Finite Group Databasebeta

← #128883

Subgroups — #128883

Order 512.

381 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21602C211yesmaximal, characteristic
2#11721C211yesmaximal, characteristic
3#11715C211yesmaximal, characteristic
4#10803C211yesmaximal, characteristic
5#8156C211yesmaximal, characteristic
6#7253C211yesmaximal, characteristic
7#7252C211yesmaximal, characteristic
8C2×C8×Q8C411yescharacteristic, complemented
9C2×#405C2211yescharacteristic
10#2396C2211yescharacteristic
11#1985C411yescharacteristic, complemented
12#1940C411yescharacteristic, complemented
13#1916—21no—
14#1850—21no—
15#1798C2211yescharacteristic
16#1771—21no—
17#1770C411yescharacteristic, complemented
18#1753—21no—
19C2×#344—21no—
20#1742C2211yescharacteristic
21#1740C2211yescharacteristic
22#1729—21no—
23#1458C2211yescharacteristic
24#1458C2211yescharacteristic
25#1450—21no—
26#1354C412yescomplemented
27#1353C412yescomplemented
28#1352C412yes—
29#1351C412yes—
30#1281—21no—
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×#117—21no—
36C2×#117C2311yescharacteristic
37C8×Q8—41no—
38C8×Q8—41no—
39#431—21no—
40#431—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C22⋊C8—21no—
46#405—41no—
47#405—41no—
48C2×#97—21no—
49C2×#97C2×C411yescharacteristic
50C2×#97D411yescharacteristic
51C2×C4×C8—21no—
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#400C2×C411yescharacteristic
55#399C2×C411yescharacteristic
56#398—21no—
57#396—21no—
58#395C2×C411yescharacteristic
59#393C2×C411yescharacteristic
60C4×C4⋊C4—21no—
61C4×C4⋊C4C2×C411yescharacteristic
62C4×C22⋊C4—21no—
63#344—42no—
64#343—21no—
65#343C2×C412yes—
66#342—21no—
67#342C2×C412yes—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—41no—
74#336—41no—
75#324—21no—
76#324C2×C412yes—
77#323—21no—
78#323C2×C412yes—
79#322—41no—
80C82—41no—
81C22×Q8C4211yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—22no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16SD1612yes—
92C2×M16Q1612yes—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8C22⋊C411yescharacteristic
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105C4×Q8—41no—
106C4×Q8—41no—
107#117—41no—
108#117—41no—
109#117—41no—
110#117—41no—
111#117—41no—
112#117—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C4211yescharacteristic
119C2×C4⋊C4C22×C411yescharacteristic
120C2×C4⋊C4#4111yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4C22×C411yescharacteristic
125C2×C22⋊C4#4111yescharacteristic
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—41no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—42no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142#97—41no—
143#97—41no—
144#97—42no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—41no—
150C4×C8—41no—
151C4×C8—42no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95C4211yescharacteristic
157#95C4211yescharacteristic
158C24C4×D411yescharacteristic
159C2×Q8—42no—
160C2×Q8—41no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4C2×C4211yescharacteristic
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4#11711yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4C4×D411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4C2×SD1611yescharacteristic
177C22×C4C2×Q1611yescharacteristic
178C22×C4#13511yescharacteristic
179C22×C4#13511yescharacteristic
180M16—42no—
181M16—42no—
182M16—42no—
183M16—42no—
184M16—81no—
185M16—81no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8#9911yescharacteristic
211C2×C8#9911yescharacteristic
212C2×C8#10312yes—
213C2×C8#10312yes—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—81no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—81no—
238C42—81no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23C4×C22⋊C411yescharacteristic
244C23C4×SD1611yescharacteristic
245C23C4×Q1611yescharacteristic
246Q8—42no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
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—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4C2×#9911yescharacteristic
282C2×C4#41011yescharacteristic
283C2×C4C2×#10311yescharacteristic
284C2×C4#41611yescharacteristic
285C2×C4#42111yescharacteristic
286C2×C4#42111yescharacteristic
287C2×C4C4×SD1611yescharacteristic
288C2×C4C4×Q1611yescharacteristic
289C8—21no—
290C8—21no—
291C8—22no—
292C8—22no—
293C8—41no—
294C8—82no—
295C8—82no—
296C8—82no—
297C8—81no—
298C8—81no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C4×#9911yescharacteristic
309C22#175911yescharacteristic
310C22C4×#10311yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—162no—
322C4#151611yescharacteristic
323C4#151811yescharacteristic
324C4#152011yescharacteristic
325C4#152111yescharacteristic
326C2—21no—
327C2—21no—
328C2—41no—
329C2—41no—
330C2#822111yescharacteristic
331C2#1262311yescharacteristic
332C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C4×#99
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×C8×Q8C4semidirect
11. #1985C4semidirect
12. #1940C4semidirect
17. #1770C4semidirect
26. #1354C4semidirect
27. #1353C4semidirect

Maximal subgroups

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

subgroupindex
#216022
#117212
#117152
#108032
#81562
#72532
#72522