Finite Group Databasebeta

← #110581

Subgroups — #110581

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21302C211yesmaximal, characteristic
2#20543C211yesmaximal, characteristic
3#17718C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#8068C211yesmaximal, characteristic
6#8060C211yesmaximal, characteristic
7#7935C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1985C414yescomplemented
16#1984—22no—
17C8×C4⋊C4—22no—
18C4×C4⋊C8—22no—
19C8×C22⋊C4—22no—
20#1729—22no—
21#1554—22no—
22#1548C412yescomplemented
23#1546C412yescomplemented
24#1542—22no—
25#1508C414yes—
26#1506—22no—
27#1503—22no—
28#1300—22no—
29#1299—22no—
30#1283—22no—
31C22×M16—21no—
32C23×C8D412yes—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×#117—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431C2×C412yes—
62#431D412yes—
63#423—21no—
64#423C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2×C412yes—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8D412yes—
72#407—21no—
73#407C2×C412yes—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C2×#97C2×C412yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—24no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4D412yes—
87C4×C22⋊C4—21no—
88C4×C22⋊C4C2×C412yes—
89#376D412yes—
90C43—21no—
91#336—24no—
92#336—28no—
93#333—24no—
94#332—24no—
95#328—28no—
96C22×Q8—21no—
97C22×Q8C4212yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—24no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—42no—
105C2×M16—41no—
106C2×M16—41no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—44no—
112C22×C8C22⋊C414yes—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121#125—41no—
122#125—41no—
123#125—41no—
124#125—41no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141#117—28no—
142#117—41no—
143#117—41no—
144#117—41no—
145#117—41no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C4212yes—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8C22⋊C414yes—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8—41no—
197C4×C8—84no—
198#95—22no—
199#95C22⋊C414yes—
200C24C4×D411yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—44no—
206C2×Q8—82no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—81no—
214C22×C4C2×C4211yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12611yescharacteristic
232M16—44no—
233M16—44no—
234M16—44no—
235M16—82no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—44no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C4⋊C4—28no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C22⋊C4—28no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—28no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—82no—
270C42C2×C22⋊C411yescharacteristic
271C42#11711yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12311yescharacteristic
282C42#12411yescharacteristic
283C42#12411yescharacteristic
284C42#12611yescharacteristic
285C42#12711yescharacteristic
286C23—22no—
287C23—22no—
288C23—41no—
289C23C4×C22⋊C411yescharacteristic
290C23#38611yescharacteristic
291C23#38811yescharacteristic
292Q8—82no—
293Q8—82no—
294Q8—82no—
295Q8—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—28no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4C4×C22⋊C411yescharacteristic
322C2×C4C4×C22⋊C411yescharacteristic
323C2×C4#38611yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#39011yescharacteristic
331C2×C4#39011yescharacteristic
332C2×C4#39011yescharacteristic
333C2×C4#44314yes—
334C2×C4#45512yes—
335C2×C4#45612yes—
336C8—44no—
337C8—44no—
338C8—44no—
339C8—82no—
340C8—82no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—44no—
349C22#143711yescharacteristic
350C22#176412yes—
351C22#188212yes—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—48no—
361C4—82no—
362C4—82no—
363C4#188212yes—
364C4#188511yescharacteristic
365C4#188611yescharacteristic
366C4#193712yes—
367C4#197412yes—
368C2—22no—
369C2—22no—
370C2—41no—
371C2#1092911yescharacteristic
372C2#1094111yescharacteristic
373C2#1095711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1985C4semidirect
22. #1548C4semidirect
23. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#213022
#205432
#177182
#175452
#80682
#80602
#79352