Finite Group Databasebeta

← #110870

Subgroups — #110870

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21833C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3#17742C211yesmaximal, characteristic
4#17324C211yesmaximal, characteristic
5#8109C211yesmaximal, characteristic
6#7987C211yesmaximal, characteristic
7#7961C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2285C2211yescharacteristic
15#1997—22no—
16#1997C414yescomplemented
17#1990—21no—
18#1990—21no—
19#1988—22no—
20#1823C414yes—
21#1821—22no—
22#1556C414yescomplemented
23#1549—22no—
24#1546—22no—
25#1454—22no—
26#1453—22no—
27#1451—21no—
28C8×M16—21no—
29#1305—22no—
30#1290—22no—
31#1286—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⋊Q8—21no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2×C412yes—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432C2×C412yes—
56#432C2×C412yes—
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#431—21no—
67#431—21no—
68#431D412yes—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72#423D412yes—
73C2×C4⋊C8D412yes—
74#407—21no—
75#407—21no—
76#407—21no—
77#407D412yes—
78C2×C22⋊C8C2×C412yes—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16C2×C412yes—
82C4×M16D412yes—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85#384—21no—
86#382—21no—
87#380—21no—
88#380C2×C412yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91#343—28no—
92#340—24no—
93#340—24no—
94#332—28no—
95#322—22no—
96#322—24no—
97C82—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—82no—
105C22×Q8—82no—
106C23×C4—22no—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—41no—
113C2×M16C22⋊C414yes—
114C22×C8C22⋊C412yes—
115C22×C8C22⋊C412yes—
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⋊Q8—41no—
131C4⋊Q8—41no—
132#117—82no—
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—22no—
140C2×C4⋊C4—22no—
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×C4⋊C4C4214yes—
148C2×C22⋊C4—22no—
149C2×C42—41no—
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
161C2×C42#4111yescharacteristic
162C4⋊C8—22no—
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⋊C8C22⋊C414yes—
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—
181C22⋊C8—41no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186C4×C8—22no—
187C4×C8—21no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—21no—
193C4×C8—41no—
194#95—22no—
195#95—41no—
196C24C4×D411yescharacteristic
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—82no—
204C2×Q8—82no—
205C2×Q8—84no—
206C2×Q8—84no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4C2×C4211yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4#11711yescharacteristic
223C22×C4#11711yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12711yescharacteristic
237M16—42no—
238M16—42no—
239M16—44no—
240M16—44no—
241M16—82no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C22⋊C4—82no—
262C42—82no—
263C42C2×C22⋊C411yescharacteristic
264C42#11711yescharacteristic
265C42C4×D411yescharacteristic
266C42C4×D411yescharacteristic
267C42C4×D411yescharacteristic
268C42C4×D411yescharacteristic
269C42C4×D411yescharacteristic
270C42C4×D411yescharacteristic
271C42#12011yescharacteristic
272C42#12111yescharacteristic
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42#12311yescharacteristic
276C42#12311yescharacteristic
277C42#12411yescharacteristic
278C42#12611yescharacteristic
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23C4×C22⋊C411yescharacteristic
284C23#38811yescharacteristic
285C23#39011yescharacteristic
286Q8—84no—
287Q8—84no—
288Q8—84no—
289Q8—84no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—84no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4C4×C22⋊C411yescharacteristic
317C2×C4C4×C22⋊C411yescharacteristic
318C2×C4#38611yescharacteristic
319C2×C4#38611yescharacteristic
320C2×C4#38611yescharacteristic
321C2×C4#38811yescharacteristic
322C2×C4#38811yescharacteristic
323C2×C4#38811yescharacteristic
324C2×C4#38811yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#39011yescharacteristic
327C2×C4#39011yescharacteristic
328C2×C4#47112yes—
329C2×C4#47212yes—
330C8—44no—
331C8—44no—
332C8—42no—
333C8—42no—
334C8—82no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—42no—
345C22—42no—
346C22#143711yescharacteristic
347C22#197711yescharacteristic
348C22#197811yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—84no—
358C4—162no—
359C4#190712yes—
360C4#190912yes—
361C4#191512yes—
362C4#193112yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2#1094011yescharacteristic
368C2#1094311yescharacteristic
369C2#1094711yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ 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
16. #1997C4semidirect
22. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#218332
#218292
#177422
#173242
#81092
#79872
#79612