Finite Group Databasebeta

← #121001

Subgroups — #121001

Order 512.

441 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21985C211yesmaximal, characteristic
2#10848C211yesmaximal, characteristic
3#8081C211yesmaximal, characteristic
4#7974C211yesmaximal, characteristic
5#7960C211yesmaximal, characteristic
6#7953C211yesmaximal, characteristic
7#7883C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18#1846—21no—
19#1835—21no—
20C2×#344—22no—
21#1301—21no—
22#1301—21no—
23#1288—22no—
24#1286—21no—
25#1286—21no—
26#1285—22no—
27#1278C414yescomplemented
28C22×M16—21no—
29C22×M16—21no—
30C2×#117D412yes—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432D412yes—
36#431—21no—
37#431—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431C2×C412yes—
43#431Q812yes—
44C2×#107—22no—
45C2×#106—22no—
46#423—21no—
47#423—21no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423Q812yes—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D411yescharacteristic
55C2×C4⋊C8D411yescharacteristic
56#407—21no—
57#407—21no—
58C2×C22⋊C8—21no—
59C4×M16—21no—
60C4×M16—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16—21no—
64C4×M16C2×C412yes—
65C2×#97—21no—
66C2×C4×C8—21no—
67C2×C4×C8C2×C412yes—
68#384—21no—
69#383—21no—
70#382—21no—
71#382D411yescharacteristic
72#382D411yescharacteristic
73#381—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C22⋊C4—21no—
81#376—21no—
82#344—44no—
83#340—22no—
84#340—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—42no—
99C2×M16—42no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—41no—
104#117—42no—
105#117—42no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—41no—
113C2×C42—22no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×Q811yescharacteristic
120C2×C42C2×Q811yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127#107—82no—
128#106—82no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—22no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8C4⋊C412yes—
140C4⋊C8C4⋊C412yes—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147#97—22no—
148#97—22no—
149#97—41no—
150#97—41no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—41no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165C24#12311yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4#11711yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4C4×Q811yescharacteristic
178C22×C4#12011yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4#12311yescharacteristic
185C22×C4#12411yescharacteristic
186C22×C4#12411yescharacteristic
187C22×C4#12511yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12611yescharacteristic
191C22×C4#12611yescharacteristic
192C22×C4#12611yescharacteristic
193M16—42no—
194M16—42no—
195M16—42no—
196M16—42no—
197M16—44no—
198M16—44no—
199M16—82no—
200M16—81no—
201M16—81no—
202C2×C8—24no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—82no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C42—82no—
226C42—82no—
227C42C2×C4⋊C411yescharacteristic
228C42#11711yescharacteristic
229C42C4×D411yescharacteristic
230C42C4×D411yescharacteristic
231C42C22⋊Q811yescharacteristic
232C42C22⋊Q811yescharacteristic
233C42C22⋊Q811yescharacteristic
234C42#12311yescharacteristic
235C42#12311yescharacteristic
236C42#12311yescharacteristic
237C42#12411yescharacteristic
238C42#12411yescharacteristic
239C42#12511yescharacteristic
240C42#12511yescharacteristic
241C42#12611yescharacteristic
242C42C4⋊Q811yescharacteristic
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23#38711yescharacteristic
248C23#39411yescharacteristic
249C23#40011yescharacteristic
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4#38211yescharacteristic
282C2×C4#38511yescharacteristic
283C2×C4#38711yescharacteristic
284C2×C4#38811yescharacteristic
285C2×C4#38811yescharacteristic
286C2×C4#38911yescharacteristic
287C2×C4#39311yescharacteristic
288C2×C4#39711yescharacteristic
289C2×C4#39811yescharacteristic
290C2×C4#39811yescharacteristic
291C2×C4#40011yescharacteristic
292C2×C4#40011yescharacteristic
293C2×C4#45311yescharacteristic
294C2×C4#45411yescharacteristic
295C2×C4#45511yescharacteristic
296C2×C4#45611yescharacteristic
297C8—44no—
298C8—42no—
299C8—42no—
300C8—82no—
301C8—82no—
302C8—82no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—81no—
312C22—81no—
313C22#144211yescharacteristic
314C22#203711yescharacteristic
315C22#203811yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—161no—
326C4—162no—
327C4—161no—
328C4#183312yes—
329C4#185911yescharacteristic
330C4#186011yescharacteristic
331C4#202711yescharacteristic
332C4#202811yescharacteristic
333C2—21no—
334C2—21no—
335C2—41no—
336C2—41no—
337C2#1126011yescharacteristic
338C2#1126511yescharacteristic
339C2#1127611yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ 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
27. #1278C4semidirect

Maximal subgroups

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

subgroupindex
#219852
#108482
#80812
#79742
#79602
#79532
#78832