Finite Group Databasebeta

← #122371

Subgroups — #122371

Order 512.

385 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 74 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#22030C211yesmaximal, characteristic
2#11487C211yesmaximal, characteristic, complemented
3#11466C211yesmaximal, characteristic, complemented
4#7064C212yesmaximal, complemented
5#7054C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C22⋊C8C411yescharacteristic
8C4×#120C411yescharacteristic
9#2162—21no—
10#2112—21no—
11#1955—21no—
12#1924—21no—
13#1923—21no—
14#1869—21no—
15#1869C411yescharacteristic
16#1850C2211yescharacteristic
17#1779C2211yescharacteristic
18#1752C411yescharacteristic
19#1322—21no—
20#1314—21no—
21#1314C2212yes—
22#1314C2212yes—
23C24×C4C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#120C811yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2×C411yescharacteristic
30C2×M32—41no—
31C8×D4—41no—
32C8×D4—41no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8D411yescharacteristic
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—41no—
41C2×C22⋊C8—41no—
42C2×C22⋊C8C2×C411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C2×C22⋊C8C2311yescharacteristic
45C2×#97—21no—
46C2×C4×C8D411yescharacteristic
47#386—21no—
48#386C811yescharacteristic
49#385—21no—
50#385C811yescharacteristic
51#379C811yescharacteristic
52C4×C22⋊C4—21no—
53C4×C22⋊C4C2×C411yescharacteristic
54C4⋊C16—41no—
55C22⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—41no—
58C4×C16—41no—
59#336—21no—
60#336—21no—
61#336—21no—
62C25C2×C811yescharacteristic
63C22×D4—21no—
64C22×D4C2×C811yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—81no—
70C23×C4—81no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×C811yescharacteristic
73C23×C4C22×C411yescharacteristic
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—41no—
81C22×C8—81no—
82C22×C8—81no—
83C22×C8—81no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88#120—42no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—81no—
92C4×D4—81no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4M1611yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4C2×C811yescharacteristic
100C2×C22⋊C4C2×C811yescharacteristic
101C2×C22⋊C4C2×C811yescharacteristic
102C2×C22⋊C4M1611yescharacteristic
103C2×C42—21no—
104C2×C42C22⋊C411yescharacteristic
105M32—82no—
106C2×C16—42no—
107C2×C16—42no—
108C4⋊C8—41no—
109C4⋊C8—81no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112C22⋊C8—42no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116C22⋊C8—81no—
117C22⋊C8—81no—
118C22⋊C8—81no—
119#97—81no—
120C4×C8—21no—
121C4×C8—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125C24—21no—
126C24—22no—
127C24—41no—
128C24—42no—
129C24—41no—
130C24—81no—
131C24C22⋊C811yescharacteristic
132C24C22×C811yescharacteristic
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—81no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
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—42no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—84no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4C22⋊C811yescharacteristic
165C22×C4C22⋊C811yescharacteristic
166C22×C4C22⋊C811yescharacteristic
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×M1611yescharacteristic
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—81no—
184C2×C8—81no—
185C2×C8—81no—
186C2×C8—82no—
187C2×C8#9911yescharacteristic
188C2×C8#9911yescharacteristic
189C4⋊C4—41no—
190C4⋊C4—81no—
191C22⋊C4—22no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—41no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C42—21no—
200C42—21no—
201C42—41no—
202C42—41no—
203C16—82no—
204C16—82no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—82no—
224C23—84no—
225C23—84no—
226C23—82no—
227C23C2×C22⋊C811yescharacteristic
228C23#50412yes—
229D4—44no—
230D4—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—84no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4#34912yes—
263C2×C4C2×#9911yescharacteristic
264C2×C4#41311yescharacteristic
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—81no—
271C8—81no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—84no—
290C22—84no—
291C22—81no—
292C22—81no—
293C22#145911yescharacteristic
294C22#211311yescharacteristic
295C22C2×#34911yescharacteristic
296C4—22no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#212111yescharacteristic
305C4#212211yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2—82no—
311C2#1146711yescharacteristic
312C2#1147311yescharacteristic
313C2#1148311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22#1459
prime radicalC2×#120C8
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11487C2semidirect
3. #11466C2semidirect
4. #7064C2semidirect
5. #7054C2semidirect

Maximal subgroups

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

subgroupindex
#220302
#114872
#114662
#70642
#70542