Finite Group Databasebeta

← #181949

Subgroups — #181949

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic
2#20533C212yesmaximal, complemented
3C2×#1458C212yesmaximal
4C4×#324C214yesmaximal, complemented
5C4×#323C214yesmaximal
6#7908C212yesmaximal, complemented
7C22×#117C411yescharacteristic
8C22×#117C2211yescharacteristic
9C23×C4⋊C4C411yescharacteristic
10C2×#431C2214yes—
11#2274—28no—
12C2×C4×C4⋊C4C412yes—
13C2×C4×C4⋊C4C411yescharacteristic
14C2×C4×C4⋊C4C411yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C22⋊C4C412yes—
19C2×C4×C22⋊C4C2212yes—
20C8×C4⋊C4—28no—
21C4×C22⋊C8C2218yes—
22#1458C4116yescomplemented
23C2×#324C2214yes—
24C2×#323C2214yes—
25#1280C2218yes—
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C412yes—
28C2×#117—24no—
29C2×#117—24no—
30C2×#117—24no—
31C2×#117—24no—
32C2×#117C818yescomplemented
33C2×#117C2×C418yes—
34C22×C4⋊C4—24no—
35C22×C4⋊C4—24no—
36C22×C4⋊C4C818yescomplemented
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2311yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C412yes—
49C22×C42C2×C412yes—
50C22×C42C2×C412yes—
51C22×C42C2×C412yes—
52C22×C42C2312yes—
53C22×C42C2311yescharacteristic
54#431—28no—
55#431C2×C4116yescomplemented
56C2×C4⋊C8—24no—
57C2×C22⋊C8C2318yes—
58C2×C4×C8—24no—
59C2×C4×C8—28no—
60C4×C4⋊C4—24no—
61C4×C4⋊C4—24no—
62C4×C4⋊C4—28no—
63C4×C4⋊C4—24no—
64C4×C4⋊C4—28no—
65C4×C4⋊C4D418yes—
66C4×C22⋊C4—28no—
67C4×C22⋊C4—28no—
68C2×#95C2×C412yes—
69C2×#95C2×C412yes—
70C2×#95C2×C412yes—
71C2×#95C2312yes—
72#336—28no—
73#324C2×C4116yescomplemented
74#323C2×C4116yes—
75C25C22×C411yescharacteristic
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—82no—
83C23×C4—82no—
84C23×C4C4214yes—
85C23×C4C2×C814yes—
86C23×C4C2×C814yes—
87C23×C4C2×C818yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C412yes—
92C23×C4C22×C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C412yes—
95C23×C4C22×C412yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C2411yescharacteristic
98C22×C8—28no—
99#117—232no—
100#117—216no—
101#117—216no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—232no—
113C2×C4⋊C4C4214yes—
114C2×C4⋊C4C22⋊C414yes—
115C2×C4⋊C4C22⋊C414yes—
116C2×C4⋊C4C2×C818yes—
117C2×C4⋊C4C2×C814yes—
118C2×C4⋊C4C2×C814yes—
119C2×C4⋊C4C2×C814yes—
120C2×C4⋊C4C2×C814yes—
121C2×C4⋊C4M1614yes—
122C2×C4⋊C4M1614yes—
123C2×C4⋊C4C22×C414yes—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4C4214yes—
131C2×C22⋊C4C2×C818yes—
132C2×C22⋊C4C22×C414yes—
133C2×C42—22no—
134C2×C42—24no—
135C2×C42—22no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—82no—
141C2×C42—82no—
142C2×C42—82no—
143C2×C42—82no—
144C2×C42C4214yes—
145C2×C42C2×C818yes—
146C2×C42C22×C414yes—
147C2×C42C2×D418yes—
148C2×C42C2×D414yes—
149C2×C42#4114yes—
150C4⋊C8—216no—
151C22⋊C8—28no—
152C22⋊C8C22×C4116yes—
153C4×C8—216no—
154C4×C8—88no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95C22⋊C418yes—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24C2×C4212yes—
168C24C22×C812yes—
169C24C22×C812yes—
170C24C23×C411yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—216no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—88no—
209C22×C4—88no—
210C22×C4C4×C8116yescomplemented
211C22×C4C2×C4212yes—
212C22×C4C2×C4212yes—
213C22×C4C2×C4212yes—
214C22×C4C2×C4212yes—
215C22×C4C2×C4212yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C414yes—
218C22×C4C2×C22⋊C414yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4#11712yes—
221C22×C4#11712yes—
222C22×C4C22×C812yes—
223C22×C4C22×C812yes—
224C22×C4C22×C814yes—
225C22×C4C22×C814yes—
226C22×C4C22×C814yes—
227C22×C4C22×C814yes—
228C22×C4C22×C812yes—
229C22×C4C22×C812yes—
230C22×C4C2×M1614yes—
231C22×C4C2×M1614yes—
232C22×C4C2×M1612yes—
233C22×C4C2×M1612yes—
234C22×C4C23×C412yes—
235C22×C4C22×D412yes—
236C22×C4C2×#4112yes—
237C2×C8—216no—
238C2×C8—48no—
239C2×C8—48no—
240C4⋊C4—28no—
241C4⋊C4—28no—
242C4⋊C4—216no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4C22⋊C8116yes—
246C4⋊C4C22⋊C8116yes—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—216no—
250C42—28no—
251C42—216no—
252C42—48no—
253C42—84no—
254C42—84no—
255C42—84no—
256C42C4×D418yes—
257C42C4×D418yes—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×C4×C814yes—
285C23C22×C4211yescharacteristic
286C23C22×C22⋊C411yescharacteristic
287C23C2×#11711yescharacteristic
288C23C23×C811yescharacteristic
289C23C23×C811yescharacteristic
290C23C22×M1611yescharacteristic
291C23C22×M1611yescharacteristic
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—232no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—816no—
336C2×C4C4×C22⋊C414yes—
337C2×C4C4×C22⋊C414yes—
338C2×C4C2×C4×C814yes—
339C2×C4C2×C4×C814yes—
340C2×C4C4×M1614yes—
341C2×C4C2×C22⋊C818yes—
342C2×C4C2×C22⋊C818yes—
343C2×C4C2×C22⋊C814yes—
344C2×C4C2×C22⋊C814yes—
345C2×C4#43114yes—
346C2×C4#43114yes—
347C2×C4C2×C4×D414yes—
348C8—416no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—24no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22C2×C4×C22⋊C411yescharacteristic
373C22C22×C4×C811yescharacteristic
374C22C2×C4×M1611yescharacteristic
375C22C22×C22⋊C811yescharacteristic
376C22C22×C22⋊C811yescharacteristic
377C22#288311yescharacteristic
378C22#288411yescharacteristic
379C22#288811yescharacteristic
380C22#288911yescharacteristic
381C22C2×#43111yescharacteristic
382C22C2×#43111yescharacteristic
383C4—24no—
384C4—24no—
385C4—24no—
386C4—28no—
387C4—28no—
388C4—42no—
389C4—42no—
390C4—44no—
391C4—44no—
392C4C4×C22⋊C818yes—
393C2—21no—
394C2—21no—
395C2—21no—
396C2—21no—
397C2—44no—
398C2#1735011yescharacteristic
399C2#1735111yescharacteristic
400C2#1735211yescharacteristic
401C2#1735311yescharacteristic
402C2C2×C4×C22⋊C811yescharacteristic
403C2#2039311yescharacteristic
404C2#2039711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22C22×C4×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20533C2semidirect
4. C4×#324C2semidirect
6. #7908C2semidirect
22. #1458C4semidirect
32. C2×#117C8semidirect
36. C22×C4⋊C4C8semidirect
55. #431C2×C4semidirect
73. #324C2×C4semidirect
210. C22×C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
#205332
C2×#14582
C4×#3242
C4×#3232
#79082