Finite Group Databasebeta

← #127004

Subgroups — #127004

Order 512.

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

normal subgroups only

This group has 754 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#13958C211yesmaximal, characteristic
4#7187C214yesmaximal
5C22×#117C2211yescharacteristic
6C2×#432—21no—
7C2×#431—21no—
8C2×#381C411yescharacteristic
9C22×#95C411yescharacteristic
10#1464—22no—
11#1464C414yescomplemented
12#1458—22no—
13#1458C414yescomplemented
14C2×#324—21no—
15C2×#324—21no—
16C2×#323C2212yes—
17C2×#323C2212yes—
18#1445—21no—
19#1445—21no—
20#1434—21no—
21#1434—21no—
22C22⋊#95C412yes—
23C22⋊#95C2212yes—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C412yes—
26C2×#117—41no—
27C2×#117—41no—
28C2×#117—42no—
29C2×#117C2×C412yes—
30C2×#117C2×C412yes—
31C2×#117D412yes—
32C2×#117Q812yes—
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2311yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40#432—24no—
41#432—42no—
42#431—24no—
43#431—42no—
44C2×C4⋊C8—41no—
45C2×C4⋊C8—41no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×#97—41no—
49C2×C4×C8—41no—
50#381—42no—
51C2×#95—22no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—41no—
60C2×#95—41no—
61#324—24no—
62#324—24no—
63#323—22no—
64#323—22no—
65#323C2×C414yes—
66#323C2×C414yes—
67C25C22×C411yescharacteristic
68C23×C4—24no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—24no—
75C23×C4—42no—
76C23×C4—44no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—82no—
80C23×C4C4212yes—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C4⋊C412yes—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—42no—
89C22×C8—42no—
90#117—42no—
91#117—42no—
92#117—42no—
93#117—42no—
94#117—42no—
95#117—42no—
96#117—42no—
97#117—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4C22⋊C412yes—
103C2×C4⋊C4C22⋊C412yes—
104C2×C4⋊C4C4⋊C412yes—
105C2×C4⋊C4C4⋊C412yes—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4C4212yes—
114C2×C22⋊C4C4⋊C412yes—
115C2×C22⋊C4C22×C411yescharacteristic
116C2×C22⋊C4C22×C411yescharacteristic
117C2×C22⋊C4C2×D411yescharacteristic
118C2×C22⋊C4C2×Q811yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42C22⋊C412yes—
126C2×C42C22⋊C412yes—
127C2×C42C4⋊C412yes—
128C2×C42C4⋊C412yes—
129C4⋊C8—44no—
130C4⋊C8—44no—
131C22⋊C8—24no—
132C22⋊C8—24no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135#97—44no—
136C4×C8—44no—
137#95—44no—
138#95—44no—
139#95—42no—
140#95—42no—
141#95—41no—
142#95—41no—
143#95—42no—
144#95—82no—
145#95—82no—
146C24—21no—
147C24—21no—
148C24—22no—
149C24—42no—
150C24—41no—
151C24—41no—
152C24C2×C4211yescharacteristic
153C24C2×C22⋊C411yescharacteristic
154C24C2×C4⋊C411yescharacteristic
155C24#13114yes—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—48no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—88no—
204C22×C4—88no—
205C22×C4—84no—
206C22×C4—84no—
207C22×C4#9512yes—
208C22×C4#9512yes—
209C22×C4#9512yes—
210C22×C4#9512yes—
211C22×C4#9912yes—
212C22×C4#9911yescharacteristic
213C22×C4#9911yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C4⋊C411yescharacteristic
217C22×C4C2×C4⋊C411yescharacteristic
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—84no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—84no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23—24no—
246C23—24no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—44no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—44no—
257C23—44no—
258C23—44no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23C2×#9511yescharacteristic
265C23#40512yes—
266C23#40812yes—
267C23C2×#9911yescharacteristic
268C23C2×#9911yescharacteristic
269C23#41312yes—
270C23#42412yes—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—88no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—88no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#34212yes—
324C2×C4#34212yes—
325C8—84no—
326C8—84no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22—42no—
338C22—42no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—44no—
343C22—44no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22#146812yes—
349C22#146812yes—
350C22#172911yescharacteristic
351C22C2×#34211yescharacteristic
352C22#174211yescharacteristic
353C22#212012yes—
354C22#212211yescharacteristic
355C22#212311yescharacteristic
356C4—22no—
357C4—22no—
358C4—41no—
359C4—42no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—41no—
364C4—42no—
365C4—84no—
366C4—162no—
367C2—22no—
368C2—21no—
369C2—21no—
370C2—42no—
371C2—42no—
372C2#791311yescharacteristic
373C2#1154612yes—
374C2#1154812yes—
375C2#1257311yescharacteristic
376C2#1257511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
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
#139582
#71872