Finite Group Databasebeta

← #110729

Subgroups — #110729

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#20551C211yesmaximal, characteristic
3#17562C211yesmaximal, characteristic
4#17496C211yesmaximal, characteristic
5#8128C211yesmaximal, characteristic
6#8087C211yesmaximal, characteristic
7#7973C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1998—22no—
16#1997C412yescomplemented
17#1997C412yescomplemented
18#1771—22no—
19#1768—22no—
20#1753—22no—
21#1742—22no—
22#1515—22no—
23#1511C414yes—
24#1503—22no—
25#1497—22no—
26#1491C414yescomplemented
27#1481—22no—
28#1309—22no—
29#1302—22no—
30#1288—22no—
31C22×M16—21no—
32C22×M16D412yes—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#432D412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423C2×C412yes—
67#423D412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D412yes—
71#407—21no—
72#407C2×C412yes—
73C2×C22⋊C8—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16C2×C412yes—
79C4×M16C2×C412yes—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85#383D412yes—
86#382—21no—
87#381C2×C412yes—
88#380—21no—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—21no—
91#376—21no—
92#343—24no—
93#343—24no—
94#336—24no—
95#328—28no—
96Q8⋊C8—28no—
97C22×Q8—21no—
98C22×Q8C4212yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—24no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—42no—
108C2×M16—41no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16C22⋊C414yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123#125—41no—
124#125—41no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—41no—
144#117—41no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C4212yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8C22⋊C414yes—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—41no—
191#97—41no—
192#97—82no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—41no—
196C4×C8—41no—
197C4×C8—82no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95C22⋊C414yes—
202C24C4×D411yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4C2×C4211yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4#11711yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12611yescharacteristic
234M16—24no—
235M16—44no—
236M16—42no—
237M16—44no—
238M16—82no—
239M16—82no—
240M16—82no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—82no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42C2×C22⋊C411yescharacteristic
272C42#11711yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12311yescharacteristic
283C42#12411yescharacteristic
284C42#12411yescharacteristic
285C42#12611yescharacteristic
286C42#12711yescharacteristic
287C23—22no—
288C23—41no—
289C23—41no—
290C23C4×C22⋊C411yescharacteristic
291C23#38611yescharacteristic
292C23#38811yescharacteristic
293Q8—44no—
294Q8—82no—
295Q8—82no—
296Q8—82no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4C4×C22⋊C411yescharacteristic
324C2×C4C4×C22⋊C411yescharacteristic
325C2×C4#38611yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#39011yescharacteristic
333C2×C4#39011yescharacteristic
334C2×C4#39011yescharacteristic
335C2×C4#45512yes—
336C2×C4#45612yes—
337C8—44no—
338C8—44no—
339C8—44no—
340C8—82no—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—82no—
351C22#143711yescharacteristic
352C22#188212yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—84no—
364C4—82no—
365C4—82no—
366C4#188212yes—
367C4#188511yescharacteristic
368C4#188611yescharacteristic
369C2—22no—
370C2—41no—
371C2—41no—
372C2#1094111yescharacteristic
373C2#1095511yescharacteristic
374C2#1095711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
16. #1997C4semidirect
17. #1997C4semidirect
26. #1491C4semidirect

Maximal subgroups

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

subgroupindex
#213372
#205512
#175622
#174962
#81282
#80872
#79732