Finite Group Databasebeta

← #415346

Subgroups — #415346

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21324C212yesmaximal
3#14161C211yesmaximal, characteristic
4#11392C214yesmaximal
5#11296C214yesmaximal
6#10851C212yesmaximal
7C4×#343C211yesmaximal, characteristic
8C2×#544C2212yes—
9C22×C4⋊Q8C2211yescharacteristic
10C2×C4×M16C2212yes—
11#2857—22no—
12#2659—22no—
13#2552—22no—
14#2551—22no—
15#2550—24no—
16#2316C2212yes—
17#2285C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1993—24no—
21#1990—22no—
22#1990—22no—
23C2×#343C2212yes—
24C2×#343C2212yes—
25#1736C2218yes—
26C2×#339C2218yes—
27#1446—22no—
28C4⋊#95—22no—
29C4⋊#95C2212yes—
30C4⋊#95C2212yes—
31C23×Q8—21no—
32C22×M16C2314yes—
33#544—28no—
34#544—44no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8Q814yescomplemented
42C2×#125—24no—
43C2×C22⋊Q8—24no—
44C2×#117C2314yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C42D412yes—
51C22×C42C2312yes—
52C22×C42C2311yescharacteristic
53C4×M16—24no—
54C4×M16—24no—
55C2×#97—22no—
56C2×C4×C8—22no—
57#398—28no—
58#395—28no—
59#391—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#389—24no—
64#384—22no—
65#384D412yes—
66#384D412yes—
67#382D414yes—
68#380—24no—
69#380—22no—
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—24no—
76C2×#95—21no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95D412yes—
80#343—24no—
81#343Q818yescomplemented
82#340—28no—
83#340—28no—
84#339—48no—
85C22×Q8—24no—
86C22×Q8—24no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—82no—
90C22×Q8—82no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2411yescharacteristic
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—24no—
103C2×M16C2×Q818yes—
104C22×C8—24no—
105C4⋊Q8—28no—
106C4⋊Q8—28no—
107C4⋊Q8—416no—
108C4⋊Q8—44no—
109C4⋊Q8—44no—
110C4⋊Q8—84no—
111#125—416no—
112C22⋊Q8—48no—
113C22⋊Q8—48no—
114#117—28no—
115#117—28no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2×D414yes—
140C2×C4⋊C4C2×Q814yes—
141C2×C4⋊C4C2×Q814yes—
142C2×C4⋊C4#4114yes—
143C2×C22⋊C4—24no—
144C2×C42—22no—
145C2×C42—24no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—44no—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C2×C42C2×Q812yes—
154C2×C42#4114yes—
155C2×C42#4112yes—
156#97—44no—
157C4×C8—44no—
158#95—28no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—42no—
163#95—84no—
164#95—84no—
165C24C22×D411yescharacteristic
166C2×Q8—48no—
167C2×Q8—48no—
168C2×Q8—84no—
169C2×Q8—84no—
170C2×Q8—88no—
171C2×Q8—84no—
172C2×Q8—84no—
173C2×Q8—88no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4#12011yescharacteristic
201C22×C4#12012yes—
202C22×C4#12011yescharacteristic
203C22×C4C22⋊Q812yes—
204C22×C4C22⋊Q812yes—
205C22×C4#12412yes—
206C22×C4#12414yes—
207C22×C4#12412yes—
208C22×C4C4⋊Q814yes—
209C22×C4C22×D412yes—
210C22×C4C22×Q811yescharacteristic
211C22×C4C22×Q812yes—
212C22×C4C2×#4112yes—
213C22×C4C2×#4111yescharacteristic
214C22×C4C2×#4112yes—
215C22×C4H(5,𝔽2)12yes—
216C22×C42−1+412yes—
217M16—28no—
218M16—28no—
219M16—44no—
220M16—44no—
221C2×C8—28no—
222C2×C8—44no—
223C2×C8—44no—
224C4⋊C4—28no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—28no—
228C4⋊C4—28no—
229C4⋊C4—48no—
230C4⋊C4—48no—
231C4⋊C4—416no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C42—48no—
244C42—44no—
245C42—48no—
246C42—82no—
247C42C22⋊Q814yes—
248C42C22⋊Q814yes—
249C42#12414yes—
250C42#12414yes—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23C2×#12011yescharacteristic
256C23C2×C22⋊Q811yescharacteristic
257C23C2×#12411yescharacteristic
258C23C2×#12411yescharacteristic
259C23C2×C4⋊Q811yescharacteristic
260C23#53511yescharacteristic
261C23#53611yescharacteristic
262Q8—816no—
263Q8—168no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#39312yes—
301C2×C4#39314yes—
302C2×C4#39312yes—
303C2×C4#45512yes—
304C2×C4#45612yes—
305C2×C4C2×C22⋊Q812yes—
306C2×C4C2×#12412yes—
307C2×C4#53314yes—
308C2×C4#54214yes—
309C2×C4#54614yes—
310C2×C4D4×Q814yes—
311C2×C4#55412yes—
312C2×C4#55612yes—
313C2×C4#55712yes—
314C2×C4Q8212yes—
315C8—48no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22C2×#39311yescharacteristic
329C22#243411yescharacteristic
330C22#243511yescharacteristic
331C22#256711yescharacteristic
332C22#256811yescharacteristic
333C22#264011yescharacteristic
334C22#286211yescharacteristic
335C22C2×#45511yescharacteristic
336C22C2×#45611yescharacteristic
337C22#302111yescharacteristic
338C22#302111yescharacteristic
339C4—22no—
340C4—24no—
341C4—44no—
342C4—44no—
343C4—84no—
344C4—84no—
345C4—84no—
346C4—84no—
347C4#202814yes—
348C4#245114yes—
349C4#249214yes—
350C2—21no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2#1519111yescharacteristic
355C2C2×#202811yescharacteristic
356C2#2346311yescharacteristic
357C2#2582311yescharacteristic
358C2#2582511yescharacteristic
359C2#2586911yescharacteristic
360C2#2587111yescharacteristic
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
41. C2×C4⋊Q8Q8semidirect
81. #343Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#213242
#141612
#113922
#112962
#108512
C4×#3432