Finite Group Databasebeta

← #107595

Subgroups — #107595

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#17584C211yesmaximal, characteristic
3C2×#1515C211yesmaximal, characteristic
4#8090C211yesmaximal, characteristic
5#8069C211yesmaximal, characteristic
6#8025C211yesmaximal, characteristic
7#7975C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986—22no—
10#2985—22no—
11#2984C414yescomplemented
12#2983—21no—
13#2982—21no—
14#2962—22no—
15#2960—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2283C2211yescharacteristic
22C2×#383C2211yescharacteristic
23#1518C414yescomplemented
24#1515—22no—
25#1515—22no—
26#1515C414yescomplemented
27#1508—22no—
28C2×#328—22no—
29#1302—21no—
30#1302—21no—
31#1300—22no—
32#1294—22no—
33#1288—22no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40#533D412yes—
41#533D412yes—
42C2×C4⋊Q8Q811yescharacteristic
43C2×C4⋊Q8Q811yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8D412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#446—24no—
52#446—24no—
53#446—24no—
54#446—28no—
55C8×Q8—24no—
56C8×Q8—24no—
57C8×Q8—24no—
58#433—28no—
59#432—24no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432C2×C412yes—
66#432C2×C412yes—
67#432D412yes—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—24no—
73#431—21no—
74#431C2×C412yes—
75#431C2×C412yes—
76#431D411yescharacteristic
77#431D411yescharacteristic
78#424—24no—
79#423—21no—
80#423—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83#407—21no—
84#407—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8D412yes—
87#405—24no—
88C4×M16—21no—
89C4×M16D412yes—
90C2×#97—21no—
91C2×C4×C8—21no—
92#383—21no—
93#383—21no—
94#383—21no—
95#383—21no—
96#383C2×C412yes—
97#382C2×C412yes—
98#381—21no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C22⋊C4—21no—
102#376—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106#328—28no—
107#328—28no—
108C22×Q8—21no—
109C22×Q8C4⋊C412yes—
110C23×C4—41no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—22no—
115C2×M16—41no—
116C22×C8—22no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8C4⋊C412yes—
123C4⋊Q8C4⋊C412yes—
124C4⋊Q8C4⋊C412yes—
125C4⋊Q8C4⋊C412yes—
126#125—22no—
127#125—22no—
128#125—22no—
129#125—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8C22⋊C414yes—
137C22⋊Q8C22⋊C414yes—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8C4214yes—
145C4×Q8C22⋊C414yes—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C4⋊C412yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—41no—
159C2×C42—22no—
160C2×C42—41no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—28no—
174C4⋊C8—28no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8C22⋊C412yes—
192C22⋊C8C22⋊C412yes—
193#97—28no—
194#97—22no—
195#97—41no—
196#97—41no—
197#97C22⋊C414yes—
198C4×C8—28no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214C24#12111yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—44no—
219C2×Q8—82no—
220C2×Q8#9514yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249M16—44no—
250M16—82no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—82no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—82no—
270C4⋊C4#9514yes—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—82no—
274C42—24no—
275C42—24no—
276C42—82no—
277C42C2×C4211yescharacteristic
278C42C2×C22⋊C411yescharacteristic
279C42C2×C22⋊C411yescharacteristic
280C42C2×C4⋊C411yescharacteristic
281C42C2×C4⋊C411yescharacteristic
282C42#11711yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42#12011yescharacteristic
288C42#12011yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42C22⋊Q811yescharacteristic
291C42#12311yescharacteristic
292C42#12311yescharacteristic
293C23—22no—
294C23—22no—
295C23—41no—
296C23#38011yescharacteristic
297C23#38611yescharacteristic
298C23#38611yescharacteristic
299Q8—44no—
300Q8—44no—
301Q8—44no—
302Q8—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—164no—
332C2×C4C2×#9511yescharacteristic
333C2×C4C4×C22⋊C411yescharacteristic
334C2×C4C4×C22⋊C411yescharacteristic
335C2×C4#37911yescharacteristic
336C2×C4#38011yescharacteristic
337C2×C4#38111yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38511yescharacteristic
340C2×C4#38511yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38611yescharacteristic
343C2×C4#38611yescharacteristic
344C8—42no—
345C8—42no—
346C8—44no—
347C8—82no—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—44no—
357C22C22⋊#9511yescharacteristic
358C22#181412yes—
359C22#190512yes—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—44no—
373C4—44no—
374C4—162no—
375C4—162no—
376C2—22no—
377C2—22no—
378C2—41no—
379C2#1076611yescharacteristic
380C2#1077311yescharacteristic
381C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
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
11. #2984C4semidirect
23. #1518C4semidirect
26. #1515C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#175842
C2×#15152
#80902
#80692
#80252
#79752