Finite Group Databasebeta

← #213639

Subgroups — #213639

Order 512.

948 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 206 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#35333C211yesmaximal, characteristic
2#21379C212yesmaximal
3#9126C214yesmaximal, complemented
4#9121C214yesmaximal, complemented
5#8722C212yesmaximal, complemented
6#8335C212yesmaximal, complemented
7C22×#125C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C2×#432C2212yes—
10C22×#97C2212yes—
11#2851C2214yes—
12#2629C414yes—
13#2579—22no—
14#2579—22no—
15#2566—22no—
16#2565—22no—
17#2317C412yes—
18#2311C412yes—
19#2285C2211yescharacteristic
20#1985—22no—
21#1955—22no—
22#1869—22no—
23#1869—22no—
24#1749C2214yes—
25#1749C2214yes—
26C2×#330C2212yes—
27C2×#330C2214yes—
28C2×#330C2212yes—
29#1295C2212yes—
30#1295C2214yes—
31#1295C2212yes—
32C22×#41C2×C411yescharacteristic
33C23×C8—22no—
34C2×#125—22no—
35C2×#125—41no—
36C2×#125—42no—
37C2×#125—41no—
38C2×#125C2×C412yes—
39C2×#125C2312yes—
40C4×#41—41no—
41C4×#41—42no—
42C4×#41—41no—
43C2×C4×Q8D412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4D412yes—
47C2×#117C2×C412yes—
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C412yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2×C412yes—
52C22×C42C2312yes—
53C22×C42C2311yescharacteristic
54#432—42no—
55#432—42no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—24no—
59C2×C22⋊C8—24no—
60C2×#97—22no—
61C2×#97—42no—
62C2×#97—42no—
63C2×#97C2314yes—
64C2×#97C2314yes—
65#398—24no—
66#398—24no—
67#396—24no—
68#396—24no—
69#391—22no—
70#390—22no—
71#389C2×C414yes—
72#389C2×C414yes—
73#387—24no—
74#387—24no—
75#386—22no—
76#386—22no—
77#385—24no—
78#385—24no—
79#385—22no—
80#385—22no—
81#384—22no—
82#384C2×C412yes—
83#382—22no—
84#382C2312yes—
85#380—22no—
86#380C2×C414yes—
87#380C2×C414yes—
88C4×C4⋊C4—22no—
89#336—24no—
90#336—24no—
91#330—42no—
92#330—44no—
93#330—42no—
94C2×#41—24no—
95C2×#41—41no—
96C2×#41—41no—
97C2×#41—42no—
98C22×Q8C22⋊C411yescharacteristic
99C22×D4C22⋊C412yes—
100C22×D4C22⋊C411yescharacteristic
101C23×C4—22no—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C22×C8—24no—
108C22×C8—24no—
109C22×C8—48no—
110C22×C8—44no—
111#125—22no—
112#125—24no—
113#125—22no—
114#125—42no—
115#125—42no—
116#125—44no—
117#125—44no—
118C4×Q8—42no—
119C4×Q8—42no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126#117—41no—
127#117—41no—
128#117—42no—
129#117—42no—
130#117—44no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4C22⋊C411yescharacteristic
150C2×C22⋊C4C22⋊C411yescharacteristic
151C2×C22⋊C4C22⋊C412yes—
152C2×C22⋊C4C22⋊C412yes—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2411yescharacteristic
167C4⋊C8—44no—
168C22⋊C8—48no—
169C22⋊C8—48no—
170C22⋊C8—44no—
171#97—24no—
172#97—24no—
173#97—24no—
174#97—24no—
175#97—44no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181C24C2×C22⋊C412yes—
182C24C2×C22⋊C411yescharacteristic
183#41—48no—
184#41—48no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—41no—
193C2×D4#10114yes—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—84no—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C23×C411yescharacteristic
221C22×C4C22×D412yes—
222C2×C8—28no—
223C2×C8—44no—
224C2×C8—416no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C4⋊C4—84no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—44no—
247C22⋊C4—84no—
248C22⋊C4—84no—
249C42—22no—
250C42—22no—
251C42—21no—
252C42—21no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—44no—
260C42H(5,𝔽2)11yescharacteristic
261C42H(5,𝔽2)12yes—
262C42H(5,𝔽2)11yescharacteristic
263C23—21no—
264C23—22no—
265C23—21no—
266C23—22no—
267C23—24no—
268C23—24no—
269C23—24no—
270C23C2×#10114yes—
271C23#41312yes—
272C23C22×C22⋊C411yescharacteristic
273Q8—44no—
274D4—44no—
275D4—48no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—88no—
307C2×C4C2×#10112yes—
308C2×C4#51811yescharacteristic
309C2×C4#51811yescharacteristic
310C2×C4#53412yes—
311C2×C4#53412yes—
312C8—48no—
313C8—84no—
314C22—28no—
315C22—24no—
316C22—22no—
317C22—24no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—44no—
323C22#230811yescharacteristic
324C22C22×#10111yescharacteristic
325C22C2×#41311yescharacteristic
326C22#290114yes—
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—22no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—84no—
337C4—84no—
338C2—44no—
339C2—42no—
340C2#1768912yes—
341C2#1769012yes—
342C2#2175311yescharacteristic
343C2#2177512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9126C2semidirect
4. #9121C2semidirect
5. #8722C2semidirect
6. #8335C2semidirect

Maximal subgroups

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

subgroupindex
#353332
#213792
#91262
#91212
#87222
#83352