Finite Group Databasebeta

← #106384

Subgroups — #106384

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#13925C211yesmaximal, characteristic
4#7959C212yesmaximal
5#7950C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7#2984—24no—
8#2983—22no—
9#2983C412yescomplemented
10#2982C412yescomplemented
11Q8×M16—24no—
12#2920—22no—
13#2920C2212yes—
14#2920C2212yes—
15C2×#382C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1821—21no—
18#1801—22no—
19#1784—21no—
20C4×#101C414yescomplemented
21#1446—21no—
22#1445—21no—
23#1436C414yes—
24C4⋊#95—22no—
25#1286—24no—
26#1285—24no—
27C23×Q8D411yescharacteristic
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8C2×C411yescharacteristic
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8C2×C412yes—
35C2×C4×Q8D412yes—
36C2×C4×Q8Q811yescharacteristic
37C2×C4×Q8Q811yescharacteristic
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4D411yescharacteristic
42C22×C42C2×C412yes—
43C22×C42D412yes—
44C22×C42C2311yescharacteristic
45#446—216no—
46#446—28no—
47C8×Q8—28no—
48#432—24no—
49#432—24no—
50#432—22no—
51#432—22no—
52#432—22no—
53#432D414yes—
54#431—24no—
55#431—24no—
56#431—22no—
57#431—22no—
58#431—22no—
59#431C2×C414yes—
60#423—24no—
61#423—22no—
62#423D414yes—
63C2×#101—24no—
64#407—22no—
65#407—22no—
66C4×M16—24no—
67C4×M16—22no—
68C4×M16C2×C414yes—
69#382—22no—
70#382—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C2×#95—24no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—24no—
77C2×#95C2×C412yes—
78#343—24no—
79C22×Q8—22no—
80C22×Q8—22no—
81C22×Q8—44no—
82C22×Q8C22⋊C412yes—
83C22×Q8C4⋊C411yescharacteristic
84C22×Q8C4⋊C411yescharacteristic
85C23×C4—22no—
86C23×C4—22no—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C2×M16—24no—
93C2×M16—24no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—24no—
99C4×Q8—24no—
100C4×Q8—24no—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×Q8—22no—
104C4×Q8—22no—
105C4×Q8—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C4212yes—
118C2×C4⋊C4C4212yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C4⋊C412yes—
121C2×C4⋊C4C4⋊C412yes—
122C2×C4⋊C4C4⋊C411yescharacteristic
123C2×C4⋊C4C4⋊C411yescharacteristic
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—24no—
129C2×C42—82no—
130C2×C42C22⋊C412yes—
131C2×C42C22⋊C412yes—
132C2×C42C4⋊C412yes—
133C2×C42C4⋊C412yes—
134C2×C42C22×C411yescharacteristic
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×Q811yescharacteristic
140C2×C42#4112yes—
141C2×C42#4112yes—
142C4⋊C8—28no—
143C4⋊C8—28no—
144C4⋊C8—24no—
145C4⋊C8—42no—
146C4⋊C8—42no—
147C4⋊C8C22⋊C418yes—
148#101—48no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153#97—28no—
154#97—24no—
155#97—24no—
156C4×C8—28no—
157C4×C8—24no—
158C4×C8—24no—
159#95—24no—
160#95—48no—
161#95—42no—
162#95—42no—
163#95—84no—
164C24C2×C22⋊C411yescharacteristic
165C2×Q8—24no—
166C2×Q8—24no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—24no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—48no—
173C2×Q8—48no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4#9512yes—
198C22×C4#9512yes—
199C22×C4#9512yes—
200C22×C4#9512yes—
201C22×C4C2×C4211yescharacteristic
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C4⋊C411yescharacteristic
205C22×C4C2×C4⋊C411yescharacteristic
206C22×C4C2×C4⋊C411yescharacteristic
207C22×C4C4×D412yes—
208C22×C4C4×D412yes—
209C22×C4#12011yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4#12311yescharacteristic
214C22×C4#12311yescharacteristic
215M16—44no—
216M16—44no—
217M16—48no—
218C2×C8—28no—
219C2×C8—44no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—84no—
240C42C2×C22⋊C412yes—
241C42#11712yes—
242C42C4×D412yes—
243C42C4×D412yes—
244C42#12012yes—
245C42C22⋊Q812yes—
246C42C22⋊Q812yes—
247C42#12312yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23C2×#9511yescharacteristic
254C23#38611yescharacteristic
255C23#38611yescharacteristic
256Q8—28no—
257Q8—48no—
258Q8—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
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—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4H(3,ℤ/4)14yes—
299C2×C4C4×C22⋊C412yes—
300C2×C4#37911yescharacteristic
301C2×C4#38012yes—
302C2×C4#38111yescharacteristic
303C2×C4#38511yescharacteristic
304C2×C4#38512yes—
305C2×C4#38511yescharacteristic
306C2×C4#38612yes—
307C2×C4#45412yes—
308C2×C4#45612yes—
309C8—48no—
310C8—84no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22C22⋊#9511yescharacteristic
322C22C2×H(3,ℤ/4)12yes—
323C22#188611yescharacteristic
324C22#188811yescharacteristic
325C22#191411yescharacteristic
326C22#191511yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—24no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—84no—
338C4—162no—
339C4#179312yes—
340C4#185712yes—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2#1071711yescharacteristic
347C2#1075011yescharacteristic
348C2#1075511yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
9. #2983C4semidirect
10. #2982C4semidirect
20. C4×#101C4semidirect

Maximal subgroups

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

subgroupindex
#602482
#207272
#139252
#79592
#79502