Finite Group Databasebeta

← #136400

Subgroups — #136400

Order 512.

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

normal subgroups only

This group has 342 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#22686C211yesmaximal, characteristic
2#12886C211yesmaximal, characteristic, complemented
3#12816C211yesmaximal, characteristic
4#11592C211yesmaximal, characteristic
5#9617C211yesmaximal, characteristic, complemented
6#7281C211yesmaximal, characteristic, complemented
7#7168C211yesmaximal, characteristic, complemented
8C2×#444—21no—
9C4×#135C2211yescharacteristic
10#2906—21no—
11#2150C2211yescharacteristic
12C2×#360—21no—
13C4×M32C2211yescharacteristic
14#1947—21no—
15#1946—21no—
16#1866—22no—
17#1848C2211yescharacteristic
18#1844C2211yescharacteristic
19#1806—21no—
20#1770C2211yescharacteristic
21C2×#335C2211yescharacteristic
22#1386—21no—
23#1383—22no—
24#1332C412yescomplemented
25#1331C412yescomplemented
26#1296—21no—
27C2×#135D411yescharacteristic
28C2×#131—22no—
29C4×#41—21no—
30C2×M32—21no—
31C2×M32—21no—
32#444—42no—
33#444—42no—
34C4×Q16C2×C411yescharacteristic
35C4×Q16Q811yescharacteristic
36C4×SD16—21no—
37C4×SD16—21no—
38C4×D8C2×C411yescharacteristic
39C4×D8Q811yescharacteristic
40C2×#108—21no—
41C2×#108D412yes—
42#427D411yescharacteristic
43C2×#107—41no—
44C2×#106—21no—
45C2×#106—21no—
46#424—22no—
47#424—42no—
48#423—21no—
49#423—41no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—41no—
52C2×C4⋊C8D411yescharacteristic
53C2×#104—22no—
54#416—21no—
55C2×#97—21no—
56C2×C4×C8C2311yescharacteristic
57#384—21no—
58C4×C4⋊C4D411yescharacteristic
59C4⋊C16—21no—
60C4⋊C16—21no—
61#360—42no—
62#346—21no—
63C4×C16C2×C412yes—
64#340—41no—
65#339—22no—
66#336—21no—
67#335—21no—
68#335C2×C412yes—
69C2×#41—21no—
70#135—21no—
71#135—21no—
72#135D812yes—
73#135Q1612yes—
74C2×Q16C4⋊C411yescharacteristic
75C2×SD16—21no—
76C2×D8C4⋊C411yescharacteristic
77#131—44no—
78#131—82no—
79C2×M16—22no—
80C2×M16—42no—
81C22×C8—21no—
82C22×C8—42no—
83C22×C8—41no—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C4×Q8—21no—
87C4×Q8—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—41no—
91C4×D4—41no—
92#117—21no—
93#117—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C42—21no—
99C2×C42—41no—
100C2×C42C2×D411yescharacteristic
101M32—41no—
102M32—41no—
103M32—41no—
104M32—41no—
105C2×C16—22no—
106C2×C16—41no—
107#108—42no—
108#108—41no—
109#108—41no—
110#107—82no—
111#107C4⋊C411yescharacteristic
112#107C4⋊C411yescharacteristic
113#106—21no—
114#106—42no—
115#106—41no—
116#106—41no—
117C4⋊C8—22no—
118C4⋊C8—21no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8SD1612yes—
125#104—44no—
126#103—21no—
127#103—21no—
128#102—21no—
129#102—21no—
130#97—41no—
131#97—41no—
132C4×C8C22×C411yescharacteristic
133C4×C8C2×Q811yescharacteristic
134C4×C8#4111yescharacteristic
135C4×C8#4111yescharacteristic
136#95—21no—
137#41—22no—
138#41—22no—
139#41—41no—
140#41—41no—
141C2×Q8—21no—
142C2×D4—21no—
143C2×D4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—81no—
149C22×C4#12011yescharacteristic
150C22×C4#13511yescharacteristic
151C22×C4AGL(1,ℤ/8)11yescharacteristic
152Q16—21no—
153Q16#10712yescomplemented
154SD16—22no—
155SD16—22no—
156D8—21no—
157D8#10712yescomplemented
158M16—42no—
159M16—42no—
160M16—84no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—22no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—84no—
169C2×C8—81no—
170C2×C8—81no—
171C2×C8C2×C4⋊C411yescharacteristic
172C2×C8C4×D411yescharacteristic
173C2×C8C22⋊Q811yescharacteristic
174C2×C8#12311yescharacteristic
175C2×C8C2×D811yescharacteristic
176C2×C8C2×Q1611yescharacteristic
177C2×C8AGL(1,ℤ/8)11yescharacteristic
178C2×C8#13711yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—161no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C42—41no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42—41no—
198C42C4×D411yescharacteristic
199C42C22⋊Q811yescharacteristic
200C42C2×SD1611yescharacteristic
201C42AGL(1,ℤ/8)11yescharacteristic
202C16—42no—
203C16—81no—
204C23—41no—
205C23#44911yescharacteristic
206Q8—22no—
207Q8—41no—
208D4—22no—
209D4—42no—
210D4—42no—
211D4—41no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—161no—
226C2×C4—161no—
227C2×C4—161no—
228C2×C4—161no—
229C2×C4#38511yescharacteristic
230C2×C4C4×SD1611yescharacteristic
231C2×C4#44211yescharacteristic
232C2×C4#44711yescharacteristic
233C2×C4#45011yescharacteristic
234C2×C4#45111yescharacteristic
235C2×C4#45411yescharacteristic
236C2×C4#45511yescharacteristic
237C2×C4#47611yescharacteristic
238C2×C4#47811yescharacteristic
239C8—21no—
240C8—21no—
241C8—42no—
242C8—82no—
243C8—82no—
244C8C2×#10711yescharacteristic
245C8#42811yescharacteristic
246C8#48011yescharacteristic
247C8#48411yescharacteristic
248C22—21no—
249C22—21no—
250C22—42no—
251C22—81no—
252C22—81no—
253C22#162511yescharacteristic
254C22#164411yescharacteristic
255C22#186411yescharacteristic
256C4—21no—
257C4—21no—
258C4—21no—
259C4—42no—
260C4—81no—
261C4—81no—
262C4—161no—
263C4—321no—
264C4#163711yescharacteristic
265C4#163911yescharacteristic
266C4#185211yescharacteristic
267C4#186011yescharacteristic
268C4#218011yescharacteristic
269C4AGL(1,ℤ/16)11yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2#889011yescharacteristic
274C2#1294811yescharacteristic
275C2#1295511yescharacteristic
subgroupG/H
center, socleC22#1864
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#135D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12886C2semidirect
5. #9617C2semidirect
6. #7281C2semidirect
7. #7168C2semidirect
24. #1332C4semidirect
25. #1331C4semidirect
153. Q16#107semidirect
157. D8#107semidirect

Maximal subgroups

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

subgroupindex
#226862
#128862
#128162
#115922
#96172
#72812
#71682