Finite Group Databasebeta

← #225522

Subgroups — #225522

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22758C212yesmaximal
2#22657C211yesmaximal, characteristic, complemented
3D4×#106C211yesmaximal, characteristic, complemented
4C2×C8×D8C211yesmaximal, characteristic
5#9619C212yesmaximal, complemented
6#8893C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8815C211yesmaximal, characteristic, complemented
9#8642C211yesmaximal, characteristic, complemented
10#8642C211yesmaximal, characteristic, complemented
11#8283C211yesmaximal, characteristic, complemented
12C2×C4×D8C2211yescharacteristic
13C2×C8×D4C2211yescharacteristic
14C2×C8×D4C2211yescharacteristic
15C2×#427—21no—
16C22×#106—21no—
17#2354C2212yes—
18D4×C4⋊C4C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1936—22no—
22#1921—22no—
23#1879—22no—
24#1854—22no—
25#1853—22no—
26#1844C2211yescharacteristic
27#1844C2211yescharacteristic
28#1834C2212yes—
29#1834C2212yes—
30#1807—21no—
31#1807—21no—
32#1802C2212yes—
33#1802C2212yes—
34C4×#106C2211yescharacteristic
35#1770C2212yes—
36#1762—22no—
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
60#427—44no—
61C2×#107—22no—
62C2×#107D411yescharacteristic
63C2×#106—21no—
64C2×#106—21no—
65C2×#106—22no—
66C2×#106—21no—
67C2×#106—21no—
68C2×#106—44no—
69C2×#106D411yescharacteristic
70C2×#106D411yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×#102—22no—
75C2×#102—22no—
76C2×#102D411yescharacteristic
77C2×#102D411yescharacteristic
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×#97—22no—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#390—22no—
85#389—22no—
86#388—42no—
87#385—42no—
88#384C2312yes—
89#384C2312yes—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2312yes—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—42no—
101#340D412yes—
102#336—22no—
103#335C2×C414yescomplemented
104#334—22no—
105D4⋊C8C2×C414yes—
106D4⋊C8C2×C414yescomplemented
107C82C2×C414yescomplemented
108C22×D4C2×D411yescharacteristic
109C22×D4C2×D411yescharacteristic
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—42no—
113C2×D8—24no—
114C2×D8C4⋊C414yes—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—24no—
119C22×C8—24no—
120C22×C8—22no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—44no—
134C4×D4—82no—
135C4×D4C22×C412yes—
136C4×D4C22×C412yes—
137C4×D4C2×Q812yes—
138C4×D4C2×Q812yes—
139#117—44no—
140#117—82no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C42—22no—
160C2×C42—42no—
161C2×C42C2411yescharacteristic
162#107—44no—
163#107C4⋊C414yes—
164#106—22no—
165#106—22no—
166#106—44no—
167#106—42no—
168#106—42no—
169#106—81no—
170#106—81no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8C22×C412yes—
176C4⋊C8C22×C412yes—
177#102—44no—
178#102—82no—
179#102C4⋊C414yes—
180#102C4⋊C414yes—
181C22⋊C8—24no—
182C22⋊C8—24no—
183#97—82no—
184C4×C8—21no—
185C4×C8C22×C412yes—
186C4×C8C22×C412yes—
187C4×C8C22×C412yes—
188C4×C8C2×Q812yes—
189C4×C8#4112yes—
190#95—22no—
191#95—42no—
192#95—42no—
193C24—21no—
194C24—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4C2×C4⋊C412yes—
201C2×D4C2×C4⋊C412yes—
202C2×D4#13514yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—162no—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4H(5,𝔽2)11yescharacteristic
227D8—28no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—48no—
241C2×C8—48no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8C2×C4⋊C412yes—
246C2×C8C2×C4⋊C412yes—
247C2×C8C4×D412yes—
248C2×C8C4×D412yes—
249C2×C8AGL(1,ℤ/8)12yes—
250C2×C8AGL(1,ℤ/8)12yes—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—84no—
268C4⋊C4C2×C4⋊C412yes—
269C4⋊C4C2×C4⋊C412yes—
270C22⋊C4—22no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—84no—
275C22⋊C4—84no—
276C22⋊C4—162no—
277C42—42no—
278C42—42no—
279C42—84no—
280C42C23×C411yescharacteristic
281C42C22×Q811yescharacteristic
282C42C2×#4111yescharacteristic
283C422−1+411yescharacteristic
284C23—22no—
285C23—24no—
286C23—22no—
287C23—41no—
288C23—41no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—42no—
293C23—42no—
294C23D4211yescharacteristic
295D4—24no—
296D4—22no—
297D4—22no—
298D4—22no—
299D4—42no—
300D4#42714yescomplemented
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—168no—
329C2×C4C22×C4⋊C411yescharacteristic
330C2×C4C2×C4×D411yescharacteristic
331C2×C4#52011yescharacteristic
332C2×C4#54711yescharacteristic
333C2×C4D4×Q811yescharacteristic
334C2×C4#55411yescharacteristic
335C2×C4C2×#13512yes—
336C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
337C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
338C2×C4#57712yes—
339C2×C4#57711yescharacteristic
340C2×C4#57711yescharacteristic
341C8—22no—
342C8—22no—
343C8—22no—
344C8—42no—
345C8—42no—
346C8—42no—
347C8—82no—
348C8#44214yes—
349C22—24no—
350C22—42no—
351C22—42no—
352C22—44no—
353C22—44no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—82no—
358C22—82no—
359C22D4×C4⋊C411yescharacteristic
360C22#327211yescharacteristic
361C22#327211yescharacteristic
362C22#328212yes—
363C22#329012yes—
364C4—21no—
365C4—21no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4—162no—
371C4—162no—
372C4C2×#42711yescharacteristic
373C4#291211yescharacteristic
374C4C2×#44211yescharacteristic
375C4#295211yescharacteristic
376C4#333712yes—
377C4#338411yescharacteristic
378C4#338411yescharacteristic
379C2—44no—
380C2—82no—
381C2#1991012yes—
382C2#1991512yes—
383C2#2265411yescharacteristic
384C2#2267011yescharacteristic
385C2#2269011yescharacteristic
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. #22657C2semidirect
3. D4×#106C2semidirect
5. #9619C2semidirect
6. #8893C2semidirect
7. #8890C2semidirect
8. #8815C2semidirect
9. #8642C2semidirect
10. #8642C2semidirect
11. #8283C2semidirect
37. C8×D8C4semidirect
59. C8×D4C2×C4semidirect
103. #335C2×C4semidirect
106. D4⋊C8C2×C4semidirect
107. C82C2×C4semidirect
300. D4#427semidirect

Maximal subgroups

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

subgroupindex
#227582
#226572
D4×#1062
C2×C8×D82
#96192
#88932
#88902
#88152
#86422
#82832