Finite Group Databasebeta

← #124334

Subgroups — #124334

Order 512.

514 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 106 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#21373C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic, complemented
3#12544C211yesmaximal, characteristic, complemented
4#7128C212yesmaximal, complemented
5#7102C212yesmaximal, complemented
6C2×C4×#41C411yescharacteristic
7C22×M32—21no—
8C2×#431C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C2×C22⋊C16—21no—
11#1984C411yescharacteristic
12#1984C2211yescharacteristic
13#1954C411yescharacteristic
14#1954C2211yescharacteristic
15#1869—21no—
16#1869—21no—
17C4×C22⋊C8—21no—
18C4×C22⋊C8—21no—
19#1374—21no—
20#1374—21no—
21#1374C2212yes—
22#1374C2212yes—
23C22×#41C811yescharacteristic
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C4×#41—42no—
27C4×#41—42no—
28C2×C4×Q8C811yescharacteristic
29C2×C4×Q8C2×C411yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4—21no—
32C2×C4×D4C811yescharacteristic
33C2×C4×D4C2×C411yescharacteristic
34C2×#117—21no—
35C2×#117C811yescharacteristic
36C22×C42—21no—
37C22×C42C2×C411yescharacteristic
38C2×M32—24no—
39C2×M32—42no—
40C2×M32—42no—
41C22×C16—22no—
42C22×C16—22no—
43#431—42no—
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C4×C8—21no—
52C2×C4×C8—42no—
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57C22⋊C16—42no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—42no—
65C2×#41—42no—
66C22×Q8C2×C811yescharacteristic
67C22×D4—21no—
68C22×D4C2×C811yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C2×C811yescharacteristic
74C23×C4M1611yescharacteristic
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—41no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C4×Q8—42no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×D4—41no—
92C4×D4—41no—
93C4×D4—42no—
94C4×D4—44no—
95C4×D4—42no—
96C4×D4—42no—
97#117—44no—
98#117—42no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2×C811yescharacteristic
102C2×C4⋊C4C2×C811yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42C22⋊C411yescharacteristic
114C2×C42C2×C811yescharacteristic
115C2×C42M1611yescharacteristic
116C2×C42C22×C411yescharacteristic
117M32—48no—
118C2×C16—24no—
119C2×C16—42no—
120C2×C16—42no—
121C2×C16—42no—
122C2×C16—42no—
123C2×C16—42no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8—41no—
135C4×C8—41no—
136C4×C8—42no—
137C24—21no—
138C24C22⋊C811yescharacteristic
139#41—48no—
140#41—48no—
141C2×Q8—21no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8M3212yescomplemented
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4M3212yescomplemented
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4C22⋊C811yescharacteristic
179C22×C4C22⋊C811yescharacteristic
180C22×C4C22⋊C811yescharacteristic
181C22×C4#9912yes—
182C22×C4#10012yes—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C22×C811yescharacteristic
185C22×C4C2×M1611yescharacteristic
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—44no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8#10412yes—
205C2×C8#13512yes—
206C2×C8AGL(1,ℤ/8)11yescharacteristic
207C2×C8#13711yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C42—22no—
216C42—22no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—42no—
222C42—42no—
223C42—41no—
224C42—41no—
225C16—44no—
226C16—82no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23#32312yes—
241C23C2×C22⋊C811yescharacteristic
242Q8—42no—
243Q8—42no—
244D4—42no—
245D4—42no—
246D4—48no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—44no—
266C2×C4—48no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—84no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4#32312yes—
281C2×C4#34912yes—
282C2×C4#35012yes—
283C2×C4C2×#9911yescharacteristic
284C2×C4C2×#10011yescharacteristic
285C2×C4#41611yescharacteristic
286C2×C4#41711yescharacteristic
287C2×C4C2×#10411yescharacteristic
288C2×C4#42111yescharacteristic
289C2×C4C2×M3211yescharacteristic
290C2×C4C2×M3211yescharacteristic
291C8—21no—
292C8—22no—
293C8—21no—
294C8—22no—
295C8—41no—
296C8—81no—
297C8—81no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22C2×#32311yescharacteristic
313C22#147611yescharacteristic
314C22#147711yescharacteristic
315C22#211211yescharacteristic
316C22C2×#34911yescharacteristic
317C22C2×#35011yescharacteristic
318C22#211711yescharacteristic
319C4—22no—
320C4—22no—
321C4—21no—
322C4—21no—
323C4—22no—
324C4—42no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4#150411yescharacteristic
329C4#151611yescharacteristic
330C4#153011yescharacteristic
331C4#155111yescharacteristic
332C2—42no—
333C2—82no—
334C2#757811yescharacteristic
335C2#1141811yescharacteristic
336C2#1142211yescharacteristic
337C2#1161511yescharacteristic
338C2#1161711yescharacteristic
339C2#1162311yescharacteristic
340C2#1162811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC22×#41C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12844C2semidirect
3. #12544C2semidirect
4. #7128C2semidirect
5. #7102C2semidirect
144. C2×Q8M32semidirect
151. C2×D4M32semidirect

Maximal subgroups

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

subgroupindex
#213732
#128442
#125442
#71282
#71022