Finite Group Databasebeta

← #379842

Subgroups — #379842

Order 512.

1721 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2623C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4#14342C211yesmaximal, characteristic, complemented
5#11203C214yesmaximal, complemented
6#11043C214yesmaximal
7#10736C212yesmaximal, complemented
8C22×#123C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C22×#99C2214yes—
12#2623—24no—
13#2623—24no—
14C2×#400C412yescomplemented
15C2×#399C412yescomplemented
16C2×#398C412yescomplemented
17C2×#397C412yescomplemented
18#2316—28no—
19C2×#389—24no—
20C2×#386C2212yes—
21C2×#385C2212yes—
22C2×#381C2212yes—
23C2×#381C2211yescharacteristic
24C2×#380C2214yes—
25#1899—28no—
26C2×#342C2214yes—
27C2×#342C2218yes—
28#1445—24no—
29C22⋊#95C2214yes—
30C23×D4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C24×C4C2312yes—
33C2×#123—22no—
34C2×#123—22no—
35C2×#123—24no—
36C2×#123—24no—
37C2×#117—24no—
38C2×#117—48no—
39C2×#117D418yes—
40C22×C4⋊C4—24no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4D414yes—
50C22×C4⋊C4D414yes—
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C414yes—
54C22×C22⋊C4C2314yes—
55C22×C22⋊C4C2311yescharacteristic
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2314yes—
58C22×C42—22no—
59C2×#99—28no—
60C2×#99—48no—
61#400—28no—
62#399—28no—
63#398—28no—
64#397—28no—
65#389—232no—
66#386—24no—
67#386—24no—
68#385—24no—
69#385—24no—
70#381—22no—
71#381—22no—
72#381—24no—
73#381—24no—
74#380—216no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—28no—
78C2×#95C2×C412yes—
79C2×#95C2×C414yes—
80C2×#95C2×C414yes—
81C2×#95C2×C412yes—
82#342—28no—
83#342—48no—
84C25C22×C411yescharacteristic
85C25C2411yescharacteristic
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—24no—
91C22×D4—24no—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—84no—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C414yes—
105C23×C4C22×C414yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D414yes—
109C23×C4C2×D414yes—
110C23×C4C2×D414yes—
111C23×C4#4114yes—
112#123—44no—
113#123—48no—
114#123—44no—
115#117—416no—
116#117—416no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—216no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—216no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—48no—
130C2×C4⋊C4—88no—
131C2×C4⋊C4—84no—
132C2×C4⋊C4C22⋊C418yes—
133C2×C4⋊C4C22⋊C418yes—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—84no—
155C2×C22⋊C4C2×D418yes—
156C2×C42—28no—
157C2×C42—48no—
158#99—416no—
159#95—216no—
160#95—24no—
161#95—24no—
162#95—28no—
163#95—28no—
164#95—48no—
165#95—88no—
166C24—21no—
167C24—22no—
168C24—24no—
169C24—42no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24C23×C411yescharacteristic
174C24C22×D411yescharacteristic
175C24C22×D411yescharacteristic
176C24C22×D412yes—
177C24C2×#4112yes—
178C24H(5,𝔽2)14yes—
179C242−1+414yes—
180C2×D4—41no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—44no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—48no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—28no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—28no—
195C22×C4—28no—
196C22×C4—216no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—28no—
211C22×C4—48no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—48no—
226C22×C4—88no—
227C22×C4—88no—
228C22×C4C2×C22⋊C418yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C4×D414yes—
232C22×C4C4×D414yes—
233C22×C4#12014yes—
234C22×C4#12118yes—
235C22×C4#12314yes—
236C4⋊C4—216no—
237C4⋊C4—216no—
238C4⋊C4—416no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—88no—
242C4⋊C4—816no—
243C22⋊C4—28no—
244C22⋊C4—28no—
245C22⋊C4—41no—
246C22⋊C4—42no—
247C22⋊C4—41no—
248C22⋊C4—48no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—48no—
257C22⋊C4—88no—
258C42—416no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—48no—
270C23—48no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—41no—
275C23—44no—
276C23—88no—
277C23—82no—
278C23—82no—
279C23C22×C22⋊C411yescharacteristic
280C23C2×C4×D412yes—
281C23#52014yes—
282C23C2×#12011yescharacteristic
283C23C2×#12112yes—
284C23C2×#12311yescharacteristic
285C23#53414yes—
286C23#53512yes—
287C23#53612yes—
288C23#53614yes—
289C23#53714yes—
290C23#53918yes—
291D4—81no—
292D4—81no—
293D4—84no—
294D4—82no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—216no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—88no—
322C2×C4—88no—
323C2×C4—816no—
324C2×C4—88no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—816no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4#38614yes—
332C2×C4#38614yes—
333C22—24no—
334C22—28no—
335C22—28no—
336C22—24no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—22no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—44no—
346C22—44no—
347C22—48no—
348C22—84no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—164no—
353C22C2×#38611yescharacteristic
354C22#231112yes—
355C22#232214yes—
356C22#261311yescharacteristic
357C22#261411yescharacteristic
358C22#261814yes—
359C22#262312yes—
360C22#288312yes—
361C22#288412yes—
362C4—24no—
363C4—24no—
364C4—48no—
365C4—88no—
366C4—84no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—164no—
371C2—21no—
372C2—24no—
373C2—24no—
374C2—21no—
375C2—22no—
376C2—161no—
377C2#1425611yescharacteristic
378C2#2098511yescharacteristic
379C2#2098611yescharacteristic
380C2#2187714yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2316C2semidirect
4. #14342C2semidirect
5. #11203C2semidirect
7. #10736C2semidirect
14. C2×#400C4semidirect
15. C2×#399C4semidirect
16. C2×#398C4semidirect
17. C2×#397C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
C2×#23162
C2×#18992
#143422
#112032
#110432
#107362