Finite Group Databasebeta

← #111166

Subgroups — #111166

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#20549C211yesmaximal, characteristic
3#17749C211yesmaximal, characteristic
4#17596C211yesmaximal, characteristic
5#7970C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7941C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—21no—
16#1994—21no—
17#1986C414yescomplemented
18#1826—22no—
19#1771—22no—
20#1767—21no—
21C4×C4⋊C8—21no—
22#1753—22no—
23#1559—22no—
24#1550—22no—
25#1548C412yescomplemented
26#1546C412yescomplemented
27#1520—22no—
28#1515C414yescomplemented
29#1506—22no—
30#1287—22no—
31#1286—22no—
32#1284—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yes—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#423D412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D412yes—
73#407—21no—
74#407—21no—
75C2×C22⋊C8D412yes—
76C4×M16—21no—
77C4×M16C2×C412yes—
78C2×#97—21no—
79C2×#97—22no—
80C2×C4×C8—22no—
81#383C2×C412yes—
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#376—21no—
93C43D411yescharacteristic
94C43D411yescharacteristic
95#339—24no—
96#339—24no—
97#336—22no—
98#336—22no—
99#333—24no—
100#332—24no—
101#328—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—41no—
111C2×M16—82no—
112C22×C8—22no—
113C22×C8—22no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8C4212yes—
121C4⋊Q8C4212yes—
122#125—22no—
123#125—22no—
124#125—41no—
125#125—41no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142#117—22no—
143#117—22no—
144#117—44no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42C22⋊C412yes—
165C2×C42C22⋊C412yes—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8C22⋊C414yes—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8C22⋊C414yes—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—41no—
198#97—82no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—82no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—41no—
208C24#12111yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×Q8—82no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12611yescharacteristic
242C22×C4#12711yescharacteristic
243M16—44no—
244M16—82no—
245M16—82no—
246M16—82no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C22⋊C4—44no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—82no—
274C42—24no—
275C42—42no—
276C42—42no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—82no—
285C42C2×C4211yescharacteristic
286C42C2×C22⋊C411yescharacteristic
287C42#11711yescharacteristic
288C42#11711yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42#12311yescharacteristic
299C42#12411yescharacteristic
300C42#12611yescharacteristic
301C23—22no—
302C23—41no—
303C23—41no—
304C23#38611yescharacteristic
305C23#38811yescharacteristic
306C23#39011yescharacteristic
307Q8—82no—
308Q8—82no—
309Q8—81no—
310Q8—81no—
311Q8—82no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4C4×C22⋊C411yescharacteristic
347C2×C4C4×C22⋊C411yescharacteristic
348C2×C4C4×C22⋊C411yescharacteristic
349C2×C4#38611yescharacteristic
350C2×C4#38611yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#38811yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#39011yescharacteristic
357C2×C4#39011yescharacteristic
358C8—44no—
359C8—44no—
360C8—82no—
361C8—82no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—82no—
371C22#143711yescharacteristic
372C22#190912yes—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—44no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4—162no—
388C4#190411yescharacteristic
389C4#190511yescharacteristic
390C4#193212yes—
391C2—22no—
392C2—41no—
393C2—41no—
394C2#1094611yescharacteristic
395C2#1094711yescharacteristic
396C2#1095411yescharacteristic
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
25. #1548C4semidirect
26. #1546C4semidirect
28. #1515C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#205492
#177492
#175962
#79702
#79622
#79412