Finite Group Databasebeta

← #11721299

Subgroups — #11721299

Order 864.

293 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 64 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×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C2×#6196C211yesmaximal, characteristic, complemented
3#88552C211yesmaximal, characteristic
4#3757⋊C3C211yesmaximal, characteristic
5#88449C211yesmaximal, characteristic, complemented
6#88448C211yesmaximal, characteristic, complemented
7#88448C211yesmaximal, characteristic, complemented
8#63982—31nomaximal
9#63964—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C2×#1084C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C2×(C3⋊Dic3)⋊C3—22no—
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#6196—22no—
19C22×C3⋊Dic3D311yescharacteristic
20C22×C3×Dic3—31no—
21C2×#627—31no—
22C2×#627—31no—
23#3757D311yescharacteristic
24C3×#217—31no—
25C3×#217—31no—
26C3×#217—31no—
27C3×#217—61no—
28#3724—31no—
29#3724—31no—
30Dic3⋊Dic3—31no—
31Dic3⋊Dic3—31no—
32Dic3⋊Dic3—31no—
33Dic3⋊Dic3—31no—
34C22×H(3,𝔽3)—21no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)Q812yescomplemented
37C22×H(3,𝔽3)C2311yescharacteristic
38#1084—21no—
39#1084—21no—
40(C3⋊Dic3)⋊C3—21no—
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3—21no—
43(C3⋊Dic3)⋊C3D412yes—
44#924—91no—
45#878—91no—
46#878—91no—
47C23×C32—21no—
48C23×C32C2×D311yescharacteristic
49C23×C32C2×D311yescharacteristic
50C2×C3⋊Dic3—22no—
51C2×C3⋊Dic3C2×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—31no—
63C2×C3×Dic3—61no—
64C2×C3×Dic3—61no—
65C2×C3×Dic3—62no—
66#627—62no—
67#627—62no—
68C2×H(3,𝔽3)—22no—
69C2×H(3,𝔽3)C2×D411yescharacteristic
70C2×H(3,𝔽3)C2×Q811yescharacteristic
71C2×H(3,𝔽3)#4111yescharacteristic
72C22×C3×C4—91no—
73C22×Dic3—31no—
74C22×Dic3—91no—
75C2×Dic6—91no—
76C2×Dic6—91no—
77C2×Dic6—91no—
78C3×C22⋊C4—91no—
79C3×C22⋊C4—91no—
80#217—31no—
81#217—31no—
82#217—31no—
83#217—61no—
84#217—91no—
85C4⋊Dic3—91no—
86C4⋊Dic3—91no—
87C4⋊Dic3—91no—
88Dic3⋊C4—91no—
89Dic3⋊C4—91no—
90Dic3⋊C4—91no—
91Dic3⋊C4—91no—
92Dic3⋊C4—91no—
93Dic3⋊C4—91no—
94C22×C32—21no—
95C22×C32—21no—
96C22×C32—21no—
97C22×C32—21no—
98C22×C32—22no—
99C22×C32—21no—
100C22×C32—41no—
101C22×C32—41no—
102C22×C32Dic612yescomplemented
103C22×C32Dic612yescomplemented
104C22×C32C22×D311yescharacteristic
105C22×C32C22×D311yescharacteristic
106C3⋊Dic3—21no—
107C3⋊Dic3—21no—
108C3⋊Dic3—21no—
109C3⋊Dic3#6712yes—
110C3×Dic3—32no—
111C3×Dic3—61no—
112C3×Dic3—61no—
113C3×Dic3—61no—
114C3×Dic3—61no—
115C3×Dic3—61no—
116C3×Dic3—61no—
117C3×Dic3—61no—
118C3×Dic3—121no—
119C3×Dic3—121no—
120C22⋊Q8—271noSylow
121H(3,𝔽3)C22⋊Q811yescharacteristic, complemented, Sylow
122C23×C3—31no—
123C23×C3—31no—
124C23×C3—61no—
125C23×C3D3211yescharacteristic
126C2×C3×C4—91no—
127C2×C3×C4—91no—
128C2×C3×C4—91no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—182no—
133C2×Dic3—31no—
134C2×Dic3—31no—
135C2×Dic3—31no—
136C2×Dic3—31no—
137C2×Dic3—31no—
138C2×Dic3—31no—
139C2×Dic3—31no—
140C2×Dic3—31no—
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—62no—
144C2×Dic3—91no—
145C2×Dic3—91no—
146C2×Dic3—91no—
147C2×Dic3—91no—
148C2×Dic3—182no—
149Dic6—182no—
150Dic6—182no—
151Dic6—182no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32—21no—
157C2×C32—42no—
158C2×C32C2×Dic611yescharacteristic
159C2×C32C2×Dic611yescharacteristic
160C2×C32#23511yescharacteristic
161C2×C32D3×D411yescharacteristic
162C2×C32#23711yescharacteristic
163C2×C32C2×#6711yescharacteristic
164C2×Q8—271no—
165C22×C4—91no—
166C4⋊C4—271no—
167C4⋊C4—271no—
168C4⋊C4—271no—
169C22⋊C4—91no—
170C22⋊C4—91no—
171C22×C3—21no—
172C22×C3—21no—
173C22×C3—32no—
174C22×C3—32no—
175C22×C3—31no—
176C22×C3—31no—
177C22×C3—61no—
178C22×C3—61no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3—62no—
182C22×C3—61no—
183C22×C3—121no—
184C22×C3—121no—
185C22×C3#62712yes—
186C22×C3C2×D3211yescharacteristic
187C3×C4—92no—
188C3×C4—181no—
189C3×C4—181no—
190C3×C4—181no—
191C3×C4—181no—
192C3×C4—181no—
193Dic3—32no—
194Dic3—61no—
195Dic3—61no—
196Dic3—61no—
197Dic3—61no—
198Dic3—61no—
199Dic3—61no—
200Dic3—61no—
201Dic3—92no—
202Dic3—121no—
203Dic3—121no—
204Dic3—181no—
205Dic3—181no—
206Dic3—181no—
207C32—21no—
208C32#87811yescharacteristic, complemented
209C32#92411yescharacteristic, complemented
210C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
211Q8—542no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—91no—
218C2×C4—182no—
219C2×C3—22no—
220C2×C3—31no—
221C2×C3—31no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—62no—
227C2×C3—62no—
228C2×C3—61no—
229C2×C3—61no—
230C2×C3—61no—
231C2×C3—122no—
232C2×C3#380411yescharacteristic
233C2×C3C2×#62711yescharacteristic
234C2×C3D3×#6711yescharacteristic
235C22—21no—
236C22—21no—
237C22#619612yescomplemented
238C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
239C4—92no—
240C4—181no—
241C4—181no—
242C4—181no—
243C4—181no—
244C4—181no—
245C3—31no—
246C3—31no—
247C3—61no—
248C3#6396411yescharacteristic
249C2—22no—
250C2C2×#619611yescharacteristic
251C2#8867111yescharacteristic
252C2#8867611yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
Fitting, prime radicalC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#6196C2semidirect
5. #88449C2semidirect
6. #88448C2semidirect
7. #88448C2semidirect
36. C22×H(3,𝔽3)Q8semidirect
102. C22×C32Dic6semidirect
103. C22×C32Dic6semidirect
121. H(3,𝔽3)C22⋊Q8semidirect
208. C32#878semidirect
209. C32#924semidirect
237. C22#6196semidirect

Maximal subgroups

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

subgroupindex
C22×(C3⋊Dic3)⋊C32
C2×#61962
#885522
#3757⋊C32
#884492
#884482
#639823
#639643