Finite Group Databasebeta

← #108203

Subgroups — #108203

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34048C211yesmaximal, characteristic
2#20717C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#7949C212yesmaximal, complemented
5#7869C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10#2314—24no—
11#2313—21no—
12#2313C412yes—
13#2312—21no—
14#2312C412yes—
15#2298—24no—
16#2285—21no—
17#2280—21no—
18C4×#117C2212yes—
19#1817—21no—
20#1798C414yescomplemented
21#1779—21no—
22#1750—22no—
23#1464C414yescomplemented
24#1463—21no—
25#1462—21no—
26#1458—22no—
27#1285—24no—
28#1276—24no—
29C24×C4D411yescharacteristic
30C2×#117—22no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C2×#117D412yes—
38C2×#117Q812yes—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C22⋊C4C2×C411yescharacteristic
42C22×C22⋊C4D411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42C2311yescharacteristic
45#432—22no—
46#432—22no—
47#432—22no—
48#432C2×C414yes—
49#431—22no—
50#431—22no—
51#431—22no—
52#431—22no—
53#423—22no—
54C2×C4⋊C8C2×C412yes—
55C2×C4⋊C8C2×C412yes—
56#407—22no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C4×M16—22no—
61C2×#97—21no—
62C2×C4×C8—21no—
63#388—28no—
64#388—28no—
65#387—28no—
66#387—28no—
67#384—22no—
68#382—22no—
69#381—22no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380—22no—
74#379—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D414yes—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4C2×C414yes—
83#376—22no—
84C43D414yes—
85#336—24no—
86#324—24no—
87#323—24no—
88C25C22⋊C411yescharacteristic
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—44no—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C4⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—42no—
103C22×C8—22no—
104C22×C8—22no—
105#117—24no—
106#117—24no—
107#117—24no—
108#117—24no—
109#117—24no—
110#117—24no—
111#117—24no—
112#117—24no—
113#117—48no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4C4212yes—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C4⋊C412yes—
124C2×C4⋊C4C4⋊C412yes—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4C4212yes—
133C2×C22⋊C4C22⋊C412yes—
134C2×C22⋊C4C4⋊C412yes—
135C2×C22⋊C4C4⋊C412yes—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42C22⋊C412yes—
143C2×C42C22⋊C418yes—
144C2×C42C4⋊C412yes—
145C2×C42C22×C411yescharacteristic
146C2×C42C22×C411yescharacteristic
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×Q811yescharacteristic
151C2×C42#4112yes—
152C2×C42#4112yes—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—24no—
156C4⋊C8—42no—
157C22⋊C8—24no—
158C22⋊C8—48no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—42no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#97—24no—
165#97—42no—
166C4×C8—42no—
167C4×C8—42no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174C24—21no—
175C24—21no—
176C24—42no—
177C24—42no—
178C24—81no—
179C24#9512yes—
180C24C2×C22⋊C411yescharacteristic
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—28no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4#9512yes—
207C22×C4#9512yes—
208C22×C4#9512yes—
209C22×C4C2×C4211yescharacteristic
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C4⋊C411yescharacteristic
214C22×C4C2×C4⋊C411yescharacteristic
215C22×C4C2×C4⋊C411yescharacteristic
216C22×C4#11712yes—
217C22×C4#12011yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12012yes—
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12312yes—
223M16—84no—
224C2×C8—44no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—44no—
228C2×C8—44no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—48no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—48no—
241C22⋊C4—84no—
242C22⋊C4—84no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—48no—
248C42—48no—
249C42—84no—
250C42C4×D412yes—
251C42C4×D412yes—
252C42C4×D412yes—
253C42C4×D412yes—
254C42#12112yes—
255C42#12112yes—
256C42C22⋊Q812yes—
257C42C22⋊Q812yes—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—41no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—88no—
273C23C2×#9511yescharacteristic
274C23#37911yescharacteristic
275C23#38111yescharacteristic
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4C4×C22⋊C412yes—
310C2×C4#38012yes—
311C2×C4#38511yescharacteristic
312C2×C4#38511yescharacteristic
313C2×C4#38512yes—
314C2×C4#38612yes—
315C2×C4#38611yescharacteristic
316C2×C4#38611yescharacteristic
317C2×C4#42114yes—
318C8—84no—
319C8—82no—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—44no—
329C22—44no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—82no—
334C22—84no—
335C22—84no—
336C22C22⋊#9511yescharacteristic
337C22#173312yes—
338C22#179112yes—
339C4—21no—
340C4—21no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—88no—
349C4#189712yes—
350C4#189712yes—
351C2—22no—
352C2—22no—
353C2—41no—
354C2—82no—
355C2#1072011yescharacteristic
356C2#1074311yescharacteristic
357C2#1074711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC25C22⋊C4
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
4. #7949C2semidirect
5. #7869C2semidirect
20. #1798C4semidirect
23. #1464C4semidirect

Maximal subgroups

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

subgroupindex
#340482
#207172
#173572
#79492
#78692