Finite Group Databasebeta

← #387107

Subgroups — #387107

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35576C211yesmaximal, characteristic
2#21338C212yesmaximal
3#11382C211yesmaximal, characteristic
4#11382C211yesmaximal, characteristic
5#11223C212yesmaximal
6#11206C212yesmaximal
7#10919C212yesmaximal
8#10916C212yesmaximal
9#10830C212yesmaximal
10C2×#544C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#2905C2212yes—
13#2873C2212yes—
14#2583C414yes—
15#2551—24no—
16#2550—24no—
17#2434C414yes—
18#2317C2212yes—
19#2296C2212yes—
20#2274C2211yescharacteristic
21#1998—24no—
22#1997—24no—
23C2×#344C2212yes—
24C2×#344C2212yes—
25C2×#344C2212yes—
26C2×#344C2212yes—
27#1742C2212yes—
28#1742C2212yes—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2211yescharacteristic
32C2×#343C2212yes—
33C2×#343C2211yescharacteristic
34#1728C2211yescharacteristic
35#1728C2212yes—
36#1728C2211yescharacteristic
37C23×Q8C2×C412yes—
38C22×M16C2312yes—
39C22×M16C2312yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42#544—41no—
43#544—42no—
44#544—41no—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8D414yes—
51C2×#125D414yes—
52C2×C22⋊Q8C2×C414yes—
53C2×#117C2312yes—
54C2×#117C2312yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#424—42no—
60#424—42no—
61#423—24no—
62#405—42no—
63#405—42no—
64C4×M16—24no—
65#398—24no—
66#395—24no—
67#395—28no—
68#391—24no—
69#389—28no—
70#389—22no—
71#389—22no—
72#389—24no—
73#385C2×C418yes—
74#385C2×C414yes—
75#385C2×C414yes—
76#384—22no—
77#382—22no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C4×C22⋊C4C2×C414yes—
81#344—24no—
82#344—24no—
83#344—24no—
84#344Q818yescomplemented
85#343—24no—
86#343—24no—
87#343—22no—
88#343—22no—
89#343—22no—
90#343—22no—
91#343—41no—
92#343—42no—
93#343—41no—
94#336—41no—
95#336—42no—
96#336—41no—
97C22×Q8—24no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—84no—
101C22×Q8C22⋊C414yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C412yes—
105C23×C4C22×C412yes—
106C23×C4C2411yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16C2×Q814yes—
130C2×M16C2×Q814yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C4⋊Q8—48no—
138C4⋊Q8—48no—
139C4⋊Q8—88no—
140#125—44no—
141#125—44no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117C2×Q814yes—
151#117#4114yes—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C22⋊C418yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4C22×C414yes—
172C2×C22⋊C4C22×C414yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—24no—
179C2×C42C2×D414yes—
180C4⋊C8—48no—
181#97—44no—
182C4×C8—44no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—28no—
187C24C23×C411yescharacteristic
188C2×Q8—44no—
189C2×Q8—48no—
190C2×Q8—48no—
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—84no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×Q8—84no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4C2×C22⋊C414yes—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C414yes—
216C22×C4C22×D412yes—
217C22×C4C22×Q812yes—
218C22×C4C2×#4112yes—
219C22×C4H(5,𝔽2)12yes—
220C22×C4H(5,𝔽2)11yescharacteristic
221C22×C4H(5,𝔽2)11yescharacteristic
222C22×C42−1+412yes—
223C22×C42−1+412yes—
224M16—24no—
225M16—24no—
226M16—24no—
227M16—24no—
228M16—42no—
229M16—42no—
230M16—42no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—88no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4C4×Q818yes—
272C42—24no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—42no—
277C42—42no—
278C42—84no—
279C23—41no—
280C23—41no—
281C23C22×C22⋊C411yescharacteristic
282C23C2×C4×Q812yes—
283C23#51811yescharacteristic
284C23#51811yescharacteristic
285C23#51912yes—
286Q8—816no—
287Q8—164no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4#53412yes—
318C2×C4#53412yes—
319C2×C4#54212yes—
320C2×C4#54212yes—
321C2×C4#54412yes—
322C2×C4#54412yes—
323C2×C4#54812yes—
324C2×C4#54812yes—
325C2×C4D4×Q812yes—
326C2×C4D4×Q812yes—
327C2×C4#55412yes—
328C2×C4#55511yescharacteristic
329C2×C4Q8211yescharacteristic
330C2×C4#56311yescharacteristic
331C2×C4#56311yescharacteristic
332C2×C4#56312yes—
333C8—44no—
334C8—44no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—42no—
344C22—42no—
345C22#230811yescharacteristic
346C22#233112yes—
347C22Q8×C22⋊C412yes—
348C22#236911yescharacteristic
349C22#238111yescharacteristic
350C4—48no—
351C4—42no—
352C4—44no—
353C4—44no—
354C4—42no—
355C4—44no—
356C4—82no—
357C4#272912yes—
358C4#272912yes—
359C4#273112yes—
360C4#273412yes—
361C2—22no—
362C2—22no—
363C2—21no—
364C2—21no—
365C2#1445011yescharacteristic
366C2#2177811yescharacteristic
367C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
84. #344Q8semidirect

Maximal subgroups

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

subgroupindex
#355762
#213382
#113822
#112232
#112062
#109192
#109162
#108302