Finite Group Databasebeta

← #136988

Subgroups — #136988

Order 512.

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

normal subgroups only

This group has 408 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#22657C211yesmaximal, characteristic
2#19105C211yesmaximal, characteristic
3#13287C211yesmaximal, characteristic, complemented
4#13278C211yesmaximal, characteristic, complemented
5#11583C211yesmaximal, characteristic
6#7278C211yesmaximal, characteristic, complemented
7#7175C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9#3101—21no—
10C2×C8×D4C2211yescharacteristic
11C2×#427—21no—
12#2354—21no—
13#2150C2211yescharacteristic
14C4×M32C2211yescharacteristic
15#1947—21no—
16#1854—22no—
17#1844C2211yescharacteristic
18#1807—21no—
19C4×#106C2211yescharacteristic
20#1675C412yescomplemented
21#1673C412yescomplemented
22#1671—22no—
23#1564C2211yescharacteristic
24#1542—21no—
25#1386—21no—
26#1386—21no—
27#1386—22no—
28D4⋊C16—22no—
29C23×C8—21no—
30#533—21no—
31C2×C4×D4D411yescharacteristic
32C2×#117—22no—
33C2×M32—21no—
34C2×M32—21no—
35#500C2×C411yescharacteristic
36#500C2×C411yescharacteristic
37#499—21no—
38#499—21no—
39#498—21no—
40#483—41no—
41#476—22no—
42#461—22no—
43#450—41no—
44C4×SD16—42no—
45C8×D4—21no—
46C8×D4—21no—
47C8×D4C2×C412yes—
48C8×D4D412yes—
49#431—21no—
50#427—21no—
51#427—21no—
52#427—42no—
53#427—42no—
54C2×#107—22no—
55C2×#106—21no—
56C2×#106D411yescharacteristic
57C2×#106D411yescharacteristic
58C2×#106D412yes—
59C2×C4⋊C8D411yescharacteristic
60C2×C22⋊C8—21no—
61C2×C4×C8C2311yescharacteristic
62#390—41no—
63#388—42no—
64#384—21no—
65#380—22no—
66C4×C4⋊C4—21no—
67C4×C4⋊C4—41no—
68C4⋊C16—21no—
69C4⋊C16—21no—
70#346—21no—
71C4×C16—21no—
72#340—22no—
73#340—21no—
74#335C2×C412yes—
75#334—21no—
76#333—22no—
77#332—22no—
78C22×D4D811yescharacteristic
79C23×C4—21no—
80C2×SD16—42no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92#125—41no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4C22⋊C412yes—
100C4×D4SD1612yes—
101#117—44no—
102#117—41no—
103#117—82no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4D811yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C42—42no—
114C2×C42—41no—
115C2×C42C2×D411yescharacteristic
116M32—22no—
117M32—41no—
118M32—41no—
119M32—41no—
120C2×C16—41no—
121C2×C16—41no—
122#107—21no—
123#107—21no—
124#107—21no—
125#107—21no—
126#107—44no—
127#106—21no—
128#106—21no—
129#106—21no—
130#106—41no—
131#106—41no—
132#106—41no—
133#106—41no—
134#106—42no—
135#106—42no—
136#106SD1612yescomplemented
137C4⋊C8—22no—
138C4⋊C8—21no—
139C4⋊C8C22⋊C412yes—
140#103—42no—
141#102—42no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—41no—
145C4×C8—41no—
146C4×C8C22×C411yescharacteristic
147C4×C8C2×D411yescharacteristic
148C4×C8#4111yescharacteristic
149C4×C8#4111yescharacteristic
150#95—42no—
151#95—41no—
152C24—21no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×D4—21no—
156C2×D4—42no—
157C2×D4#10212yes—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4#12011yescharacteristic
169C22×C4C2×D811yescharacteristic
170C22×C4AGL(1,ℤ/8)11yescharacteristic
171SD16—84no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8C4×D411yescharacteristic
183C2×C8C4×D411yescharacteristic
184C2×C8#12111yescharacteristic
185C2×C8#12111yescharacteristic
186C2×C8C2×SD1611yescharacteristic
187C2×C8#13511yescharacteristic
188C2×C8AGL(1,ℤ/8)11yescharacteristic
189C2×C8#13711yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—82no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—161no—
205C4⋊C4#10212yes—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—84no—
209C22⋊C4—81no—
210C22⋊C4—162no—
211C42—82no—
212C42—84no—
213C42—81no—
214C42C2×C22⋊C411yescharacteristic
215C42#12311yescharacteristic
216C42C2×SD1611yescharacteristic
217C42#13711yescharacteristic
218C16—42no—
219C16—81no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23#44711yescharacteristic
226Q8—82no—
227Q8—161no—
228D4—22no—
229D4—22no—
230D4—42no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4#38611yescharacteristic
252C2×C4C2×#10211yescharacteristic
253C2×C4#41711yescharacteristic
254C2×C4#44811yescharacteristic
255C2×C4#45011yescharacteristic
256C2×C4#45211yescharacteristic
257C2×C4#45511yescharacteristic
258C2×C4#45611yescharacteristic
259C2×C4#48011yescharacteristic
260C2×C4#48111yescharacteristic
261C8—21no—
262C8—21no—
263C8—42no—
264C8—82no—
265C8C4×SD1611yescharacteristic
266C8#44011yescharacteristic
267C8#46111yescharacteristic
268C8#46311yescharacteristic
269C22—21no—
270C22—21no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—82no—
275C22—82no—
276C22#163911yescharacteristic
277C22#164711yescharacteristic
278C22#187411yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—42no—
283C4—82no—
284C4—162no—
285C4—161no—
286C4—161no—
287C4—161no—
288C4#164311yescharacteristic
289C4#165111yescharacteristic
290C4#187711yescharacteristic
291C4#188211yescharacteristic
292C4#214512yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#898511yescharacteristic
297C2#1300211yescharacteristic
298C2#1300511yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13287C2semidirect
4. #13278C2semidirect
6. #7278C2semidirect
7. #7175C2semidirect
20. #1675C4semidirect
21. #1673C4semidirect
136. #106SD16semidirect

Maximal subgroups

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

subgroupindex
#226572
#191052
#132872
#132782
#115832
#72782
#71752