Finite Group Databasebeta

← #139148

Subgroups — #139148

Order 512.

328 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 84 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#33486C211yesmaximal, characteristic
2#19259C211yesmaximal, characteristic
3#11863C211yesmaximal, characteristic, complemented
4#11650C211yesmaximal, characteristic, complemented
5#8660C211yesmaximal, characteristic
6#7289C211yesmaximal, characteristic, complemented
7#7220C211yesmaximal, characteristic, complemented
8#3170C2211yescharacteristic
9#3110—21no—
10C2×C8×D4C2211yescharacteristic
11D4×C16—22no—
12D4×C16—22no—
13#2162C2211yescharacteristic
14#2162C2211yescharacteristic
15C2×C22⋊C16—22no—
16#1837C2211yescharacteristic
17#1679—22no—
18#1673C412yes—
19#1669C412yes—
20#1568C2211yescharacteristic
21#1564C2211yescharacteristic
22#1552—21no—
23#1486—21no—
24#1350—22no—
25#1332—22no—
26#1301—21no—
27C23×C8—21no—
28#533—21no—
29C2×C4×D4D411yescharacteristic
30C22×C16—24no—
31#500C2×C411yescharacteristic
32#494C2×C411yescharacteristic
33#487—21no—
34#484—41no—
35#474—22no—
36#467—21no—
37#467—21no—
38#461—22no—
39#450—41no—
40#440—42no—
41C4×Q16—21no—
42C4×Q16—21no—
43C8×D4—21no—
44C8×D4—21no—
45C8×D4C2×C412yes—
46C8×D4D412yes—
47#432—21no—
48#431—21no—
49#423—21no—
50C2×C4⋊C8D411yescharacteristic
51C2×C22⋊C8—21no—
52C2×C4×C8C2311yescharacteristic
53C4⋊C16—21no—
54C4⋊C16—21no—
55C22⋊C16—24no—
56C22⋊C16—21no—
57C22⋊C16D412yes—
58C4×C16—21no—
59C4×C16D412yes—
60#341—21no—
61#341—21no—
62#335—21no—
63#335C2×C412yes—
64#334—21no—
65#334—21no—
66#332—22no—
67#332—22no—
68D4⋊C8—42no—
69C22×D4SD1611yescharacteristic
70C23×C4—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—42no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—82no—
79C22×C8C2×D411yescharacteristic
80C22×C8#4111yescharacteristic
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83#125—41no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C4×Q8—41no—
87C4×Q8—41no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4C22⋊C412yes—
91C4×D4D812yes—
92#117—41no—
93C2×C4⋊C4SD1611yescharacteristic
94C2×C22⋊C4—21no—
95C2×C42C2×D411yescharacteristic
96C2×C16—22no—
97C2×C16—22no—
98C2×C16—84no—
99#107—21no—
100#107—21no—
101#107—42no—
102C4⋊C8—22no—
103C4⋊C8—21no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8C22⋊C412yes—
108#103—41no—
109#103—41no—
110#103—42no—
111#102—42no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116#97—41no—
117C4×C8—41no—
118C4×C8C22×C411yescharacteristic
119C4×C8C2×D411yescharacteristic
120C4×C8C2×D411yescharacteristic
121C4×C8#4111yescharacteristic
122C24—21no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×D4—24no—
126C2×D4—21no—
127C2×D4#10212yes—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4#12111yescharacteristic
134C22×C4C2×SD1611yescharacteristic
135C22×C4#13511yescharacteristic
136Q16—42no—
137Q16—81no—
138SD16—84no—
139M16—81no—
140M16—81no—
141C2×C8—22no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—44no—
145C2×C8—42no—
146C2×C8—82no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8C4×D411yescharacteristic
151C2×C8C4×D411yescharacteristic
152C2×C8#12011yescharacteristic
153C2×C8#12311yescharacteristic
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C4⋊C4#10212yes—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—81no—
165C42—81no—
166C42C2×C22⋊C411yescharacteristic
167C42#12111yescharacteristic
168C42C2×D811yescharacteristic
169C42#13511yescharacteristic
170C16—42no—
171C16—42no—
172C23—24no—
173C23—22no—
174C23—81no—
175C23—81no—
176C23#46511yescharacteristic
177Q8—81no—
178Q8—82no—
179Q8—81no—
180D4—24no—
181D4—22no—
182D4#35914yes—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4#38611yescharacteristic
195C2×C4C2×#10211yescharacteristic
196C2×C4#41611yescharacteristic
197C2×C4#46511yescharacteristic
198C2×C4#46611yescharacteristic
199C2×C4#46611yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—42no—
205C8—82no—
206C8—161no—
207C8—161no—
208C22—21no—
209C22—24no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—84no—
215C22#189311yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—42no—
221C4—161no—
222C4—161no—
223C4#188111yescharacteristic
224C4#190411yescharacteristic
225C4C2×#35912yes—
226C2—41no—
227C2—44no—
228C2#930511yescharacteristic
229C2#1306311yescharacteristic
230C2#1306911yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
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. #11863C2semidirect
4. #11650C2semidirect
6. #7289C2semidirect
7. #7220C2semidirect

Maximal subgroups

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

subgroupindex
#334862
#192592
#118632
#116502
#86602
#72892
#72202