Finite Group Databasebeta

← #124198

Subgroups — #124198

Order 512.

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

normal subgroups only

This group has 482 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#14318C211yesmaximal, characteristic
2C4×#324C211yesmaximal, characteristic
3#7908C211yesmaximal, characteristic
4#7132C212yesmaximal
5#7123C212yesmaximal
6C2×#431—21no—
7C2×#400C411yescharacteristic
8C2×#389C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#324—21no—
12#1445C2211yescharacteristic
13#1444—21no—
14#1444C2211yescharacteristic
15#1442—21no—
16#1441C411yescharacteristic
17#1436—21no—
18#1434—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#400—22no—
40#400C814yescomplemented
41#389—41no—
42#389—41no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C2×#95—21no—
46C2×#95—21no—
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×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95C2311yescharacteristic
59#336—41no—
60#324—24no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C22×C8—42no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4C2×C812yes—
91C2×C4⋊C4C2×C812yes—
92C2×C4⋊C4M1612yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—81no—
101C2×C42—81no—
102C2×C42SD1612yes—
103C2×C42Q1612yes—
104C4⋊C8—44no—
105C22⋊C8—24no—
106C22⋊C8—24no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C4×C8—44no—
110C4×C8—82no—
111#95—24no—
112#95—24no—
113#95—21no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—42no—
121#95—81no—
122#95—81no—
123#95—81no—
124#95—81no—
125#95—81no—
126#95C2×C812yes—
127C24C2×C22⋊C411yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
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—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
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×C4C22⋊C812yes—
184C22×C4C22⋊C812yes—
185C22×C4#9911yescharacteristic
186C22×C4#9911yescharacteristic
187C22×C4#10312yes—
188C22×C4#10312yes—
189C22×C4#10412yes—
190C22×C4#10412yes—
191C22×C4C22×C811yescharacteristic
192C22×C4C2×M1611yescharacteristic
193C22×C4C2×SD1611yescharacteristic
194C22×C4C2×Q1611yescharacteristic
195C22×C4#13512yes—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—84no—
209C4⋊C4—84no—
210C42—22no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—42no—
215C42—81no—
216C42—81no—
217C42—81no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23C2×C22⋊C811yescharacteristic
223C23C2×#9911yescharacteristic
224C23#41311yescharacteristic
225C23C2×#10311yescharacteristic
226C23#41611yescharacteristic
227C23C2×#10411yescharacteristic
228C23C2×#10411yescharacteristic
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
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×C4Q8⋊C814yes—
299C2×C4#32812yes—
300C2×C4#32812yes—
301C8—84no—
302C8—84no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22#145911yescharacteristic
316C22C2×Q8⋊C811yescharacteristic
317C22#147611yescharacteristic
318C22C2×#32811yescharacteristic
319C22#150111yescharacteristic
320C22#152411yescharacteristic
321C22#154211yescharacteristic
322C22#212111yescharacteristic
323C22#212311yescharacteristic
324C22#212811yescharacteristic
325C22#213311yescharacteristic
326C4—24no—
327C4—22no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—42no—
335C4—82no—
336C4—81no—
337C4—82no—
338C4—161no—
339C4—161no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#756611yescharacteristic
345C2#1147411yescharacteristic
346C2#1148011yescharacteristic
347C2#1171811yescharacteristic
348C2#1172211yescharacteristic
349C2#1172911yescharacteristic
350C2#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. #400C8semidirect

Maximal subgroups

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

subgroupindex
#143182
C4×#3242
#79082
#71322
#71232