Finite Group Databasebeta

← #11721150

Subgroups — #11721150

Order 864.

293 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 56 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×#6199C211yesmaximal, characteristic
2C2×#6196C211yesmaximal, characteristic, complemented
3#88542C211yesmaximal, characteristic
4C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5#3750⋊C3C211yesmaximal, characteristic, complemented
6#88451C211yesmaximal, characteristic
7#88448C211yesmaximal, characteristic, complemented
8#63899—31nomaximal
9#63893—31nomaximal
10C22×#272C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#6199—22no—
18#6196—22no—
19C2×#627—31no—
20C2×#627—31no—
21C2×D3×Dic3—31no—
22C2×D3×Dic3—31no—
23C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
24#3750D311yescharacteristic, complemented
25C3×(C2×D3)⋊C4—31no—
26C3×(C2×D3)⋊C4—31no—
27C3×(C2×D3)⋊C4—61no—
28C3×C4⋊Dic3—31no—
29C3×Dic3⋊C4—31no—
30#3724—31no—
31Dic3⋊Dic3—31no—
32(C2×D3)⋊Dic3—31no—
33(C2×D3)⋊Dic3—31no—
34C22×H(3,𝔽3)C2311yescharacteristic
35C2×#272—21no—
36C2×#272—21no—
37C2×#272Q812yescomplemented
38C4×H(3,𝔽3)—21no—
39#1084—21no—
40(C3⋊Dic3)⋊C3—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×D3—31no—
48C22×C3×D3—31no—
49C22×C3×D3—61no—
50C2×C32×C4—21no—
51C2×C32×C4C2×D311yescharacteristic
52C2×C32×C4C2×D311yescharacteristic
53C2×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊Dic3C2×D311yescharacteristic
55C2×C3⋊Dic3C2×D311yescharacteristic
56C2×C3⋊Dic3C2×D311yescharacteristic
57C2×C3×Dic3—31no—
58C2×C3×Dic3—31no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—61no—
64#627—62no—
65#627—62no—
66D3×Dic3—62no—
67D3×Dic3—62no—
68C2×H(3,𝔽3)C2×D411yescharacteristic
69C2×H(3,𝔽3)C2×Q811yescharacteristic
70C2×H(3,𝔽3)#4111yescharacteristic
71#272—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—31no—
84(C2×D3)⋊C4—61no—
85(C2×D3)⋊C4—91no—
86(C2×D3)⋊C4—91no—
87C4⋊Dic3—31no—
88C4⋊Dic3—91no—
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—32no—
98C2×C3×D3—32no—
99C2×C3×D3—62no—
100C2×C3×D3—61no—
101C2×C3×D3—61no—
102C2×C3×D3—61no—
103C2×C3×D3—61no—
104C2×C3×D3—121no—
105C2×C3×D3—121no—
106C32×C4—21no—
107C32×C4—21no—
108C32×C4—41no—
109C3⋊Dic3—21no—
110C3⋊Dic3—21no—
111C3⋊Dic3—21no—
112C3⋊Dic3D1212yes—
113C3×Dic3—32no—
114C3×Dic3—61no—
115C3×Dic3—61no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118C3×Dic3—61no—
119C3×Dic3—121no—
120C22⋊Q8—271noSylow
121H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
122C23×C3—91no—
123C22×D3—31no—
124C22×D3—31no—
125C22×D3—61no—
126C2×C3×C4—31no—
127C2×C3×C4—61no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—91no—
133C2×C3×C4—121no—
134C2×C3×C4D3211yescharacteristic
135C2×Dic3—31no—
136C2×Dic3—31no—
137C2×Dic3—31no—
138C2×Dic3—31no—
139C2×Dic3—31no—
140C2×Dic3—31no—
141C2×Dic3—61no—
142C2×Dic3—91no—
143C2×Dic3—91no—
144C2×Dic3—181no—
145C2×Dic3—181no—
146C2×Dic3—182no—
147C4×D3—182no—
148C4×D3—182no—
149Dic6—182no—
150Dic6—182no—
151Dic6—182no—
152C2×C32—21no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32C2×D1211yescharacteristic
156C2×C32#23511yescharacteristic
157C2×C32D3×D411yescharacteristic
158C2×C32#23711yescharacteristic
159C2×C32D3×Q811yescharacteristic
160C2×C32D3×Q811yescharacteristic
161C3×D3—62no—
162C3×D3—62no—
163C3×D3—122no—
164C2×Q8—271no—
165C22×C4—271no—
166C4⋊C4—91no—
167C4⋊C4—91no—
168C4⋊C4—271no—
169C22⋊C4—91no—
170C22⋊C4—271no—
171C22×C3—31no—
172C22×C3—61no—
173C22×C3—92no—
174C22×C3—121no—
175C22×C3—181no—
176C22×C3—181no—
177C22×C3C2×D3211yescharacteristic
178C2×D3—32no—
179C2×D3—32no—
180C2×D3—62no—
181C2×D3—61no—
182C2×D3—61no—
183C2×D3—61no—
184C2×D3—61no—
185C2×D3—121no—
186C2×D3—121no—
187C3×C4—21no—
188C3×C4—61no—
189C3×C4—92no—
190C3×C4—121no—
191C3×C4—121no—
192C3×C4—181no—
193C3×C4—181no—
194C3×C4—181no—
195C3×C4—181no—
196Dic3—32no—
197Dic3—61no—
198Dic3—61no—
199Dic3—61no—
200Dic3—61no—
201Dic3—61no—
202Dic3—121no—
203Dic3—182no—
204Dic3—181no—
205Dic3—181no—
206Dic3—361no—
207C32—21no—
208C32#89611yescharacteristic, complemented
209C32#89711yescharacteristic, complemented
210C23—91no—
211Q8—542no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—542no—
218C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
219C2×C3—31no—
220C2×C3—31no—
221C2×C3—31no—
222C2×C3—61no—
223C2×C3—61no—
224C2×C3—61no—
225C2×C3—121no—
226C2×C3—121no—
227C2×C3—121no—
228C2×C3—182no—
229C2×C3#379711yescharacteristic
230C2×C3D3×D1211yescharacteristic
231C2×C3#380411yescharacteristic
232D3—62no—
233D3—62no—
234D3—122no—
235C22—92no—
236C22—181no—
237C22—181no—
238C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
239C4—21no—
240C4—92no—
241C4—181no—
242C4—181no—
243C4—181no—
244C4—181no—
245C3—31no—
246C3—61no—
247C3—121no—
248C3#6389511yescharacteristic
249C2—182no—
250C2#8865111yescharacteristic
251C2#8865411yescharacteristic
252C2#8867111yescharacteristic
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×#272C22
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
4. C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. #3750⋊C3C2semidirect
7. #88448C2semidirect
23. C4⋊(C3⋊Dic3)D3semidirect
24. #3750D3semidirect
37. C2×#272Q8semidirect
121. H(3,𝔽3)C22⋊Q8semidirect
208. C32#896semidirect
209. C32#897semidirect

Maximal subgroups

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

subgroupindex
C2×#61992
C2×#61962
#885422
C4⋊((C3⋊Dic3)⋊C3)2
#3750⋊C32
#884512
#884482
#638993
#638933