Finite Group Databasebeta

← #413274

Subgroups — #413274

Order 512.

1588 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2062C214yesmaximal
3#14149C211yesmaximal, characteristic
4#13965C212yesmaximal
5#11344C214yesmaximal
6#10854C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C2×#543C2212yes—
9#2857—22no—
10#2659—22no—
11#2549—24no—
12#2440—24no—
13#2315C2212yes—
14C2×#385C2212yes—
15C2×#385C2218yes—
16C2×#380C2211yescharacteristic
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#2062—216no—
22C2×#342C2218yes—
23C2×#342C2212yes—
24C2×#342C2212yes—
25#1444—24no—
26#1441—22no—
27#1436—22no—
28C4⋊#95—24no—
29C22⋊#95C2212yes—
30C24×C4C2312yes—
31C24×C4C2311yescharacteristic
32#543—28no—
33#543—44no—
34C2×C4⋊Q8—22no—
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—28no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—28no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D414yes—
41C22×C22⋊C4C2314yes—
42C22×C22⋊C4C2312yes—
43C22×C22⋊C4C2311yescharacteristic
44C22×C22⋊C4C2311yescharacteristic
45C22×C22⋊C4C2314yes—
46C22×C42—22no—
47C22×C42D412yes—
48#398—28no—
49#395—28no—
50#391—24no—
51#389—24no—
52#389—24no—
53#385—28no—
54#385—216no—
55#385—28no—
56#385—48no—
57#384—22no—
58#380—24no—
59#380—22no—
60#380—22no—
61#380—22no—
62#380—22no—
63#380—22no—
64#380Q814yescomplemented
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—24no—
68C4×C22⋊C4Q818yescomplemented
69C2×#95—21no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—28no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95D412yes—
80C2×#95D414yes—
81#342—216no—
82#342D418yes—
83#342Q818yescomplemented
84C25C2411yescharacteristic
85C22×Q8—24no—
86C23×C4—24no—
87C23×C4—22no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—44no—
93C23×C4—82no—
94C23×C4—82no—
95C23×C4C2×D418yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×Q814yes—
103C23×C4C2×Q812yes—
104C23×C4#4112yes—
105C4⋊Q8—84no—
106C22⋊Q8—416no—
107C22⋊Q8—416no—
108C22⋊Q8—48no—
109C22⋊Q8—48no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—416no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—416no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—84no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4—88no—
124C2×C4⋊C4—82no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4C2×D414yes—
140C2×C22⋊C4C2×D418yes—
141C2×C22⋊C4C2×Q814yes—
142C2×C22⋊C4C2×Q814yes—
143C2×C22⋊C4C2×Q818yes—
144C2×C22⋊C4C2×Q818yes—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—28no—
148C2×C42—82no—
149C2×C42—84no—
150#95—28no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—28no—
155#95—24no—
156#95—416no—
157#95—44no—
158#95—41no—
159#95—42no—
160#95—41no—
161#95—44no—
162#95—82no—
163C24—42no—
164C24—42no—
165C24C22×D411yescharacteristic
166C24C22×D412yes—
167C24C22×D412yes—
168C24C22×Q812yes—
169C24C22×Q811yescharacteristic
170C24C22×Q812yes—
171C24C2×#4111yescharacteristic
172C24H(5,𝔽2)12yes—
173C242−1+412yes—
174C2×Q8—48no—
175C2×Q8—84no—
176C2×Q8—84no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—48no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—84no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—88no—
221C22×C4—88no—
222C22×C4#12011yescharacteristic
223C22×C4#12012yes—
224C22×C4#12011yescharacteristic
225C22×C4C22⋊Q812yes—
226C22×C4C22⋊Q814yes—
227C22×C4C22⋊Q814yes—
228C22×C4C22⋊Q812yes—
229C22×C4#12714yes—
230C22×C4C4⋊Q814yes—
231C22×C4C4⋊Q812yes—
232C22×C4C4⋊Q812yes—
233C22×C4C4⋊Q814yes—
234C22×C4C4⋊Q814yes—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—416no—
238C4⋊C4—84no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C4⋊C4—84no—
242C4⋊C4—832no—
243C4⋊C4—88no—
244C22⋊C4—28no—
245C22⋊C4—28no—
246C22⋊C4—28no—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—28no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—48no—
253C22⋊C4—44no—
254C22⋊C4C4⋊Q8116yescomplemented
255C42—44no—
256C42—82no—
257C42—88no—
258C23—22no—
259C23—24no—
260C23—22no—
261C23—24no—
262C23—44no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—44no—
267C23—42no—
268C23—41no—
269C23—44no—
270C23—41no—
271C23—44no—
272C23C2×#12011yescharacteristic
273C23C2×C22⋊Q811yescharacteristic
274C23C2×C22⋊Q812yes—
275C23C2×#12711yescharacteristic
276C23C2×C4⋊Q811yescharacteristic
277C23C2×C4⋊Q811yescharacteristic
278C23C2×C4⋊Q812yes—
279C23C2×C4⋊Q814yes—
280C23#53511yescharacteristic
281C23#53611yescharacteristic
282C23#54414yes—
283C23D4214yes—
284C23#54712yes—
285C23D4×Q814yes—
286C23D4×Q812yes—
287C23#55412yes—
288C23Q8212yes—
289Q8—168no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—216no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—48no—
307C2×C4—416no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—416no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—88no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—88no—
336C2×C4—88no—
337C2×C4#39512yes—
338C2×C4#39514yes—
339C2×C4#39512yes—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—28no—
350C22—22no—
351C22—22no—
352C22—28no—
353C22—22no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22C2×#39511yescharacteristic
361C22#243311yescharacteristic
362C22#243411yescharacteristic
363C22#244014yes—
364C22#254211yescharacteristic
365C22#255114yes—
366C22#256711yescharacteristic
367C22#266011yescharacteristic
368C22#286811yescharacteristic
369C22#302512yes—
370C22#302612yes—
371C4—22no—
372C4—22no—
373C4—24no—
374C4—416no—
375C4—44no—
376C4—44no—
377C4—84no—
378C4—84no—
379C4—88no—
380C2—24no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—24no—
386C2#1514311yescharacteristic
387C2#2356411yescharacteristic
388C2#2356511yescharacteristic
389C2#2577312yes—
390C2#2585412yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
64. #380Q8semidirect
68. C4×C22⋊C4Q8semidirect
83. #342Q8semidirect
254. C22⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#20622
#141492
#139652
#113442
#108542
C4×#3422