Finite Group Databasebeta

← #122255

Subgroups — #122255

Order 512.

381 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 80 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#21848C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic, complemented
3C2×#1322C211yesmaximal, characteristic, complemented
4#7057C212yesmaximal, complemented
5#7056C212yesmaximal, complemented
6C2×#529C411yescharacteristic
7C2×#431C411yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9#2112—21no—
10C2×C22⋊C16—21no—
11#1984C411yescharacteristic
12#1984C2211yescharacteristic
13#1954C411yescharacteristic
14#1954C2211yescharacteristic
15#1869—21no—
16#1869—21no—
17#1798—21no—
18#1798—21no—
19#1322—21no—
20#1322—21no—
21#1322C2212yes—
22#1322C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25#529—41no—
26#529—41no—
27#529—41no—
28#529—41no—
29C2×#126—21no—
30C2×#125C811yescharacteristic
31C2×#124C811yescharacteristic
32C2×#123—21no—
33C2×C22⋊Q8C811yescharacteristic
34C2×#121C811yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4C2×C411yescharacteristic
38C2×#117—21no—
39C22×C42C2×C411yescharacteristic
40C2×M32—41no—
41C22×C16—41no—
42#431—41no—
43#431—41no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—41no—
46C2×C4⋊C8—41no—
47C2×C4⋊C8C2×C411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8C2311yescharacteristic
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C4×C8C2×C411yescharacteristic
57C2×C4×C8C2×C411yescharacteristic
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16—41no—
61C22⋊C16—41no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66C22×Q8C2×C811yescharacteristic
67C22×D4—21no—
68C22×D4C2×C811yescharacteristic
69C23×C4—21no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C2×C811yescharacteristic
72C23×C4M1611yescharacteristic
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—81no—
80C22×C8—81no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85#126—42no—
86#126—42no—
87#125—41no—
88#125—41no—
89#124—41no—
90#124—41no—
91#123—42no—
92#123—42no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95#121—41no—
96#121—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—41no—
105#117—42no—
106#117—42no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4C2×C811yescharacteristic
112C2×C4⋊C4C2×C811yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C22⋊C411yescharacteristic
124C2×C42C2×C811yescharacteristic
125C2×C42M1611yescharacteristic
126C2×C42C22×C411yescharacteristic
127M32—82no—
128C2×C16—42no—
129C2×C16—42no—
130C2×C16—82no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—42no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—81no—
139C22⋊C8—81no—
140C22⋊C8—81no—
141C22⋊C8—81no—
142C4×C8—41no—
143C4×C8—41no—
144C24—21no—
145C24C22⋊C811yescharacteristic
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×D4—22no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—41no—
155C2×D4—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4C22⋊C811yescharacteristic
175C22×C4C22⋊C811yescharacteristic
176C22×C4C22⋊C811yescharacteristic
177C22×C4#9911yescharacteristic
178C22×C4#9911yescharacteristic
179C22×C4#10011yescharacteristic
180C22×C4#10011yescharacteristic
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C22×C811yescharacteristic
183C22×C4C2×M1611yescharacteristic
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—81no—
201C2×C8—81no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—41no—
225C42—41no—
226C16—82no—
227C16—82no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23#32311yescharacteristic
240C23#32311yescharacteristic
241C23C2×C22⋊C811yescharacteristic
242Q8—82no—
243D4—82no—
244D4—82no—
245D4—82no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
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—21no—
258C2×C4—21no—
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—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4#32311yescharacteristic
282C2×C4#32311yescharacteristic
283C2×C4C2×#9911yescharacteristic
284C2×C4C2×#10011yescharacteristic
285C2×C4#50412yes—
286C8—41no—
287C8—41no—
288C8—41no—
289C8—41no—
290C8—81no—
291C8—81no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22#132112yes—
309C22C2×#32311yescharacteristic
310C22#211311yescharacteristic
311C22#211511yescharacteristic
312C4—22no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4#212711yescharacteristic
323C4#213311yescharacteristic
324C2—21no—
325C2—21no—
326C2—22no—
327C2—41no—
328C2—161no—
329C2#1141911yescharacteristic
330C2#1143211yescharacteristic
331C2C2×#132111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11440C2semidirect
3. C2×#1322C2semidirect
4. #7057C2semidirect
5. #7056C2semidirect

Maximal subgroups

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

subgroupindex
#218482
#114402
C2×#13222
#70572
#70562