Finite Group Databasebeta

← #124348

Subgroups — #124348

Order 512.

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

normal subgroups only

This group has 424 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#21377C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic, complemented
3C2×#1377C211yesmaximal, characteristic, complemented
4#7133C212yesmaximal, complemented
5#7105C212yesmaximal, complemented
6C2×#529C411yescharacteristic
7C22×M32—21no—
8C2×#431C411yescharacteristic
9C22×#97C2211yescharacteristic
10#2112—21no—
11#1985C411yescharacteristic
12#1985C2211yescharacteristic
13#1955C411yescharacteristic
14#1955C2211yescharacteristic
15#1869—21no—
16#1869—21no—
17#1750—21no—
18#1750—21no—
19#1377—21no—
20#1377—21no—
21#1377C2212yes—
22#1377C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25#529—41no—
26#529—41no—
27#529—41no—
28#529—41no—
29C2×#126—21no—
30C2×#125C811yescharacteristic
31C2×#124C811yescharacteristic
32C2×#123—21no—
33C2×C22⋊Q8C811yescharacteristic
34C2×#121C811yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4C2×C411yescharacteristic
38C2×#117—21no—
39C22×C42C2×C411yescharacteristic
40C2×M32—24no—
41C2×M32—22no—
42C2×M32—22no—
43C2×M32—41no—
44C2×M32—42no—
45C22×C16—41no—
46#431—41no—
47#431—41no—
48C2×C4⋊C8C2×C411yescharacteristic
49C2×C4⋊C8C2×C411yescharacteristic
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—41no—
56C2×#97—41no—
57C2×#97C2×C411yescharacteristic
58C2×#97D411yescharacteristic
59C2×#97D411yescharacteristic
60C2×#97C2311yescharacteristic
61C2×C4×C8—21no—
62C22⋊C16—41no—
63C22⋊C16—41no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×Q8C2×C811yescharacteristic
69C22×D4—21no—
70C22×D4C2×C811yescharacteristic
71C23×C4—21no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C2×C811yescharacteristic
74C23×C4M1611yescharacteristic
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—41no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90#126—42no—
91#126—42no—
92#125—41no—
93#125—41no—
94#124—41no—
95#124—41no—
96#123—42no—
97#123—42no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100#121—41no—
101#121—41no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×D4—41no—
105C4×D4—41no—
106C4×D4—41no—
107C4×D4—41no—
108C4×D4—41no—
109C4×D4—41no—
110#117—42no—
111#117—42no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×C811yescharacteristic
117C2×C4⋊C4C2×C811yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42C22⋊C411yescharacteristic
129C2×C42C2×C811yescharacteristic
130C2×C42M1611yescharacteristic
131C2×C42C22×C411yescharacteristic
132M32—24no—
133M32—44no—
134M32—42no—
135M32—42no—
136M32—42no—
137C2×C16—44no—
138C2×C16—42no—
139C2×C16—42no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C22⋊C8—42no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#97—41no—
150#97—41no—
151#97—41no—
152#97—41no—
153C4×C8—42no—
154C24—21no—
155C24C22⋊C811yescharacteristic
156C2×Q8—41no—
157C2×Q8—81no—
158C2×D4—41no—
159C2×D4—44no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—81no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—81no—
182C22×C4C22⋊C811yescharacteristic
183C22×C4C22⋊C811yescharacteristic
184C22×C4C22⋊C811yescharacteristic
185C22×C4#9911yescharacteristic
186C22×C4#9911yescharacteristic
187C22×C4#10011yescharacteristic
188C22×C4#10011yescharacteristic
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C22×C811yescharacteristic
191C22×C4C2×M1611yescharacteristic
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8#10412yes—
214C2×C8#13512yes—
215C2×C8AGL(1,ℤ/8)11yescharacteristic
216C2×C8#13711yescharacteristic
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—41no—
241C16—44no—
242C16—82no—
243C23—22no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23#32311yescharacteristic
253C23#32311yescharacteristic
254C23C2×C22⋊C811yescharacteristic
255Q8—82no—
256D4—82no—
257D4—82no—
258D4—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4#32311yescharacteristic
293C2×C4#32311yescharacteristic
294C2×C4C2×#9911yescharacteristic
295C2×C4C2×#10011yescharacteristic
296C2×C4#41611yescharacteristic
297C2×C4#41711yescharacteristic
298C2×C4C2×#10411yescharacteristic
299C2×C4#42111yescharacteristic
300C8—21no—
301C8—22no—
302C8—21no—
303C8—22no—
304C8—41no—
305C8—81no—
306C8—81no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—42no—
315C22—42no—
316C22—82no—
317C22—82no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#132112yes—
322C22C2×#32311yescharacteristic
323C22#147611yescharacteristic
324C22#147711yescharacteristic
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#150411yescharacteristic
335C4#151611yescharacteristic
336C4#153011yescharacteristic
337C4#155111yescharacteristic
338C2—21no—
339C2—21no—
340C2—22no—
341C2—41no—
342C2—161no—
343C2#757811yescharacteristic
344C2#1143611yescharacteristic
345C2C2×#132111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#323
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12846C2semidirect
3. C2×#1377C2semidirect
4. #7133C2semidirect
5. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#213772
#128462
C2×#13772
#71332
#71052