Finite Group Databasebeta

← #241329

Subgroups — #241329

Order 512.

804 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37642C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21981C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#10578C214yesmaximal, complemented
6#10575C214yesmaximal, complemented
7#8038C211yesmaximal, characteristic
8#7896C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2844—22no—
14#2628—22no—
15#2628—22no—
16#2628—22no—
17#2571—21no—
18#2571—21no—
19#2571C2212yes—
20#2551—21no—
21#2551—21no—
22#2551C2212yes—
23#2285C2211yescharacteristic
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#1948C2214yes—
28#1947C2214yes—
29#1853—22no—
30#1852—22no—
31#1845—21no—
32#1844—21no—
33#1836—22no—
34#1834—21no—
35#1834—21no—
36#1834C2212yes—
37#1834C2212yes—
38#1736—22no—
39#1296C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×M16—21no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117D412yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C42C2311yescharacteristic
54#431—22no—
55#431—22no—
56#431Q814yescomplemented
57#431Q814yescomplemented
58C2×#107—24no—
59C2×#106—24no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C4×M16—22no—
65C4×M16Q814yescomplemented
66C2×#97—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8C2312yes—
70#400—24no—
71#400—24no—
72#400—24no—
73#399—24no—
74#395—24no—
75#395—24no—
76#395—24no—
77#393—24no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384D411yescharacteristic
88#384D411yescharacteristic
89#384D411yescharacteristic
90#384D411yescharacteristic
91#384C2312yes—
92#384C2311yescharacteristic
93#384C2311yescharacteristic
94#382—22no—
95#382—22no—
96#382—22no—
97#382—21no—
98#382—21no—
99#382D412yes—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380D412yes—
105#380D412yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2312yes—
111C4×C4⋊C4C2312yes—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116C22×Q8—24no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—24no—
122C2×M16—42no—
123C2×M16—42no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C4⋊Q8—84no—
128C4⋊Q8—84no—
129C4⋊Q8—84no—
130#117—44no—
131#117—82no—
132#117—82no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q812yes—
173C2×C42C2×Q812yes—
174C2×C42C2×Q812yes—
175C2×C42#4112yes—
176C2×C42C2411yescharacteristic
177#107—84no—
178#106—84no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8C2×Q814yes—
182C4⋊C8C2×Q814yes—
183C22⋊C8—24no—
184C22⋊C8—24no—
185#97—24no—
186C4×C8—24no—
187C4×C8—24no—
188C4×C8—22no—
189C4×C8C2×Q814yes—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200C24#12011yescharacteristic
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—84no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4C22⋊Q812yes—
224C22×C4C22⋊Q812yes—
225C22×C4C22⋊Q812yes—
226C22×C4C4⋊Q812yes—
227C22×C4C4⋊Q812yes—
228C22×C4C4⋊Q812yes—
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232M16—48no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—84no—
268C4⋊C4—82no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C42—44no—
273C42—44no—
274C42—44no—
275C42C22⋊Q812yes—
276C42C22⋊Q812yes—
277C42C22⋊Q812yes—
278C42C4⋊Q812yes—
279C42C4⋊Q812yes—
280C42C4⋊Q812yes—
281C42C22×Q811yescharacteristic
282C42C22×Q811yescharacteristic
283C42C22×Q811yescharacteristic
284C42C2×#4111yescharacteristic
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#39512yes—
292C23C2×#12011yescharacteristic
293Q8—168no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—42no—
307C2×C4—42no—
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—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—164no—
328C2×C4#39512yes—
329C2×C4#39512yes—
330C2×C4#39512yes—
331C2×C4#45412yes—
332C2×C4#45512yes—
333C2×C4C2×C22⋊Q811yescharacteristic
334C2×C4C2×C22⋊Q811yescharacteristic
335C2×C4C2×C22⋊Q811yescharacteristic
336C2×C4C2×C4⋊Q811yescharacteristic
337C2×C4C2×C4⋊Q811yescharacteristic
338C2×C4C2×C4⋊Q811yescharacteristic
339C2×C4#57612yes—
340C2×C4#57712yes—
341C2×C4#57712yes—
342C2×C4#57812yes—
343C8—44no—
344C8—44no—
345C8—44no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22C2×#39511yescharacteristic
354C22#300811yescharacteristic
355C22#300911yescharacteristic
356C22#301011yescharacteristic
357C22#301111yescharacteristic
358C22C2×#45511yescharacteristic
359C22C2×#45411yescharacteristic
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—82no—
366C4—82no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4—162no—
371C4#206114yes—
372C4#308312yes—
373C4#308412yes—
374C4#316312yes—
375C4#316412yes—
376C2—42no—
377C2#1872211yescharacteristic
378C2#1872511yescharacteristic
379C2#1872911yescharacteristic
380C2#1873011yescharacteristic
381C2#2355511yescharacteristic
382C2#2355611yescharacteristic
383C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10578C2semidirect
6. #10575C2semidirect
56. #431Q8semidirect
57. #431Q8semidirect
65. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#219822
#219812
#207642
#105782
#105752
#80382
#78962