Finite Group Databasebeta

← #127010

Subgroups — #127010

Order 512.

700 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 146 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×#1464C211yesmaximal, characteristic
2C2×#1458C211yesmaximal, characteristic
3#13939C211yesmaximal, characteristic
4#7189C214yesmaximal, complemented
5C22×#117C2211yescharacteristic
6C2×#432—21no—
7C2×#431—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10#1464—22no—
11#1464C414yescomplemented
12#1458—22no—
13#1458C414yescomplemented
14C2×#324C2212yes—
15C2×#324C2212yes—
16C2×#323—21no—
17C2×#323—21no—
18#1440—21no—
19#1440—21no—
20C4⋊#95C412yes—
21C4⋊#95C2212yes—
22#1434—21no—
23#1434—21no—
24C24×C4C2×C411yescharacteristic
25C2×#117—41no—
26C2×#117—41no—
27C2×#117—42no—
28C2×#117C2×C412yes—
29C2×#117C2×C412yes—
30C2×#117D412yes—
31C2×#117Q812yes—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2311yescharacteristic
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4D411yescharacteristic
38C22×C22⋊C4D411yescharacteristic
39C22×C42C2×C412yes—
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42#432—24no—
43#432—42no—
44#431—24no—
45#431—42no—
46C2×C4⋊C8—41no—
47C2×C4⋊C8—41no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×#97—41no—
51C2×C4×C8—41no—
52#380—41no—
53#380—41no—
54C4×C22⋊C4—42no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—22no—
61#324—22no—
62#324—22no—
63#324C2×C414yes—
64#324C2×C414yes—
65#323—24no—
66#323—24no—
67C25C22⋊C411yescharacteristic
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—24no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—42no—
75C23×C4C4212yes—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C4⋊C412yes—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C22×C8—42no—
84C22×C8—42no—
85#117—42no—
86#117—42no—
87#117—42no—
88#117—42no—
89#117—42no—
90#117—42no—
91#117—42no—
92#117—42no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C4212yes—
101C2×C4⋊C4C4⋊C412yes—
102C2×C4⋊C4C22×C411yescharacteristic
103C2×C4⋊C4C22×C411yescharacteristic
104C2×C4⋊C4C2×D411yescharacteristic
105C2×C4⋊C4C2×Q811yescharacteristic
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—82no—
114C2×C22⋊C4C22⋊C412yes—
115C2×C22⋊C4C22⋊C412yes—
116C2×C22⋊C4C4⋊C412yes—
117C2×C22⋊C4C4⋊C412yes—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—24no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42C22⋊C412yes—
128C2×C42C22⋊C412yes—
129C2×C42C4⋊C412yes—
130C2×C42C4⋊C412yes—
131C4⋊C8—44no—
132C4⋊C8—44no—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#97—44no—
138C4×C8—44no—
139#95—44no—
140#95—44no—
141#95—42no—
142#95—41no—
143#95—41no—
144#95—82no—
145#95—82no—
146C24—21no—
147C24—21no—
148C24—22no—
149C24—22no—
150C24—42no—
151C24—41no—
152C24—41no—
153C24#9512yes—
154C24C2×C22⋊C411yescharacteristic
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—48no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
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—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4#9512yes—
200C22×C4#9512yes—
201C22×C4#9512yes—
202C22×C4#9912yes—
203C22×C4#9911yescharacteristic
204C22×C4#9911yescharacteristic
205C22×C4C2×C4211yescharacteristic
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C4⋊C411yescharacteristic
209C22×C4C2×C4⋊C411yescharacteristic
210C22×C4C2×C4⋊C411yescharacteristic
211C22×C4#13114yes—
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8—82no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—84no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—48no—
230C22⋊C4—84no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—84no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—82no—
256C23—82no—
257C23—82no—
258C23—82no—
259C23C2×#9511yescharacteristic
260C23#40812yes—
261C23C2×#9911yescharacteristic
262C23C2×#9911yescharacteristic
263C23#41312yes—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4#34212yes—
316C2×C4#34212yes—
317C2×C4#40512yes—
318C2×C4#42412yes—
319C8—84no—
320C8—84no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—82no—
333C22—82no—
334C22—84no—
335C22—82no—
336C22—82no—
337C22#146812yes—
338C22#146812yes—
339C22#172911yescharacteristic
340C22C2×#34211yescharacteristic
341C22#174211yescharacteristic
342C22#212811yescharacteristic
343C22#212911yescharacteristic
344C22#213212yes—
345C4—22no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—42no—
355C4—84no—
356C4—162no—
357C2—22no—
358C2—21no—
359C2—21no—
360C2—82no—
361C2#791311yescharacteristic
362C2#1155412yes—
363C2#1155612yes—
364C2#1258311yescharacteristic
365C2#1258511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7189C2semidirect
11. #1464C4semidirect
13. #1458C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14642
C2×#14582
#139392
#71892