Finite Group Databasebeta

← #13147282

Subgroups — #13147282

Order 1728.

433 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720765C212yesmaximal, complemented
3#10596283C311yesmaximal, characteristic, complemented
4#5812⋊C3—31nomaximal
5C2×Q8⋊H(3,𝔽3)C412yescomplemented
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#3757⋊C3—41nomaximal
8C22×C32×Q8C2×C311yescharacteristic
9C22×C3×SL(2,𝔽3)—21no—
10C22×C3×SL(2,𝔽3)D311yescharacteristic
11Q8×C3⋊Dic3—32no—
12Q8×C3⋊Dic3C2×C312yescomplemented
13#64135—31no—
14#64135—31no—
15Dic3×SL(2,𝔽3)—32no—
16C23×H(3,𝔽3)—41no—
17C2×(C3⋊Dic3)⋊C3—42no—
18Q8⋊H(3,𝔽3)C2×C411yescharacteristic
19#5812—31no—
20#5812—91no—
21#519⋊C3—91no—
22C2×C32×Q8—32no—
23C2×C32×Q8—31no—
24C2×C32×Q8C3×C412yescomplemented
25C2×C32×Q8C22×C311yescharacteristic
26C22×C32×C4—31no—
27C2×C3×SL(2,𝔽3)—22no—
28C2×C3×SL(2,𝔽3)—21no—
29C2×C3×SL(2,𝔽3)Dic312yescomplemented
30C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
31#3757—31no—
32#3757A411yescharacteristic
33C4⋊(C3⋊Dic3)—32no—
34C4⋊(C3⋊Dic3)—32no—
35C4×C3⋊Dic3—32no—
36C4×C3⋊Dic3—32no—
37C3×#217—121no—
38C22×H(3,𝔽3)—41no—
39C22×H(3,𝔽3)—42no—
40C22×H(3,𝔽3)—82no—
41(C3⋊Dic3)⋊C3—82no—
42C22×C3×Q8—31no—
43C22×C3×Q8C3×D311yescharacteristic
44C22×SL(2,𝔽3)—31no—
45C22×SL(2,𝔽3)—61no—
46Q8×Dic3—32no—
47Q8×Dic3—92no—
48Q8×Dic3—96no—
49Q8×Dic3—92no—
50#926—91no—
51#926—93no—
52#926—93no—
53#926—91no—
54C4×SL(2,𝔽3)—92no—
55C23×C32—41no—
56C23×C32—81no—
57C23×C32C2×A411yescharacteristic
58C32×Q8—32no—
59C32×Q8—31no—
60C32×Q8—32no—
61C32×Q8C2×C3×C411yescharacteristic
62C2×C32×C4—32no—
63C2×C32×C4—32no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C3⋊Dic3—32no—
67C2×C3⋊Dic3C2×A412yes—
68C2×C3×Dic3—122no—
69C3×SL(2,𝔽3)—21no—
70C3×SL(2,𝔽3)C2×Dic311yescharacteristic
71#519—91noSylow
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—42no—
74C2×H(3,𝔽3)—82no—
75C2×C3×Q8—33no—
76C2×C3×Q8—36no—
77C2×C3×Q8—31no—
78C2×C3×Q8—32no—
79C2×C3×Q8—32no—
80C2×C3×Q8—31no—
81C2×C3×Q8C3×Dic312yes—
82C2×C3×Q8C2×C3×D311yescharacteristic
83C22×C3×C4—33no—
84C22×C3×C4—31no—
85C2×SL(2,𝔽3)—32no—
86C2×SL(2,𝔽3)—31no—
87C2×SL(2,𝔽3)—62no—
88C2×SL(2,𝔽3)—61no—
89C3×C22⋊C4—361no—
90#217—31no—
91#217—91no—
92#217—93no—
93#217—91no—
94C4⋊Dic3—92no—
95C4⋊Dic3—96no—
96C4⋊Dic3—96no—
97C4⋊Dic3—92no—
98C4×Dic3—92no—
99C4×Dic3—96no—
100C4×Dic3—96no—
101C4×Dic3—92no—
102C22×C32—22no—
103C22×C32—41no—
104C22×C32—42no—
105C22×C32—82no—
106C22×C32—84no—
107C22×C32—81no—
108C22×C32—82no—
109C22×C32C4×A412yes—
110C22×C32C22×A411yescharacteristic
111C32×C4—31no—
112C32×C4—32no—
113C32×C4—31no—
114C3⋊Dic3—22no—
115C3⋊Dic3—62no—
116C3×Dic3—242no—
117C22×Q8(C3⋊D3)⋊C311yescharacteristic
118C4×Q8—92no—
119C4×Q8—272no—
120#117—271no—
121#117—271no—
122H(3,𝔽3)—41noSylow
123C23×C3—31no—
124C23×C3—121no—
125C23×C3—241no—
126C23×C3D3×A411yescharacteristic
127C3×Q8—32no—
128C3×Q8—31no—
129C3×Q8—36no—
130C3×Q8—33no—
131C3×Q8—36no—
132C3×Q8—32no—
133C3×Q8—31no—
134C3×Q8C2×C3×Dic311yescharacteristic
135C2×C3×C4—33no—
136C2×C3×C4—33no—
137C2×C3×C4—36no—
138C2×C3×C4—36no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—362no—
144C2×Dic3—32no—
145C2×Dic3—92no—
146C2×Dic3—96no—
147C2×Dic3—92no—
148SL(2,𝔽3)—31no—
149SL(2,𝔽3)—61no—
150C2×C32—22no—
151C2×C32—42no—
152C2×C32—41no—
153C2×C32—82no—
154C2×C32—84no—
155C2×C32—82no—
156C2×C32—81no—
157C2×C32C2×C4×A411yescharacteristic
158C2×C322−1+4⋊C312yes—
159C2×Q8—31no—
160C2×Q8—32no—
161C2×Q8(C3⋊Dic3)⋊C312yescomplemented
162C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
163C22×C4—31no—
164C4⋊C4—272no—
165C4⋊C4—272no—
166C22⋊C4—91no—
167C22⋊C4—271no—
168C42—272no—
169C42—272no—
170C22×C3—22no—
171C22×C3—32no—
172C22×C3—31no—
173C22×C3—62no—
174C22×C3—122no—
175C22×C3—121no—
176C22×C3—242no—
177C22×C3—244no—
178C22×C3—242no—
179C22×C3—241no—
180C22×C3Dic3×A412yes—
181C22×C3C2×D3×A411yescharacteristic
182C3×C4—33no—
183C3×C4—36no—
184C3×C4—33no—
185C3×C4—31no—
186C3×C4—32no—
187C3×C4—31no—
188C3×C4—722no—
189Dic3—62no—
190Dic3—182no—
191Dic3—186no—
192Dic3—182no—
193C32—41no—
194C32—81no—
195C32#519⋊C311yescharacteristic, complemented
196C23(C3⋊D3)⋊A411yescharacteristic
197Q8—32no—
198Q8—31no—
199Q8—32no—
200Q8C2×(C3⋊Dic3)⋊C311yescharacteristic
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—92no—
206C2×C4—272no—
207C2×C3—22no—
208C2×C3—32no—
209C2×C3—31no—
210C2×C3—62no—
211C2×C3—122no—
212C2×C3—121no—
213C2×C3—242no—
214C2×C3—244no—
215C2×C3—242no—
216C2×C3—241no—
217C2×C3#1007⋊C312yes—
218C2×C3C2×Dic3×A411yescharacteristic
219C22—22no—
220C22(C3⋊Dic3)⋊A412yes—
221C22C2×(C3⋊D3)⋊A411yescharacteristic
222C4—31no—
223C4—32no—
224C4—31no—
225C4—182no—
226C4—542no—
227C3—31no—
228C3—121no—
229C3—241no—
230C3#5812⋊C311yescharacteristic
231C2—22no—
232C2#1172243212yes—
233C2C2×(C3⋊Dic3)⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, hypocenterC32×Q8C2×C3×C4
Frattini, socleC22×C3C2×D3×A4
FittingC22×C32×Q8C2×C3
prime radicalC22×Q8⋊H(3,𝔽3)C2
cosocleC2×C32×Q8C22×C3
hypercenterC23(C3⋊D3)⋊A4
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11720765C2semidirect
3. #10596283C3semidirect
5. C2×Q8⋊H(3,𝔽3)C4semidirect
12. Q8×C3⋊Dic3C2×C3semidirect
24. C2×C32×Q8C3×C4semidirect
29. C2×C3×SL(2,𝔽3)Dic3semidirect
161. C2×Q8(C3⋊Dic3)⋊C3semidirect
195. C32#519⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
#117207652
#105962833
#5812⋊C33
#3757⋊C34