Finite Group Databasebeta

← #106604

Subgroups — #106604

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3C2×#1464C211yesmaximal, characteristic
4#8036C212yesmaximal, complemented
5#7898C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7C2×#432C2212yes—
8C22×C22⋊C8—21no—
9C22×#97C2211yescharacteristic
10C22×C4×C8C2211yescharacteristic
11#2343—24no—
12#2343—24no—
13#2315C412yes—
14#2313—21no—
15#2313C412yes—
16#2283—21no—
17#2283C2212yes—
18C2×#381—21no—
19#2273—21no—
20#1750—22no—
21#1750C414yescomplemented
22C2×#336—21no—
23#1464—22no—
24#1464C414yescomplemented
25C2×#324—21no—
26C2×#323—21no—
27#1295—24no—
28#1279—24no—
29C24×C4D411yescharacteristic
30C23×C8—21no—
31C23×C8—21no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117D412yes—
39C2×#117Q812yes—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C22⋊C4—22no—
43C22×C22⋊C4C2×C411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432C2×C414yescomplemented
51C2×C4⋊C8—22no—
52C2×C22⋊C8—28no—
53C2×C22⋊C8—24no—
54C2×C22⋊C8—22no—
55C2×#97—22no—
56C2×#97—22no—
57C2×#97—22no—
58C2×#97—21no—
59C2×#97C2×C412yescomplemented
60C2×C4×C8—22no—
61C2×C4×C8—22no—
62C2×C4×C8—21no—
63C2×C4×C8C2×C412yescomplemented
64#387—28no—
65#387—28no—
66#385—28no—
67#383—22no—
68#383—22no—
69#382—22no—
70#382D414yes—
71#381—22no—
72#381—22no—
73#381—22no—
74#381—22no—
75#380—22no—
76#380C2×C414yes—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D414yes—
79C4×C22⋊C4—28no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82#376—22no—
83#376—22no—
84C2×#95—22no—
85C2×#95—22no—
86#336—24no—
87#324—24no—
88#323—24no—
89C25C22⋊C411yescharacteristic
90C23×C4—24no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—24no—
96C23×C4—42no—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C4⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C22×C8—24no—
106C22×C8—24no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—42no—
116#117—24no—
117#117—24no—
118#117—24no—
119#117—24no—
120#117—24no—
121#117—24no—
122#117—24no—
123#117—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4C4212yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C418yes—
132C2×C4⋊C4C4⋊C412yes—
133C2×C4⋊C4C4⋊C412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4C4212yes—
143C2×C22⋊C4C22⋊C412yes—
144C2×C22⋊C4C4⋊C412yes—
145C2×C22⋊C4C4⋊C412yes—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42C22⋊C412yes—
153C2×C42C4⋊C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×Q811yescharacteristic
160C2×C42#4112yes—
161C2×C42#4112yes—
162C4⋊C8—24no—
163C4⋊C8—42no—
164C22⋊C8—216no—
165C22⋊C8—24no—
166C22⋊C8—42no—
167#97—24no—
168#97—24no—
169#97—42no—
170#97—42no—
171C4×C8—24no—
172C4×C8—42no—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181C24—21no—
182C24—21no—
183C24—22no—
184C24—42no—
185C24—42no—
186C24#9512yes—
187C24#10414yes—
188C24C2×C22⋊C411yescharacteristic
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—28no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—48no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4#9512yes—
220C22×C4#9512yes—
221C22×C4#9512yes—
222C22×C4C2×C4211yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C4⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4C2×C4⋊C411yescharacteristic
229C22×C4#11712yes—
230C22×C4#12011yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12012yes—
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12312yes—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—44no—
247C2×C8—84no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—48no—
253C4⋊C4—84no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—48no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—84no—
266C42—84no—
267C42C4×D412yes—
268C42C4×D412yes—
269C42C4×D412yes—
270C42C4×D412yes—
271C42#12112yes—
272C42#12112yes—
273C42C22⋊Q812yes—
274C42C22⋊Q812yes—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—24no—
280C23—22no—
281C23—42no—
282C23—42no—
283C23—48no—
284C23—42no—
285C23—42no—
286C23—81no—
287C23—84no—
288C23—81no—
289C23#33914yes—
290C23C2×#9511yescharacteristic
291C23#37911yescharacteristic
292C23#38111yescharacteristic
293C23C2×#10412yes—
294C23#42112yes—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—168no—
332C2×C4C4×C22⋊C412yes—
333C2×C4#38012yes—
334C2×C4#38511yescharacteristic
335C2×C4#38511yescharacteristic
336C2×C4#38512yes—
337C2×C4#38611yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38612yes—
340C8—44no—
341C8—44no—
342C8—84no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—24no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—48no—
353C22—42no—
354C22—84no—
355C22—82no—
356C22C22⋊#9511yescharacteristic
357C22C2×#33911yescharacteristic
358C22#173311yescharacteristic
359C22#178912yes—
360C22#179112yes—
361C22#179412yes—
362C22#182612yes—
363C4—22no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—22no—
370C4—22no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4—88no—
375C2—21no—
376C2—22no—
377C2—21no—
378C2—44no—
379C2#834814yes—
380C2#1073311yescharacteristic
381C2#1074711yescharacteristic
382C2#1075211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8036C2semidirect
5. #7898C2semidirect
21. #1750C4semidirect
24. #1464C4semidirect
50. #432C2×C4semidirect
59. C2×#97C2×C4semidirect
63. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341772
C2×#17502
C2×#14642
#80362
#78982