Finite Group Databasebeta

← Q8×C3⋊(#41⋊C3) #11923289

Subgroups — Q8×C3⋊(#41⋊C3) #11923289

Order 1152.

561 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5570⋊C3C213yesmaximal, complemented
3C4×C3⋊(#41⋊C3)C213yesmaximal
4Q8×#239C311yesmaximal, characteristic, complemented
5Q8×#41⋊C3—31nomaximal
6C2×C3⋊(#41⋊C3)C2211yescharacteristic
7C3×Q8×Dic3—41nomaximal
8C3×C4×SL(2,𝔽3)C2213yes—
9Dic3×SL(2,𝔽3)C2213yes—
10C3×Q82C2×C311yescharacteristic
11#5730—33no—
12#5728—33no—
13Q8×D12—31no—
14C4×#239C2×C313yes—
15C4×D3×Q8—31no—
16#5570C2×C313yes—
17Q8×SL(2,𝔽3)—21no—
18Q8×SL(2,𝔽3)D311yescharacteristic, complemented
19#557⋊C3—33no—
20C4×#41⋊C3—33no—
21C2×C32×Q8—41no—
22C2×C3×SL(2,𝔽3)C2311yescharacteristic
23C3⋊(#41⋊C3)Q812yesdirect product factor
24C3×C4⋊Dic3—43no—
25C3×C4×Dic3—43no—
26Q8×#41—31noSylow
27C2×#239C22×C311yescharacteristic
28C2×D3×Q8—31no—
29C2×#41⋊C3—31no—
30C3×C4⋊Q8—33no—
31C3×C4×Q8—31no—
32C3×C4×Q8—121no—
33C3×C4×Q8C22×C313yes—
34#946—36no—
35Q8×Dic3—32no—
36Q8×Dic3A411yescharacteristic
37Q8×Dic3C22×C313yes—
38#897—36no—
39#893—33no—
40#892—33no—
41D3×C4⋊C4—33no—
42#890—36no—
43#887—33no—
44C4×D12—33no—
45C42×D3—33no—
46#869—33no—
47C4×Dic6—33no—
48C4×SL(2,𝔽3)—23no—
49C4×SL(2,𝔽3)—33no—
50C4×SL(2,𝔽3)C2×D313yes—
51C32×Q8—44no—
52C2×C32×C4—43no—
53C2×C3×Dic3—41no—
54C2×C3×Dic3—43no—
55C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
56Q82C3×D311yescharacteristic
57#557—33no—
58#554—93no—
59D4×Q8—91no—
60#533—93no—
61C4×#41—33no—
62C2×C4×Q8—91no—
63C2×C3×Q8—32no—
64C2×C3×Q8—41no—
65C2×C3×Q8—81no—
66C2×C3×Q8C2×A411yescharacteristic
67C2×C3×Q8C23×C311yescharacteristic
68#239—32no—
69#239C3×Q812yescomplemented
70D3×Q8—64no—
71C2×D12—31no—
72C2×C4×D3—31no—
73C2×C4×D3—33no—
74C2×C4×D3—33no—
75C2×Dic6—33no—
76#41⋊C3—32no—
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)C22×D311yescharacteristic
79C3×C4⋊C4—33no—
80C3×C4⋊C4—33no—
81C3×C4⋊C4—123no—
82C3×C42—33no—
83C3×C42—123no—
84(C2×D3)⋊C4—36no—
85C4⋊Dic3—36no—
86C4⋊Dic3—33no—
87C4⋊Dic3C2×A413yes—
88Dic3⋊C4—36no—
89C4×Dic3—33no—
90C4×Dic3—33no—
91C4×Dic3C2×A413yes—
92C22×C32—41no—
93C32×C4—46no—
94C3×Dic3—42no—
95C3×Dic3—83no—
96C2×#41—31no—
97C22×Q8—91no—
98C4⋊Q8—33no—
99C4⋊Q8—93no—
100#125—96no—
101C22⋊Q8—96no—
102C22⋊Q8—96no—
103C4×Q8—31no—
104C4×Q8—31no—
105C4×Q8—33no—
106C4×Q8—93no—
107C4×Q8—93no—
108C4×Q8—92no—
109C4×Q8C2×C3×D313yes—
110C4×D4—93no—
111C4×D4—93no—
112#117—93no—
113C2×C4⋊C4—93no—
114C2×C42—93no—
115C22×D3—31no—
116C3×Q8—31no—
117C3×Q8—44no—
118C3×Q8—84no—
119C3×Q8—122no—
120C3×Q8#41⋊C314yescomplemented
121C3×Q8C2×C3×Q811yescharacteristic
122C2×C3×C4—31no—
123C2×C3×C4—33no—
124C2×C3×C4—43no—
125C2×C3×C4—83no—
126C2×C3×C4—121no—
127C2×C3×C4—123no—
128C2×C3×C4C22×A413yes—
129C2×Dic3—33no—
130C2×Dic3C22×A413yes—
131C2×Dic3C22×A411yescharacteristic
132D12—32no—
133D12—32no—
134C4×D3—32no—
135C4×D3—32no—
136C4×D3—66no—
137C4×D3—123no—
138Dic6—66no—
139SL(2,𝔽3)—21no—
140SL(2,𝔽3)D3×Q811yescharacteristic, complemented
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144#41—32no—
145#41—92no—
146C2×Q8—32no—
147C2×Q8—93no—
148C2×Q8—184no—
149C2×Q8D3×A411yescharacteristic
150C2×Q8C22×C3×D311yescharacteristic
151C2×D4—91no—
152C22×C4—91no—
153C22×C4—93no—
154C22×C4—93no—
155C4⋊C4—33no—
156C4⋊C4—33no—
157C4⋊C4—33no—
158C4⋊C4—96no—
159C4⋊C4—93no—
160C4⋊C4—96no—
161C22⋊C4—96no—
162C42—33no—
163C42—33no—
164C42—93no—
165C42—93no—
166C22×C3—41no—
167C22×C3—81no—
168C22×C3C23×A411yescharacteristic
169C2×D3—32no—
170C2×D3—61no—
171C2×D3—61no—
172C3×C4—31no—
173C3×C4—31no—
174C3×C4—46no—
175C3×C4—63no—
176C3×C4—86no—
177C3×C4—122no—
178C3×C4—243no—
179C3×C4C2×#41⋊C316yes—
180Dic3—23no—
181Dic3—63no—
182Dic3Q8×A412yes—
183C32—41noSylow
184C23—91no—
185Q8—31no—
186Q8—122no—
187Q8—186no—
188Q8C3⋊(#41⋊C3)14yesdirect product factor
189Q8C3×D3×Q811yescharacteristic
190D4—92no—
191D4—92no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—33no—
195C2×C4—33no—
196C2×C4—92no—
197C2×C4—92no—
198C2×C4—93no—
199C2×C4—186no—
200C2×C4—363no—
201C2×C4C2×D3×A413yes—
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—81no—
206C2×C3—81no—
207C2×C3—81no—
208C2×C3C2×Q8×A411yescharacteristic
209C2×C3C22×#41⋊C311yescharacteristic
210C2×C3#594511yescharacteristic
211D3—62no—
212C22—92no—
213C22—181no—
214C22—181no—
215C22C22×D3×A411yescharacteristic
216C4—31no—
217C4—31no—
218C4—32no—
219C4—63no—
220C4—63no—
221C4—183no—
222C4C2×C3⋊(#41⋊C3)16yes—
223C3—41no—
224C3—81no—
225C3Q8×#41⋊C311yescharacteristic, complemented
226C2—182no—
227C2C22×C3⋊(#41⋊C3)11yescharacteristic
228C2D3×Q8×A411yescharacteristic
229C2#1059662311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×Q82C2×C3
socleC22×C3C23×A4
prime radicalC2×C3⋊(#41⋊C3)C22
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. #5570⋊C3C2semidirect
4. Q8×#239C3semidirect
18. Q8×SL(2,𝔽3)D3semidirect
23. C3⋊(#41⋊C3)Q8direct (N × H)
55. C3×SL(2,𝔽3)C2×Q8semidirect
69. #239C3×Q8semidirect
120. C3×Q8#41⋊C3semidirect
140. SL(2,𝔽3)D3×Q8semidirect
188. Q8C3⋊(#41⋊C3)direct (N × H)
225. C3Q8×#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
#5570⋊C32
C4×C3⋊(#41⋊C3)2
Q8×#2393
Q8×#41⋊C33
C3×Q8×Dic34