Finite Group Databasebeta

← #93048

Subgroups — #93048

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#11095C211yesmaximal, characteristic
3#8008C211yesmaximal, characteristic
4C22×#106C411yescharacteristic
5C2×#407—21no—
6C22×C22⋊C8C2211yescharacteristic
7C2×#380—21no—
8#2112—21no—
9C2×#340—21no—
10C2×#340C411yescharacteristic
11#1314—21no—
12#1314C412yescomplemented
13#1293—21no—
14#1293C412yescomplemented
15C24×C4D411yescharacteristic
16C22×M16—41no—
17C23×C8C2×C411yescharacteristic
18C23×C8D411yescharacteristic
19C22×C4⋊C4—21no—
20C22×C4⋊C4—21no—
21C22×C22⋊C4—41no—
22C2×M32—41no—
23C2×#106—41no—
24C2×#106—42no—
25#407—22no—
26#407—22no—
27#407—41no—
28#407—41no—
29C2×C22⋊C8—22no—
30C2×C22⋊C8—41no—
31C2×C22⋊C8—41no—
32C2×C22⋊C8—41no—
33C2×C22⋊C8C2×C411yescharacteristic
34C2×C22⋊C8C2×C411yescharacteristic
35C2×C22⋊C8D411yescharacteristic
36C2×C22⋊C8Q811yescharacteristic
37#380—42no—
38#380—42no—
39C2×#95—41no—
40C22⋊C16—22no—
41C22⋊C16—41no—
42#340—42no—
43#340—41no—
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—41no—
54C23×C4—41no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C411yescharacteristic
57C23×C4C4⋊C411yescharacteristic
58C23×C4D811yescharacteristic
59C23×C4SD1611yescharacteristic
60C23×C4Q1611yescharacteristic
61C2×M16—42no—
62C2×M16—42no—
63C2×M16—81no—
64C2×M16—81no—
65C2×M16—81no—
66C2×M16—81no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8C4211yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C4⋊C411yescharacteristic
79C22×C8C4⋊C411yescharacteristic
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—82no—
94C2×C22⋊C4—82no—
95M32—81no—
96M32—81no—
97C2×C16—42no—
98#106—42no—
99#106—42no—
100C22⋊C8—24no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#95—44no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24—41no—
113C24—41no—
114C24—41no—
115C24—41no—
116C24—41no—
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—42no—
149C22×C4—42no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4#9511yescharacteristic
169C22×C4#9911yescharacteristic
170C22×C4#9911yescharacteristic
171C22×C4#10211yescharacteristic
172C22×C4#10311yescharacteristic
173C22×C4#10411yescharacteristic
174C22×C4#10411yescharacteristic
175C22×C4#10711yescharacteristic
176M16—82no—
177M16—82no—
178M16—82no—
179M16—82no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—42no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—44no—
194C2×C8—81no—
195C2×C8—82no—
196C2×C8—82no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C22⋊C4—88no—
206C16—82no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23#32711yescharacteristic
239C23#32711yescharacteristic
240C23#34011yescharacteristic
241C23#36512yes—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
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—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—161no—
278C2×C4—161no—
279C2×C4—161no—
280C2×C4—161no—
281C2×C4—161no—
282C2×C4—161no—
283C2×C4—161no—
284C2×C4—161no—
285C2×C4#32811yescharacteristic
286C2×C4#32811yescharacteristic
287C2×C4#33911yescharacteristic
288C2×C4#34211yescharacteristic
289C2×C4AΣL(1,𝔽16)11yescharacteristic
290C2×C4#35211yescharacteristic
291C2×C4#35311yescharacteristic
292C2×C4#35411yescharacteristic
293C8—41no—
294C8—41no—
295C8—41no—
296C8—41no—
297C8—84no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—81no—
326C22#129411yescharacteristic
327C22#134111yescharacteristic
328C22#134211yescharacteristic
329C22#134311yescharacteristic
330C22#134411yescharacteristic
331C22#138311yescharacteristic
332C22#138911yescharacteristic
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—162no—
338C4—162no—
339C4#135111yescharacteristic
340C4#135211yescharacteristic
341C4#139011yescharacteristic
342C4#139311yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—81no—
351C2#725011yescharacteristic
352C2#725511yescharacteristic
353C2#726311yescharacteristic
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
12. #1314C4semidirect
14. #1293C4semidirect

Maximal subgroups

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

subgroupindex
#114182
#110952
#80082