Finite Group Databasebeta

← #213567

Subgroups — #213567

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2#20717C212yesmaximal
3#9086C214yesmaximal, complemented
4#9082C214yesmaximal, complemented
5#8702C212yesmaximal
6#8299C212yesmaximal
7C22×#117C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#432C2212yes—
10C2×#431C2212yes—
11#2484—22no—
12#2484—22no—
13#2484—22no—
14#2484—22no—
15C2×#400C412yes—
16C2×#398C412yes—
17C2×#387C2214yes—
18C2×#385C412yes—
19C2×#381C412yes—
20C2×#381C2211yescharacteristic
21#1817C2214yes—
22#1798—22no—
23#1798—22no—
24#1779C2214yes—
25#1750—22no—
26#1750—22no—
27C2×#324C2212yes—
28C2×#324C2214yes—
29C2×#324C2212yes—
30#1280C2212yes—
31#1280C2214yes—
32#1280C2212yes—
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C2×#117—41no—
37C2×#117—42no—
38C2×#117—41no—
39C2×#117D412yes—
40C2×#117D412yes—
41C2×#117D412yes—
42C2×#117D412yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—42no—
46C22×C4⋊C4—41no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C42C2312yes—
56#432—42no—
57#432—42no—
58#431—42no—
59#431—42no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8C2314yes—
65C2×C22⋊C8C2314yes—
66C2×#97—22no—
67C2×C4×C8—22no—
68#400—22no—
69#400—24no—
70#400—22no—
71#398—22no—
72#398—24no—
73#398—22no—
74#387—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#385—22no—
79#385—22no—
80#385—22no—
81#385—22no—
82#381—22no—
83#381—22no—
84#381—22no—
85#381—22no—
86#381—22no—
87#381—22no—
88C2×#95C2×C412yes—
89C2×#95C2×C414yes—
90C2×#95C2×C414yes—
91C2×#95C2×C414yes—
92C2×#95C2312yes—
93#336—24no—
94#336—24no—
95#324—42no—
96#324—44no—
97#324—42no—
98C25C22×C411yescharacteristic
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—42no—
108C23×C4—44no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—44no—
112C23×C4C22×C412yes—
113C23×C4C22×C412yes—
114C23×C4C22×C414yes—
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C22×C8—24no—
123#117—42no—
124#117—42no—
125#117—42no—
126#117—42no—
127#117—42no—
128#117—42no—
129#117—42no—
130#117—42no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C4⋊C8—44no—
175C4⋊C8—44no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—24no—
179C22⋊C8—24no—
180C22⋊C8—48no—
181C22⋊C8—48no—
182#97—44no—
183C4×C8—44no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—22no—
189#95—22no—
190#95—24no—
191#95—24no—
192#95—24no—
193C24—21no—
194C24—21no—
195C24—24no—
196C24—22no—
197C24—22no—
198C24—42no—
199C24C2×C22⋊C412yes—
200C24C23×C411yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4#13114yes—
255C22×C4C22×D412yes—
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C4H(5,𝔽2)12yes—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—44no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C23—22no—
292C23—22no—
293C23—24no—
294C23—24no—
295C23—22no—
296C23—24no—
297C23—22no—
298C23—24no—
299C23—48no—
300C23—48no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—41no—
305C23—41no—
306C23#41014yes—
307C23C22×C22⋊C411yescharacteristic
308C23#51811yescharacteristic
309C23#51811yescharacteristic
310C23C2×#13112yes—
311C23#56812yes—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—416no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—88no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#53412yes—
359C2×C4#53412yes—
360C8—84no—
361C8—84no—
362C22—24no—
363C22—24no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—28no—
369C22—24no—
370C22—48no—
371C22—44no—
372C22—44no—
373C22—44no—
374C22—42no—
375C22—42no—
376C22#230811yescharacteristic
377C22#288012yes—
378C22C2×#41012yes—
379C22#288912yes—
380C22#297612yes—
381C22#297711yescharacteristic
382C22#297811yescharacteristic
383C4—22no—
384C4—22no—
385C4—42no—
386C4—42no—
387C4—44no—
388C4—41no—
389C4—41no—
390C4—42no—
391C4—42no—
392C4—84no—
393C4—82no—
394C2—22no—
395C2—21no—
396C2—21no—
397C2—44no—
398C2#1741714yes—
399C2#2174411yescharacteristic
400C2#2174711yescharacteristic
401C2#2175611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9086C2semidirect
4. #9082C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
#207172
#90862
#90822
#87022
#82992