Finite Group Databasebeta

← #129685

Subgroups — #129685

Order 512.

767 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2974C211yesmaximal, characteristic
2#11634C212yesmaximal, complemented
3#11631C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic, complemented
5#11459C211yesmaximal, characteristic, complemented
6C22×C4×D4C411yescharacteristic
7#2974—24no—
8#2974—24no—
9C2×#436C411yescharacteristic
10C2×#436C2211yescharacteristic
11C2×#435—21no—
12C2×#435—21no—
13C2×C8×D4C411yescharacteristic
14C2×C8×D4C2211yescharacteristic
15C2×#432C411yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#407—21no—
18C22×C22⋊C8—21no—
19#2163C2212yescomplemented
20#2162C2212yescomplemented
21#1330—24no—
22D4⋊C16—24no—
23#1322—21no—
24#1322—21no—
25#1314—21no—
26#1314—21no—
27C23×D4C811yescharacteristic
28C24×C4—21no—
29C22×M16—21no—
30C22×M16—21no—
31C23×C8—21no—
32C23×C8—21no—
33C2×C4×D4—22no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—22no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4C811yescharacteristic
40C2×C4×D4C811yescharacteristic
41C2×C4×D4C2×C411yescharacteristic
42C2×C4×D4C2×C411yescharacteristic
43C22×C4⋊C4C811yescharacteristic
44C22×C22⋊C4—21no—
45C22×C42C2×C411yescharacteristic
46#436—22no—
47#436—22no—
48#436—22no—
49#436D414yes—
50#435—28no—
51#435—28no—
52C8×D4—22no—
53C8×D4—22no—
54C8×D4—22no—
55C8×D4D414yes—
56#432—22no—
57#432—22no—
58#431—22no—
59#431—22no—
60C2×C4⋊C8C2×C411yescharacteristic
61C2×C4⋊C8C2×C411yescharacteristic
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64#407—28no—
65C2×C22⋊C8—28no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8D411yescharacteristic
72C2×C22⋊C8D411yescharacteristic
73C2×#97—21no—
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C4⋊C16—22no—
77C4⋊C16—22no—
78C22⋊C16—22no—
79C22⋊C16—22no—
80#346—22no—
81C4×C16—22no—
82C25—21no—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4—21no—
86C22×D4—44no—
87C22×D4C2×C811yescharacteristic
88C22×D4C2×C811yescharacteristic
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—82no—
97C23×C4—82no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C2×C811yescharacteristic
100C23×C4M1611yescharacteristic
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—82no—
104C2×M16—82no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—24no—
108C22×C8—24no—
109C22×C8—82no—
110C22×C8—82no—
111C22×C8C22⋊C411yescharacteristic
112C22×C8C22⋊C411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—24no—
118C4×D4—24no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—21no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4C22⋊C412yes—
128C4×D4C22⋊C412yes—
129C4×D4M1612yes—
130C4×D4M1612yes—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C2×C811yescharacteristic
135C2×C4⋊C4C2×C811yescharacteristic
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42C22⋊C411yescharacteristic
147C2×C42C2×C811yescharacteristic
148C2×C42M1611yescharacteristic
149C2×C42C22×C411yescharacteristic
150C2×C16—42no—
151C2×C16—42no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8C22⋊C412yes—
157C4⋊C8C22⋊C412yes—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—22no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—22no—
165#97—24no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8C2×D412yes—
169C4×C8C2×D412yes—
170C24—21no—
171C24—24no—
172C24—21no—
173C24—44no—
174C24—42no—
175C24—42no—
176C24—82no—
177C24C22⋊C811yescharacteristic
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—42no—
183C2×D4—48no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4C22⋊C812yes—
188C2×D4C22⋊C812yes—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—88no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4C22⋊C811yescharacteristic
220C22×C4C22⋊C811yescharacteristic
221C22×C4C22⋊C811yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C22×C811yescharacteristic
224C22×C4C2×M1611yescharacteristic
225M16—84no—
226M16—84no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—84no—
238C2×C8—84no—
239C2×C8#12012yes—
240C2×C8#12012yes—
241C4⋊C4—24no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4C22⋊C812yes—
245C4⋊C4C22⋊C812yes—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42C2×C22⋊C411yescharacteristic
259C42C2×C22⋊C411yescharacteristic
260C42C2×M1611yescharacteristic
261C42C2×M1611yescharacteristic
262C16—82no—
263C16—82no—
264C23—24no—
265C23—21no—
266C23—22no—
267C23—21no—
268C23—22no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—48no—
275C23—48no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—84no—
281C23—88no—
282C23—88no—
283C23#35012yes—
284C23C2×C22⋊C811yescharacteristic
285D4—24no—
286D4—22no—
287D4—42no—
288D4—48no—
289D4—42no—
290D4#34914yes—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4#37911yescharacteristic
324C2×C4C2×C22⋊C811yescharacteristic
325C2×C4C2×C22⋊C811yescharacteristic
326C2×C4#40711yescharacteristic
327C2×C4#40711yescharacteristic
328C2×C4#40711yescharacteristic
329C8—22no—
330C8—22no—
331C8—42no—
332C8—42no—
333C8—41no—
334C8—41no—
335C22—21no—
336C22—21no—
337C22—24no—
338C22—22no—
339C22—42no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—82no—
351C22—82no—
352C22—816no—
353C22#177911yescharacteristic
354C22C2×#35011yescharacteristic
355C22#211711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4#179511yescharacteristic
366C4#179511yescharacteristic
367C4C2×#34912yes—
368C2—21no—
369C2—21no—
370C2—44no—
371C2—42no—
372C2—82no—
373C2#832211yescharacteristic
374C2#1268111yescharacteristic
375C2#1268311yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC23×D4C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11634C2semidirect
3. #11631C2semidirect
4. #11460C2semidirect
5. #11459C2semidirect
19. #2163C22semidirect
20. #2162C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29742
#116342
#116312
#114602
#114592