Finite Group Databasebeta

← #124397

Subgroups — #124397

Order 512.

496 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 92 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
1#14241C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7123C212yesmaximal
6C2×#395C411yescharacteristic
7C2×#391C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1817—21no—
10C4×C22⋊C8—21no—
11C2×#324—21no—
12#1446C2211yescharacteristic
13#1444C2211yescharacteristic
14#1441C411yescharacteristic
15#1436—21no—
16#1436—21no—
17#1436—21no—
18C4⋊#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C4×C8—41no—
36#395—22no—
37#395C814yescomplemented
38#391—41no—
39#391—41no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95D411yescharacteristic
54C2×#95C2311yescharacteristic
55#336—41no—
56#336—41no—
57#336—41no—
58#324—24no—
59C22×Q8—21no—
60C22×Q8—41no—
61C22×Q8—41no—
62C22×Q8—81no—
63C22×Q8—81no—
64C22×Q8C2×C812yes—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4C2×C812yes—
92C2×C4⋊C4M1612yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—41no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42—81no—
104C2×C42—81no—
105C2×C42—81no—
106C22⋊C8—24no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C4×C8—82no—
111#95—24no—
112#95—24no—
113#95—21no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—41no—
121#95—41no—
122#95—81no—
123#95—81no—
124#95—81no—
125#95C2×C812yes—
126C24C2×C22⋊C411yescharacteristic
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—82no—
130C2×Q8—82no—
131C2×Q8—84no—
132C2×Q8—82no—
133C2×Q8—82no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4C22⋊C812yes—
193C22×C4C22⋊C812yes—
194C22×C4#9911yescharacteristic
195C22×C4#9911yescharacteristic
196C22×C4#10412yes—
197C22×C4C22×C811yescharacteristic
198C22×C4C2×M1611yescharacteristic
199C22×C4#13512yes—
200C22×C4#13712yes—
201C2×C8—44no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—84no—
211C4⋊C4—84no—
212C4⋊C4—82no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42—81no—
217C42—81no—
218C42—81no—
219C42—84no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23C2×C22⋊C811yescharacteristic
225C23C2×#9911yescharacteristic
226C23#41311yescharacteristic
227C23#41611yescharacteristic
228C23#41911yescharacteristic
229C23C2×#10411yescharacteristic
230C23#42111yescharacteristic
231Q8—84no—
232Q8—84no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#35612yes—
303C8—84no—
304C8—162no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22#145911yescharacteristic
318C22#147611yescharacteristic
319C22#147911yescharacteristic
320C22#150811yescharacteristic
321C22#151611yescharacteristic
322C22#152911yescharacteristic
323C22#155411yescharacteristic
324C22#212111yescharacteristic
325C22#212211yescharacteristic
326C22#212611yescharacteristic
327C22C2×#35611yescharacteristic
328C4—22no—
329C4—44no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—42no—
338C4—82no—
339C4—82no—
340C4—161no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2#758311yescharacteristic
346C2#1147311yescharacteristic
347C2#1147811yescharacteristic
348C2#1178911yescharacteristic
349C2#1179511yescharacteristic
350C2#1184411yescharacteristic
351C2#1185411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #395C8semidirect

Maximal subgroups

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

subgroupindex
#142412
#87012
C4×#3242
#71312
#71232