Finite Group Databasebeta

← #98816

Subgroups — #98816

Order 512.

670 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×C22⋊C4C211yesmaximal, characteristic
2#8595C212yesmaximal, complemented
3#8557C212yesmaximal
4#7020C212yesmaximal, complemented
5C24×C8C411yescharacteristic
6C22×C22⋊C8C411yescharacteristic
7C22×C4×C8C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797C2214yes—
11C8×C22⋊C4—22no—
12C8×C22⋊C4—22no—
13C2×#336C411yescharacteristic
14C2×#336C2211yescharacteristic
15C2×#334—22no—
16C2×#323—22no—
17#1280—22no—
18#1277—22no—
19C24×C4C2×C411yescharacteristic
20C23×C8—21no—
21C23×C8—21no—
22C23×C8—22no—
23C23×C8—42no—
24C23×C8—42no—
25C23×C8C814yescomplemented
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2×C411yescharacteristic
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4Q811yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2311yescharacteristic
34C2×C4⋊C8—24no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—24no—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—42no—
42C2×C4×C8—42no—
43C2×C4×C8D412yes—
44C2×C4×C8D412yes—
45C4×C22⋊C4—22no—
46C4×C22⋊C4—22no—
47C2×#95D411yescharacteristic
48C2×#95Q811yescharacteristic
49#336—22no—
50#336—22no—
51#336—24no—
52#334—44no—
53#323—44no—
54C25C4⋊C411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C4⋊C411yescharacteristic
64C23×C4C4⋊C411yescharacteristic
65C23×C4C4⋊C411yescharacteristic
66C23×C4C2×C812yes—
67C23×C4M1612yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×Q811yescharacteristic
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—48no—
86C22×C8—48no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—44no—
94C22×C8C22⋊C412yes—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C2×C812yes—
98C22×C8C2×C812yes—
99C22×C8M1612yes—
100C22×C8M1612yes—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4SD1614yes—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C2×D412yes—
114C2×C42#4112yes—
115C4⋊C8—44no—
116C4⋊C8—44no—
117C22⋊C8—28no—
118C22⋊C8—44no—
119C22⋊C8—44no—
120C4×C8—24no—
121C4×C8—42no—
122C4×C8—42no—
123C4×C8—42no—
124#95—22no—
125#95—22no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—41no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24C4⋊C812yes—
135C24#10612yes—
136C24#10812yes—
137C24C2×C4⋊C411yescharacteristic
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—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
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—42no—
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×C4C4⋊C812yes—
179C22×C4#10612yes—
180C22×C4#10812yes—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4#11711yescharacteristic
183C22×C4#12112yes—
184C22×C4C22⋊Q812yes—
185C22×C4C22×C811yescharacteristic
186C22×C4C2×M1611yescharacteristic
187C22×C4C2×M1611yescharacteristic
188C22×C4C2×M1611yescharacteristic
189C22×C4C2×SD1612yes—
190C22×C4#13512yes—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—48no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—48no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—84no—
213C2×C8—84no—
214C2×C8—84no—
215C2×C8C22⋊C812yes—
216C2×C8C22⋊C812yes—
217C2×C8#9912yes—
218C2×C8#10012yes—
219C22⋊C4—28no—
220C22⋊C4—28no—
221C42—22no—
222C42—22no—
223C42—42no—
224C42—42no—
225C42—41no—
226C42—41no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#33414yescomplemented
254C23#38011yescharacteristic
255C23C2×C4⋊C811yescharacteristic
256C23#42311yescharacteristic
257C23C2×#10611yescharacteristic
258C23#42711yescharacteristic
259C23C2×#10811yescharacteristic
260C23C2×#10811yescharacteristic
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4C2×C22⋊C811yescharacteristic
302C2×C4#40711yescharacteristic
303C2×C4C2×#9911yescharacteristic
304C2×C4#41011yescharacteristic
305C2×C4C2×#10011yescharacteristic
306C2×C4#41311yescharacteristic
307C2×C4#43111yescharacteristic
308C2×C4#43211yescharacteristic
309C2×C4#46514yes—
310C8—22no—
311C8—42no—
312C8—42no—
313C8—42no—
314C8—164no—
315C8#32314yescomplemented
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—24no—
329C22—44no—
330C22—44no—
331C22—44no—
332C22—44no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22C2×#33411yescharacteristic
340C22#156411yescharacteristic
341C22#156611yescharacteristic
342C22#156811yescharacteristic
343C22#179711yescharacteristic
344C22#179911yescharacteristic
345C22#180011yescharacteristic
346C22#180711yescharacteristic
347C22#181011yescharacteristic
348C22#181211yescharacteristic
349C22#181311yescharacteristic
350C4—21no—
351C4—21no—
352C4—24no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4C2×#32311yescharacteristic
358C4#145811yescharacteristic
359C4#146511yescharacteristic
360C4#146611yescharacteristic
361C2—21no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—44no—
366C2#855711yescharacteristic
367C2#859511yescharacteristic
368C2#860111yescharacteristic
369C2#860811yescharacteristic
370C2#861111yescharacteristic
371C2#861411yescharacteristic
372C2#862011yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C4⋊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
2. #8595C2semidirect
4. #7020C2semidirect
25. C23×C8C8semidirect
253. C23#334semidirect
315. C8#323semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C22⋊C42
#85952
#85572
#70202