Finite Group Databasebeta

← #11898355

Subgroups — #11898355

Order 1152.

1080 conjugacy classes of subgroups; only the 233 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 233 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#10596203C212yesmaximal, complemented
2C32×#494C211yesmaximal, characteristic, complemented
3#10592890C212yesmaximal
4#10592884C214yesmaximal, complemented
5Q16×C3⋊Dic3C212yesmaximal, complemented
6#10592782C212yesmaximal, complemented
7C4×#3745C211yesmaximal, characteristic
8C8×#634C211yesmaximal, characteristic, complemented
9#81795—34nomaximal
10C2×C32×Q16C2212yes—
11C2×C32×Q16C2212yes—
12C32×C4⋊Q8C2212yes—
13#64156C2214yes—
14Q8×C3⋊Dic3C2214yes—
15C2×#3755C2214yes—
16C2×#3745C2211yescharacteristic
17#64107—24no—
18#64106—22no—
19#64101—22no—
20C4×C4⋊(C3⋊D3)C2212yes—
21C4×#634C2211yescharacteristic
22C32×C4×C8C2211yescharacteristic
23#63662C2214yes—
24#63648C2212yes—
25#63645C2211yescharacteristic
26#63641C2212yes—
27C8×C3⋊Dic3C2212yes—
28C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
29#5732—38no—
30C3×#494D314yescomplemented
31#5191—38no—
32#5185—316no—
33Dic3×Q16—38no—
34#5083—38no—
35C4×D24—34no—
36C8×Dic6—34no—
37C2×C32×Q8C2314yes—
38C2×C4⋊(C3⋊D3)C2312yes—
39C2×C4×C3⋊D3—22no—
40C2×#634D411yescharacteristic
41C32×Q16—24no—
42C32×Q16—42no—
43C2×C32×C8C2311yescharacteristic
44C2×C32×C8C2311yescharacteristic
45C32×C4⋊C4—22no—
46C32×C4⋊C4—22no—
47C32×C42C2311yescharacteristic
48#3755—44no—
49#3752—22no—
50#3752—24no—
51#3752—24no—
52C4⋊(C3⋊Dic3)—24no—
53C4⋊(C3⋊Dic3)D411yescharacteristic
54C4⋊(C3⋊Dic3)C2312yes—
55#3750D412yes—
56C4×C3⋊Dic3—24no—
57C4×C3⋊Dic3C2312yes—
58C2×C3⋊(C3⋊C8)C2312yes—
59#3745—22no—
60#3393—91noSylow
61C2×C3×Q16C2×D318yes—
62C2×C3×Q16C2×D318yes—
63C3×C4⋊Q8C2×D318yes—
64#947—316no—
65Q8×Dic3—316no—
66C2×#215—316no—
67C2×D24—34no—
68#898—616no—
69#897—68no—
70#892—68no—
71C4×D12—38no—
72C4×Dic6—34no—
73C3×C4×C8C2×D314yes—
74#835—316no—
75#821—38no—
76#818—34no—
77Dic3⋊C8—38no—
78C8×Dic3—38no—
79C4⋊(C3⋊C8)—34no—
80C22×C3⋊D3—22no—
81C32×Q8—24no—
82C32×Q8—24no—
83C2×C32×C4—24no—
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2411yescharacteristic
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3C2×D412yes—
89C2×C3⋊Dic3C2×D412yes—
90C4⋊(C3⋊D3)—22no—
91C4⋊(C3⋊D3)—22no—
92C4×C3⋊D3—44no—
93#634—22no—
94C32×C8—21no—
95C32×C8#4112yes—
96C3⋊(C3⋊C8)—22no—
97#556—92no—
98#495—92no—
99#494C3⋊D311yescharacteristic, complemented
100#464—94no—
101#461—92no—
102C8×Q8—91no—
103C4×Q16—92no—
104C4×D8—91no—
105C2×C3×Q8C22×D3116yes—
106C2×D12—38no—
107C2×C4×D3—68no—
108C2×Dic6—34no—
109C3×Q16—216no—
110C3×Q16—48no—
111C2×C3×C8C22×D314yes—
112C2×C3×C8C22×D314yes—
113C3×C4⋊C4—28no—
114C3×C4⋊C4—28no—
115C3×C42C22×D314yes—
116#215—1216no—
117(C2×D3)⋊C4—616no—
118(C2×D3)⋊C4—616no—
119(C2×D3)⋊C4—68no—
120C4⋊Dic3—34no—
121C4⋊Dic3—38no—
122C4⋊Dic3—616no—
123Dic3⋊C4—38no—
124C4×Dic3—38no—
125C4×Dic3—616no—
126C2×C3⋊C8—38no—
127D24—68no—
128C22×C32C22×D411yescharacteristic
129C2×C3⋊D3—42no—
130C2×C3⋊D3—42no—
131C2×C3⋊D3—42no—
132C32×C4—21no—
133C32×C4—44no—
134C32×C4#13512yes—
135C32×C4C2×#4111yescharacteristic
136C32×C4H(5,𝔽2)11yescharacteristic
137C3⋊Dic3—22no—
138C3⋊Dic3—22no—
139C3⋊Dic3—42no—
140C2×Q16C2×C3⋊D312yes—
141C2×Q16C2×C3⋊D312yes—
142C2×SD16—94no—
143C2×D8—91no—
144C4⋊Q8C2×C3⋊D312yes—
145#126—184no—
146#124—94no—
147C22⋊Q8—182no—
148C4×Q8—91no—
149C4×Q8—94no—
150C4×D4—92no—
151#117—182no—
152#107—91no—
153C4⋊C8—92no—
154C4⋊C8—91no—
155#103—94no—
156#102—92no—
157C4×C8—92no—
158C4×C8C2×C3⋊D311yescharacteristic
159C22×D3—68no—
160C3×Q8—216no—
161C3×Q8—216no—
162C2×C3×C4—216no—
163C2×C3×C4D3×D414yes—
164C2×C3×C4D3×D414yes—
165C2×C3×C4C23×D314yes—
166C2×Dic3—38no—
167C2×Dic3—38no—
168C2×Dic3—68no—
169D12—68no—
170D12—68no—
171C4×D3—1216no—
172Dic6—68no—
173C3×C8—24no—
174C3×C8#23718yes—
175C3⋊C8—68no—
176C2×C32#55011yescharacteristic
177C2×C32C2×#13511yescharacteristic
178C2×C32#57811yescharacteristic
179C3⋊D3—82no—
180C2×Q8—91no—
181C2×Q8C22×C3⋊D314yes—
182C2×D4—92no—
183C22×C4—182no—
184Q16—24no—
185Q16—42no—
186SD16—364no—
187D8—182no—
188C2×C8—92no—
189C2×C8C22×C3⋊D311yescharacteristic
190C2×C8C22×C3⋊D311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—91no—
194C4⋊C4—92no—
195C4⋊C4—92no—
196C4⋊C4—184no—
197C22⋊C4—184no—
198C22⋊C4—184no—
199C22⋊C4—182no—
200C42—92no—
201C42—184no—
202C42C22×C3⋊D311yescharacteristic
203C22×C3C2×D3×D414yes—
204C2×D3—128no—
205C2×D3—128no—
206C2×D3—128no—
207C3×C4—24no—
208C3×C4—416no—
209C3×C4#91918yes—
210C3×C4C2×#23714yes—
211C3×C4#100414yes—
212Dic3—68no—
213Dic3—68no—
214Dic3—128no—
215C32#339311yescharacteristic, complemented, Sylow
216C23—182no—
217Q8—24no—
218Q8—24no—
219Q8—182no—
220D4—182no—
221D4—182no—
222C2×C4—24no—
223C2×C4—92no—
224C2×C4—92no—
225C2×C4—182no—
226C2×C4—364no—
227C2×C4D4×C3⋊D311yescharacteristic
228C2×C4D4×C3⋊D311yescharacteristic
229C2×C4C23×C3⋊D311yescharacteristic
230C8—21no—
231C8—182no—
232C8#383012yes—
233C2×C3#571814yes—
234C2×C3C2×#91914yes—
235C2×C3#576614yes—
236D3—248no—
237C22—362no—
238C22—362no—
239C22—362no—
240C22C2×D4×C3⋊D311yescharacteristic
241C4—21no—
242C4—44no—
243C4—182no—
244C4—182no—
245C4—362no—
246C4#6412812yes—
247C4C2×#383011yescharacteristic
248C4#6436111yescharacteristic
249C3#8179514yescomplemented
250C2—722no—
251C2#1059618911yescharacteristic
252C2C2×#6412811yescharacteristic
253C2#1059623711yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#494C2
socleC22×C32C22×D4
prime radicalC2×#3745C22
hypercenter#494C3⋊D3
hypocenterC32#3393
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #10596203C2semidirect
2. C32×#494C2semidirect
4. #10592884C2semidirect
5. Q16×C3⋊Dic3C2semidirect
6. #10592782C2semidirect
8. C8×#634C2semidirect
30. C3×#494D3semidirect
99. #494C3⋊D3semidirect
215. C32#3393semidirect
249. C3#81795semidirect

Maximal subgroups

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

subgroupindex
#105962032
C32×#4942
#105928902
#105928842
Q16×C3⋊Dic32
#105927822
C4×#37452
C8×#6342
#817953