Finite Group Databasebeta

← #134456

Subgroups — #134456

Order 512.

395 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 98 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#21601C211yesmaximal, characteristic
2#11715C212yesmaximal
3#10866C211yesmaximal, characteristic
4#8601C211yesmaximal, characteristic
5#7252C212yesmaximal
6C2×C8×Q8C411yescharacteristic, complemented
7C2×#427—21no—
8C2×#405C2211yescharacteristic
9#2396C2211yescharacteristic
10#1985C411yescharacteristic, complemented
11#1940C411yescharacteristic, complemented
12#1917—21no—
13#1851—21no—
14#1798C2211yescharacteristic
15C8×C4⋊C4—21no—
16C8×C4⋊C4C411yescharacteristic, complemented
17C8×C22⋊C4—21no—
18#1740C2212yes—
19C2×H(3,ℤ/4)—21no—
20#1564—21no—
21#1458C2212yes—
22#1352—22no—
23#1351—22no—
24#1275—21no—
25C22×M16D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×#117—22no—
29C2×#117C2311yescharacteristic
30C8×Q8—41no—
31C8×Q8—41no—
32#431—22no—
33#427—24no—
34#427—42no—
35C2×#107—41no—
36C2×#106—41no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—21no—
42#405—41no—
43#405—41no—
44C2×#97C2×C411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8—21no—
47C2×C4×C8—21no—
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#400Q811yescharacteristic
51#399D411yescharacteristic
52#398—21no—
53#396—21no—
54#395Q811yescharacteristic
55#393D411yescharacteristic
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2×C411yescharacteristic
58C4×C22⋊C4—21no—
59#342—22no—
60#342—22no—
61H(3,ℤ/4)—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#335—41no—
68#334—41no—
69#324—22no—
70#323—22no—
71C22×Q8C4⋊C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×M16—21no—
74C2×M16—21no—
75C2×M16—41no—
76C2×M16—41no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8SD1612yes—
87C22×C8Q1612yes—
88C4×Q8—41no—
89C4×Q8—41no—
90#117—24no—
91#117—41no—
92#117—41no—
93#117—42no—
94#117—42no—
95#117—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C4⋊C411yescharacteristic
102C2×C4⋊C4C2×Q811yescharacteristic
103C2×C4⋊C4#4111yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4C2×D411yescharacteristic
108C2×C22⋊C4#4111yescharacteristic
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—42no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114#107—44no—
115#106—44no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—42no—
124#97—41no—
125#97—41no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—42no—
131C4×C8—41no—
132C4×C8—41no—
133C4×C8—42no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95C4⋊C411yescharacteristic
139#95C4⋊C411yescharacteristic
140C24#12111yescharacteristic
141C2×Q8—42no—
142C2×Q8—41no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—82no—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C4⋊C411yescharacteristic
154C22×C4#11711yescharacteristic
155C22×C4#12111yescharacteristic
156C22×C4C22⋊Q811yescharacteristic
157C22×C4C22⋊Q811yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159C22×C4C2×Q1611yescharacteristic
160C22×C4AGL(1,ℤ/8)11yescharacteristic
161C22×C4#13711yescharacteristic
162M16—42no—
163M16—42no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—82no—
183C2×C8#9911yescharacteristic
184C2×C8#9911yescharacteristic
185C2×C8#10312yes—
186C2×C8#10312yes—
187C2×C8#10412yes—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—82no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—44no—
202C22⋊C4—82no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—44no—
208C42—41no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42—82no—
213C23—21no—
214C23—21no—
215C23—41no—
216C23—41no—
217C23#38011yescharacteristic
218C23#46011yescharacteristic
219C23#46211yescharacteristic
220Q8—42no—
221Q8—42no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—84no—
253C2×C4—162no—
254C2×C4—162no—
255C2×C4C2×#9911yescharacteristic
256C2×C4#41011yescharacteristic
257C2×C4C2×#10311yescharacteristic
258C2×C4#41711yescharacteristic
259C2×C4C2×#10411yescharacteristic
260C2×C4#42111yescharacteristic
261C2×C4#47511yescharacteristic
262C2×C4#47711yescharacteristic
263C8—22no—
264C8—41no—
265C8—41no—
266C8—81no—
267C8—81no—
268C8—162no—
269C8#32812yes—
270C8#32812yes—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—81no—
280C22#179911yescharacteristic
281C22#180311yescharacteristic
282C22#180611yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—41no—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—164no—
294C4C2×#32811yescharacteristic
295C4#150911yescharacteristic
296C4#151311yescharacteristic
297C4#151811yescharacteristic
298C2—21no—
299C2—21no—
300C2—41no—
301C2—41no—
302C2#857211yescharacteristic
303C2#1280511yescharacteristic
304C2#1280711yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC24#121
LCSG ▷ C2×C42 ▷ 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
6. C2×C8×Q8C4semidirect
10. #1985C4semidirect
11. #1940C4semidirect
16. C8×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#216012
#117152
#108662
#86012
#72522