Finite Group Databasebeta

← #93051

Subgroups — #93051

Order 512.

438 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 68 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#11418C211yesmaximal, characteristic
2#11097C211yesmaximal, characteristic
3C2×#1289C211yesmaximal, characteristic
4C22×#108C411yescharacteristic, complemented
5C2×#407—21no—
6C22×C22⋊C8C2211yescharacteristic
7C2×#380—21no—
8#2112—21no—
9C2×#341—21no—
10C2×#341C411yescharacteristic, complemented
11#1314—21no—
12#1314C412yescomplemented
13#1289—21no—
14#1289C412yes—
15C24×C4D411yescharacteristic
16C22×M16—21no—
17C22×M16—21no—
18C23×C8C2×C411yescharacteristic
19C23×C8D411yescharacteristic
20C22×C4⋊C4—41no—
21C22×C22⋊C4—41no—
22C2×M32—41no—
23C2×#108—41no—
24C2×#108—42no—
25#407—42no—
26#407—42no—
27C2×C22⋊C8—41no—
28C2×C22⋊C8—41no—
29C2×C22⋊C8—41no—
30C2×C22⋊C8—41no—
31C2×C22⋊C8C2×C411yescharacteristic
32C2×C22⋊C8C2×C411yescharacteristic
33C2×C22⋊C8D411yescharacteristic
34C2×C22⋊C8Q811yescharacteristic
35#380—22no—
36#380—22no—
37#380—41no—
38#380—41no—
39C2×#95—22no—
40C22⋊C16—22no—
41C22⋊C16—41no—
42#341—41no—
43#341—42no—
44C25SD1611yescharacteristic
45C23×C4—21no—
46C23×C4—21no—
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—41no—
50C23×C4—41no—
51C23×C4—41no—
52C23×C4—41no—
53C23×C4—42no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C411yescharacteristic
56C23×C4C4⋊C411yescharacteristic
57C23×C4D811yescharacteristic
58C23×C4SD1611yescharacteristic
59C23×C4Q1611yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—41no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8C4211yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C4⋊C411yescharacteristic
85C22×C8C4⋊C411yescharacteristic
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—82no—
95C2×C22⋊C4—82no—
96M32—81no—
97M32—81no—
98C2×C16—42no—
99#108—42no—
100#108—42no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—44no—
106C22⋊C8—44no—
107#95—24no—
108#95—42no—
109C24—21no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—22no—
114C24—41no—
115C24—41no—
116C24—81no—
117C24#10211yescharacteristic
118C24#10311yescharacteristic
119C24#10611yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
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—21no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—44no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—84no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—84no—
159C22×C4#9511yescharacteristic
160C22×C4#9911yescharacteristic
161C22×C4#9911yescharacteristic
162C22×C4#10211yescharacteristic
163C22×C4#10311yescharacteristic
164C22×C4#10411yescharacteristic
165C22×C4#10411yescharacteristic
166C22×C4#10711yescharacteristic
167M16—42no—
168M16—42no—
169M16—42no—
170M16—42no—
171M16—81no—
172M16—81no—
173M16—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—42no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—81no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—84no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C16—82no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—82no—
230C23—82no—
231C23#32811yescharacteristic
232C23#32811yescharacteristic
233C23#34011yescharacteristic
234C23#36111yescharacteristic
235C23#36211yescharacteristic
236C23#36512yes—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—164no—
270C2×C4—164no—
271C2×C4#32711yescharacteristic
272C2×C4#32711yescharacteristic
273C2×C4#33911yescharacteristic
274C2×C4#34211yescharacteristic
275C2×C4AΣL(1,𝔽16)11yescharacteristic
276C2×C4#35211yescharacteristic
277C2×C4#35311yescharacteristic
278C2×C4#35411yescharacteristic
279C8—41no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—82no—
284C8—82no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22—81no—
312C22—82no—
313C22—82no—
314C22#129411yescharacteristic
315C22#135111yescharacteristic
316C22#135211yescharacteristic
317C22#138311yescharacteristic
318C22#138711yescharacteristic
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—164no—
324C4#134311yescharacteristic
325C4#134411yescharacteristic
326C4#139011yescharacteristic
327C4#139311yescharacteristic
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2—41no—
333C2—41no—
334C2—41no—
335C2—42no—
336C2#725011yescharacteristic
337C2#725611yescharacteristic
338C2#726411yescharacteristic
subgroupG/H
center, socleC22#1294
commutatorC22×C8C42
Frattini, cosocleC22×C22⋊C8C22
prime radicalC25SD16
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C22×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×#108C4semidirect
10. C2×#341C4semidirect
12. #1314C4semidirect

Maximal subgroups

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

subgroupindex
#114182
#110972
C2×#12892