Finite Group Databasebeta

← #380992

Subgroups — #380992

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35639C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#20358C211yesmaximal, characteristic
4#11383C212yesmaximal, complemented
5#11352C212yesmaximal, complemented
6#11223C212yesmaximal, complemented
7#11206C212yesmaximal, complemented
8#10755C212yesmaximal
9#10746C212yesmaximal
10C2×#568C2212yes—
11C22×#125C2211yescharacteristic
12C2×#520C2211yescharacteristic
13#2905C2212yes—
14C2×#413C2212yes—
15#2579C414yes—
16#2487C414yes—
17#2454—22no—
18#2453—22no—
19#2442—22no—
20#2441—22no—
21#2317C2211yescharacteristic
22#2316C2211yescharacteristic
23#2311C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#1801—22no—
27#1800—22no—
28C2×#344C2214yes—
29C2×#344C2214yes—
30#1742—22no—
31#1742C2212yes—
32C2×#343—22no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2214yes—
36#1728C2214yes—
37C22×#41C2311yescharacteristic
38#568—24no—
39#568—44no—
40#568Q818yes—
41C2×#131—44no—
42C22×M16—24no—
43C22×M16C2314yes—
44C22×M16C2314yes—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—44no—
48#520—44no—
49C2×C4×Q8D412yes—
50C2×C4×D4C2×C414yes—
51C2×C4×D4D412yes—
52C2×#117D412yes—
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42C2311yescharacteristic
62#424—44no—
63#423—24no—
64#413—24no—
65#413—24no—
66C2×#101—22no—
67C2×#100—22no—
68#400—24no—
69#399—24no—
70#398—24no—
71#396—24no—
72#391—21no—
73#391—21no—
74#390—21no—
75#390—21no—
76#389—22no—
77#389—22no—
78#389—21no—
79#389—21no—
80#389—24no—
81#389—22no—
82#387—24no—
83#386C2×C414yes—
84#386C2×C412yes—
85#386C2×C412yes—
86#385—24no—
87#385C2×C414yes—
88#385C2×C412yes—
89#385C2×C412yes—
90#384—21no—
91#382—21no—
92#380—24no—
93#380—22no—
94#380C2×C414yes—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—22no—
97#344—44no—
98#344—44no—
99#343—24no—
100#343—22no—
101#343—24no—
102#343—22no—
103#343—44no—
104#336—44no—
105C2×#41—24no—
106C2×#41—22no—
107C2×#41—22no—
108C2×#41C2×Q814yes—
109C2×#41#4114yes—
110C22×Q8C2×D411yescharacteristic
111C22×D4C22×C412yes—
112C22×D4C2×D411yescharacteristic
113C23×C4C22⋊C414yes—
114C23×C4C22×C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C23×C4C2×D412yes—
119C23×C4C2411yescharacteristic
120#131—416no—
121#131—416no—
122C2×M16—24no—
123C2×M16—216no—
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—24no—
133C2×M16—24no—
134C2×M16—44no—
135C2×M16—48no—
136C2×M16C2×Q818yes—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—44no—
140#125—48no—
141#125—48no—
142C4×Q8—44no—
143C4×D4—44no—
144C4×D4—48no—
145#117—24no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—44no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4C22⋊C414yes—
175C2×C22⋊C4C22×C412yes—
176C2×C22⋊C4C22×C412yes—
177C2×C22⋊C4C22×C412yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—24no—
183C2×C42C22⋊C414yes—
184C2×C42C2×D412yes—
185C4⋊C8—48no—
186#101—44no—
187#100—44no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—24no—
193#95—24no—
194#95—24no—
195C24C2×C22⋊C412yes—
196C24C23×C411yescharacteristic
197#41—28no—
198#41—28no—
199#41—28no—
200#41—28no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8#12412yes—
204C2×Q8C4⋊Q812yes—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4C4×Q818yes—
210C2×D4#12412yes—
211C2×D4C4⋊Q812yes—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—44no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—84no—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C414yes—
234C22×C4C22⋊Q818yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C22×Q812yes—
238C22×C4C2×#4112yes—
239C22×C4H(5,𝔽2)14yes—
240C22×C4H(5,𝔽2)12yes—
241C22×C42−1+412yes—
242M16—28no—
243M16—28no—
244M16—416no—
245M16—44no—
246M16—44no—
247M16—44no—
248M16—44no—
249M16—44no—
250M16—44no—
251C2×C8—28no—
252C2×C8—416no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—44no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—82no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—82no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—44no—
278C22⋊C4—84no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C42—24no—
282C42—44no—
283C42—42no—
284C42—42no—
285C42—41no—
286C42—41no—
287C42—42no—
288C23—22no—
289C23—22no—
290C23—41no—
291C23—41no—
292C23—42no—
293C23—42no—
294C23#39114yes—
295C23C22×C22⋊C411yescharacteristic
296C23C2×C4×Q812yes—
297C23#51812yes—
298C23#52012yes—
299Q8—24no—
300Q8—24no—
301D4—216no—
302D4—24no—
303D4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—28no—
307C2×C4—28no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—164no—
333C2×C4#39114yes—
334C2×C4C2×C22⋊Q812yes—
335C2×C4C2×#12411yescharacteristic
336C2×C4C2×C4⋊Q811yescharacteristic
337C2×C4#53412yes—
338C2×C4#53412yes—
339C2×C4#53511yescharacteristic
340C2×C4#53611yescharacteristic
341C2×C4#53712yes—
342C2×C4#53814yes—
343C2×C4#53914yes—
344C2×C4#54314yes—
345C2×C4#54414yes—
346C8—48no—
347C8—84no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—24no—
353C22—42no—
354C22—42no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22C2×#39111yescharacteristic
360C22#230811yescharacteristic
361C22#231111yescharacteristic
362C22#232914yes—
363C4—24no—
364C4—44no—
365C4—42no—
366C4—42no—
367C4—82no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—82no—
372C4#262612yes—
373C4#262712yes—
374C4#263912yes—
375C4#265912yes—
376C2—21no—
377C2—21no—
378C2—24no—
379C2—44no—
380C2#1428111yescharacteristic
381C2#2177911yescharacteristic
382C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
4. #11383C2semidirect
5. #11352C2semidirect
6. #11223C2semidirect
7. #11206C2semidirect

Maximal subgroups

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

subgroupindex
#356392
#206402
#203582
#113832
#113522
#112232
#112062
#107552
#107462