Finite Group Databasebeta

← #99173

Subgroups — #99173

Order 512.

516 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 112 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#13806C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7028C212yesmaximal
5#7024C212yesmaximal
6C2×#383C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8C2×#376C411yescharacteristic
9#1797C2212yes—
10C4×C22⋊C8C2212yes—
11C2×#324—21no—
12C2×#324—21no—
13#1436—21no—
14#1436—21no—
15#1434—21no—
16#1434C411yescharacteristic
17#1434C2211yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20C4×#95C2211yescharacteristic
21#1280—21no—
22#1280—21no—
23C22×C4⋊C4D411yescharacteristic
24C22×C4⋊C4D411yescharacteristic
25C22×C42—21no—
26C22×C42—21no—
27C22×C42—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×C4⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C4×C8—22no—
35#383—22no—
36#383—22no—
37C4×C4⋊C4—22no—
38C4×C4⋊C4—22no—
39#376—22no—
40#376C814yescomplemented
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95D411yescharacteristic
55C2×#95D411yescharacteristic
56#336—22no—
57#336—22no—
58#324—24no—
59#324—42no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C22×C8—42no—
72C22×C8—42no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—24no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—42no—
88C2×C42—42no—
89C2×C42—42no—
90C2×C42—42no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42—81no—
94C2×C42—81no—
95C2×C42—81no—
96C2×C42—81no—
97C2×C42C2×C812yes—
98C2×C42M1612yes—
99C2×C42#4112yes—
100C2×C42#4112yes—
101C4⋊C8—82no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C4×C8—82no—
107#95—24no—
108#95—24no—
109#95—24no—
110#95—22no—
111#95—22no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—41no—
120#95—41no—
121#95—81no—
122#95—81no—
123#95C2×C812yes—
124#95C2×C812yes—
125#95M1612yes—
126#95M1612yes—
127C24C2×C22⋊C411yescharacteristic
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4C22⋊C812yes—
184C22×C4C22⋊C812yes—
185C22×C4#10412yes—
186C22×C4#11711yescharacteristic
187C22×C4#11711yescharacteristic
188C22×C4#12312yes—
189C22×C4#12312yes—
190C22×C4C22×C811yescharacteristic
191C22×C4C2×M1611yescharacteristic
192C22×C4C2×M1611yescharacteristic
193C22×C4C2×M1611yescharacteristic
194C22×C4#13512yes—
195C22×C4AGL(1,ℤ/8)11yescharacteristic
196C22×C4#13711yescharacteristic
197C2×C8—44no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C42—22no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—42no—
211C42—42no—
212C42—48no—
213C42—81no—
214C42—81no—
215C42—84no—
216C42—84no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23#38111yescharacteristic
222C23C2×C22⋊C811yescharacteristic
223C23#40711yescharacteristic
224C23#41611yescharacteristic
225C23#41711yescharacteristic
226C23C2×#10411yescharacteristic
227C23#42111yescharacteristic
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4#43111yescharacteristic
297C2×C4#43111yescharacteristic
298C2×C4#43211yescharacteristic
299C2×C4#43211yescharacteristic
300C2×C4#48212yes—
301C2×C4#48512yes—
302C8—84no—
303C8—162no—
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#147611yescharacteristic
317C22#147711yescharacteristic
318C22#148811yescharacteristic
319C22#148911yescharacteristic
320C22#181711yescharacteristic
321C22#182511yescharacteristic
322C22#182611yescharacteristic
323C22#182711yescharacteristic
324C22#182711yescharacteristic
325C22#182811yescharacteristic
326C22#183011yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—22no—
332C4—22no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—82no—
339C4—82no—
340C4—82no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2#870811yescharacteristic
347C2#873011yescharacteristic
348C2#873511yescharacteristic
349C2#876311yescharacteristic
350C2#876911yescharacteristic
351C2#877411yescharacteristic
352C2#877611yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
40. #376C8semidirect

Maximal subgroups

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

subgroupindex
#138062
#85582
C4×#3242
#70282
#70242