Finite Group Databasebeta

← #111157

Subgroups — #111157

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2#17746C211yesmaximal, characteristic
3#17742C211yesmaximal, characteristic
4#17324C211yesmaximal, characteristic
5#8007C211yesmaximal, characteristic
6#7961C211yesmaximal, characteristic
7#7953C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2285C2211yescharacteristic
15#1990—21no—
16#1990—21no—
17#1987C414yes—
18#1824—22no—
19#1558C414yescomplemented
20#1556—22no—
21#1549—22no—
22#1548—22no—
23#1548—22no—
24#1546C414yescomplemented
25#1454—22no—
26#1453—22no—
27#1451—21no—
28C8×M16—21no—
29#1293—22no—
30#1286—22no—
31#1285—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⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432C2×C412yes—
55#432C2×C412yes—
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#423—21no—
71#423C2×C412yes—
72#423D412yes—
73C2×C4⋊C8C2×C412yes—
74#407—21no—
75#407—21no—
76#407—21no—
77#407—21no—
78C2×C22⋊C8D412yes—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85#384—21no—
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C412yes—
91#340—24no—
92#340—24no—
93#333—28no—
94#332—28no—
95#322—22no—
96#322—24no—
97C82—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—84no—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—41no—
113C22×C8C22⋊C412yes—
114C22×C8C22⋊C412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8C4214yes—
131#117—82no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C42—22no—
149C2×C42C22×C411yescharacteristic
150C2×C42C22×C411yescharacteristic
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8C22⋊C414yes—
170C4⋊C8C22⋊C414yes—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—41no—
185C4×C8—22no—
186C4×C8—21no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—21no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—41no—
193#95—22no—
194#95—41no—
195C24#12411yescharacteristic
196C2×Q8—24no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—84no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×Q8—84no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—84no—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4#11711yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12611yescharacteristic
231C22×C4#12611yescharacteristic
232C22×C4#12711yescharacteristic
233M16—42no—
234M16—42no—
235M16—82no—
236M16—82no—
237M16—82no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C22⋊C4—82no—
258C42—82no—
259C42C2×C4211yescharacteristic
260C42C2×C22⋊C411yescharacteristic
261C42C2×C22⋊C411yescharacteristic
262C42#11711yescharacteristic
263C42C4×D411yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42C4×D411yescharacteristic
267C42C4×D411yescharacteristic
268C42C4×D411yescharacteristic
269C42#12011yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42#12311yescharacteristic
273C42#12411yescharacteristic
274C42#12411yescharacteristic
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23#38811yescharacteristic
280C23#38811yescharacteristic
281C23#39011yescharacteristic
282Q8—84no—
283Q8—84no—
284Q8—84no—
285Q8—84no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
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—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4C4×C22⋊C411yescharacteristic
312C2×C4C4×C22⋊C411yescharacteristic
313C2×C4C4×C22⋊C411yescharacteristic
314C2×C4#38611yescharacteristic
315C2×C4#38611yescharacteristic
316C2×C4#38611yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#38811yescharacteristic
319C2×C4#38811yescharacteristic
320C2×C4#38811yescharacteristic
321C2×C4#39011yescharacteristic
322C2×C4#39011yescharacteristic
323C2×C4#47112yes—
324C2×C4#47312yes—
325C8—44no—
326C8—44no—
327C8—42no—
328C8—42no—
329C8—82no—
330C8—82no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—42no—
340C22—42no—
341C22#143711yescharacteristic
342C22#197711yescharacteristic
343C22#197911yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—84no—
353C4—162no—
354C4#190312yes—
355C4#190812yes—
356C4#190912yes—
357C4#193012yes—
358C2—21no—
359C2—21no—
360C2—41no—
361C2—41no—
362C2#1094211yescharacteristic
363C2#1094711yescharacteristic
364C2#1095311yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
19. #1558C4semidirect
24. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#218342
#177462
#177422
#173242
#80072
#79612
#79532