Finite Group Databasebeta

← #122639

Subgroups — #122639

Order 512.

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

normal subgroups only

This group has 456 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
1C2×#2076C211yesmaximal, characteristic
2#11418C211yesmaximal, characteristic
3#8008C211yesmaximal, characteristic
4#7085C214yesmaximal, complemented
5C22×#107C411yescharacteristic
6C2×#407—21no—
7C22×C22⋊C8C2211yescharacteristic
8C2×#380—21no—
9#2112—21no—
10#2076—21no—
11#2076—21no—
12#2076C412yes—
13#2076C2212yes—
14C2×#340—21no—
15C2×#340C411yescharacteristic
16#1314—21no—
17#1314C2212yes—
18#1293—21no—
19#1293C2212yes—
20C24×C4D411yescharacteristic
21C22×M16—41no—
22C23×C8C2×C411yescharacteristic
23C23×C8D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C22⋊C4—41no—
27C2×M32—41no—
28C2×#107—21no—
29C2×#107—41no—
30C2×#107—41no—
31C2×#107C2×C412yes—
32#407—22no—
33#407—22no—
34#407—41no—
35#407—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—41no—
38C2×C22⋊C8—41no—
39C2×C22⋊C8—41no—
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C2×C22⋊C8C2311yescharacteristic
44#380—22no—
45#380—22no—
46#380—41no—
47#380—41no—
48C2×#95—41no—
49C22⋊C16—22no—
50C22⋊C16—41no—
51#340—21no—
52#340—22no—
53#340—22no—
54#340C2×C412yes—
55C25SD1611yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4D811yescharacteristic
68C23×C4SD1611yescharacteristic
69C23×C4C2×D411yescharacteristic
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—82no—
73C2×M16—82no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—42no—
79C22×C8—42no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—84no—
98M32—82no—
99C2×C16—42no—
100#107—42no—
101#107—42no—
102C22⋊C8—24no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—41no—
106C22⋊C8—41no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#95—42no—
111#95—42no—
112C24—22no—
113C24—21no—
114C24—21no—
115C24—41no—
116C24—42no—
117C24—42no—
118C24#10211yescharacteristic
119C24#10211yescharacteristic
120C24C2×SD1611yescharacteristic
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—82no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4—82no—
148C22×C4—82no—
149C22×C4—82no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4#9911yescharacteristic
156C22×C4#9911yescharacteristic
157C22×C4#10211yescharacteristic
158C22×C4#10211yescharacteristic
159C22×C4#10411yescharacteristic
160C22×C4#10411yescharacteristic
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4C2×D811yescharacteristic
163M16—84no—
164M16—82no—
165M16—82no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—44no—
178C2×C8—81no—
179C2×C8—84no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—82no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C22⋊C4—84no—
188C22⋊C4—84no—
189C16—82no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—48no—
206C23—42no—
207C23—42no—
208C23—81no—
209C23—81no—
210C23—84no—
211C23#32711yescharacteristic
212C23#32711yescharacteristic
213C23#36212yes—
214C23C2×#10211yescharacteristic
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—164no—
245C2×C4—162no—
246C2×C4—162no—
247C2×C4#32711yescharacteristic
248C2×C4#32711yescharacteristic
249C2×C4#35212yes—
250C2×C4#35412yes—
251C2×C4C2×#9911yescharacteristic
252C2×C4C2×#10411yescharacteristic
253C8—41no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8—84no—
258C22—24no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—84no—
276C22—84no—
277C22—81no—
278C22#134112yes—
279C22#134412yes—
280C22C2×#32711yescharacteristic
281C22#214311yescharacteristic
282C22C2×#36211yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—162no—
288C4—162no—
289C4#134412yes—
290C4C2×#35211yescharacteristic
291C4C2×#35411yescharacteristic
292C2—21no—
293C2—21no—
294C2—22no—
295C2—22no—
296C2—42no—
297C2—81no—
298C2C2×#134111yescharacteristic
299C2C2×#134411yescharacteristic
300C2#1171211yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C22⋊C8C23
prime radicalC25SD16
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7085C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
#114182
#80082
#70852