Finite Group Databasebeta

← #117544

Subgroups — #117544

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17810C211yesmaximal, characteristic
2#17718C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4#17385C211yesmaximal, characteristic
5#8090C211yesmaximal, characteristic
6#8072C211yesmaximal, characteristic
7C2×#1274C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1573—22no—
16#1572—22no—
17#1565—22no—
18#1554—22no—
19#1548C412yescomplemented
20#1546C412yescomplemented
21#1542—22no—
22#1506—22no—
23#1506—22no—
24C2×#328—22no—
25#1472—22no—
26#1470—22no—
27#1461—22no—
28#1302—22no—
29#1300—22no—
30#1274—22no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8D412yes—
44C2×C4×Q8—21no—
45C2×C4×Q8Q812yes—
46C2×#117—21no—
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432D412yes—
55#432D412yes—
56#432D412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431D412yes—
68#431D412yes—
69#423—21no—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C4×M16—21no—
80C4×M16—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#376—21no—
85#376—21no—
86#376—21no—
87#376D412yes—
88C2×#95—21no—
89C2×#95—21no—
90#335—24no—
91#334—24no—
92#333—24no—
93#332—24no—
94#328—28no—
95#324—28no—
96C22×Q8—21no—
97C22×Q8C4⋊C412yes—
98C23×C4—41no—
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115#125—41no—
116#125—41no—
117#125—41no—
118#125—41no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138#117—41no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C4⋊C412yes—
145C2×C22⋊C4—22no—
146C2×C42—24no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—42no—
150C2×C42—41no—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×Q811yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8C22⋊C414yes—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—41no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—41no—
191C4×C8—41no—
192C4×C8—41no—
193#95—22no—
194#95—22no—
195#95—41no—
196#95—41no—
197C24#12311yescharacteristic
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—82no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4C2×C4⋊C411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12611yescharacteristic
232C22×C4#12611yescharacteristic
233M16—42no—
234M16—42no—
235M16—82no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—24no—
259C42—44no—
260C42—42no—
261C42—42no—
262C42—42no—
263C42C2×C22⋊C411yescharacteristic
264C42#11711yescharacteristic
265C42C4×D411yescharacteristic
266C42C4×D411yescharacteristic
267C42#12011yescharacteristic
268C42#12111yescharacteristic
269C42#12111yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42#12311yescharacteristic
277C42#12411yescharacteristic
278C42#12711yescharacteristic
279C23—22no—
280C23—22no—
281C23—41no—
282C23#38511yescharacteristic
283C23#39411yescharacteristic
284C23#39711yescharacteristic
285Q8—82no—
286Q8—82no—
287Q8—82no—
288Q8—81no—
289Q8—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
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—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—88no—
316C2×C4—82no—
317C2×C4#38011yescharacteristic
318C2×C4#38511yescharacteristic
319C2×C4#38611yescharacteristic
320C2×C4#38811yescharacteristic
321C2×C4#38811yescharacteristic
322C2×C4#39011yescharacteristic
323C2×C4#39211yescharacteristic
324C2×C4#39311yescharacteristic
325C2×C4#39411yescharacteristic
326C2×C4#39611yescharacteristic
327C2×C4#39611yescharacteristic
328C2×C4#39911yescharacteristic
329C2×C4#44312yes—
330C2×C4#44412yes—
331C8—44no—
332C8—82no—
333C8—82no—
334C8—82no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—44no—
344C22#143911yescharacteristic
345C22#186612yes—
346C22#208212yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—84no—
358C4—82no—
359C4—82no—
360C4#187612yes—
361C4#193711yescharacteristic
362C4#193811yescharacteristic
363C2—22no—
364C2—22no—
365C2—41no—
366C2#1106811yescharacteristic
367C2#1113411yescharacteristic
368C2#1113811yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×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
19. #1548C4semidirect
20. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#178102
#177182
C2×#15062
#173852
#80902
#80722
C2×#12742