Finite Group Databasebeta

← #135152

Subgroups — #135152

Order 512.

238 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 76 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#17974C211yesmaximal, characteristic
2#12005C211yesmaximal, characteristic, complemented
3#11979C211yesmaximal, characteristic
4#11978C211yesmaximal, characteristic
5#8796C211yesmaximal, characteristic, complemented
6#7244C211yesmaximal, characteristic, complemented
7#7243C211yesmaximal, characteristic, complemented
8#3168C2211yescharacteristic
9#2993—21no—
10#2150C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12#1594C411yescharacteristic
13#1593C411yescharacteristic
14#1590—21no—
15#1580C411yescharacteristic
16#1579C411yescharacteristic
17C8×SD16—21no—
18#1568C2211yescharacteristic
19#1568C2211yescharacteristic
20#1566C2211yescharacteristic
21#1495—21no—
22#1450C2211yescharacteristic
23#1370—21no—
24#1370—21no—
25#1363—21no—
26#1363—21no—
27#1360—21no—
28#1360—21no—
29#1287—21no—
30#1287—21no—
31#529—21no—
32C2×M32—21no—
33C22×C16—21no—
34#499D411yescharacteristic
35#495D411yescharacteristic
36#488—21no—
37#467Q811yescharacteristic
38#466Q811yescharacteristic
39#465—21no—
40#446—21no—
41C8×Q8—21no—
42C4×Q16C2×C411yescharacteristic
43C4×SD16—21no—
44C4×D8C2×C411yescharacteristic
45#436—21no—
46#436—41no—
47#435—41no—
48C8×D4—21no—
49#433—41no—
50#432—21no—
51#432—21no—
52#431D411yescharacteristic
53#431D411yescharacteristic
54#424—41no—
55#423—21no—
56C2×C4×C8C2311yescharacteristic
57C4⋊C16—21no—
58C4⋊C16—21no—
59C4⋊C16—21no—
60C4⋊C16—21no—
61#335—21no—
62#335—21no—
63#334—21no—
64#334—21no—
65#334—21no—
66#334C2×C411yescharacteristic
67#334C2×C411yescharacteristic
68Q8⋊C8—21no—
69Q8⋊C8—21no—
70D4⋊C8—21no—
71D4⋊C8—21no—
72#322C2×C411yescharacteristic
73#322C2×C411yescharacteristic
74C82D411yescharacteristic
75C82D411yescharacteristic
76C2×Q16C4⋊C411yescharacteristic
77C2×SD16—21no—
78C2×D8C4⋊C411yescharacteristic
79C2×M16—41no—
80C2×M16—41no—
81C22×C8—41no—
82C22×C8C2×Q811yescharacteristic
83C22×C8#4111yescharacteristic
84#126—41no—
85#125—21no—
86#124—21no—
87#123—41no—
88C22⋊Q8—21no—
89#121—21no—
90C4×Q8—21no—
91C4×D4—21no—
92C4×D4—41no—
93#117—41no—
94C2×C42C2×D411yescharacteristic
95M32—81no—
96C2×C16—41no—
97C2×C16—41no—
98C2×C16—81no—
99#107C4⋊C411yescharacteristic
100#107C4⋊C411yescharacteristic
101#106—21no—
102C4⋊C8—21no—
103C4⋊C8—21no—
104C4⋊C8—21no—
105C4⋊C8—21no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111#103—21no—
112#103—21no—
113#102—21no—
114#102—21no—
115C22⋊C8—21no—
116C22⋊C8—21no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121#97—41no—
122#97—41no—
123#97—41no—
124C4×C8—41no—
125C4×C8C22⋊C411yescharacteristic
126C4×C8C22⋊C411yescharacteristic
127C4×C8C22⋊C411yescharacteristic
128C4×C8C22⋊C411yescharacteristic
129C4×C8C22×C411yescharacteristic
130C4×C8C2×D411yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8#4111yescharacteristic
133C2×Q8—21no—
134C2×D4—21no—
135C2×D4—41no—
136C22×C4—41no—
137C22×C4C22⋊Q811yescharacteristic
138C22×C4#13511yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140Q16—41no—
141SD16—41no—
142SD16—41no—
143D8—41no—
144M16—161no—
145M16—161no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—81no—
156C2×C8—81no—
157C2×C8—161no—
158C2×C8C2×C4⋊C411yescharacteristic
159C2×C8#11711yescharacteristic
160C2×C8#12111yescharacteristic
161C2×C8C22⋊Q811yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—41no—
170C22⋊C4—41no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C42—41no—
174C42C2×C22⋊C411yescharacteristic
175C42#12111yescharacteristic
176C42#13511yescharacteristic
177C42AGL(1,ℤ/8)11yescharacteristic
178C16—81no—
179C16—81no—
180C23—41no—
181C23#47811yescharacteristic
182Q8—41no—
183Q8—41no—
184D4—41no—
185D4—41no—
186D4—161no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4#38011yescharacteristic
197C2×C4#41611yescharacteristic
198C2×C4#41811yescharacteristic
199C2×C4#46311yescharacteristic
200C2×C4#46311yescharacteristic
201C2×C4#47811yescharacteristic
202C8—21no—
203C8—21no—
204C8—21no—
205C8—41no—
206C8—41no—
207C8—41no—
208C8—41no—
209C8—81no—
210C8—81no—
211C8—161no—
212C8—161no—
213C8#47211yescharacteristic
214C8#47311yescharacteristic
215C22—21no—
216C22—21no—
217C22—21no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22#180411yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—81no—
228C4—81no—
229C4—81no—
230C4#181311yescharacteristic
231C4#181611yescharacteristic
232C2—41no—
233C2—161no—
234C2#868811yescharacteristic
235C2#1282811yescharacteristic
236C2#1283011yescharacteristic
subgroupG/H
center, socleC22#1804
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#466Q8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12005C2semidirect
5. #8796C2semidirect
6. #7244C2semidirect
7. #7243C2semidirect

Maximal subgroups

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

subgroupindex
#179742
#120052
#119792
#119782
#87962
#72442
#72432