Finite Group Databasebeta

← #124174

Subgroups — #124174

Order 512.

496 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 108 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#14314C211yesmaximal, characteristic
2C4×#324C211yesmaximal, characteristic
3#7908C211yesmaximal, characteristic
4#7122C212yesmaximal
5#7104C212yesmaximal
6C2×#431—21no—
7C2×#398C411yescharacteristic
8C2×#389C411yescharacteristic
9C2×#387—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#324—21no—
13#1445C2211yescharacteristic
14#1443—21no—
15#1442C2211yescharacteristic
16#1441—21no—
17#1441C411yescharacteristic
18#1436—21no—
19#1434C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32#431—24no—
33#431—42no—
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C4×C8—41no—
38C2×C4×C8—41no—
39#398—22no—
40#398C814yescomplemented
41#389—41no—
42#389—41no—
43#387—42no—
44#387—42no—
45C4×C4⋊C4—41no—
46C4×C4⋊C4—41no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95C2311yescharacteristic
61#336—41no—
62#324—24no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
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—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4C2×C812yes—
94C2×C4⋊C4C2×C812yes—
95C2×C4⋊C4M1612yes—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—81no—
104C2×C42—81no—
105C2×C42D812yes—
106C2×C42SD1612yes—
107C4⋊C8—44no—
108C22⋊C8—24no—
109C22⋊C8—24no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C4×C8—44no—
113C4×C8—82no—
114#95—24no—
115#95—21no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—42no—
127#95—81no—
128#95—81no—
129#95—81no—
130#95C2×C812yes—
131C24C2×C22⋊C411yescharacteristic
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
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×C4C22⋊C812yes—
188C22×C4C22⋊C812yes—
189C22×C4#9911yescharacteristic
190C22×C4#9911yescharacteristic
191C22×C4#10212yes—
192C22×C4#10212yes—
193C22×C4#10412yes—
194C22×C4#10412yes—
195C22×C4C22×C811yescharacteristic
196C22×C4C2×M1611yescharacteristic
197C22×C4C2×D811yescharacteristic
198C22×C4C2×SD1611yescharacteristic
199C22×C4#13512yes—
200C2×C8—44no—
201C2×C8—44no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8—82no—
205C2×C8—82no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—48no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—42no—
220C42—81no—
221C42—81no—
222C42—81no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23C2×C22⋊C811yescharacteristic
228C23C2×#9911yescharacteristic
229C23#41311yescharacteristic
230C23C2×#10211yescharacteristic
231C23#41611yescharacteristic
232C23C2×#10411yescharacteristic
233C23C2×#10411yescharacteristic
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
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—81no—
303C2×C4D4⋊C814yes—
304C2×C4#32712yes—
305C2×C4#32712yes—
306C8—84no—
307C8—84no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22#145911yescharacteristic
321C22C2×D4⋊C811yescharacteristic
322C22#147611yescharacteristic
323C22C2×#32711yescharacteristic
324C22#150111yescharacteristic
325C22#152411yescharacteristic
326C22#154011yescharacteristic
327C22#212111yescharacteristic
328C22#212311yescharacteristic
329C22#212911yescharacteristic
330C22#213311yescharacteristic
331C4—24no—
332C4—22no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—42no—
340C4—82no—
341C4—82no—
342C4—81no—
343C4—161no—
344C4—161no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2#756511yescharacteristic
350C2#1147411yescharacteristic
351C2#1148211yescharacteristic
352C2#1169411yescharacteristic
353C2#1169811yescharacteristic
354C2#1173111yescharacteristic
355C2#1174011yescharacteristic
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
40. #398C8semidirect

Maximal subgroups

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

subgroupindex
#143142
C4×#3242
#79082
#71222
#71042