Finite Group Databasebeta

← #225524

Subgroups — #225524

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22758C212yesmaximal
2#22707C211yesmaximal, characteristic, complemented
3#22703C211yesmaximal, characteristic, complemented
4C2×C8×D8C211yesmaximal, characteristic
5#9619C212yesmaximal, complemented
6#8885C212yesmaximal, complemented
7#8882C212yesmaximal, complemented
8#8816C211yesmaximal, characteristic, complemented
9#8646C211yesmaximal, characteristic, complemented
10#8646C211yesmaximal, characteristic, complemented
11#8283C211yesmaximal, characteristic, complemented
12C2×C4×D8C2211yescharacteristic
13C2×C8×D4C2211yescharacteristic
14C2×C8×D4C2211yescharacteristic
15#2354C2212yes—
16D4×C4⋊C4C2212yes—
17#1948C2212yes—
18#1947C2212yes—
19#1936—22no—
20#1921—22no—
21#1879—22no—
22#1854—22no—
23#1853—22no—
24#1845C2212yes—
25#1834C2212yes—
26#1834C2211yescharacteristic
27#1834C2211yescharacteristic
28#1808—21no—
29#1807—21no—
30#1802C2212yes—
31#1802C2212yes—
32C4×#106C2211yescharacteristic
33#1770C2212yes—
34#1762—22no—
35#1735—21no—
36C2×#340—21no—
37C8×D8C418yescomplemented
38C2×#335C2211yescharacteristic
39C2×#334C2211yescharacteristic
40C2×D4⋊C8C2211yescharacteristic
41C2×D4⋊C8C2211yescharacteristic
42C2×C82C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1279C2212yes—
46C22×D8D411yescharacteristic
47C23×C8—21no—
48C23×C8—21no—
49C2×C4×D4—22no—
50C2×C4×D4C2311yescharacteristic
51C2×C4×D4C2311yescharacteristic
52C2×#117—22no—
53C22×C4⋊C4—22no—
54C4×D8C2×C414yes—
55C4×D8Q814yes—
56C8×D4—22no—
57C8×D4—22no—
58C8×D4C2×C414yes—
59C8×D4C2×C414yescomplemented
60C2×#107—22no—
61C2×#107—42no—
62C2×#107D411yescharacteristic
63C2×#106—22no—
64C2×#106D412yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×#102—22no—
69C2×#102—22no—
70C2×#102D411yescharacteristic
71C2×#102D411yescharacteristic
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×#97—22no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#390—22no—
79#389—22no—
80#388—42no—
81#385—42no—
82#384C2312yes—
83#384C2312yes—
84#380—22no—
85#380—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2312yes—
88#340—21no—
89#340—21no—
90#340—21no—
91#340—21no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340D411yescharacteristic
99#340D411yescharacteristic
100#336—22no—
101#335C2×C414yescomplemented
102#334—22no—
103D4⋊C8C2×C414yes—
104D4⋊C8C2×C414yescomplemented
105C82C2×C414yescomplemented
106C22×D4C2×D411yescharacteristic
107C22×D4C2×D411yescharacteristic
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—42no—
111C2×D8—24no—
112C2×D8C4⋊C414yes—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—22no—
119C22×C8—82no—
120C22×C8—82no—
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C4×D4—21no—
126C4×D4—21no—
127C4×D4—21no—
128C4×D4—21no—
129C4×D4—44no—
130C4×D4—82no—
131C4×D4C22×C412yes—
132C4×D4C22×C412yes—
133C4×D4C2×Q812yes—
134C4×D4C2×Q812yes—
135#117—44no—
136#117—82no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D411yescharacteristic
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C42—22no—
156C2×C42—42no—
157C2×C42C2411yescharacteristic
158#107—82no—
159#107—84no—
160#107C4⋊C414yes—
161#106—42no—
162#106—82no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8C22×C412yes—
168C4⋊C8C22×C412yes—
169#102—44no—
170#102—82no—
171#102C4⋊C414yes—
172#102C4⋊C414yes—
173C22⋊C8—24no—
174C22⋊C8—24no—
175#97—82no—
176C4×C8—21no—
177C4×C8C22×C412yes—
178C4×C8C22×C412yes—
179C4×C8C22×C412yes—
180C4×C8C2×Q812yes—
181C4×C8#4112yes—
182#95—22no—
183#95—42no—
184#95—42no—
185C24—21no—
186C24—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4C2×C4⋊C412yes—
193C2×D4C2×C4⋊C412yes—
194C2×D4#13514yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—84no—
210C22×C4—84no—
211C22×C4—162no—
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4H(5,𝔽2)11yescharacteristic
215D8—28no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—84no—
231C2×C8—84no—
232C2×C8C2×C4⋊C412yes—
233C2×C8C2×C4⋊C412yes—
234C2×C8C4×D412yes—
235C2×C8C4×D412yes—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4C2×C4⋊C412yes—
254C4⋊C4C2×C4⋊C412yes—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—84no—
260C22⋊C4—84no—
261C22⋊C4—162no—
262C42—42no—
263C42—42no—
264C42—84no—
265C42C23×C411yescharacteristic
266C42C22×Q811yescharacteristic
267C42C2×#4111yescharacteristic
268C422−1+411yescharacteristic
269C23—22no—
270C23—24no—
271C23—22no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23D4211yescharacteristic
280D4—24no—
281D4—22no—
282D4—22no—
283D4—22no—
284D4—42no—
285D4#42714yes—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—168no—
311C2×C4C22×C4⋊C411yescharacteristic
312C2×C4C2×C4×D411yescharacteristic
313C2×C4#52011yescharacteristic
314C2×C4#54711yescharacteristic
315C2×C4D4×Q811yescharacteristic
316C2×C4#55411yescharacteristic
317C2×C4C2×#13512yes—
318C2×C4#57512yes—
319C2×C4#57612yes—
320C2×C4#57712yes—
321C8—24no—
322C8—22no—
323C8—22no—
324C8—22no—
325C8—22no—
326C8—42no—
327C8—42no—
328C8—82no—
329C22—24no—
330C22—42no—
331C22—42no—
332C22—44no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22—82no—
339C22D4×C4⋊C411yescharacteristic
340C22#327312yes—
341C22#328212yes—
342C22#329211yescharacteristic
343C22#329311yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—82no—
349C4—82no—
350C4—162no—
351C4—162no—
352C4C2×#42711yescharacteristic
353C4#291211yescharacteristic
354C4#294311yescharacteristic
355C4#294811yescharacteristic
356C4#333712yes—
357C4#338912yes—
358C2—44no—
359C2—82no—
360C2#1991112yes—
361C2#1991412yes—
362C2#2265411yescharacteristic
363C2#2267211yescharacteristic
364C2#2269211yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22707C2semidirect
3. #22703C2semidirect
5. #9619C2semidirect
6. #8885C2semidirect
7. #8882C2semidirect
8. #8816C2semidirect
9. #8646C2semidirect
10. #8646C2semidirect
11. #8283C2semidirect
37. C8×D8C4semidirect
59. C8×D4C2×C4semidirect
101. #335C2×C4semidirect
104. D4⋊C8C2×C4semidirect
105. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#227582
#227072
#227032
C2×C8×D82
#96192
#88852
#88822
#88162
#86462
#82832