Finite Group Databasebeta

← #380746

Subgroups — #380746

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1989C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1987C211yesmaximal, characteristic
6C2×#1832C211yesmaximal, characteristic
7#13797C211yesmaximal, characteristic
8#11358C214yesmaximal
9#11357C214yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2660—24no—
15#2660C418yescomplemented
16C2×#398—22no—
17C2×#395—22no—
18C2×#384C2212yes—
19C2×#382C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1989—24no—
24#1989—24no—
25#1988—24no—
26#1987—24no—
27#1832—24no—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2214yes—
35C4⋊#95C2214yes—
36#1434—22no—
37#1434—22no—
38C23×Q8—21no—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C414yes—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125C2×C414yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72#398—216no—
73#395—216no—
74#384—24no—
75#384C2×C418yes—
76#382—24no—
77#382C2×C418yes—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4D414yes—
81C4×C4⋊C4D414yes—
82#376C2×C414yes—
83#376D414yes—
84C2×#95—24no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95D412yes—
90C2×#95D412yes—
91#340—28no—
92#340—28no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#336—28no—
98C22×Q8—24no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8—82no—
102C22×Q8—82no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D414yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—28no—
119C22×C8—28no—
120C4⋊Q8—28no—
121C4⋊Q8—28no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124#125—28no—
125#125—28no—
126#125—44no—
127#125—44no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—216no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—82no—
171C2×C42—82no—
172C2×C42C22×C414yes—
173C2×C42C22×C412yes—
174C2×C42C2×D414yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180C4⋊C8—44no—
181C4⋊C8—44no—
182#95—216no—
183#95—24no—
184#95—24no—
185#95—84no—
186#95—84no—
187#95C22⋊C418yes—
188#95C22⋊C418yes—
189C24C22×D411yescharacteristic
190C2×Q8—48no—
191C2×Q8—88no—
192C2×Q8—84no—
193C2×Q8—84no—
194C2×Q8—88no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C2×C22⋊C418yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4#12112yes—
234C22×C4#12112yes—
235C22×C4#12112yes—
236C22×C4#12112yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4111yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C22×C42−1+414yes—
244C22×C42−1+414yes—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—44no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—816no—
275C42—28no—
276C42—24no—
277C42—24no—
278C42—84no—
279C42—84no—
280C42C4×D414yes—
281C42C4×D414yes—
282C42#12414yes—
283C42#12714yes—
284C23C22×C22⋊C411yescharacteristic
285C23C2×#12111yescharacteristic
286C23C2×#12111yescharacteristic
287C23#53614yes—
288C23#57612yes—
289C23#57712yes—
290C23#57712yes—
291C23#57812yes—
292Q8—816no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—82no—
333C2×C4#39014yes—
334C2×C4#39014yes—
335C2×C4C2×C4×D411yescharacteristic
336C2×C4C2×C4×D411yescharacteristic
337C2×C4#51914yes—
338C2×C4C2×#12411yescharacteristic
339C2×C4C2×#12711yescharacteristic
340C2×C4#53612yes—
341C2×C4#53612yes—
342C2×C4#53612yes—
343C2×C4#53612yes—
344C2×C4#53612yes—
345C2×C4#53612yes—
346C2×C4#54114yes—
347C2×C4#54114yes—
348C8—82no—
349C8—82no—
350C22C2×#39011yescharacteristic
351C22#231012yes—
352C22#262112yes—
353C22#262112yes—
354C22#289712yes—
355C22#289812yes—
356C22#304111yescharacteristic
357C22#304211yescharacteristic
358C22#304311yescharacteristic
359C22#304411yescharacteristic
360C22#304512yes—
361C22#304612yes—
362C22#324114yes—
363C22#324214yes—
364C22#324314yes—
365C22#324414yes—
366C4—24no—
367C4—24no—
368C4—48no—
369C4—82no—
370C4—82no—
371C4—84no—
372C4—84no—
373C4—84no—
374C4#232712yes—
375C4#232712yes—
376C4#264212yes—
377C4#265811yescharacteristic
378C4#265811yescharacteristic
379C2#1427411yescharacteristic
380C2#2126211yescharacteristic
381C2#2126311yescharacteristic
382C2#2184314yes—
383C2#2987212yes—
384C2#2987312yes—
385C2#2987412yes—
386C2#2987512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2660C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#19892
C2×#19882
C2×#19872
C2×#18322
#137972
#113582
#113572