Finite Group Databasebeta

← #11721304

Subgroups — #11721304

Order 864.

300 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 54 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×#638⋊C3C211yesmaximal, characteristic
2C2×#6196C211yesmaximal, characteristic, complemented
3#88552C211yesmaximal, characteristic, complemented
4#3757⋊C3C211yesmaximal, characteristic, complemented
5#88450C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic, complemented
7#88447C211yesmaximal, characteristic, complemented
8#63959—31nomaximal
9#63953—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14#638⋊C3—22no—
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#6196—22no—
19C2×#638D311yescharacteristic
20C2×C3×#67—31no—
21C2×#627—31no—
22C2×#627—31no—
23#3757D311yescharacteristic, complemented
24C3×#217—31no—
25C3×#217—31no—
26C3×#217—31no—
27C3×#217—61no—
28#3722—31no—
29#3722—31no—
30(C2×D3)⋊Dic3—31no—
31(C2×D3)⋊Dic3—31no—
32Dic32—31no—
33Dic32—31no—
34C22×H(3,𝔽3)—21no—
35C22×H(3,𝔽3)—21no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)C2311yescharacteristic
38C2×(C3⋊D3)⋊C3—21no—
39C2×(C3⋊D3)⋊C3—21no—
40C2×(C3⋊D3)⋊C3—21no—
41#1084—21no—
42#1084—21no—
43(C3⋊Dic3)⋊C3—21no—
44(C3⋊Dic3)⋊C3—21no—
45(C3⋊Dic3)⋊C3D412yes—
46#936—91no—
47#885—91no—
48#885—91no—
49C23×C32—21no—
50C23×C32C2×D311yescharacteristic
51C23×C32C2×D311yescharacteristic
52C22×C3⋊D3C2×D311yescharacteristic
53C22×C3×D3—31no—
54#638—22no—
55C2×C3⋊Dic3C2×D311yescharacteristic
56C2×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊Dic3C2×D311yescharacteristic
58C3×#67—62no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—31no—
62C2×C3×Dic3—31no—
63C2×C3×Dic3—31no—
64C2×C3×Dic3—31no—
65C2×C3×Dic3—31no—
66C2×C3×Dic3—61no—
67C2×C3×Dic3—61no—
68#627—62no—
69#627—62no—
70C2×H(3,𝔽3)—41no—
71C2×H(3,𝔽3)C2×D411yescharacteristic
72C2×H(3,𝔽3)#4111yescharacteristic
73C2×H(3,𝔽3)#4111yescharacteristic
74(C3⋊D3)⋊C3—41no—
75C2×C3×D4—91no—
76C2×#67—31no—
77C2×#67—91no—
78C2×Dic6—91no—
79C2×Dic6—91no—
80C2×Dic6—91no—
81C3×C22⋊C4—91no—
82C3×C22⋊C4—91no—
83#217—31no—
84#217—31no—
85#217—31no—
86#217—61no—
87#217—91no—
88#217—91no—
89#217—91no—
90(C2×D3)⋊C4—91no—
91(C2×D3)⋊C4—91no—
92(C2×D3)⋊C4—91no—
93(C2×D3)⋊C4—91no—
94C4×Dic3—91no—
95C4×Dic3—91no—
96C4×Dic3—91no—
97C22×C32—22no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—21no—
105C22×C32—41no—
106C22×C32—41no—
107C22×C32C22×D311yescharacteristic
108C22×C32C22×D311yescharacteristic
109C2×C3⋊D3—21no—
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3×D3—61no—
113C2×C3×D3—61no—
114C2×C3×D3—61no—
115C3⋊Dic3—21no—
116C3⋊Dic3—21no—
117C3⋊Dic3#6712yes—
118C3×Dic3—32no—
119C3×Dic3—61no—
120C3×Dic3—61no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123C3×Dic3—61no—
124C3×Dic3—61no—
125C3×Dic3—121no—
126C3×Dic3—121no—
127#124—271noSylow
128H(3,𝔽3)#12411yescharacteristic, complemented, Sylow
129C23×C3—31no—
130C23×C3—31no—
131C23×C3—61no—
132C23×C3—91no—
133C23×C3D3211yescharacteristic
134C22×D3—31no—
135C22×D3—91no—
136C3×D4—182no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142#67—62no—
143#67—182no—
144C2×Dic3—31no—
145C2×Dic3—31no—
146C2×Dic3—31no—
147C2×Dic3—31no—
148C2×Dic3—31no—
149C2×Dic3—31no—
150C2×Dic3—31no—
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—91no—
154C2×Dic3—91no—
155C2×Dic3—91no—
156Dic6—182no—
157Dic6—182no—
158Dic6—182no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32—21no—
162C2×C32—42no—
163C2×C32—41no—
164C2×C32—41no—
165C2×C32#23511yescharacteristic
166C2×C32D3×D411yescharacteristic
167C2×C32#23711yescharacteristic
168C2×C32#23711yescharacteristic
169C2×C32#23711yescharacteristic
170C2×C32C2×#6711yescharacteristic
171C3⋊D3—41no—
172C3×D3—121no—
173C2×Q8—271no—
174C2×D4—91no—
175C22⋊C4—91no—
176C22⋊C4—91no—
177C22⋊C4—271no—
178C22⋊C4—271no—
179C42—271no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—61no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—61no—
189C22×C3—61no—
190C22×C3—61no—
191C22×C3—62no—
192C22×C3—61no—
193C22×C3—121no—
194C22×C3—121no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3—181no—
198C22×C3C2×D3211yescharacteristic
199C2×D3—61no—
200C2×D3—61no—
201C2×D3—61no—
202C2×D3—181no—
203C2×D3—181no—
204C2×D3—181no—
205C3×C4—92no—
206C3×C4—181no—
207C3×C4—181no—
208C3×C4—181no—
209C3×C4—181no—
210Dic3—32no—
211Dic3—61no—
212Dic3—61no—
213Dic3—61no—
214Dic3—61no—
215Dic3—61no—
216Dic3—61no—
217Dic3—92no—
218Dic3—121no—
219Dic3—121no—
220Dic3—181no—
221Dic3—181no—
222C32—21no—
223C32#88511yescharacteristic, complemented
224C32#93611yescharacteristic, complemented
225C23—91no—
226C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
227Q8—542no—
228D4—182no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C3—31no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—41no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—121no—
245C2×C3—121no—
246C2×C3—122no—
247C2×C3—361no—
248C2×C3#380411yescharacteristic
249C2×C3#380511yescharacteristic
250C2×C3D3×#6711yescharacteristic
251D3—121no—
252D3—361no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—181no—
257C22—181no—
258C22—181no—
259C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
260C4—92no—
261C4—181no—
262C4—181no—
263C4—181no—
264C4—181no—
265C3—31no—
266C3—31no—
267C3—61no—
268C3#6395911yescharacteristic
269C2—41no—
270C2—361no—
271C2#8867111yescharacteristic
272C2#8867211yescharacteristic
273C2#8867611yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
prime radicalC2×#638⋊C3C2
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#124
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
3. #88552C2semidirect
4. #3757⋊C3C2semidirect
5. #88450C2semidirect
6. #88450C2semidirect
7. #88447C2semidirect
23. #3757D3semidirect
128. H(3,𝔽3)#124semidirect
223. C32#885semidirect
224. C32#936semidirect

Maximal subgroups

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

subgroupindex
C2×#638⋊C32
C2×#61962
#885522
#3757⋊C32
#884502
#884472
#639593
#639533