Finite Group Databasebeta

← #214645

Subgroups — #214645

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37422C211yesmaximal, characteristic
2#21756C211yesmaximal, characteristic, complemented
3#21751C211yesmaximal, characteristic, complemented
4#9466C214yesmaximal, complemented
5#9445C214yesmaximal, complemented
6#8741C211yesmaximal, characteristic, complemented
7#8740C211yesmaximal, characteristic, complemented
8#8518C211yesmaximal, characteristic, complemented
9#8518C211yesmaximal, characteristic, complemented
10C2×#540C2211yescharacteristic
11C2×#536C2211yescharacteristic
12#2879C2212yes—
13#2861C411yescharacteristic, complemented
14#2858C411yescharacteristic, complemented
15#2703—24no—
16#2696—24no—
17#2657C411yescharacteristic, complemented
18#2642C411yescharacteristic, complemented
19#2547C412yescomplemented
20#2544C412yescomplemented
21#2436C2212yes—
22#2431C2212yes—
23#2310C2212yes—
24#2298C2211yescharacteristic
25#1901—24no—
26#1899—24no—
27#1822C2211yescharacteristic
28#1822C2211yescharacteristic
29#1820C2211yescharacteristic
30#1820C2211yescharacteristic
31#1784C2212yes—
32#1782C2212yes—
33#1510C2214yes—
34#1510C2214yes—
35#1284C2214yes—
36#1284C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16—22no—
40#540—44no—
41#536—44no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#127C2×C412yes—
46C2×#125C2×C411yescharacteristic
47C2×#124C2×C412yes—
48C2×#124C2×C412yes—
49C2×#124C2×C412yes—
50C2×#124C2×C411yescharacteristic
51C2×#124C2×C411yescharacteristic
52C2×#123—22no—
53C2×#123D414yes—
54C2×C22⋊Q8D414yes—
55C2×#121—22no—
56C2×#121C2312yes—
57C2×C4×D4C2×C412yes—
58C2×#117—22no—
59C2×#117C2311yescharacteristic
60C2×#101—24no—
61C2×#99—24no—
62#408—44no—
63#407—22no—
64#407C2312yes—
65#407C2312yes—
66#400—24no—
67#397—24no—
68#394—24no—
69#393—24no—
70#392—24no—
71#392—24no—
72#391—44no—
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2×C412yes—
76#390—24no—
77#390C2×C412yes—
78#388—24no—
79#388—22no—
80#387—22no—
81#386—24no—
82#386—24no—
83#386—24no—
84#381—22no—
85#381C2312yes—
86#381C2312yes—
87#380—24no—
88C4×C4⋊C4C2×C412yes—
89C4×C22⋊C4C2312yes—
90#376C2×C411yescharacteristic
91C43C2×C411yescharacteristic
92#343—22no—
93#343—22no—
94#342—22no—
95#342—22no—
96#327—44no—
97#327—44no—
98C2×#41—24no—
99C2×#41—82no—
100C22×Q8—24no—
101C22×Q8—82no—
102C22×Q8C22×C411yescharacteristic
103C22×Q8C22×C411yescharacteristic
104C22×Q8C22×C411yescharacteristic
105C22×Q8C2×D412yes—
106C22×D4—22no—
107C22×D4—24no—
108C22×D4—22no—
109C22×D4C22×C412yes—
110C22×D4C22×C411yescharacteristic
111C22×D4C2×D412yes—
112C23×C4—22no—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2411yescharacteristic
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—42no—
119C2×M16—84no—
120C22×C8—42no—
121C4⋊Q8—42no—
122C4⋊Q8—42no—
123#127—42no—
124#127—82no—
125#125—42no—
126#124—42no—
127#124—42no—
128#124—82no—
129#124—82no—
130#124—82no—
131#123—44no—
132#123—44no—
133#123—48no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136#121—24no—
137#121—24no—
138#121—48no—
139C4×D4—42no—
140C4×D4—42no—
141#117—24no—
142#117—84no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4C22⋊C414yes—
160C2×C22⋊C4C22⋊C414yes—
161C2×C22⋊C4C2×D412yes—
162C2×C42—22no—
163C2×C42—42no—
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C412yes—
168#101—84no—
169#99—84no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172C22⋊C8—44no—
173#95—22no—
174#95—22no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—24no—
180#95—24no—
181#95—44no—
182C24—22no—
183C24C2×C22⋊C412yes—
184C24C22×D411yescharacteristic
185#41—84no—
186#41—84no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—24no—
191C2×Q8—44no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—84no—
196C2×Q8—84no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—44no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—84no—
208C2×D4—84no—
209C2×D4—84no—
210C2×D4—84no—
211C2×D4—84no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—84no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4H(5,𝔽2)14yes—
234M16—44no—
235M16—44no—
236M16—84no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—84no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—82no—
247C4⋊C4—84no—
248C22⋊C4—24no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—84no—
256C22⋊C4—84no—
257C22⋊C4—84no—
258C22⋊C4—84no—
259C22⋊C4—82no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42—42no—
264C42—82no—
265C42—84no—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—42no—
270C23—44no—
271C23—44no—
272C23—44no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23C22×C22⋊C411yescharacteristic
277C23#53412yes—
278Q8—84no—
279Q8—82no—
280Q8—82no—
281Q8—82no—
282D4—82no—
283D4—82no—
284D4—82no—
285D4—84no—
286D4—164no—
287D4—164no—
288D4—168no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—164no—
321C2×C4#51812yes—
322C2×C4#53412yes—
323C8—82no—
324C8—82no—
325C22—22no—
326C22—24no—
327C22—22no—
328C22—44no—
329C22—42no—
330C22—82no—
331C22—84no—
332C22—84no—
333C22—84no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22#230811yescharacteristic
338C22#288312yes—
339C4—22no—
340C4—22no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—82no—
347C4—82no—
348C4—82no—
349C4—162no—
350C4—162no—
351C2—22no—
352C2—42no—
353C2—82no—
354C2—162no—
355C2#2175011yescharacteristic
356C2#2178512yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#37422C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21756C2semidirect
3. #21751C2semidirect
4. #9466C2semidirect
5. #9445C2semidirect
6. #8741C2semidirect
7. #8740C2semidirect
8. #8518C2semidirect
9. #8518C2semidirect
13. #2861C4semidirect
14. #2858C4semidirect
17. #2657C4semidirect
18. #2642C4semidirect
19. #2547C4semidirect
20. #2544C4semidirect

Maximal subgroups

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

subgroupindex
#374222
#217562
#217512
#94662
#94452
#87412
#87402
#85182