Finite Group Databasebeta

← #107332

Subgroups — #107332

Order 512.

712 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2C2×#1506C211yesmaximal, characteristic
3#17496C211yesmaximal, characteristic
4#8125C211yesmaximal, characteristic
5#8072C211yesmaximal, characteristic
6#7989C211yesmaximal, characteristic
7#7910C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987C414yescomplemented
10#2985—22no—
11#2984—22no—
12#2983—22no—
13#2962—22no—
14#2961—22no—
15#2924—22no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#1506—22no—
23#1506—22no—
24C2×#328C414yes—
25#1497C414yescomplemented
26#1491—22no—
27#1481—22no—
28#1308—22no—
29#1300—22no—
30#1290—22no—
31#1280—22no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533C2×C412yes—
37#533C2×C412yes—
38#533D412yes—
39#533D412yes—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8C2×C412yes—
45C2×C4×Q8—21no—
46C2×C4×Q8D412yes—
47C2×#117—21no—
48C22×C42—21no—
49C22×C42C2311yescharacteristic
50#446—28no—
51#446—24no—
52#446—28no—
53#446—28no—
54C8×Q8—24no—
55#433—28no—
56#432—24no—
57#432—21no—
58#432—21no—
59#432—21no—
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—24no—
74#431—21no—
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#407—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8C2×C412yes—
88C2×C22⋊C8D412yes—
89#405—24no—
90C4×M16—21no—
91C4×M16D412yes—
92C2×#97—21no—
93C2×#97—21no—
94C2×#97—21no—
95#376—21no—
96#376—21no—
97#376—21no—
98#376—21no—
99C2×#95—21no—
100C2×#95C2×C412yes—
101#328—28no—
102Q8⋊C8—28no—
103C22×Q8—21no—
104C22×Q8C4⋊C412yes—
105C23×C4—22no—
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—22no—
110C2×M16—41no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C4⋊C412yes—
119C4⋊Q8C4⋊C412yes—
120C4⋊Q8C4⋊C412yes—
121C4⋊Q8C4⋊C412yes—
122#125—22no—
123#125—22no—
124#125—22no—
125#125—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8C4214yes—
133C22⋊Q8C22⋊C414yes—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8C22⋊C414yes—
141C4×Q8C22⋊C414yes—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C4⋊C412yes—
152C2×C22⋊C4—22no—
153C2×C42—22no—
154C2×C42—44no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×Q811yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C4⋊C8—28no—
170C4⋊C8—28no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8C22⋊C414yes—
191#97—28no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—41no—
196#97—41no—
197#97—41no—
198#97—41no—
199#97C22⋊C414yes—
200C4×C8—28no—
201C4×C8—22no—
202C4×C8—22no—
203#95—22no—
204#95—22no—
205#95—41no—
206#95—41no—
207C24C4×D411yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8#9514yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—81no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4C2×C4211yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242M16—44no—
243M16—82no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—44no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—82no—
251C2×C8—82no—
252C4⋊C4—24no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4#9514yes—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C42—24no—
267C42—24no—
268C42—44no—
269C42—44no—
270C42C2×C22⋊C411yescharacteristic
271C42C2×C22⋊C411yescharacteristic
272C42C2×C22⋊C411yescharacteristic
273C42C2×C4⋊C411yescharacteristic
274C42C2×C4⋊C411yescharacteristic
275C42#11711yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42#12011yescharacteristic
279C42#12011yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42#12311yescharacteristic
285C42#12311yescharacteristic
286C23—22no—
287C23—22no—
288C23—41no—
289C23C4×C22⋊C411yescharacteristic
290C23#38511yescharacteristic
291C23#38611yescharacteristic
292Q8—44no—
293Q8—44no—
294Q8—82no—
295Q8—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
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—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4C2×#9511yescharacteristic
324C2×C4C4×C22⋊C411yescharacteristic
325C2×C4#37911yescharacteristic
326C2×C4#38011yescharacteristic
327C2×C4#38011yescharacteristic
328C2×C4#38111yescharacteristic
329C2×C4#38511yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38611yescharacteristic
335C8—44no—
336C8—44no—
337C8—82no—
338C8—82no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—42no—
348C22—42no—
349C22C22⋊#9511yescharacteristic
350C22#186112yes—
351C22#190512yes—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—44no—
365C4—44no—
366C4—84no—
367C2—22no—
368C2—22no—
369C2—41no—
370C2#1075711yescharacteristic
371C2#1078011yescharacteristic
372C2#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
9. #2987C4semidirect
25. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#602512
C2×#15062
#174962
#81252
#80722
#79892
#79102