Finite Group Databasebeta

← #107582

Subgroups — #107582

Order 512.

796 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34365C211yesmaximal, characteristic
2#17742C211yesmaximal, characteristic
3#17742C211yesmaximal, characteristic
4#8116C211yesmaximal, characteristic
5#8106C211yesmaximal, characteristic
6#8070C211yesmaximal, characteristic
7#7976C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2381—22no—
14#2369—22no—
15#2344—22no—
16#2344C414yes—
17#2331—22no—
18#2331—22no—
19#2310—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#1556—22no—
23#1556C414yescomplemented
24#1549—22no—
25#1549C414yescomplemented
26#1546—22no—
27#1546—22no—
28#1306—22no—
29#1305—22no—
30#1300—22no—
31#1288—22no—
32C23×Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8C2×C412yes—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8D412yes—
42C2×C4⋊Q8D412yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8Q812yes—
45C2×#117—24no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2×C412yes—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432C2×C412yes—
56#432C2×C412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#407—21no—
70#407—21no—
71#407—21no—
72#407—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77#391—28no—
78#391—28no—
79#389—24no—
80#389—24no—
81#387—28no—
82#387—28no—
83#383—21no—
84#383D412yes—
85#382—21no—
86#382—21no—
87#381—21no—
88#381—24no—
89#381D412yes—
90#380C2×C412yes—
91#380D412yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4C2×C412yes—
98#376—21no—
99#376D412yes—
100#332—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×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×M16—22no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C4⋊Q8—22no—
118C4⋊Q8—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⋊Q8C22⋊C414yes—
130C4⋊Q8C22⋊C414yes—
131C4⋊Q8C4⋊C414yes—
132C4⋊Q8C4⋊C414yes—
133#117—88no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
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—28no—
146C2×C4⋊C4C4214yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C4⋊C414yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4C22⋊C414yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—28no—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182#97—22no—
183#97—22no—
184#97—41no—
185#97—41no—
186C4×C8—22no—
187C4×C8—41no—
188C4×C8—41no—
189C4×C8—41no—
190#95—28no—
191#95—28no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95C22⋊C414yes—
200C24C2×C22⋊C411yescharacteristic
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—82no—
206C2×Q8—84no—
207C2×Q8—82no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—84no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—48no—
221C22×C4C2×C4211yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C4⋊C411yescharacteristic
225C22×C4#11711yescharacteristic
226C22×C4#11711yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239M16—82no—
240M16—82no—
241M16—82no—
242M16—82no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—88no—
257C4⋊C4#9518yes—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C42—44no—
261C42—44no—
262C42—88no—
263C42C2×C22⋊C411yescharacteristic
264C42C2×C22⋊C411yescharacteristic
265C42C2×C4⋊C411yescharacteristic
266C42C2×C4⋊C411yescharacteristic
267C42C4×D411yescharacteristic
268C42C4×D411yescharacteristic
269C42C4×D411yescharacteristic
270C42C4×D411yescharacteristic
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42#12311yescharacteristic
276C42#12311yescharacteristic
277C42#12311yescharacteristic
278C42#12311yescharacteristic
279C23—22no—
280C23—41no—
281C23—41no—
282C23C4×C22⋊C411yescharacteristic
283C23#37911yescharacteristic
284C23#38011yescharacteristic
285Q8—84no—
286Q8—84no—
287Q8—84no—
288Q8—84no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—28no—
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—48no—
309C2×C4—48no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4C2×#9511yescharacteristic
319C2×C4C4×C22⋊C411yescharacteristic
320C2×C4#38011yescharacteristic
321C2×C4#38111yescharacteristic
322C2×C4#38511yescharacteristic
323C2×C4#38511yescharacteristic
324C2×C4#38511yescharacteristic
325C2×C4#38511yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38611yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38611yescharacteristic
330C8—82no—
331C8—82no—
332C8—82no—
333C8—82no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—44no—
343C22C22⋊#9511yescharacteristic
344C22#179512yes—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—48no—
356C4—84no—
357C4—84no—
358C4#183012yes—
359C4#190512yes—
360C2—22no—
361C2—41no—
362C2—41no—
363C2#1077011yescharacteristic
364C2#1077611yescharacteristic
365C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
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
23. #1556C4semidirect
25. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#343652
#177422
#81162
#81062
#80702
#79762