Finite Group Databasebeta

← #111001

Subgroups — #111001

Order 512.

627 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
1#21844C211yesmaximal, characteristic
2#20543C211yesmaximal, characteristic
3C2×#1552C211yesmaximal, characteristic
4#17385C211yesmaximal, characteristic
5#8082C211yesmaximal, characteristic
6#7949C211yesmaximal, characteristic
7#7922C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—22no—
16#1986—22no—
17#1826C414yes—
18C8×C4⋊C4—22no—
19C4×C4⋊C8—22no—
20C8×C22⋊C4—22no—
21#1552—22no—
22#1552—22no—
23C2×#332C412yescomplemented
24C2×#332C412yescomplemented
25#1472—22no—
26#1470C414yescomplemented
27#1461—22no—
28#1301—22no—
29#1285—22no—
30#1281—22no—
31C23×C8—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C2×#117C2×C412yes—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431D412yes—
66#423—21no—
67#423—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C412yes—
73C2×C4⋊C8C2×C412yes—
74C2×C4⋊C8D412yes—
75C2×C4⋊C8D412yes—
76#407—21no—
77#407C2×C412yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8D412yes—
80C4×M16—21no—
81C4×M16—21no—
82C2×#97—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#376—21no—
90C43D412yes—
91#339—28no—
92#336—24no—
93#332—24no—
94#332—24no—
95#324—28no—
96C22×Q8—21no—
97C22×Q8—21no—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—41no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—82no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117#125—22no—
118#125—22no—
119#125—22no—
120#125—22no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137#117—28no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C4212yes—
148C2×C4⋊C4C4212yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C22⋊C414yes—
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8C22⋊C414yes—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8C22⋊C414yes—
187#97—22no—
188#97—41no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8—84no—
196#95—22no—
197#95—22no—
198C24C4×D411yescharacteristic
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—44no—
204C2×Q8—81no—
205C2×Q8—81no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4C2×C4211yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4#11711yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4#12611yescharacteristic
231M16—44no—
232M16—81no—
233M16—81no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—42no—
237C2×C8—44no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—44no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—28no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C22⋊C4—28no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C42—28no—
262C42—44no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42C2×C22⋊C411yescharacteristic
268C42#11711yescharacteristic
269C42C4×D411yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12311yescharacteristic
279C42#12411yescharacteristic
280C42#12411yescharacteristic
281C42#12611yescharacteristic
282C42#12711yescharacteristic
283C23—22no—
284C23—22no—
285C23—41no—
286C23C4×C22⋊C411yescharacteristic
287C23#38611yescharacteristic
288C23#38811yescharacteristic
289Q8—82no—
290Q8—81no—
291Q8—81no—
292Q8—82no—
293Q8—82no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—28no—
308C2×C4—41no—
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—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—82no—
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—
334C8—44no—
335C8—44no—
336C8—82no—
337C8—82no—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22#143711yescharacteristic
348C22#176412yes—
349C22#191512yes—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—48no—
360C4—82no—
361C4—82no—
362C4#190411yescharacteristic
363C4#190511yescharacteristic
364C4#193712yes—
365C4#197412yes—
366C2—22no—
367C2—22no—
368C2—41no—
369C2#1092911yescharacteristic
370C2#1094011yescharacteristic
371C2#1095411yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. C2×#332C4semidirect
24. C2×#332C4semidirect
26. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#205432
C2×#15522
#173852
#80822
#79492
#79222