Finite Group Databasebeta

← #106531

Subgroups — #106531

Order 512.

821 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34059C211yesmaximal, characteristic
2#21820C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8069C212yesmaximal
5#7956C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7#2920C2212yes—
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10#2320—24no—
11#2316—21no—
12#2313—21no—
13#2313C412yes—
14#2310C412yes—
15#2295—24no—
16#2283—21no—
17#2283C2212yes—
18C4×#117—21no—
19#1985—22no—
20#1984C414yescomplemented
21#1817—21no—
22#1817—21no—
23#1467C414yescomplemented
24#1463—21no—
25#1463—21no—
26#1461—22no—
27#1300—24no—
28#1286—24no—
29C23×Q8C2×C411yescharacteristic
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8C2×C412yes—
35C2×C4×Q8C2×C412yes—
36C2×C4×Q8D412yes—
37C2×C4×Q8Q812yes—
38C2×#117—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42D411yescharacteristic
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⋊C8—21no—
55C2×C4⋊C8C2×C412yes—
56#407—22no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C4×M16—22no—
60C2×#97—21no—
61C2×C4×C8C2×C412yes—
62#391—28no—
63#389—28no—
64#387—28no—
65#387—28no—
66#383—22no—
67#383—22no—
68#382—22no—
69#382D414yes—
70#381—22no—
71#381C2×C414yes—
72#380—22no—
73#380—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80#376—22no—
81#376—22no—
82#376D414yes—
83C43—22no—
84#336—24no—
85#336—24no—
86#324—24no—
87#324—24no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—44no—
91C22×Q8C4212yes—
92C22×Q8C4⋊C412yes—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—42no—
101C22×C8—22no—
102C22×C8—22no—
103C4×Q8—24no—
104C4×Q8—24no—
105C4×Q8—24no—
106C4×Q8—24no—
107C4×Q8—24no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—24no—
111#117—48no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
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⋊C4C22⋊C412yes—
124C2×C4⋊C4C4⋊C412yes—
125C2×C4⋊C4C4⋊C412yes—
126C2×C4⋊C4C4⋊C412yes—
127C2×C22⋊C4—24no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42C22⋊C412yes—
140C2×C42C22⋊C412yes—
141C2×C42C4⋊C412yes—
142C2×C42C4⋊C412yes—
143C2×C42C22×C411yescharacteristic
144C2×C42C22×C411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×Q811yescharacteristic
149C2×C42#4112yes—
150C2×C42#4112yes—
151C4⋊C8—24no—
152C4⋊C8—24no—
153C4⋊C8—42no—
154C4⋊C8—42no—
155C22⋊C8—24no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—42no—
161#97—24no—
162#97—42no—
163C4×C8—24no—
164C4×C8—42no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95C22⋊C418yes—
173C24C2×C22⋊C411yescharacteristic
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—24no—
177C2×Q8—24no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—48no—
181C2×Q8—48no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—28no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4#9512yes—
199C22×C4#9512yes—
200C22×C4#9512yes—
201C22×C4#9512yes—
202C22×C4C2×C4211yescharacteristic
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C4⋊C411yescharacteristic
207C22×C4C2×C4⋊C411yescharacteristic
208C22×C4C2×C4⋊C411yescharacteristic
209C22×C4#11712yes—
210C22×C4#12012yes—
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12311yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12312yes—
216M16—84no—
217C2×C8—44no—
218C2×C8—44no—
219C2×C8—42no—
220C2×C8—44no—
221C2×C8—42no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—48no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C22⋊C4—48no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—48no—
241C42—48no—
242C42—84no—
243C42C4×D412yes—
244C42C4×D412yes—
245C42C4×D412yes—
246C42C4×D412yes—
247C42#12112yes—
248C42#12112yes—
249C42C22⋊Q812yes—
250C42C22⋊Q812yes—
251C23—22no—
252C23—22no—
253C23—41no—
254C23C2×#9511yescharacteristic
255C23#37911yescharacteristic
256C23#38111yescharacteristic
257Q8—48no—
258Q8—48no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4#34414yes—
294C2×C4C4×C22⋊C412yes—
295C2×C4#38012yes—
296C2×C4#38511yescharacteristic
297C2×C4#38511yescharacteristic
298C2×C4#38512yes—
299C2×C4#38611yescharacteristic
300C2×C4#38612yes—
301C2×C4#38611yescharacteristic
302C8—82no—
303C8—84no—
304C8—82no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22C22⋊#9511yescharacteristic
314C22C2×#34412yes—
315C22#179412yes—
316C4—22no—
317C4—22no—
318C4—24no—
319C4—22no—
320C4—22no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—42no—
326C4—42no—
327C4—88no—
328C4#191212yes—
329C2—22no—
330C2—22no—
331C2—41no—
332C2#1073811yescharacteristic
333C2#1074111yescharacteristic
334C2#1075211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24C2×C22⋊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
20. #1984C4semidirect
23. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#340592
#218202
#173652
#80692
#79562