Finite Group Databasebeta

← #127020

Subgroups — #127020

Order 512.

676 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 154 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×#1467C211yesmaximal, characteristic
2C2×#1461C211yesmaximal, characteristic
3#13942C211yesmaximal, characteristic
4#7189C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C2×#432—21no—
7C2×#431—21no—
8C2×#391C412yes—
9C2×#382C411yescharacteristic
10C2×C4×C4⋊C4C411yescharacteristic
11#1467—22no—
12#1467C414yescomplemented
13#1461—22no—
14#1461C414yescomplemented
15C2×#324—21no—
16C2×#324—21no—
17C2×#324C2212yes—
18C2×#324C2212yes—
19#1441—21no—
20#1441—21no—
21C4⋊#95C2212yes—
22#1434—21no—
23#1434—21no—
24C23×Q8C2×C411yescharacteristic
25C2×C4×Q8—42no—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C2×C4×Q8C2×C412yes—
29C2×C4×Q8C2×C412yes—
30C2×C4×Q8D412yes—
31C2×C4×Q8Q812yes—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4C2311yescharacteristic
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C412yes—
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41#432—24no—
42#432—42no—
43#431—24no—
44#431—42no—
45C2×C4⋊C8—41no—
46C2×C4⋊C8—41no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×#97—41no—
50C2×C4×C8—41no—
51#391—44no—
52#382—41no—
53#382—41no—
54C4×C4⋊C4—42no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62#324—24no—
63#324—24no—
64#324—22no—
65#324—22no—
66#324C2×C414yes—
67#324C2×C414yes—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—22no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8C4212yes—
75C22×Q8C4⋊C412yes—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—42no—
86C22×C8—42no—
87C4×Q8—42no—
88C4×Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91C4×Q8—42no—
92C4×Q8—42no—
93C4×Q8—42no—
94C4×Q8—42no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C22⋊C412yes—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C4⋊C412yes—
109C2×C4⋊C4C4⋊C412yes—
110C2×C4⋊C4C22×C411yescharacteristic
111C2×C4⋊C4C22×C411yescharacteristic
112C2×C4⋊C4C2×D411yescharacteristic
113C2×C4⋊C4C2×Q811yescharacteristic
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—24no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42C4212yes—
127C2×C42C22⋊C412yes—
128C2×C42C22⋊C412yes—
129C2×C42C4⋊C412yes—
130C2×C42C4⋊C412yes—
131C2×C42C4⋊C412yes—
132C4⋊C8—44no—
133C4⋊C8—44no—
134C22⋊C8—24no—
135C22⋊C8—24no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—44no—
139C4×C8—44no—
140#95—44no—
141#95—44no—
142#95—42no—
143#95—44no—
144#95—42no—
145#95—41no—
146#95—41no—
147#95—44no—
148C24C2×C22⋊C411yescharacteristic
149C2×Q8—24no—
150C2×Q8—24no—
151C2×Q8—24no—
152C2×Q8—24no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C22×C4—21no—
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—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
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#9512yes—
198C22×C4#9512yes—
199C22×C4#9512yes—
200C22×C4#9512yes—
201C22×C4#9911yescharacteristic
202C22×C4#9911yescharacteristic
203C22×C4#9912yes—
204C22×C4#10012yes—
205C22×C4#10112yes—
206C22×C4C2×C4211yescharacteristic
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C4⋊C411yescharacteristic
210C22×C4C2×C4⋊C411yescharacteristic
211C22×C4C2×C4⋊C411yescharacteristic
212C22×C4#13112yes—
213C22×C4#13112yes—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—84no—
217C2×C8—84no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—48no—
227C4⋊C4—84no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—84no—
241C23—22no—
242C23—22no—
243C23C2×#9511yescharacteristic
244C23#40811yescharacteristic
245C23#40811yescharacteristic
246C23C2×#9911yescharacteristic
247C23C2×#9911yescharacteristic
248C23C2×#10011yescharacteristic
249C23C2×#10111yescharacteristic
250Q8—48no—
251Q8—48no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#34212yes—
308C2×C4#34212yes—
309C2×C4#34314yes—
310C2×C4#40512yes—
311C2×C4#42412yes—
312C8—84no—
313C8—84no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22#147111yescharacteristic
323C22#147111yescharacteristic
324C22#147211yescharacteristic
325C22#147211yescharacteristic
326C22#172911yescharacteristic
327C22C2×#34211yescharacteristic
328C22C2×#34311yescharacteristic
329C22#212612yes—
330C22#213312yes—
331C4—22no—
332C4—22no—
333C4—24no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—42no—
342C4—84no—
343C4—162no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2#791511yescharacteristic
349C2#1156512yes—
350C2#1157112yes—
351C2#1258111yescharacteristic
352C2#1258611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1467C4semidirect
14. #1461C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14672
C2×#14612
#139422
#71892