Finite Group Databasebeta

← #107589

Subgroups — #107589

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#17748C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#8125C211yesmaximal, characteristic
5#8113C211yesmaximal, characteristic
6#8032C211yesmaximal, characteristic
7#7989C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987—22no—
10#2985—22no—
11#2984C414yescomplemented
12#2983—22no—
13#2962—22no—
14#2961—22no—
15#2924—22no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21#2283C2211yescharacteristic
22#1558C412yescomplemented
23#1556C412yescomplemented
24#1554—22no—
25#1550—22no—
26#1520C414yes—
27#1515—22no—
28#1506—22no—
29#1308—22no—
30#1306—22no—
31#1295—22no—
32#1290—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533D412yes—
38#533D412yes—
39#533D412yes—
40#533Q812yes—
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—
49C22×C42C2311yescharacteristic
50#446—28no—
51#446—28no—
52#446—24no—
53#446—28no—
54C8×Q8—24no—
55#433—28no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—24no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432C2×C412yes—
69#432C2×C412yes—
70#432C2×C412yes—
71#432D412yes—
72#432D412yes—
73#431—21no—
74#431—24no—
75#431—21no—
76#431—21no—
77#431C2×C412yes—
78#424—24no—
79#423—21no—
80#423—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84#407—21no—
85#407D412yes—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8C2×C412yes—
89#405—24no—
90C4×M16—21no—
91C4×M16—21no—
92C2×#97—21no—
93C2×#97—21no—
94C2×#97D412yes—
95#383—21no—
96#382—21no—
97#381—21no—
98#380C2×C412yes—
99C4×C4⋊C4—21no—
100C4×C22⋊C4—21no—
101#376—21no—
102#333—24no—
103#332—24no—
104#328—28no—
105C22×Q8—21no—
106C22×Q8C4212yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—41no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8C22⋊C412yes—
120C4⋊Q8C22⋊C412yes—
121C4⋊Q8C4⋊C412yes—
122C4⋊Q8C4⋊C412yes—
123#125—22no—
124#125—22no—
125#125—22no—
126#125—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8C22⋊C414yes—
134C22⋊Q8C4⋊C414yes—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8C22⋊C414yes—
142C4×Q8C4⋊C414yes—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C4212yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C42—22no—
157C2×C42—41no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×Q811yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C4⋊C8—28no—
171C4⋊C8—28no—
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⋊C8—41no—
181C4⋊C8—41no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8C22⋊C414yes—
192#97—28no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—41no—
198#97—41no—
199#97—41no—
200#97C22⋊C414yes—
201C4×C8—28no—
202C4×C8—22no—
203C4×C8—41no—
204#95—22no—
205#95—41no—
206#95—41no—
207#95—41no—
208C24C4×D411yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8#9514yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4C2×C4211yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C4⋊C411yescharacteristic
225C22×C4#11711yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239M16—44no—
240M16—82no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—82no—
261C4⋊C4#9514yes—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—82no—
265C42—24no—
266C42—24no—
267C42—82no—
268C42C2×C22⋊C411yescharacteristic
269C42C2×C22⋊C411yescharacteristic
270C42C2×C22⋊C411yescharacteristic
271C42C2×C4⋊C411yescharacteristic
272C42C2×C4⋊C411yescharacteristic
273C42#11711yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42#12011yescharacteristic
277C42#12011yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42#12311yescharacteristic
283C42#12311yescharacteristic
284C23—22no—
285C23—41no—
286C23—41no—
287C23C4×C22⋊C411yescharacteristic
288C23#38511yescharacteristic
289C23#38611yescharacteristic
290Q8—44no—
291Q8—82no—
292Q8—82no—
293Q8—82no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4C2×#9511yescharacteristic
320C2×C4C4×C22⋊C411yescharacteristic
321C2×C4#37911yescharacteristic
322C2×C4#38011yescharacteristic
323C2×C4#38011yescharacteristic
324C2×C4#38111yescharacteristic
325C2×C4#38511yescharacteristic
326C2×C4#38511yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38611yescharacteristic
331C8—44no—
332C8—44no—
333C8—82no—
334C8—82no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—82no—
345C22C22⋊#9511yescharacteristic
346C22#190512yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—44no—
360C4—44no—
361C4—162no—
362C2—22no—
363C2—41no—
364C2—41no—
365C2#1077111yescharacteristic
366C2#1077811yescharacteristic
367C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
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
11. #2984C4semidirect
22. #1558C4semidirect
23. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#602512
#177482
#175962
#81252
#81132
#80322
#79892