Finite Group Databasebeta

← #373776

Subgroups — #373776

Order 512.

1190 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#34187C211yesmaximal, characteristic
2#21170C212yesmaximal
3C2×#1801C211yesmaximal, characteristic
4#13786C211yesmaximal, characteristic
5#11136C214yesmaximal
6#10899C214yesmaximal
7#10840C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9C2×#428C2214yes—
10C2×#423C2212yes—
11C22×#101C2212yes—
12#2356—24no—
13#2355—22no—
14#2355—22no—
15#2313—22no—
16#2310C2212yes—
17C2×#391C2212yes—
18#2285C2212yes—
19#2283—22no—
20#2274C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1949—24no—
24#1948—22no—
25#1947—22no—
26#1801C418yescomplemented
27C2×#343C2212yes—
28#1736C2214yes—
29C2×#339C2214yes—
30C2×#339C2214yes—
31#1436—22no—
32C4⋊#95C2212yes—
33#1434—22no—
34C23×Q8C2311yescharacteristic
35C22×M16C2314yes—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—24no—
39C2×C4×Q8C2×C414yes—
40C2×C4×Q8Q814yes—
41C2×#117C2314yes—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C42—21no—
46C22×C42D412yes—
47C22×C42C2312yes—
48C22×C42C2311yescharacteristic
49#428—28no—
50#428—44no—
51C2×#107—24no—
52C2×#106—24no—
53#423—24no—
54#423C2×C418yes—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×#101—44no—
58C2×#101C2×C418yes—
59#391—24no—
60#391—24no—
61#391—24no—
62#391—24no—
63#387—28no—
64#387—28no—
65#384D412yes—
66#384D412yes—
67#383—24no—
68#382—24no—
69#382D414yes—
70#381—24no—
71#380—22no—
72#380—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C22⋊C4—24no—
82#376—24no—
83#376—24no—
84C2×#95—24no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—24no—
89#343C2×C418yes—
90#340—24no—
91#340—24no—
92#339—28no—
93#339—44no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8C22×C412yes—
101C22×Q8C2×Q812yes—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16C22×C418yes—
115C22×C8—24no—
116C4×Q8—28no—
117C4×Q8—28no—
118C4×Q8—28no—
119C4×Q8—28no—
120#117—24no—
121#117—24no—
122#117—24no—
123#117—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4C4⋊C418yes—
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C2×D414yes—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2×Q812yes—
142C2×C4⋊C4#4114yes—
143C2×C22⋊C4—24no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—42no—
156C2×C42—82no—
157C2×C42C4⋊C418yes—
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×Q812yes—
162C2×C42#4114yes—
163C2×C42#4114yes—
164#107—48no—
165#106—48no—
166C4⋊C8—28no—
167C4⋊C8—28no—
168#101—48no—
169#95—216no—
170#95—28no—
171#95—28no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—41no—
177#95—41no—
178#95—42no—
179#95—42no—
180#95—84no—
181C24C22×D411yescharacteristic
182C2×Q8—24no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—24no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×C4⋊C414yes—
224C22×C4C2×C4⋊C414yes—
225C22×C4C2×C4⋊C412yes—
226C22×C4C2×C4⋊C412yes—
227C22×C4#12011yescharacteristic
228C22×C4#12012yes—
229C22×C4#12011yescharacteristic
230C22×C4#12312yes—
231C22×C4#12312yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C22×C4C22×Q811yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4C2×#4112yes—
237C22×C4C2×#4112yes—
238C22×C4H(5,𝔽2)12yes—
239C22×C42−1+412yes—
240M16—28no—
241M16—28no—
242M16—44no—
243M16—44no—
244C2×C8—28no—
245C2×C8—44no—
246C2×C8—44no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—216no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C42—28no—
265C42—28no—
266C42—24no—
267C42—24no—
268C42—44no—
269C42—44no—
270C42—42no—
271C42—42no—
272C42—84no—
273C42—84no—
274C42—84no—
275C42C4×D414yes—
276C42C4×D414yes—
277C42C22⋊Q814yes—
278C42C22⋊Q814yes—
279C23—22no—
280C23—21no—
281C23—21no—
282C23C22×C4⋊C411yescharacteristic
283C23C2×#12011yescharacteristic
284C23C2×#12311yescharacteristic
285C23#52512yes—
286C23#53712yes—
287Q8—44no—
288Q8—44no—
289Q8—44no—
290Q8—44no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—48no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#38512yes—
328C2×C4#38512yes—
329C2×C4#38514yes—
330C2×C4C2×C4×D412yes—
331C2×C4C4×#4112yes—
332C2×C4#51811yescharacteristic
333C2×C4#51911yescharacteristic
334C2×C4C2×C22⋊Q812yes—
335C2×C4#53311yescharacteristic
336C2×C4#53311yescharacteristic
337C2×C4#54412yes—
338C2×C4#54612yes—
339C2×C4#54712yes—
340C2×C4#54812yes—
341C2×C4#55012yes—
342C2×C4#55212yes—
343C2×C4#55312yes—
344C2×C4#55512yes—
345C2×C4#55612yes—
346C8—48no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22C2×#38511yescharacteristic
356C22#229611yescharacteristic
357C22#231611yescharacteristic
358C22#240712yes—
359C22#248512yes—
360C22#302012yes—
361C22#302111yescharacteristic
362C22#302211yescharacteristic
363C4—22no—
364C4—22no—
365C4—24no—
366C4—44no—
367C4—44no—
368C4—84no—
369C4—84no—
370C4—84no—
371C4—84no—
372C4#236212yes—
373C4#236312yes—
374C4#243912yes—
375C4#244112yes—
376C2—22no—
377C2—21no—
378C2—21no—
379C2#1412911yescharacteristic
380C2#2085711yescharacteristic
381C2#2085811yescharacteristic
382C2#2418112yes—
383C2#2418312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1801C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#211702
C2×#18012
#137862
#111362
#108992
#108402