Finite Group Databasebeta

← #127654

Subgroups — #127654

Order 512.

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

normal subgroups only

This group has 756 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×#1820C211yesmaximal, characteristic
2C2×#1458C211yesmaximal, characteristic
3#13938C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×#117C2211yescharacteristic
6C2×#431—21no—
7C2×#383C411yescharacteristic, complemented
8C2×#381—21no—
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10#1820—22no—
11#1820C414yescomplemented
12C2×#342—21no—
13C2×#342C2212yes—
14#1458—22no—
15#1458C414yescomplemented
16C2×#324—21no—
17C2×#323C2212yes—
18#1441C412yescomplemented
19#1439C2212yes—
20#1437—21no—
21#1436—21no—
22#1434—21no—
23C22⋊#95—21no—
24C24×C4D411yescharacteristic
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⋊C4D411yescharacteristic
35C22×C22⋊C4—22no—
36C22×C22⋊C4—22no—
37C22×C22⋊C4D411yescharacteristic
38C22×C22⋊C4C2311yescharacteristic
39C22×C42—21no—
40C22×C42C2×C411yescharacteristic
41C22×C42D411yescharacteristic
42#431—24no—
43#431—42no—
44C2×C4⋊C8—41no—
45C2×C22⋊C8—22no—
46C2×C4×C8—41no—
47#383—42no—
48#381—24no—
49#381—42no—
50C4×C4⋊C4—41no—
51C4×C4⋊C4—41no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—41no—
59C2×#95—41no—
60C2×#95C2×C412yes—
61#342—24no—
62#342—22no—
63#342C2×C414yes—
64#324—24no—
65#323—22no—
66#323C2×C414yes—
67C25C2×D411yescharacteristic
68C23×C4—24no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C412yes—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C4⋊C412yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
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—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4C4212yes—
100C2×C4⋊C4C22⋊C412yes—
101C2×C4⋊C4C4⋊C412yes—
102C2×C4⋊C4C4⋊C412yes—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—82no—
114C2×C22⋊C4C22⋊C412yes—
115C2×C22⋊C4C4⋊C412yes—
116C2×C22⋊C4C22×C411yescharacteristic
117C2×C22⋊C4C22×C411yescharacteristic
118C2×C22⋊C4C2×D411yescharacteristic
119C2×C22⋊C4C2×Q811yescharacteristic
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—22no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—42no—
127C2×C42—82no—
128C2×C42C4212yes—
129C2×C42C22⋊C412yes—
130C2×C42C4⋊C412yes—
131C2×C42C4⋊C412yes—
132C4⋊C8—44no—
133C22⋊C8—24no—
134C22⋊C8—42no—
135C4×C8—44no—
136#95—42no—
137#95—44no—
138#95—44no—
139#95—42no—
140#95—44no—
141#95—44no—
142#95—41no—
143#95—41no—
144#95—42no—
145#95—82no—
146C24—21no—
147C24—21no—
148C24—22no—
149C24—42no—
150C24—41no—
151C24—41no—
152C24C2×C22⋊C411yescharacteristic
153C24C2×C22⋊C411yescharacteristic
154C24C2×C4⋊C411yescharacteristic
155C24#13514yes—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
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—44no—
180C22×C4—44no—
181C22×C4—48no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4#9512yes—
207C22×C4#9512yes—
208C22×C4#9512yes—
209C22×C4#9512yes—
210C22×C4#9912yes—
211C22×C4#9911yescharacteristic
212C22×C4#9911yescharacteristic
213C22×C4C2×C4211yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C4⋊C411yescharacteristic
216C22×C4C2×C4⋊C411yescharacteristic
217C2×C8—44no—
218C2×C8—82no—
219C2×C8—82no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—88no—
235C22⋊C4—84no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—84no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—24no—
251C23—21no—
252C23—21no—
253C23—24no—
254C23—44no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—44no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—81no—
264C23—81no—
265C23—82no—
266C23—82no—
267C23C2×#9511yescharacteristic
268C23C2×#9911yescharacteristic
269C23C2×#9911yescharacteristic
270C23#41612yes—
271C23#41612yes—
272C23#42112yes—
273C23#42712yes—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—82no—
327C2×C4—164no—
328C2×C4—164no—
329C2×C4#34212yes—
330C2×C4#34212yes—
331C8—84no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—44no—
349C22—44no—
350C22—84no—
351C22—81no—
352C22—81no—
353C22#151612yes—
354C22#151612yes—
355C22#173311yescharacteristic
356C22#173511yescharacteristic
357C22C2×#34211yescharacteristic
358C22#212012yes—
359C22#212811yescharacteristic
360C22#212911yescharacteristic
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—42no—
366C4—41no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4—84no—
371C4—162no—
372C4—164no—
373C2—22no—
374C2—21no—
375C2—21no—
376C2—42no—
377C2—42no—
378C2#802411yescharacteristic
379C2#1184012yes—
380C2#1184212yes—
381C2#1257311yescharacteristic
382C2#1258311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
7. C2×#383C4semidirect
9. C2×C4×C4⋊C4C4semidirect
11. #1820C4semidirect
15. #1458C4semidirect
18. #1441C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18202
C2×#14582
#139382
#72502