Finite Group Databasebeta

← #229221

Subgroups — #229221

Order 512.

527 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23170C211yesmaximal, characteristic
2#22733C211yesmaximal, characteristic
3#19150C211yesmaximal, characteristic
4#17998C212yesmaximal
5#9861C211yesmaximal, characteristic
6#9627C211yesmaximal, characteristic
7#9589C212yesmaximal
8#8990C212yesmaximal
9#8933C212yesmaximal
10#8294C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12#3160C2211yescharacteristic
13#3104C2212yes—
14Q8×M16C2212yes—
15#2962C2212yes—
16C2×#446C2211yescharacteristic
17#1986C2212yes—
18#1950—22no—
19#1950C2212yes—
20#1945—21no—
21#1922—22no—
22#1849—21no—
23#1848C2211yescharacteristic
24#1778—21no—
25C4×#108C2211yescharacteristic
26#1772—21no—
27C8×C4⋊C4C2211yescharacteristic
28C4×#103C2211yescharacteristic
29#1696C412yescomplemented
30#1694C412yescomplemented
31#1692C412yescomplemented
32#1692C412yescomplemented
33#1591—22no—
34#1584—22no—
35#1584—22no—
36C8×Q16—22no—
37#1571C2212yes—
38#1564C2211yescharacteristic
39#1542C2212yes—
40#1493C2212yes—
41#1493C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C8×M16C2212yes—
44C2×#322C2211yescharacteristic
45#1303C2212yes—
46#1279C2211yescharacteristic
47#1278C2212yes—
48#1277C2211yescharacteristic
49#533C2312yes—
50C2×C4×Q8C2311yescharacteristic
51#500C2×C411yescharacteristic
52#500C2×C411yescharacteristic
53#499—21no—
54#499—21no—
55#498C2×C412yes—
56#484—22no—
57#477C2×C412yes—
58#477C2×C412yes—
59#462C2×C412yes—
60#462C2×C412yes—
61#451—22no—
62#446—22no—
63#446—22no—
64#446—21no—
65#446—21no—
66#446—42no—
67#446—42no—
68#446C2×C412yes—
69#446C2×C412yes—
70C8×Q8—22no—
71C8×Q8—22no—
72C4×Q16—22no—
73C4×Q16—22no—
74#431C2312yes—
75C2×#108—21no—
76C2×#108—22no—
77C2×#108—21no—
78C2×#108D411yescharacteristic
79C2×#108D411yescharacteristic
80#427—21no—
81#427—21no—
82#424—22no—
83#423—22no—
84#423C2312yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—21no—
89#405—22no—
90C4×M16—22no—
91C4×M16C2312yes—
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C4×C4⋊C4C2311yescharacteristic
98#341—22no—
99#341D412yes—
100#340—21no—
101#339—22no—
102#339—22no—
103#336—21no—
104#335—22no—
105#335—22no—
106#335C2×C411yescharacteristic
107#335C2×C411yescharacteristic
108#334C2×C412yes—
109#333C2×C412yes—
110#333C2×C412yes—
111#332C2×C412yes—
112#332C2×C412yes—
113Q8⋊C8—22no—
114Q8⋊C8—22no—
115Q8⋊C8—42no—
116Q8⋊C8D412yes—
117Q8⋊C8D412yes—
118#322—22no—
119#322—21no—
120#322—21no—
121C82—22no—
122C22×Q8#4111yescharacteristic
123C2×Q16—22no—
124C2×Q16—22no—
125C2×M16—22no—
126C2×M16C2×D412yes—
127C2×M16C2×D412yes—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C4⋊Q8C22×C412yes—
135C4⋊Q8C22×C412yes—
136#125—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8C22×C412yes—
144C4×Q8C2×D412yes—
145#117—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4#4111yescharacteristic
149C2×C42—21no—
150C2×C42—21no—
151C2×C42C2411yescharacteristic
152#108—22no—
153#108—42no—
154#108—81no—
155#108—82no—
156#108Q1614yescomplemented
157#107—21no—
158#107—21no—
159#107—21no—
160#107—21no—
161#107—22no—
162#107—22no—
163#106—22no—
164#106—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174C4⋊C8—42no—
175C4⋊C8C22×C412yes—
176C4⋊C8C22×C412yes—
177C4⋊C8C22×C412yes—
178C4⋊C8C2×D411yescharacteristic
179C4⋊C8C2×D411yescharacteristic
180#103—22no—
181#103—22no—
182#103—22no—
183#103—22no—
184#103—42no—
185C22⋊C8—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—42no—
190#97—42no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—42no—
197C4×C8—42no—
198C4×C8—42no—
199C4×C8C22×C411yescharacteristic
200C4×C8C22×C411yescharacteristic
201C4×C8C2×D412yes—
202C4×C8#4111yescharacteristic
203C4×C8#4111yescharacteristic
204#95—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8C4×D412yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4C22×D411yescharacteristic
215C22×C4C2×#4111yescharacteristic
216C22×C42−1+411yescharacteristic
217Q16—44no—
218Q16—82no—
219M16—22no—
220M16—22no—
221M16—22no—
222M16—42no—
223M16—42no—
224M16C2×Q1614yes—
225C2×C8—22no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—22no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8—41no—
243C2×C8—81no—
244C2×C8C4×D411yescharacteristic
245C2×C8C4×D411yescharacteristic
246C2×C8C4×D411yescharacteristic
247C2×C8C4×D411yescharacteristic
248C2×C8C2×Q1612yes—
249C2×C8#13512yes—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4C4×D412yes—
262C22⋊C4—42no—
263C42—22no—
264C42—22no—
265C42—42no—
266C42—41no—
267C42—41no—
268C42C23×C411yescharacteristic
269C42C22×D411yescharacteristic
270C42C2×#4111yescharacteristic
271C42H(5,𝔽2)11yescharacteristic
272C23#54811yescharacteristic
273Q8—22no—
274Q8—22no—
275Q8—42no—
276Q8—42no—
277Q8—81no—
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4C2×C4×D411yescharacteristic
292C2×C4C2×C4×D411yescharacteristic
293C2×C4#52011yescharacteristic
294C2×C4D4211yescharacteristic
295C2×C4#54711yescharacteristic
296C2×C4#55311yescharacteristic
297C2×C4C22×Q1611yescharacteristic
298C2×C4C2×#13511yescharacteristic
299C2×C4#57711yescharacteristic
300C2×C4#57811yescharacteristic
301C8—21no—
302C8—21no—
303C8—24no—
304C8—42no—
305C8—42no—
306C8—42no—
307C8—42no—
308C8—41no—
309C8—41no—
310C8—42no—
311C8—81no—
312C8—82no—
313C8C4×Q1612yes—
314C8C4×Q1612yes—
315C22—21no—
316C22—21no—
317C22#236111yescharacteristic
318C22#336311yescharacteristic
319C22#336411yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4—42no—
325C4—82no—
326C4—82no—
327C4—82no—
328C4C2×C4×Q1611yescharacteristic
329C4#294711yescharacteristic
330C4#295511yescharacteristic
331C4#295611yescharacteristic
332C4#328411yescharacteristic
333C4D4×Q1611yescharacteristic
334C2—21no—
335C2—21no—
336C2#2299811yescharacteristic
337C2#2302011yescharacteristic
338C2#2304311yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radicalC23#548
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
29. #1696C4semidirect
30. #1694C4semidirect
31. #1692C4semidirect
32. #1692C4semidirect
156. #108Q16semidirect

Maximal subgroups

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

subgroupindex
#231702
#227332
#191502
#179982
#98612
#96272
#95892
#89902
#89332
#82942
#82112