Finite Group Databasebeta

← #111174

Subgroups — #111174

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17747C211yesmaximal, characteristic
2#17672C211yesmaximal, characteristic
3#17501C211yesmaximal, characteristic
4#17334C211yesmaximal, characteristic
5#7998C211yesmaximal, characteristic
6#7993C211yesmaximal, characteristic
7#7969C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1559—22no—
16#1554—22no—
17#1549C412yes—
18#1548C412yes—
19#1537C414yes—
20#1531—22no—
21#1528—22no—
22#1493—21no—
23#1493—21no—
24#1491—22no—
25#1485C414yes—
26#1455—22no—
27#1453—22no—
28#1452—21no—
29#1450—21no—
30#1291—22no—
31#1291—22no—
32#1287—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432C2×C412yes—
61#432D412yes—
62#432D412yes—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431C2×C412yes—
69#431C2×C412yes—
70#431D412yes—
71#423—21no—
72#423—21no—
73#423—21no—
74#423C2×C412yes—
75C2×C4⋊C8—21no—
76#407—21no—
77#407—21no—
78#407—21no—
79#407—21no—
80C2×C22⋊C8—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16C2×C412yes—
87C4×M16D412yes—
88C4×M16D412yes—
89C2×#97—21no—
90C2×#97—21no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93#333—24no—
94#332—24no—
95#330—28no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98#322—22no—
99#322—24no—
100C82—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—44no—
111C2×M16—41no—
112C2×M16—41no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—41no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8C4212yes—
123C4⋊Q8C4212yes—
124#125—22no—
125#125—22no—
126#125—41no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144#117—22no—
145#117—22no—
146#117—41no—
147#117—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
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×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8C22⋊C414yes—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188#97—22no—
189#97—22no—
190#97—22no—
191#97—22no—
192#97—41no—
193#97—41no—
194#97—41no—
195#97C22⋊C414yes—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—21no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—21no—
202C4×C8C22⋊C412yes—
203C4×C8C22⋊C412yes—
204C24#12611yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—82no—
210C2×Q8—82no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—81no—
216C22×C4#11711yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12411yescharacteristic
234M16—44no—
235M16—44no—
236M16—82no—
237M16—82no—
238M16—82no—
239M16—82no—
240C2×C8—24no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—82no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—42no—
265C42—42no—
266C42C2×C4211yescharacteristic
267C42C2×C22⋊C411yescharacteristic
268C42C2×C22⋊C411yescharacteristic
269C42C2×C22⋊C411yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42#12011yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12611yescharacteristic
281C42#12711yescharacteristic
282C23—41no—
283C23—41no—
284C23—41no—
285C23#38811yescharacteristic
286C23#38811yescharacteristic
287C23#38811yescharacteristic
288Q8—44no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
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—42no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4C4×C22⋊C411yescharacteristic
315C2×C4C4×C22⋊C411yescharacteristic
316C2×C4C4×C22⋊C411yescharacteristic
317C2×C4#38611yescharacteristic
318C2×C4#38611yescharacteristic
319C2×C4#38611yescharacteristic
320C2×C4#38811yescharacteristic
321C2×C4#38811yescharacteristic
322C2×C4#38811yescharacteristic
323C2×C4#39011yescharacteristic
324C2×C4#39011yescharacteristic
325C2×C4#39011yescharacteristic
326C8—24no—
327C8—42no—
328C8—44no—
329C8—44no—
330C8—82no—
331C8—82no—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—82no—
343C22#143711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—82no—
355C4—82no—
356C4#190212yes—
357C4#193212yes—
358C2—41no—
359C2—41no—
360C2—41no—
361C2#1094611yescharacteristic
362C2#1095011yescharacteristic
363C2#1095211yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#126
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#177472
#176722
#175012
#173342
#79982
#79932
#79692