Finite Group Databasebeta

← #134272

Subgroups — #134272

Order 512.

646 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×C22⋊C4C211yesmaximal, characteristic
2#12783C212yesmaximal, complemented
3C2×#1314C212yesmaximal
4#7183C212yesmaximal, complemented
5C24×C8C411yescharacteristic
6C22×C22⋊C8C2211yescharacteristic
7C22×C4×C8C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10C2×C4⋊C16—22no—
11C2×C22⋊C16C2214yes—
12C8×C22⋊C4—22no—
13C8×C22⋊C4—22no—
14C2×#336C411yescharacteristic
15C2×#336C2211yescharacteristic
16#1374—22no—
17#1314—24no—
18C24×C4C2×C411yescharacteristic
19C23×C8—22no—
20C23×C8—21no—
21C23×C8—21no—
22C23×C8—22no—
23C23×C8—42no—
24C23×C8—42no—
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2311yescharacteristic
29C22×C22⋊C4C811yescharacteristic
30C22×C22⋊C4C811yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C16—24no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8D412yes—
37C2×C22⋊C8Q812yes—
38C2×C4×C8—21no—
39C2×C4×C8—21no—
40C2×C4×C8—42no—
41C2×C4×C8—42no—
42C2×C4×C8D412yes—
43C2×C4×C8D412yes—
44C4×C22⋊C4—22no—
45C4×C22⋊C4—22no—
46C2×#95C811yescharacteristic
47C2×#95C811yescharacteristic
48C4⋊C16—44no—
49C22⋊C16—24no—
50C22⋊C16—24no—
51#336—22no—
52#336—22no—
53C25C2×C811yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—41no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4C4⋊C412yes—
63C23×C4C2×C811yescharacteristic
64C23×C4C2×C811yescharacteristic
65C23×C4C2×C811yescharacteristic
66C23×C4C2×C811yescharacteristic
67C23×C4C2×C811yescharacteristic
68C23×C4M1612yes—
69C23×C4C22×C411yescharacteristic
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—24no—
75C22×C8—24no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—48no—
83C22×C8—48no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C4⋊C412yes—
95C22×C8M1612yes—
96C22×C8C2×D412yes—
97C22×C8C2×D411yescharacteristic
98C22×C8C2×Q811yescharacteristic
99C22×C8#4112yes—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4C1614yescomplemented
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42C22⋊C412yes—
113C2×C42M1612yes—
114C2×C16—44no—
115C2×C16—44no—
116C2×C16—44no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119C4×C8—24no—
120C4×C8—42no—
121C4×C8—42no—
122C4×C8—42no—
123#95—22no—
124#95—22no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—41no—
130C24—41no—
131C24—41no—
132C24—41no—
133C24C4⋊C812yes—
134C24C2×C1612yes—
135C24M3212yes—
136C24C22×C811yescharacteristic
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
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—22no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
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×C4C22⋊C812yes—
178C22×C4C22⋊C812yes—
179C22×C4C4⋊C812yes—
180C22×C4C2×C1612yes—
181C22×C4C2×C1612yes—
182C22×C4M3212yes—
183C22×C4M3212yes—
184C22×C4C2×C22⋊C411yescharacteristic
185C22×C4C2×C4⋊C411yescharacteristic
186C22×C4#11711yescharacteristic
187C22×C4C2×M1611yescharacteristic
188C22×C4C2×M1611yescharacteristic
189C22×C4C2×M1611yescharacteristic
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—21no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—48no—
204C2×C8—48no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8#9912yes—
214C2×C8#10012yes—
215C2×C8#12112yes—
216C2×C8C22⋊Q812yes—
217C22⋊C4—28no—
218C22⋊C4—28no—
219C42—22no—
220C42—22no—
221C42—42no—
222C42—42no—
223C42—41no—
224C42—41no—
225C16—84no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C4⋊C1614yescomplemented
253C23C2×C22⋊C811yescharacteristic
254C23C2×C4⋊C811yescharacteristic
255C23#43111yescharacteristic
256C23C22×C1611yescharacteristic
257C23C2×M3211yescharacteristic
258C23C2×M3211yescharacteristic
259C23C2×M3211yescharacteristic
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4C22⋊C1614yes—
301C2×C4#38011yescharacteristic
302C2×C4#40711yescharacteristic
303C2×C4C2×#9911yescharacteristic
304C2×C4#41011yescharacteristic
305C2×C4C2×#10011yescharacteristic
306C2×C4#41311yescharacteristic
307C2×C4#42311yescharacteristic
308C2×C4#43211yescharacteristic
309C8—22no—
310C8—22no—
311C8—22no—
312C8—22no—
313C8—42no—
314C8—42no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—24no—
328C22—44no—
329C22—44no—
330C22—44no—
331C22—44no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22#145911yescharacteristic
339C22#146011yescharacteristic
340C22#179711yescharacteristic
341C22C2×C22⋊C1611yescharacteristic
342C22#211211yescharacteristic
343C22#211511yescharacteristic
344C22#211511yescharacteristic
345C22C2×C4⋊C1611yescharacteristic
346C22#215011yescharacteristic
347C22#216211yescharacteristic
348C22#216311yescharacteristic
349C4—21no—
350C4—21no—
351C4—24no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4#146211yescharacteristic
357C4#146411yescharacteristic
358C4#179911yescharacteristic
359C4#180011yescharacteristic
360C2—21no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2—44no—
365C2#855911yescharacteristic
366C2#1146511yescharacteristic
367C2#1146611yescharacteristic
368C2#1148711yescharacteristic
369C2#1148811yescharacteristic
370C2#1278311yescharacteristic
371C2#1278611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C16
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC25C2×C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
4. #7183C2semidirect
105. C2×C22⋊C4C16semidirect
252. C23C4⋊C16semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C22⋊C42
#127832
C2×#13142
#71832