Finite Group Databasebeta

← #111164

Subgroups — #111164

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#20467C211yesmaximal, characteristic
3#17748C211yesmaximal, characteristic
4#17596C211yesmaximal, characteristic
5#7970C211yesmaximal, characteristic
6#7954C211yesmaximal, characteristic
7#7940C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—21no—
16#1994—21no—
17#1986C414yescomplemented
18#1826—22no—
19#1768—22no—
20#1750—22no—
21#1745—21no—
22C4×#97—21no—
23#1558C412yescomplemented
24#1556C412yescomplemented
25#1554—22no—
26#1550—22no—
27#1520—22no—
28#1515—22no—
29#1506C414yescomplemented
30#1287—22no—
31#1285—22no—
32#1284—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yes—
59#432C2×C412yes—
60#432C2×C412yes—
61#432D412yes—
62#432D412yes—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#423—21no—
68#423—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D412yes—
71#407—21no—
72#407—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C4×M16—21no—
76C4×M16C2×C412yes—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—22no—
80C2×C4×C8—22no—
81#383—21no—
82#382—21no—
83#381—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4D412yes—
91#376—21no—
92#376C2×C412yes—
93C43D411yescharacteristic
94C43D411yescharacteristic
95#339—24no—
96#339—24no—
97#336—24no—
98#333—24no—
99#332—24no—
100#328—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—41no—
108C22×C8—22no—
109C22×C8—22no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8C4212yes—
117C4⋊Q8C4212yes—
118#125—22no—
119#125—22no—
120#125—41no—
121#125—41no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138#117—22no—
139#117—22no—
140#117—44no—
141#117—41no—
142#117—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8C22⋊C414yes—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8C22⋊C414yes—
190#97—22no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—44no—
197C4×C8—22no—
198C4×C8—44no—
199C4×C8—41no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206C24#12111yescharacteristic
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—82no—
212C2×Q8—82no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—81no—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4#11711yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12711yescharacteristic
239M16—44no—
240M16—82no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—82no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—82no—
267C42—24no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—81no—
274C42—81no—
275C42—82no—
276C42C2×C4211yescharacteristic
277C42C2×C22⋊C411yescharacteristic
278C42#11711yescharacteristic
279C42#11711yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12311yescharacteristic
290C42#12411yescharacteristic
291C42#12611yescharacteristic
292C23—22no—
293C23—41no—
294C23—41no—
295C23#38611yescharacteristic
296C23#38811yescharacteristic
297C23#39011yescharacteristic
298Q8—82no—
299Q8—82no—
300Q8—82no—
301Q8—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4C4×C22⋊C411yescharacteristic
334C2×C4C4×C22⋊C411yescharacteristic
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4#38611yescharacteristic
337C2×C4#38611yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#38811yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#39011yescharacteristic
344C2×C4#39011yescharacteristic
345C8—44no—
346C8—44no—
347C8—82no—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—82no—
358C22#143711yescharacteristic
359C22#190912yes—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—44no—
372C4—82no—
373C4—82no—
374C4—82no—
375C4—162no—
376C4#190612yes—
377C2—22no—
378C2—41no—
379C2—41no—
380C2#1094511yescharacteristic
381C2#1094711yescharacteristic
382C2#1095511yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C42 ▷ 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
17. #1986C4semidirect
23. #1558C4semidirect
24. #1556C4semidirect
29. #1506C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#204672
#177482
#175962
#79702
#79542
#79402