Finite Group Databasebeta

← #137527

Subgroups — #137527

Order 512.

305 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 90 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#33728C211yesmaximal, characteristic
2#23171C211yesmaximal, characteristic
3#13274C211yesmaximal, characteristic
4#11939C211yesmaximal, characteristic
5#9752C211yesmaximal, characteristic
6#7177C211yesmaximal, characteristic
7#7170C211yesmaximal, characteristic
8#3160C2211yescharacteristic
9#3160C2211yescharacteristic
10#2962—21no—
11#2255—21no—
12#2255—21no—
13#2255C412yescomplemented
14#2255C412yescomplemented
15#2158—22no—
16C2×C4⋊C16C2211yescharacteristic
17C2×#346C2211yescharacteristic
18#1986—21no—
19#1946—21no—
20#1922—22no—
21C4×#108C2211yescharacteristic
22#1773—21no—
23#1770C2211yescharacteristic
24#1564C2211yescharacteristic
25#1503—21no—
26#1481—21no—
27#1382—21no—
28#1382—21no—
29#1365—21no—
30#1364—21no—
31#533—21no—
32#533—21no—
33C22×C16—21no—
34C22×C16—21no—
35#500C2×C411yescharacteristic
36#500C2×C411yescharacteristic
37#500D411yescharacteristic
38#500D411yescharacteristic
39#499—21no—
40#499—21no—
41#499—21no—
42#499C2×C412yes—
43#498—21no—
44#498—21no—
45#446—41no—
46#446—42no—
47C8×Q8—41no—
48#431—21no—
49C2×#108D411yescharacteristic
50C2×#108D411yescharacteristic
51#427—21no—
52#427—21no—
53#427—21no—
54#427D412yes—
55#424—22no—
56C2×C4⋊C8D411yescharacteristic
57#405—22no—
58C4×M16—21no—
59C2×#97—21no—
60C2×C4×C8C2311yescharacteristic
61C4×C4⋊C4D411yescharacteristic
62#367—22no—
63#367—22no—
64#366—22no—
65#366—22no—
66C4⋊C16—21no—
67C4⋊C16—21no—
68#346—21no—
69#346C2×C412yes—
70#341—21no—
71#339—22no—
72#339—22no—
73#336—21no—
74#335—21no—
75#334—21no—
76#328—42no—
77Q8⋊C8—42no—
78C2×M16—21no—
79C2×M16—21no—
80C22×C8—21no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87#125—22no—
88#125—41no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97#117—22no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C42—21no—
101C2×C42C2×D411yescharacteristic
102C2×C16—22no—
103C2×C16—22no—
104C2×C16—22no—
105C2×C16—41no—
106C2×C16—41no—
107C2×C16—41no—
108#108—21no—
109#108—21no—
110#108—21no—
111#108Q1612yes—
112#107—21no—
113#107—21no—
114#107—21no—
115#107—21no—
116#107—21no—
117#107—21no—
118#107C22⋊C412yes—
119#107C22⋊C412yes—
120#106—22no—
121#106—22no—
122#106—41no—
123#106—41no—
124C4⋊C8—21no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127C4⋊C8—41no—
128C4⋊C8D812yes—
129C22⋊C8—41no—
130#97—42no—
131#97—41no—
132#97—41no—
133#97—41no—
134C4×C8—42no—
135C4×C8—41no—
136C4×C8—41no—
137C4×C8C22×C411yescharacteristic
138C4×C8C2×D411yescharacteristic
139C4×C8#4111yescharacteristic
140C4×C8#4111yescharacteristic
141#95—21no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C22×C4—41no—
147C22×C4#12011yescharacteristic
148C22×C4#13511yescharacteristic
149C22×C4AGL(1,ℤ/8)11yescharacteristic
150M16—22no—
151M16—41no—
152M16—41no—
153M16—41no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—81no—
162C2×C8C2×C22⋊C411yescharacteristic
163C2×C8C4×D411yescharacteristic
164C2×C8#12111yescharacteristic
165C2×C8#12311yescharacteristic
166C2×C8C2×Q1611yescharacteristic
167C2×C8#13511yescharacteristic
168C2×C8#13711yescharacteristic
169C2×C8#13711yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C22⋊C4—42no—
184C22⋊C4—81no—
185C42—42no—
186C42—41no—
187C42—41no—
188C42—81no—
189C42C4×D411yescharacteristic
190C42#12111yescharacteristic
191C42C2×D811yescharacteristic
192C42AGL(1,ℤ/8)11yescharacteristic
193C16—24no—
194C16—81no—
195C23#44911yescharacteristic
196Q8—82no—
197Q8—82no—
198Q8—161no—
199Q8—161no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4#38611yescharacteristic
210C2×C4C4×D811yescharacteristic
211C2×C4#44211yescharacteristic
212C2×C4#44711yescharacteristic
213C2×C4#45111yescharacteristic
214C2×C4#45211yescharacteristic
215C2×C4#45611yescharacteristic
216C2×C4#45611yescharacteristic
217C2×C4#45911yescharacteristic
218C2×C4#46311yescharacteristic
219C8—21no—
220C8—21no—
221C8—42no—
222C8—42no—
223C8—81no—
224C8—161no—
225C8C4×Q1611yescharacteristic
226C8#44111yescharacteristic
227C8#46211yescharacteristic
228C8#46411yescharacteristic
229C22—21no—
230C22—21no—
231C22#164811yescharacteristic
232C22#165011yescharacteristic
233C22#187911yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—82no—
239C4—82no—
240C4—161no—
241C4—161no—
242C4—161no—
243C4#164711yescharacteristic
244C4#164911yescharacteristic
245C4#187811yescharacteristic
246C4#188611yescharacteristic
247C2—21no—
248C2—21no—
249C2#905811yescharacteristic
250C2#1302111yescharacteristic
251C2#1302411yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1879
prime radicalC23#449
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
13. #2255C4semidirect
14. #2255C4semidirect

Maximal subgroups

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

subgroupindex
#337282
#231712
#132742
#119392
#97522
#71772
#71702