Finite Group Databasebeta

← #120839

Subgroups — #120839

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17807C211yesmaximal, characteristic
2#8074C211yesmaximal, characteristic
3#8060C211yesmaximal, characteristic
4#7958C211yesmaximal, characteristic
5C2×#1283C211yesmaximal, characteristic
6#7895C211yesmaximal, characteristic
7#7887C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1573—22no—
16#1571—21no—
17#1570—21no—
18#1565—22no—
19#1300—22no—
20#1299—21no—
21#1299—21no—
22#1286C412yes—
23#1286C412yes—
24#1283—22no—
25#1279—22no—
26#1278—22no—
27C22×M16—21no—
28C22×M16—21no—
29C23×C8—21no—
30C22×C42C2311yescharacteristic
31#432—21no—
32#432—21no—
33#432—21no—
34#432Q812yes—
35#431—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431C2×C412yes—
42#431D412yes—
43#423C2×C412yes—
44#423D411yescharacteristic
45#423D411yescharacteristic
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53#407—21no—
54#407—21no—
55C2×C22⋊C8—21no—
56C4×M16—21no—
57C4×M16—21no—
58C4×M16—21no—
59C4×M16—21no—
60C4×M16—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16—21no—
64C4×M16D412yes—
65C4×M16Q812yes—
66C2×#97—21no—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71#383—21no—
72#382C2×C411yescharacteristic
73#382C2×C411yescharacteristic
74#381—21no—
75#380—21no—
76C4×C4⋊C4—21no—
77C4×C22⋊C4—21no—
78#376—21no—
79#335—24no—
80#334—24no—
81C23×C4#4111yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C2×M16—22no—
85C2×M16—24no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—44no—
91C2×M16—42no—
92C2×M16—41no—
93C2×M16—41no—
94C22×C8—22no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—82no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C22⋊C4—41no—
104C2×C42—41no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×Q811yescharacteristic
111C2×C42C2×Q811yescharacteristic
112C2×C42#4111yescharacteristic
113C2×C42#4111yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C2×C42#4111yescharacteristic
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—21no—
120C4⋊C8—22no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—22no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8C4⋊C414yes—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135#97—22no—
136#97—22no—
137#97—22no—
138#97—41no—
139#97—41no—
140#97—41no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—41no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—41no—
154#95—41no—
155C24#12611yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#11711yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4C4×D411yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12311yescharacteristic
170C22×C4#12311yescharacteristic
171C22×C4#12311yescharacteristic
172C22×C4#12411yescharacteristic
173C22×C4#12411yescharacteristic
174C22×C4#12411yescharacteristic
175C22×C4#12511yescharacteristic
176C22×C4#12511yescharacteristic
177C22×C4#12511yescharacteristic
178C22×C4#12511yescharacteristic
179C22×C4#12611yescharacteristic
180C22×C4#12611yescharacteristic
181M16—24no—
182M16—24no—
183M16—42no—
184M16—42no—
185M16—44no—
186M16—44no—
187M16—42no—
188M16—42no—
189M16—81no—
190M16—81no—
191C2×C8—24no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C22⋊C4—82no—
207C42—81no—
208C42—81no—
209C42C2×C4⋊C411yescharacteristic
210C42#11711yescharacteristic
211C42C4×D411yescharacteristic
212C42C4×Q811yescharacteristic
213C42#12011yescharacteristic
214C42#12111yescharacteristic
215C42#12111yescharacteristic
216C42C22⋊Q811yescharacteristic
217C42C22⋊Q811yescharacteristic
218C42C22⋊Q811yescharacteristic
219C42#12311yescharacteristic
220C42#12311yescharacteristic
221C42#12411yescharacteristic
222C42#12511yescharacteristic
223C42#12611yescharacteristic
224C42C4⋊Q811yescharacteristic
225C23—22no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23#38811yescharacteristic
230C23#40011yescharacteristic
231C23#40011yescharacteristic
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4#38211yescharacteristic
260C2×C4#38511yescharacteristic
261C2×C4#38711yescharacteristic
262C2×C4#38711yescharacteristic
263C2×C4#38811yescharacteristic
264C2×C4#38911yescharacteristic
265C2×C4#39311yescharacteristic
266C2×C4#39411yescharacteristic
267C2×C4#39711yescharacteristic
268C2×C4#39811yescharacteristic
269C2×C4#39811yescharacteristic
270C2×C4#40011yescharacteristic
271C2×C4#44312yes—
272C2×C4#44412yes—
273C8—44no—
274C8—44no—
275C8—44no—
276C8—82no—
277C8—82no—
278C8—81no—
279C8—81no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—42no—
288C22—42no—
289C22#144211yescharacteristic
290C22#193711yescharacteristic
291C22#193811yescharacteristic
292C22#210212yes—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—161no—
303C4—161no—
304C4#186612yes—
305C4#192212yes—
306C4#204811yescharacteristic
307C4#204911yescharacteristic
308C2—21no—
309C2—21no—
310C2—22no—
311C2—41no—
312C2#1127411yescharacteristic
313C2#1128611yescharacteristic
314C2#1129111yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1442
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#178072
#80742
#80602
#79582
C2×#12832
#78952
#78872