Finite Group Databasebeta

← #127022

Subgroups — #127022

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1467C211yesmaximal, characteristic
2C2×#1461C211yesmaximal, characteristic
3#13926C211yesmaximal, characteristic
4#7189C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C2×#432—21no—
7C2×#431—21no—
8C2×#391C412yes—
9C2×#384C411yescharacteristic
10C2×C4×C4⋊C4C411yescharacteristic
11#1467—22no—
12#1467C414yescomplemented
13#1461—22no—
14#1461C414yescomplemented
15C2×#324—21no—
16C2×#324—21no—
17C2×#324C2212yes—
18C2×#324C2212yes—
19#1441—21no—
20#1441—21no—
21C4⋊#95—21no—
22C4⋊#95—21no—
23C4⋊#95C2212yes—
24C23×Q8C2×C411yescharacteristic
25C2×C4×Q8—42no—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C2×C4×Q8C2×C412yes—
29C2×C4×Q8C2×C412yes—
30C2×C4×Q8D412yes—
31C2×C4×Q8Q812yes—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4D411yescharacteristic
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4C2311yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C412yes—
41C22×C42D411yescharacteristic
42C22×C42D411yescharacteristic
43#432—24no—
44#432—42no—
45#431—24no—
46#431—42no—
47C2×C4⋊C8—41no—
48C2×C4⋊C8—41no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×#97—41no—
52C2×C4×C8—41no—
53#391—44no—
54#384—41no—
55#384—41no—
56C4×C4⋊C4—42no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62#324—24no—
63#324—24no—
64#324—22no—
65#324—22no—
66#324C2×C414yes—
67#324C2×C414yes—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—22no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8C4212yes—
75C22×Q8C4⋊C412yes—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—42no—
86C22×C8—42no—
87C4×Q8—42no—
88C4×Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91C4×Q8—42no—
92C4×Q8—42no—
93C4×Q8—42no—
94C4×Q8—42no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C22⋊C412yes—
115C2×C4⋊C4C4⋊C412yes—
116C2×C4⋊C4C4⋊C412yes—
117C2×C4⋊C4C22×C411yescharacteristic
118C2×C4⋊C4C22×C411yescharacteristic
119C2×C4⋊C4C2×D411yescharacteristic
120C2×C4⋊C4C2×Q811yescharacteristic
121C2×C42—22no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—24no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42C4212yes—
134C2×C42C22⋊C412yes—
135C2×C42C22⋊C412yes—
136C2×C42C4⋊C412yes—
137C2×C42C4⋊C412yes—
138C2×C42C4⋊C412yes—
139C4⋊C8—44no—
140C4⋊C8—44no—
141C22⋊C8—24no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145#97—44no—
146C4×C8—44no—
147#95—42no—
148#95—44no—
149#95—44no—
150#95—82no—
151#95—82no—
152C24C2×C22⋊C411yescharacteristic
153C2×Q8—24no—
154C2×Q8—24no—
155C2×Q8—24no—
156C2×Q8—24no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—44no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4#9512yes—
206C22×C4#9512yes—
207C22×C4#9512yes—
208C22×C4#9512yes—
209C22×C4#9912yes—
210C22×C4#9911yescharacteristic
211C22×C4#9911yescharacteristic
212C22×C4#10012yes—
213C22×C4#10112yes—
214C22×C4C2×C4211yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C4⋊C411yescharacteristic
218C22×C4C2×C4⋊C411yescharacteristic
219C22×C4C2×C4⋊C411yescharacteristic
220C22×C4#13112yes—
221C22×C4#13112yes—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—48no—
237C4⋊C4—84no—
238C4⋊C4—82no—
239C4⋊C4—84no—
240C4⋊C4—82no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—84no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23C2×#9511yescharacteristic
259C23#40811yescharacteristic
260C23#40811yescharacteristic
261C23C2×#9911yescharacteristic
262C23C2×#9911yescharacteristic
263C23C2×#10011yescharacteristic
264C23C2×#10111yescharacteristic
265Q8—48no—
266Q8—48no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#34212yes—
326C2×C4#34212yes—
327C2×C4#34314yes—
328C2×C4#35512yes—
329C2×C4#35612yes—
330C2×C4#40512yes—
331C2×C4#42412yes—
332C8—84no—
333C8—84no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22#147111yescharacteristic
347C22#147111yescharacteristic
348C22#147211yescharacteristic
349C22#147211yescharacteristic
350C22#172911yescharacteristic
351C22C2×#34211yescharacteristic
352C22C2×#34311yescharacteristic
353C22#212811yescharacteristic
354C22#212911yescharacteristic
355C22C2×#35511yescharacteristic
356C22C2×#35611yescharacteristic
357C4—22no—
358C4—24no—
359C4—22no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—42no—
364C4—41no—
365C4—41no—
366C4—42no—
367C4—84no—
368C4—162no—
369C4#139414yes—
370C2—21no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2#791511yescharacteristic
375C2#1156711yescharacteristic
376C2#1156811yescharacteristic
377C2#1157311yescharacteristic
378C2#1157411yescharacteristic
379C2#1258311yescharacteristic
380C2C2×#139411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1467C4semidirect
14. #1461C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14672
C2×#14612
#139262
#71892