Finite Group Databasebeta

← #381125

Subgroups — #381125

Order 512.

1119 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35672C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic, complemented
3#20399C211yesmaximal, characteristic, complemented
4#11206C212yesmaximal, complemented
5#11171C212yesmaximal, complemented
6#11071C212yesmaximal
7#10968C212yesmaximal
8#10937C212yesmaximal
9#10916C212yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2905C2212yes—
13C22×#101C2212yes—
14C22×#101C2212yes—
15C2×#407C2214yes—
16#2873C2212yes—
17#2484C414yes—
18#2482C414yes—
19#2423—22no—
20#2423—22no—
21#2421—22no—
22#2421—22no—
23#2317C2211yescharacteristic
24#2317C2211yescharacteristic
25#2310C2211yescharacteristic
26#2280C2211yescharacteristic
27#2280C2211yescharacteristic
28#1801—22no—
29#1801—22no—
30C4×#101—22no—
31C4×#101—22no—
32C2×#344C2214yes—
33C2×#344C2214yes—
34#1742C2212yes—
35#1742C2212yes—
36C2×#343C2214yes—
37C23×Q8C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C22×M16C2314yes—
40C22×M16C2314yes—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—44no—
44#519—44no—
45C2×C4×Q8D412yes—
46C2×C4×Q8D412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×#117C2×C414yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4C2×C412yes—
57#424—44no—
58#423—24no—
59C2×#101—24no—
60C2×#101—28no—
61C2×#101—24no—
62C2×#101—44no—
63#407—28no—
64#407—44no—
65C2×C22⋊C8—44no—
66#405—44no—
67C4×M16—24no—
68#400—24no—
69#399—24no—
70#398—24no—
71#397—24no—
72#391—22no—
73#391—22no—
74#389—22no—
75#389—22no—
76#388—24no—
77#388—24no—
78#387—24no—
79#387—24no—
80#385C2×C414yes—
81#385C2×C414yes—
82#384—21no—
83#384—21no—
84#382—21no—
85#382—21no—
86#381C2×C414yes—
87#381C2×C412yes—
88#381C2×C412yes—
89#380—21no—
90#380—21no—
91#380—21no—
92#380—21no—
93#379C2×C412yes—
94#379C2×C412yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97#344—28no—
98#344—44no—
99#343—22no—
100#343—22no—
101#343—22no—
102#343—22no—
103#343—44no—
104C25C22×C411yescharacteristic
105C22×Q8—24no—
106C22×Q8—24no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8C2×D412yes—
110C22×Q8C2×D412yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C412yes—
119C23×C4C22×C412yes—
120C23×C4C2411yescharacteristic
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16#4118yes—
133C22×C8—24no—
134C22×C8—24no—
135C22⋊Q8—48no—
136C22⋊Q8—48no—
137C4×Q8—44no—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—44no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—22no—
145#117—22no—
146#117—48no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4C22⋊C414yes—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C22⋊C4—24no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4C22×C412yes—
169C2×C22⋊C4C22×C412yes—
170C2×C22⋊C4C22×C412yes—
171C2×C22⋊C4C22×C412yes—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C22⋊C414yes—
175C2×C42C22⋊C414yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C4⋊C8—48no—
179#101—216no—
180#101—48no—
181C22⋊C8—416no—
182#97—44no—
183C4×C8—44no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192C24—21no—
193C24—22no—
194C24—22no—
195C24—22no—
196C24—42no—
197C24—81no—
198C24C23×C411yescharacteristic
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—24no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—44no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—82no—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C2×#4114yes—
241C22×C4H(5,𝔽2)14yes—
242C22×C4H(5,𝔽2)12yes—
243C22×C42−1+412yes—
244M16—28no—
245M16—28no—
246M16—44no—
247M16—44no—
248M16—44no—
249M16—44no—
250M16—44no—
251M16—44no—
252C2×C8—28no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—48no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—84no—
271C4⋊C4—84no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—48no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C42—24no—
282C42—24no—
283C42—44no—
284C42—44no—
285C42—42no—
286C42—42no—
287C23—22no—
288C23—24no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—44no—
296C23—41no—
297C23—41no—
298C23—44no—
299C23—41no—
300C23—84no—
301C23—84no—
302C23C22×C22⋊C411yescharacteristic
303C23C4×#4112yes—
304C23#51812yes—
305C23#52012yes—
306Q8—48no—
307Q8—44no—
308Q8—44no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4#52512yes—
338C2×C4#53212yes—
339C2×C4#53412yes—
340C2×C4#53412yes—
341C2×C4#53511yescharacteristic
342C2×C4#53611yescharacteristic
343C2×C4#53712yes—
344C2×C4#53814yes—
345C2×C4#53914yes—
346C2×C4#54014yes—
347C2×C4#54214yes—
348C8—48no—
349C8—84no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—22no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—48no—
359C22—42no—
360C22—81no—
361C22—81no—
362C22—84no—
363C22—81no—
364C22—81no—
365C22#229711yescharacteristic
366C22#230811yescharacteristic
367C22#231111yescharacteristic
368C22#233514yes—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—44no—
374C4—82no—
375C4—82no—
376C4—82no—
377C4—81no—
378C4#263014yes—
379C4#266112yes—
380C4#266312yes—
381C2—21no—
382C2—21no—
383C2—24no—
384C2—82no—
385C2#1428711yescharacteristic
386C2#2178011yescharacteristic
387C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20639C2semidirect
3. #20399C2semidirect
4. #11206C2semidirect
5. #11171C2semidirect
9. #10916C2semidirect

Maximal subgroups

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

subgroupindex
#356722
#206392
#203992
#112062
#111712
#110712
#109682
#109372
#109162