex
#109212
#87002
#81522