Finite Group Databasebeta

← #99170

Subgroups — #99170

Order 512.

560 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 124 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#13804C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7028C212yesmaximal
5#7024C212yesmaximal
6C2×#382C411yescharacteristic
7C2×C4×C4⋊C4—21no—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797C2212yes—
11C4×C22⋊C8C2212yes—
12C2×#324—21no—
13C2×#324—21no—
14#1436—21no—
15#1436—21no—
16C4⋊#95—21no—
17#1434C411yescharacteristic
18#1434C2211yescharacteristic
19C4×#95C411yescharacteristic
20C4×#95C2211yescharacteristic
21#1280—21no—
22#1280—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C42—21no—
28C22×C42—21no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C4×C8—22no—
37#382—22no—
38#382C814yescomplemented
39C4×C4⋊C4—22no—
40C4×C4⋊C4—22no—
41C4×C4⋊C4—22no—
42C4×C4⋊C4—22no—
43C4×C4⋊C4—42no—
44C4×C4⋊C4—42no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57C2×#95D411yescharacteristic
58#336—22no—
59#336—22no—
60#324—24no—
61#324—42no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—42no—
74C22×C8—42no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4D812yes—
87C2×C4⋊C4SD1612yes—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—24no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—42no—
96C2×C42—42no—
97C2×C42—42no—
98C2×C42—42no—
99C2×C42—42no—
100C2×C42—41no—
101C2×C42—41no—
102C2×C42—81no—
103C2×C42—81no—
104C2×C42—81no—
105C2×C42—81no—
106C2×C42C2×C812yes—
107C2×C42M1612yes—
108C2×C42#4112yes—
109C2×C42#4112yes—
110C4⋊C8—82no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C4×C8—82no—
116#95—24no—
117#95—22no—
118#95—22no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95—41no—
127#95—41no—
128#95—81no—
129#95—81no—
130#95C2×C812yes—
131#95C2×C812yes—
132#95M1612yes—
133#95M1612yes—
134C24C2×C22⋊C411yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
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×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4C22⋊C812yes—
189C22×C4C22⋊C812yes—
190C22×C4#10212yes—
191C22×C4#10212yes—
192C22×C4#10412yes—
193C22×C4#11711yescharacteristic
194C22×C4#11711yescharacteristic
195C22×C4#12312yes—
196C22×C4#12312yes—
197C22×C4C22×C811yescharacteristic
198C22×C4C2×M1611yescharacteristic
199C22×C4C2×M1611yescharacteristic
200C22×C4C2×M1611yescharacteristic
201C22×C4C2×D811yescharacteristic
202C22×C4C2×SD1611yescharacteristic
203C22×C4#13712yes—
204C2×C8—44no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—48no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42—42no—
220C42—42no—
221C42—48no—
222C42—81no—
223C42—81no—
224C42—82no—
225C42—82no—
226C42—84no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#38111yescharacteristic
232C23C2×C22⋊C811yescharacteristic
233C23#40711yescharacteristic
234C23C2×#10211yescharacteristic
235C23#41911yescharacteristic
236C23C2×#10411yescharacteristic
237C23#42111yescharacteristic
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4D4⋊C814yes—
306C2×C4#43111yescharacteristic
307C2×C4#43111yescharacteristic
308C2×C4#43211yescharacteristic
309C2×C4#43211yescharacteristic
310C2×C4#48012yes—
311C2×C4#48312yes—
312C8—84no—
313C8—162no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22C2×D4⋊C811yescharacteristic
327C22#147911yescharacteristic
328C22#148611yescharacteristic
329C22#149111yescharacteristic
330C22#181711yescharacteristic
331C22#182311yescharacteristic
332C22#182611yescharacteristic
333C22#182711yescharacteristic
334C22#182711yescharacteristic
335C22#182811yescharacteristic
336C22#183011yescharacteristic
337C4—24no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—22no—
344C4—44no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—82no—
349C4—82no—
350C4—82no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—21no—
355C2—21no—
356C2#870611yescharacteristic
357C2#873111yescharacteristic
358C2#873511yescharacteristic
359C2#876011yescharacteristic
360C2#876611yescharacteristic
361C2#877511yescharacteristic
362C2#877911yescharacteristic
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
38. #382C8semidirect

Maximal subgroups

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

subgroupindex
#138042
#85582
C4×#3242
#70282
#70242