Finite Group Databasebeta

← #120757

Subgroups — #120757

Order 512.

449 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21361C211yesmaximal, characteristic
2#8093C211yesmaximal, characteristic
3C2×#1303C211yesmaximal, characteristic
4#8054C211yesmaximal, characteristic
5#7950C211yesmaximal, characteristic
6#7934C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1853—21no—
17#1852—21no—
18C4×#107—21no—
19C4×#106—21no—
20#1303—22no—
21#1303—21no—
22#1303—21no—
23#1298—21no—
24#1298—21no—
25#1285—22no—
26#1283C414yescomplemented
27#1274—22no—
28C22×M16—21no—
29C22×C4⋊C4—21no—
30C22×C42—21no—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431C2×C412yes—
46#431D412yes—
47#431Q812yes—
48C2×#107—22no—
49C2×#106—22no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423Q812yes—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2×C412yes—
60#407—21no—
61#407—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8D411yescharacteristic
64C2×C22⋊C8D411yescharacteristic
65C4×M16—21no—
66C4×M16C2×C412yes—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69#383—21no—
70#382—21no—
71#381—21no—
72#380—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4D412yes—
77C4×C4⋊C4D411yescharacteristic
78C4×C4⋊C4D411yescharacteristic
79C4×C22⋊C4—21no—
80#376—21no—
81C2×#95—21no—
82#340—22no—
83#340—22no—
84C23×C4—41no—
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—81no—
93C2×M16—81no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—22no—
97C22×C8—41no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—41no—
108C2×C42—81no—
109C2×C42—81no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×Q811yescharacteristic
115C2×C42C2×Q811yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122#107—44no—
123#106—44no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8C4⋊C412yes—
145C22⋊C8C4⋊C412yes—
146#97—22no—
147#97—41no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—41no—
153C4×C8—41no—
154#95—22no—
155#95—21no—
156#95—21no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161C24#12411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4C2×C4⋊C411yescharacteristic
175C22×C4#11711yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4#12011yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4C22⋊Q811yescharacteristic
182C22×C4#12311yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12311yescharacteristic
185C22×C4#12411yescharacteristic
186C22×C4#12511yescharacteristic
187C22×C4#12511yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12611yescharacteristic
191C22×C4#12611yescharacteristic
192M16—44no—
193M16—82no—
194M16—81no—
195M16—81no—
196M16—82no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—82no—
206C2×C8—82no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C22⋊C4—82no—
213C42—41no—
214C42—41no—
215C42—41no—
216C42—41no—
217C42—82no—
218C42—81no—
219C42—81no—
220C42C4×D411yescharacteristic
221C42C4×D411yescharacteristic
222C42C4×D411yescharacteristic
223C42C4×Q811yescharacteristic
224C42C22⋊Q811yescharacteristic
225C42C22⋊Q811yescharacteristic
226C42C22⋊Q811yescharacteristic
227C42C22⋊Q811yescharacteristic
228C42#12311yescharacteristic
229C42#12311yescharacteristic
230C42#12411yescharacteristic
231C42#12411yescharacteristic
232C42#12511yescharacteristic
233C42#12611yescharacteristic
234C42#12611yescharacteristic
235C42C4⋊Q811yescharacteristic
236C23—21no—
237C23—21no—
238C23—22no—
239C23—41no—
240C23#38211yescharacteristic
241C23#39411yescharacteristic
242C23#40011yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—82no—
267C2×C4—88no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4#38511yescharacteristic
276C2×C4#38711yescharacteristic
277C2×C4#38711yescharacteristic
278C2×C4#38811yescharacteristic
279C2×C4#38811yescharacteristic
280C2×C4#38911yescharacteristic
281C2×C4#39311yescharacteristic
282C2×C4#39711yescharacteristic
283C2×C4#39811yescharacteristic
284C2×C4#39811yescharacteristic
285C2×C4#40011yescharacteristic
286C2×C4#40011yescharacteristic
287C2×C4#47111yescharacteristic
288C2×C4#47111yescharacteristic
289C2×C4#47211yescharacteristic
290C2×C4#47311yescharacteristic
291C8—42no—
292C8—42no—
293C8—82no—
294C8—82no—
295C8—82no—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—42no—
305C22—42no—
306C22#144211yescharacteristic
307C22#183312yes—
308C22#205111yescharacteristic
309C22#205211yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—84no—
320C4—161no—
321C4—161no—
322C4#193011yescharacteristic
323C4#193111yescharacteristic
324C4#203112yes—
325C4#208911yescharacteristic
326C4#209011yescharacteristic
327C2—21no—
328C2—21no—
329C2—22no—
330C2—41no—
331C2#1126011yescharacteristic
332C2#1126311yescharacteristic
333C2#1129011yescharacteristic
subgroupG/H
centerC2×C4#388
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1442
prime radicalC24#124
LCSG ▷ C2×C42 ▷ 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
26. #1283C4semidirect

Maximal subgroups

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

subgroupindex
#213612
#80932
C2×#13032
#80542
#79502
#79342
#78582