Finite Group Databasebeta

← #93047

Subgroups — #93047

Order 512.

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

normal subgroups only

This group has 458 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#7982C211yesmaximal, characteristic
4C22×#106C411yescharacteristic, complemented
5C22×C22⋊C8C2211yescharacteristic
6#2280—21no—
7C2×#380—21no—
8#2112—21no—
9C2×#340—21no—
10C2×#340C411yescharacteristic, complemented
11#1314—21no—
12#1314C412yescomplemented
13#1289—21no—
14#1289C412yescomplemented
15C24×C4D411yescharacteristic
16C23×C8C2×C411yescharacteristic
17C23×C8D411yescharacteristic
18C2×#117—41no—
19C22×C4⋊C4—21no—
20C22×C4⋊C4—21no—
21C22×C22⋊C4—41no—
22C2×M32—41no—
23C2×#106—41no—
24C2×#106—42no—
25C2×C22⋊C8—41no—
26C2×C22⋊C8—41no—
27C2×C22⋊C8C2×C411yescharacteristic
28C2×C22⋊C8C2×C411yescharacteristic
29C2×C22⋊C8D411yescharacteristic
30C2×C22⋊C8Q811yescharacteristic
31#381—22no—
32#380—22no—
33#380—42no—
34#380—42no—
35#380—41no—
36#379—41no—
37C4×C22⋊C4—22no—
38C4×C22⋊C4—41no—
39C2×#95—41no—
40C22⋊C16—22no—
41C22⋊C16—41no—
42#340—42no—
43#340—41no—
44C25D811yescharacteristic
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×C4SD1611yescharacteristic
59C23×C4SD1611yescharacteristic
60C23×C4Q1611yescharacteristic
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C4211yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C4⋊C411yescharacteristic
72C22×C8C4⋊C411yescharacteristic
73#117—81no—
74#117—81no—
75#117—81no—
76#117—81no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
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—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—82no—
95C2×C22⋊C4—82no—
96C2×C42—42no—
97M32—81no—
98M32—81no—
99C2×C16—42no—
100#106—42no—
101#106—42no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—41no—
106#95—24no—
107#95—42no—
108#95—44no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24—41no—
113C24—41no—
114C24—41no—
115C24—41no—
116C24—41no—
117C24#10211yescharacteristic
118C24#10211yescharacteristic
119C24#10711yescharacteristic
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—42no—
150C22×C4—42no—
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—81no—
169C22×C4#9511yescharacteristic
170C22×C4#9911yescharacteristic
171C22×C4#9911yescharacteristic
172C22×C4#10311yescharacteristic
173C22×C4#10311yescharacteristic
174C22×C4#10411yescharacteristic
175C22×C4#10411yescharacteristic
176C22×C4#10611yescharacteristic
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—81no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—81no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—82no—
204C22⋊C4—88no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C42—82no—
209C42—82no—
210C16—82no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—42no—
223C23—42no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23#32711yescharacteristic
243C23#32711yescharacteristic
244C23#34011yescharacteristic
245C23#36012yes—
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—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—82no—
284C2×C4—161no—
285C2×C4—161no—
286C2×C4—161no—
287C2×C4—161no—
288C2×C4—161no—
289C2×C4—161no—
290C2×C4—161no—
291C2×C4—161no—
292C2×C4#32811yescharacteristic
293C2×C4#32811yescharacteristic
294C2×C4#33911yescharacteristic
295C2×C4#34211yescharacteristic
296C2×C4AΣL(1,𝔽16)11yescharacteristic
297C2×C4#35211yescharacteristic
298C2×C4#35311yescharacteristic
299C2×C4#35411yescharacteristic
300C8—41no—
301C8—41no—
302C8—41no—
303C8—41no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—81no—
332C22#129411yescharacteristic
333C22#133911yescharacteristic
334C22#134011yescharacteristic
335C22#134311yescharacteristic
336C22#134411yescharacteristic
337C22#138411yescharacteristic
338C22#138611yescharacteristic
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—162no—
344C4—162no—
345C4—164no—
346C4#135111yescharacteristic
347C4#135211yescharacteristic
348C4#139011yescharacteristic
349C4#139311yescharacteristic
350C2—21no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—81no—
358C2#725011yescharacteristic
359C2#725411yescharacteristic
360C2#726211yescharacteristic
subgroupG/H
center, socleC22#1294
commutatorC22×C8C42
Frattini, cosocleC22×C22⋊C8C22
prime radicalC25D8
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×#106C4semidirect
10. C2×#340C4semidirect
12. #1314C4semidirect
14. #1289C4semidirect

Maximal subgroups

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

subgroupindex
#114182
#110952
#79822