Finite Group Databasebeta

← #413334

Subgroups — #413334

Order 512.

1568 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×#2030C214yesmaximal
3#13904C211yesmaximal, characteristic
4#13904C212yesmaximal
5#11070C214yesmaximal
6#10854C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C2×#543C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2857—22no—
11#2659—22no—
12#2549—24no—
13#2440—24no—
14#2315C2212yes—
15C2×#381C2212yes—
16C2×#381C2212yes—
17C2×#381C2218yes—
18C2×#380C2211yescharacteristic
19C2×#380C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#2030—216no—
23C2×#342C2212yes—
24C2×#342C2218yes—
25C2×#342C2212yes—
26#1445—24no—
27#1445—22no—
28C4⋊#95—22no—
29C4⋊#95—24no—
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—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D414yes—
45C22×C4⋊C4Q814yescomplemented
46C22×C22⋊C4C2312yes—
47C22×C22⋊C4C2314yes—
48C22×C22⋊C4C2314yes—
49C22×C22⋊C4C2311yescharacteristic
50C22×C22⋊C4C2311yescharacteristic
51C22×C42—22no—
52C22×C42D412yes—
53#398—28no—
54#395—28no—
55#391—24no—
56#389—24no—
57#389—24no—
58#385—28no—
59#384—22no—
60#381—216no—
61#381—28no—
62#381—28no—
63#381—48no—
64#380—24no—
65#380—22no—
66#380—22no—
67#380—22no—
68#380—22no—
69#380—24no—
70C4×C22⋊C4—22no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—24no—
74C2×#95—216no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95D412yes—
80C2×#95D414yes—
81#342—216no—
82#342—24no—
83#342D418yes—
84C25C2411yescharacteristic
85C22×Q8—24no—
86C23×C4—24no—
87C23×C4—22no—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—44no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D418yes—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×Q814yes—
105C23×C4C2×Q812yes—
106C23×C4#4112yes—
107C23×C4#4114yes—
108C4⋊Q8—84no—
109C22⋊Q8—432no—
110C22⋊Q8—48no—
111C22⋊Q8—48no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—48no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4C2×D418yes—
149C2×C22⋊C4C2×D414yes—
150C2×C22⋊C4#4114yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—44no—
154C2×C42—48no—
155C2×C42—44no—
156#95—28no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—432no—
161#95—44no—
162#95—42no—
163#95—48no—
164#95—48no—
165C24—42no—
166C24—42no—
167C24C22×D411yescharacteristic
168C24C22×D412yes—
169C24C22×D412yes—
170C24C22×Q811yescharacteristic
171C24C2×#4112yes—
172C24C2×#4111yescharacteristic
173C24C2×#4112yes—
174C24H(5,𝔽2)12yes—
175C242−1+412yes—
176C2×Q8—48no—
177C2×Q8—84no—
178C2×Q8—84no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—28no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—28no—
191C22×C4—216no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—48no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—48no—
213C22×C4—48no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—48no—
217C22×C4—44no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4#12011yescharacteristic
226C22×C4#12012yes—
227C22×C4#12011yescharacteristic
228C22×C4C22⋊Q812yes—
229C22×C4C22⋊Q814yes—
230C22×C4C22⋊Q812yes—
231C22×C4C22⋊Q814yes—
232C22×C4#12414yes—
233C22×C4#12412yes—
234C22×C4#12414yes—
235C22×C4#12412yes—
236C22×C4#12414yes—
237C22×C4#12714yes—
238C4⋊C4—48no—
239C4⋊C4—416no—
240C4⋊C4—44no—
241C4⋊C4—48no—
242C4⋊C4—44no—
243C4⋊C4—48no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C22⋊C4—28no—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—28no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—48no—
256C22⋊C4—44no—
257C22⋊C4—48no—
258C42—44no—
259C42—82no—
260C42—88no—
261C23—22no—
262C23—24no—
263C23—22no—
264C23—24no—
265C23—41no—
266C23—42no—
267C23—41no—
268C23—44no—
269C23—41no—
270C23—44no—
271C23—42no—
272C23—41no—
273C23—44no—
274C23—44no—
275C23C2×#12011yescharacteristic
276C23C2×C22⋊Q811yescharacteristic
277C23C2×C22⋊Q812yes—
278C23C2×#12411yescharacteristic
279C23C2×#12411yescharacteristic
280C23C2×#12412yes—
281C23C2×#12711yescharacteristic
282C23#53214yes—
283C23#53511yescharacteristic
284C23#53611yescharacteristic
285C23#54214yes—
286C23D4214yes—
287C23#54712yes—
288C23#54814yes—
289C23D4×Q812yes—
290C23#55412yes—
291C23#55512yes—
292Q8—168no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—28no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—28no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—416no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—416no—
316C2×C4—48no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—44no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—88no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—88no—
330C2×C4—816no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#39312yes—
337C2×C4#39314yes—
338C2×C4#39312yes—
339C2×C4#45814yes—
340C22—22no—
341C22—22no—
342C22—28no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—28no—
349C22—22no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22C2×#39311yescharacteristic
361C22#243311yescharacteristic
362C22#243611yescharacteristic
363C22#245214yes—
364C22#249114yes—
365C22#254211yescharacteristic
366C22#256311yescharacteristic
367C22#264011yescharacteristic
368C22#285811yescharacteristic
369C22C2×#45812yes—
370C22#302812yes—
371C4—22no—
372C4—24no—
373C4—44no—
374C4—44no—
375C4—88no—
376C4—84no—
377C4—88no—
378C4—162no—
379C4#203014yescomplemented
380C2—24no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2—24no—
385C2—21no—
386C2#1514411yescharacteristic
387C2C2×#203011yescharacteristic
388C2#2347011yescharacteristic
389C2#2577212yes—
390C2#2594812yes—
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
45. C22×C4⋊C4Q8semidirect
379. C4#2030semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#20302
#139042
#110702
#108542
C4×#3422