Finite Group Databasebeta

← #11721133

Subgroups — #11721133

Order 864.

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

normal subgroups only

This group has 285 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
1C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
2C2×#6196C211yesmaximal, characteristic, complemented
3C4⋊#1084C211yesmaximal, characteristic, complemented
4#3752⋊C3C211yesmaximal, characteristic
5#3750⋊C3C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic
7#88448C211yesmaximal, characteristic, complemented
8#63899—31nomaximal
9#63895—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17((C3⋊D3)⋊C3)⋊C4—22no—
18#6196—22no—
19C2×#627—31no—
20C2×#627—31no—
21C2×#624—31no—
22C2×D3×Dic3—31no—
23#3752D311yescharacteristic
24#3750D311yescharacteristic, complemented
25C3×(C2×D3)⋊C4—31no—
26C3×C4⋊Dic3—31no—
27C3×C4⋊Dic3—31no—
28C3×C4⋊Dic3—61no—
29C3×Dic3⋊C4—31no—
30#3724—31no—
31Dic3⋊Dic3—31no—
32#3722—31no—
33(C2×D3)⋊Dic3—31no—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊D3)⋊C3—21no—
36C2×(C3⋊D3)⋊C3—21no—
37C2×(C3⋊D3)⋊C3Q812yescomplemented
38C4×H(3,𝔽3)—21no—
39#1084—21no—
40#1084—21no—
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3—21no—
43(C3⋊Dic3)⋊C3D412yes—
44#897—91no—
45#896—91no—
46#878—91no—
47C22×C3⋊D3C2×D311yescharacteristic
48C22×C3×D3—31no—
49C2×C32×C4—21no—
50C2×C32×C4C2×D311yescharacteristic
51C2×C32×C4C2×D311yescharacteristic
52C2×C3⋊Dic3C2×D311yescharacteristic
53C2×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊Dic3C2×D311yescharacteristic
55C2×C3×Dic3—31no—
56C2×C3×Dic3—31no—
57C2×C3×Dic3—31no—
58C2×C3×Dic3—31no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—61no—
63C2×C3×Dic3—61no—
64#627—62no—
65#627—62no—
66#624—62no—
67D3×Dic3—62no—
68C2×H(3,𝔽3)C2×D411yescharacteristic
69C2×H(3,𝔽3)C2×Q811yescharacteristic
70C2×H(3,𝔽3)#4111yescharacteristic
71(C3⋊D3)⋊C3—22no—
72C22×Dic3—91no—
73C2×C4×D3—91no—
74C2×C4×D3—91no—
75C2×Dic6—91no—
76C2×Dic6—91no—
77C2×Dic6—91no—
78C3×C4⋊C4—91no—
79C3×C4⋊C4—91no—
80C3×C22⋊C4—91no—
81#217—91no—
82(C2×D3)⋊C4—31no—
83(C2×D3)⋊C4—91no—
84(C2×D3)⋊C4—91no—
85(C2×D3)⋊C4—91no—
86C4⋊Dic3—31no—
87C4⋊Dic3—31no—
88C4⋊Dic3—61no—
89C4⋊Dic3—91no—
90Dic3⋊C4—31no—
91Dic3⋊C4—91no—
92Dic3⋊C4—91no—
93Dic3⋊C4—181no—
94C22×C32—21no—
95C22×C32C22×D311yescharacteristic
96C22×C32C22×D311yescharacteristic
97C2×C3⋊D3—21no—
98C2×C3⋊D3—21no—
99C2×C3⋊D3Dic612yescomplemented
100C2×C3×D3—32no—
101C2×C3×D3—61no—
102C2×C3×D3—61no—
103C32×C4—21no—
104C32×C4—22no—
105C32×C4—21no—
106C3⋊Dic3—21no—
107C3⋊Dic3—21no—
108C3⋊Dic3D1212yes—
109C3×Dic3—32no—
110C3×Dic3—61no—
111C3×Dic3—61no—
112C3×Dic3—61no—
113C3×Dic3—61no—
114C3×Dic3—61no—
115C3×Dic3—61no—
116C3×Dic3—121no—
117C3×Dic3—121no—
118C22⋊Q8—271noSylow
119H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
120C23×C3—91no—
121C22×D3—31no—
122C22×D3—91no—
123C2×C3×C4—31no—
124C2×C3×C4—61no—
125C2×C3×C4—91no—
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—121no—
131C2×C3×C4D3211yescharacteristic
132C2×Dic3—31no—
133C2×Dic3—31no—
134C2×Dic3—31no—
135C2×Dic3—31no—
136C2×Dic3—31no—
137C2×Dic3—31no—
138C2×Dic3—31no—
139C2×Dic3—61no—
140C2×Dic3—61no—
141C2×Dic3—91no—
142C2×Dic3—181no—
143C2×Dic3—181no—
144C2×Dic3—182no—
145C4×D3—182no—
146C4×D3—182no—
147Dic6—182no—
148Dic6—182no—
149Dic6—182no—
150C2×C32—21no—
151C2×C32—21no—
152C2×C32—21no—
153C2×C32C2×Dic611yescharacteristic
154C2×C32C2×D1211yescharacteristic
155C2×C32D3×D411yescharacteristic
156C2×C32#23711yescharacteristic
157C2×C32#23711yescharacteristic
158C2×C32D3×Q811yescharacteristic
159C3⋊D3—22no—
160C3×D3—62no—
161C2×Q8—271no—
162C22×C4—271no—
163C4⋊C4—91no—
164C4⋊C4—91no—
165C4⋊C4—271no—
166C22⋊C4—91no—
167C22⋊C4—271no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—92no—
171C22×C3—121no—
172C22×C3—181no—
173C22×C3—181no—
174C22×C3C2×D3211yescharacteristic
175C2×D3—32no—
176C2×D3—61no—
177C2×D3—61no—
178C2×D3—92no—
179C2×D3—181no—
180C2×D3—181no—
181C3×C4—21no—
182C3×C4—61no—
183C3×C4—92no—
184C3×C4—121no—
185C3×C4—122no—
186C3×C4—181no—
187C3×C4—181no—
188C3×C4—181no—
189C3×C4—181no—
190Dic3—32no—
191Dic3—61no—
192Dic3—61no—
193Dic3—61no—
194Dic3—61no—
195Dic3—61no—
196Dic3—61no—
197Dic3—121no—
198Dic3—121no—
199Dic3—182no—
200Dic3—181no—
201Dic3—361no—
202C32—21no—
203C32#87811yescharacteristic, complemented
204C32#89711yescharacteristic, complemented
205C23—91no—
206Q8—542no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—542no—
213C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
214C2×C3—31no—
215C2×C3—31no—
216C2×C3—31no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3—121no—
221C2×C3—121no—
222C2×C3—121no—
223C2×C3—182no—
224C2×C3D3×Dic611yescharacteristic
225C2×C3D3×D1211yescharacteristic
226C2×C3#380511yescharacteristic
227D3—62no—
228D3—182no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C4—21no—
234C4—92no—
235C4—181no—
236C4—181no—
237C4—181no—
238C4—181no—
239C3—31no—
240C3—61no—
241C3—121no—
242C3#6389311yescharacteristic
243C2—182no—
244C2((C3⋊D3)⋊C3)⋊Q811yescharacteristic
245C2#8865411yescharacteristic
246C2#8867211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#6196C2semidirect
3. C4⋊#1084C2semidirect
5. #3750⋊C3C2semidirect
7. #88448C2semidirect
24. #3750D3semidirect
37. C2×(C3⋊D3)⋊C3Q8semidirect
99. C2×C3⋊D3Dic6semidirect
119. H(3,𝔽3)C22⋊Q8semidirect
203. C32#878semidirect
204. C32#897semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊C42
C2×#61962
C4⋊#10842
#3752⋊C32
#3750⋊C32
#884502
#884482
#638993
#638953