Finite Group Databasebeta

← #107496

Subgroups — #107496

Order 512.

849 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34365C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3#17742C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#8052C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7941C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12#2381C414yescomplemented
13#2369—22no—
14#2344—22no—
15#2344—22no—
16#2331—22no—
17#2331—22no—
18#2310—22no—
19#2283C2211yescharacteristic
20#2283C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#1988C414yescomplemented
23#1987—22no—
24#1736—22no—
25#1556—22no—
26#1549C414yescomplemented
27#1546—22no—
28#1303—22no—
29#1297—22no—
30#1286—22no—
31#1284—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8D412yes—
45C2×C4⋊Q8Q812yes—
46C2×#117—24no—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D412yes—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431C2×C412yes—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2×C412yes—
67#407—21no—
68#407—21no—
69C4×M16—21no—
70C4×M16—21no—
71#391—28no—
72#391—28no—
73#389—24no—
74#389—24no—
75#387—28no—
76#387—28no—
77#383C2×C412yes—
78#383C2×C412yes—
79#382—21no—
80#382D412yes—
81#381—24no—
82#381—21no—
83#381D412yes—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C412yes—
91C4×C4⋊C4D412yes—
92C4×C22⋊C4—21no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4D412yes—
96#376—21no—
97#376—21no—
98#376—21no—
99C43—21no—
100#340—28no—
101#332—28no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—84no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×M16—22no—
114C2×M16—44no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C22×C8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8C4214yes—
132C4⋊Q8C22⋊C414yes—
133C4⋊Q8C4⋊C414yes—
134C4⋊Q8C4⋊C414yes—
135#117—24no—
136#117—42no—
137#117—88no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4C22⋊C414yes—
152C2×C4⋊C4C22⋊C414yes—
153C2×C4⋊C4C4⋊C414yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—41no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—28no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×Q811yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C22⋊C8—22no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187#97—41no—
188#97—41no—
189C4×C8—22no—
190C4×C8—41no—
191#95—28no—
192#95—28no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—41no—
201#95C22⋊C414yes—
202#95C22⋊C414yes—
203C24#12011yescharacteristic
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—84no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—48no—
225C22×C4—42no—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244M16—82no—
245M16—82no—
246M16—82no—
247M16—82no—
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—88no—
265C4⋊C4#9518yes—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C42—44no—
270C42—44no—
271C42—42no—
272C42—42no—
273C42—82no—
274C42—82no—
275C42—88no—
276C42C2×C4211yescharacteristic
277C42C2×C22⋊C411yescharacteristic
278C42C2×C4⋊C411yescharacteristic
279C42C2×C4⋊C411yescharacteristic
280C42#11711yescharacteristic
281C42#11711yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42#12111yescharacteristic
287C42#12111yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42#12311yescharacteristic
291C42#12311yescharacteristic
292C23—22no—
293C23—22no—
294C23—41no—
295C23#37911yescharacteristic
296C23#38511yescharacteristic
297C23#38611yescharacteristic
298Q8—84no—
299Q8—84no—
300Q8—84no—
301Q8—84no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—28no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C2×#9511yescharacteristic
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4C4×C22⋊C411yescharacteristic
338C2×C4#38011yescharacteristic
339C2×C4#38011yescharacteristic
340C2×C4#38111yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38611yescharacteristic
346C2×C4#38611yescharacteristic
347C2×C4#45412yes—
348C2×C4#45612yes—
349C8—82no—
350C8—82no—
351C8—82no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—44no—
360C22C22⋊#9511yescharacteristic
361C22#179312yes—
362C22#190412yes—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—48no—
373C4—84no—
374C4—84no—
375C4—84no—
376C4#185712yes—
377C4#188611yescharacteristic
378C4#188811yescharacteristic
379C4#190512yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2#1075011yescharacteristic
384C2#1078511yescharacteristic
385C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
12. #2381C4semidirect
22. #1988C4semidirect
26. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#343652
#213132
#177422
#80932
#80522
#79622
#79412