Finite Group Databasebeta

← #110252

Subgroups — #110252

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17746C211yesmaximal, characteristic
2#17742C211yesmaximal, characteristic
3#17742C211yesmaximal, characteristic
4#17338C211yesmaximal, characteristic
5#7996C211yesmaximal, characteristic
6#7996C211yesmaximal, characteristic
7#7996C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15#1558—22no—
16#1556—22no—
17#1556—22no—
18#1549—22no—
19#1549C414yes—
20#1548—22no—
21#1548C414yes—
22#1546—22no—
23#1546C414yes—
24#1453—22no—
25#1453—22no—
26#1453—22no—
27#1291—22no—
28#1291—22no—
29#1291—22no—
30C23×Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊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⋊Q8—21no—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C412yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C42C2311yescharacteristic
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432D412yes—
56#432D412yes—
57#432D412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431C2×C412yes—
66#431C2×C412yes—
67#431D412yes—
68#431D412yes—
69#431D412yes—
70#423—21no—
71#423—21no—
72#423—21no—
73#423C2×C412yes—
74#423C2×C412yes—
75#423C2×C412yes—
76#407—21no—
77#407—21no—
78#407—21no—
79#407—21no—
80#407—21no—
81#407—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88#333—28no—
89#332—28no—
90#332—28no—
91#322—26no—
92C82—22no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—84no—
100C23×C4—41no—
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8C4214yes—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C42C22×C411yescharacteristic
139C2×C42C22×C411yescharacteristic
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8C22⋊C414yes—
160C4⋊C8C22⋊C414yes—
161C4⋊C8C22⋊C414yes—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174#97—22no—
175#97—22no—
176#97—22no—
177#97—41no—
178#97—41no—
179#97—41no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—22no—
186C24#12611yescharacteristic
187C2×Q8—24no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—84no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×Q8—84no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—84no—
204C22×C4#11711yescharacteristic
205C22×C4#11711yescharacteristic
206C22×C4#11711yescharacteristic
207C22×C4C4×D411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12411yescharacteristic
220C22×C4#12411yescharacteristic
221C22×C4#12411yescharacteristic
222M16—82no—
223M16—82no—
224M16—82no—
225M16—82no—
226M16—82no—
227M16—82no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C42C2×C4211yescharacteristic
243C42C2×C22⋊C411yescharacteristic
244C42C2×C22⋊C411yescharacteristic
245C42C2×C22⋊C411yescharacteristic
246C42C4×D411yescharacteristic
247C42C4×D411yescharacteristic
248C42C4×D411yescharacteristic
249C42C4×D411yescharacteristic
250C42C4×D411yescharacteristic
251C42C4×D411yescharacteristic
252C42#12011yescharacteristic
253C42#12111yescharacteristic
254C42#12111yescharacteristic
255C42#12111yescharacteristic
256C42#12611yescharacteristic
257C42#12711yescharacteristic
258C23—41no—
259C23—41no—
260C23—41no—
261C23#38811yescharacteristic
262C23#38811yescharacteristic
263C23#38811yescharacteristic
264Q8—84no—
265Q8—84no—
266Q8—84no—
267Q8—84no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—84no—
290C2×C4C4×C22⋊C411yescharacteristic
291C2×C4C4×C22⋊C411yescharacteristic
292C2×C4C4×C22⋊C411yescharacteristic
293C2×C4#38611yescharacteristic
294C2×C4#38611yescharacteristic
295C2×C4#38611yescharacteristic
296C2×C4#38811yescharacteristic
297C2×C4#38811yescharacteristic
298C2×C4#38811yescharacteristic
299C2×C4#39011yescharacteristic
300C2×C4#39011yescharacteristic
301C2×C4#39011yescharacteristic
302C8—44no—
303C8—44no—
304C8—44no—
305C8—82no—
306C8—82no—
307C8—82no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—44no—
318C22#143711yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—84no—
329C4#190212yes—
330C4#190212yes—
331C4#190312yes—
332C2—41no—
333C2—41no—
334C2—41no—
335C2#1095211yescharacteristic
336C2#1095211yescharacteristic
337C2#1095311yescharacteristic
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
#177462
#177422
#173382
#79962