Finite Group Databasebeta

← #139096

Subgroups — #139096

Order 512.

321 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 74 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#33772C211yesmaximal, characteristic
2#17989C211yesmaximal, characteristic
3#11965C211yesmaximal, characteristic, complemented
4#11864C211yesmaximal, characteristic
5#9779C211yesmaximal, characteristic, complemented
6#7220C211yesmaximal, characteristic, complemented
7#7147C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9C2×#494C2211yescharacteristic
10#2981—21no—
11#2248C412yes—
12#2241C412yes—
13#2208—22no—
14#2185—22no—
15#2163C2211yescharacteristic
16#2162C2211yescharacteristic
17#1837C2211yescharacteristic
18#1586—22no—
19#1581—22no—
20#1568C2211yescharacteristic
21#1546—21no—
22#1546—21no—
23#1496—21no—
24#1452C2211yescharacteristic
25#1362—22no—
26#1349—22no—
27C22×Q16—21no—
28C22×Q16—21no—
29#532—21no—
30C2×C4⋊Q8—21no—
31C2×SD32—24no—
32#500C2×C411yescharacteristic
33#494—21no—
34#494—21no—
35#494—21no—
36#494C2×C412yes—
37#493C2×C411yescharacteristic
38#486—21no—
39#466—21no—
40#466D412yes—
41C4×D8—21no—
42C4×D8D412yes—
43#436—22no—
44#435—42no—
45C8×D4—22no—
46#432—21no—
47#431D411yescharacteristic
48#431D411yescharacteristic
49#423—21no—
50#408—42no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16—21no—
54#358—24no—
55C22⋊C16—21no—
56C22⋊C16—21no—
57#346—21no—
58C4×C16C2×C412yes—
59#341—21no—
60#341—21no—
61#335D412yes—
62#334—21no—
63#332—42no—
64#332—42no—
65D4⋊C8—24no—
66#322—21no—
67#322—21no—
68C2×#41—41no—
69C22×Q8—41no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16—82no—
75C2×Q16—82no—
76C2×D8—21no—
77C2×D8C22⋊C412yes—
78C2×M16—42no—
79C2×M16—41no—
80C22×C8—42no—
81C22×C8C2×D411yescharacteristic
82C22×C8#4111yescharacteristic
83C4⋊Q8—21no—
84C4⋊Q8—22no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88#127—21no—
89#124—41no—
90#121—22no—
91#121—41no—
92C4×D4—22no—
93C4×D4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C42C2×D411yescharacteristic
97SD32—84no—
98C2×C16—22no—
99C2×C16—41no—
100#107—21no—
101#107C22⋊C412yes—
102C4⋊C8—22no—
103C4⋊C8—21no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8D812yes—
107#102—22no—
108#102—22no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—41no—
114#97—41no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8C22×C411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8C2×D411yescharacteristic
120C4×C8#4111yescharacteristic
121#41—162no—
122C2×Q8—41no—
123C2×Q8—42no—
124C2×Q8—82no—
125C2×Q8—82no—
126C2×Q8—81no—
127C2×D4—22no—
128C2×D4—42no—
129C2×D4—41no—
130C2×D4—41no—
131C22×C4—42no—
132C22×C4—81no—
133C22×C4#12111yescharacteristic
134C22×C4#13511yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136Q16—44no—
137Q16—82no—
138Q16—82no—
139Q16—82no—
140D8—42no—
141D8—81no—
142M16—81no—
143M16—81no—
144M16—162no—
145C2×C8—22no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—42no—
149C2×C8—41no—
150C2×C8—81no—
151C2×C8—162no—
152C2×C8C2×C22⋊C411yescharacteristic
153C2×C8C4×D411yescharacteristic
154C2×C8#12111yescharacteristic
155C2×C8#12311yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42C4×D411yescharacteristic
165C42#12011yescharacteristic
166C42C2×D811yescharacteristic
167C42AGL(1,ℤ/8)11yescharacteristic
168C16—42no—
169C16—81no—
170C23—42no—
171C23#46311yescharacteristic
172Q8—82no—
173Q8—82no—
174Q8—161no—
175Q8—162no—
176D4—42no—
177D4—42no—
178D4—161no—
179D4—161no—
180D4—162no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—162no—
190C2×C4#38611yescharacteristic
191C2×C4C4×D811yescharacteristic
192C2×C4#44211yescharacteristic
193C2×C4#44711yescharacteristic
194C2×C4#44911yescharacteristic
195C2×C4#45911yescharacteristic
196C8—21no—
197C8—21no—
198C8—21no—
199C8—21no—
200C8—42no—
201C8—81no—
202C8—82no—
203C8—161no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22#187911yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—82no—
216C4—162no—
217C4#190311yescharacteristic
218C4#191011yescharacteristic
219C4#219412yes—
220C2—41no—
221C2—162no—
222C2#929311yescharacteristic
223C2#1301911yescharacteristic
224C2#1303511yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1452 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11965C2semidirect
5. #9779C2semidirect
6. #7220C2semidirect
7. #7147C2semidirect

Maximal subgroups

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

subgroupindex
#337722
#179892
#119652
#118642
#97792
#72202
#71472