Finite Group Databasebeta

← #415994

Subgroups — #415994

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#20647C212yesmaximal
3#13786C211yesmaximal, characteristic
4#11276C214yesmaximal
5#11064C214yesmaximal
6#10847C211yesmaximal, characteristic
7#10840C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9C2×#423C2212yes—
10C2×#419C2214yes—
11#2851—22no—
12#2646—22no—
13#2571—22no—
14#2570—24no—
15#2552—22no—
16#2317C2214yes—
17C2×#391C2212yes—
18#2285C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1805—24no—
22#1803—22no—
23#1803—22no—
24C2×#343C2214yes—
25#1736C2214yes—
26C2×#339C2214yes—
27C2×#339C2214yes—
28#1436—22no—
29C4⋊#95C2212yes—
30#1434—22no—
31C23×Q8C2311yescharacteristic
32C22×M16C2314yes—
33C2×#125—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—24no—
38C2×C4×Q8Q814yes—
39C2×#117C2314yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2312yes—
42C22×C4⋊C4C2311yescharacteristic
43C22×C42—21no—
44C22×C42D412yes—
45C22×C42C2312yes—
46C22×C42C2311yescharacteristic
47#423—24no—
48#423—24no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51#419—28no—
52#419—44no—
53C2×#103—24no—
54C2×#103—24no—
55#400—28no—
56#398—28no—
57#391—24no—
58#391—24no—
59#389—24no—
60#389—24no—
61#389—24no—
62#387—28no—
63#385—28no—
64#384—24no—
65#384D412yes—
66#384D412yes—
67#382—22no—
68#382—24no—
69#382D414yes—
70#380—22no—
71#380—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76#376—24no—
77C2×#95—24no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—24no—
81C2×#95D412yes—
82#343—44no—
83#340—24no—
84#340—24no—
85#339—28no—
86#339—44no—
87C22×Q8—21no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8C2×Q812yes—
94C22×Q8#4114yes—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4C2×D414yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2411yescharacteristic
102C2×M16—24no—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—24no—
108C22×C8—24no—
109#125—84no—
110C4×Q8—28no—
111C4×Q8—28no—
112C4×Q8—44no—
113C4×Q8—44no—
114#117—24no—
115#117—24no—
116#117—24no—
117#117—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2×Q812yes—
140C2×C22⋊C4—24no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42—82no—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C2×C42C2×Q812yes—
156C2×C42#4114yes—
157C2×C42#4114yes—
158C2×C42#4112yes—
159C4⋊C8—44no—
160C4⋊C8—44no—
161#103—48no—
162#103—48no—
163#95—216no—
164#95—28no—
165#95—28no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—42no—
170#95—42no—
171#95—84no—
172C24C22×D411yescharacteristic
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—24no—
176C2×Q8—24no—
177C2×Q8—24no—
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8#12514yescomplemented
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4#12011yescharacteristic
217C22×C4#12012yes—
218C22×C4#12011yescharacteristic
219C22×C4C22⋊Q812yes—
220C22×C4C22⋊Q812yes—
221C22×C4#12412yes—
222C22×C4#12412yes—
223C22×C4C22×D412yes—
224C22×C4C22×Q811yescharacteristic
225C22×C4C2×#4112yes—
226C22×C4C2×#4111yescharacteristic
227C22×C4C2×#4112yes—
228C22×C4C2×#4112yes—
229C22×C4H(5,𝔽2)12yes—
230C22×C42−1+412yes—
231M16—44no—
232M16—44no—
233M16—44no—
234M16—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—88no—
255C4⋊C4—88no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C42—24no—
259C42—24no—
260C42—44no—
261C42—44no—
262C42—44no—
263C42—42no—
264C42—42no—
265C42—48no—
266C42—82no—
267C42C22⋊Q814yes—
268C42C22⋊Q814yes—
269C42#12414yes—
270C42#12414yes—
271C23—21no—
272C23—21no—
273C23—22no—
274C23C2×#12011yescharacteristic
275C23C2×C22⋊Q811yescharacteristic
276C23#52512yes—
277C23C2×#12411yescharacteristic
278C23#53712yes—
279Q8—28no—
280Q8—44no—
281Q8—44no—
282Q8—44no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—84no—
321C2×C4#39314yes—
322C2×C4#39312yes—
323C2×C4#39312yes—
324C2×C4C2×C22⋊Q812yes—
325C2×C4C2×#12412yes—
326C2×C4C2×#12511yescharacteristic
327C2×C4#52912yes—
328C2×C4#53311yescharacteristic
329C2×C4#54011yescharacteristic
330C2×C4#54111yescharacteristic
331C2×C4#54412yes—
332C2×C4#54612yes—
333C2×C4#54712yes—
334C2×C4#54812yes—
335C2×C4#55012yes—
336C2×C4#55212yes—
337C2×C4#55212yes—
338C2×C4#55312yes—
339C2×C4#55312yes—
340C8—84no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22C2×#39311yescharacteristic
350C22#240712yes—
351C22#256212yes—
352C22#262911yescharacteristic
353C22#285111yescharacteristic
354C22#301911yescharacteristic
355C22#302011yescharacteristic
356C22#302212yes—
357C4—22no—
358C4—22no—
359C4—24no—
360C4—44no—
361C4—84no—
362C4—84no—
363C4—84no—
364C4—84no—
365C4—82no—
366C4#244112yes—
367C4#244212yes—
368C4#248912yes—
369C4#249012yes—
370C2—22no—
371C2—21no—
372C2—21no—
373C2#1521611yescharacteristic
374C2#2346211yescharacteristic
375C2#2346411yescharacteristic
376C2#2418212yes—
377C2#2418312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×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
188. C2×Q8#125semidirect

Maximal subgroups

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

subgroupindex
#351812
#206472
#137862
#112762
#110642
#108472
#108402