Finite Group Databasebeta

← #98827

Subgroups — #98827

Order 512.

486 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1771C211yesmaximal, characteristic
2#8596C211yesmaximal, characteristic
3#8595C211yesmaximal, characteristic
4#8558C212yesmaximal
5#7020C212yesmaximal
6C22×C4⋊C8C411yescharacteristic
7C22×#97C411yescharacteristic
8C22×#97C411yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797C2212yes—
12#1797C2212yes—
13#1771—21no—
14#1771—21no—
15#1771—21no—
16#1771—21no—
17C2×#336C411yescharacteristic
18C2×#336C2211yescharacteristic
19C2×#335—21no—
20C2×#334—21no—
21C2×#324—22no—
22#1280—22no—
23#1277—21no—
24#1277—21no—
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4Q811yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2311yescharacteristic
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8C814yescomplemented
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×#97—21no—
43C2×#97—21no—
44C2×#97—22no—
45C2×#97—22no—
46C2×#97—21no—
47C2×#97—21no—
48C2×#97—42no—
49C2×C4×C8—42no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—21no—
58C2×#95D411yescharacteristic
59C2×#95Q811yescharacteristic
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—22no—
65#336—22no—
66#335—42no—
67#334—42no—
68#324—44no—
69C23×C4C4⋊C411yescharacteristic
70C23×C4C4⋊C411yescharacteristic
71C23×C4C4⋊C411yescharacteristic
72C23×C4C4⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×Q811yescharacteristic
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8C2×C812yes—
97C22×C8C2×C812yes—
98C22×C8M1612yes—
99C22×C8M1612yes—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—42no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42C2×C812yes—
118C2×C42M1612yes—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C4⋊C8—24no—
124C4⋊C8—24no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128C4⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#97—24no—
134#97—24no—
135#97—42no—
136#97—42no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—41no—
140C4×C8—41no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24C2×C4⋊C411yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
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×C4C4⋊C812yes—
172C22×C4C4⋊C812yes—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4#11711yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179C22×C4C22×C811yescharacteristic
180C22×C4C2×M1611yescharacteristic
181C22×C4C2×M1611yescharacteristic
182C22×C4C2×M1611yescharacteristic
183C22×C4AGL(1,ℤ/8)12yes—
184C22×C4#13712yes—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8C22⋊C812yes—
214C2×C8C22⋊C812yes—
215C2×C8#9912yes—
216C2×C8#10112yes—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C42—22no—
223C42—22no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—22no—
229C42—22no—
230C42—41no—
231C42—41no—
232C42—42no—
233C42—41no—
234C42—41no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23#38011yescharacteristic
240C23C2×C4⋊C811yescharacteristic
241C23#42311yescharacteristic
242C23#42812yes—
243C23#43012yes—
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—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4C2×C22⋊C811yescharacteristic
286C2×C4#40711yescharacteristic
287C2×C4C2×#9911yescharacteristic
288C2×C4#41011yescharacteristic
289C2×C4C2×#10111yescharacteristic
290C2×C4#41311yescharacteristic
291C2×C4#43111yescharacteristic
292C2×C4#43211yescharacteristic
293C2×C4#46812yes—
294C2×C4#46911yescharacteristic
295C2×C4#47011yescharacteristic
296C8—22no—
297C8—22no—
298C8—22no—
299C8—22no—
300C8—42no—
301C8—42no—
302C8—162no—
303C8—162no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22#156512yes—
313C22#156912yes—
314C22#179711yescharacteristic
315C22#179911yescharacteristic
316C22#180111yescharacteristic
317C22#180911yescharacteristic
318C22#181111yescharacteristic
319C22#181211yescharacteristic
320C22#181311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—22no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4#145911yescharacteristic
333C4#146111yescharacteristic
334C4#146311yescharacteristic
335C4#146411yescharacteristic
336C2—22no—
337C2—22no—
338C2#856011yescharacteristic
339C2#859711yescharacteristic
340C2#860111yescharacteristic
341C2#861011yescharacteristic
342C2#861311yescharacteristic
343C2#861911yescharacteristic
344C2#862511yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C4⋊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
39. C2×C4⋊C8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#17712
#85962
#85952
#85582
#70202