Finite Group Databasebeta

← #13166266

Subgroups — #13166266

Order 1728.

408 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#11724255C212yesmaximal
2D3×C8×C3⋊D3C211yesmaximal, characteristic
3#11723096C211yesmaximal, characteristic, complemented
4C3×#63771C211yesmaximal, characteristic, complemented
5#11722717C212yesmaximal, complemented
6#10593771—31nomaximal
7C4×D3×C3⋊D3C2211yescharacteristic
8#88989C2212yes—
9C3×#3788C2212yes—
10#88925—22no—
11#88924—22no—
12C8×C3⋊(C3⋊D3)C411yescharacteristic
13C3×C8×C3⋊D3C2211yescharacteristic
14C32×D3×C8C411yescharacteristic
15#88785C411yescharacteristic
16C3⋊D3×C3⋊C8C2211yescharacteristic
17D3×C3⋊(C3⋊C8)C411yescharacteristic
18#64283—32no—
19C2×C8×C3⋊D3—31no—
20D32×C8—62no—
21#63771—31no—
22#63771D311yescharacteristic, complemented
23#63770—32no—
24C2×D3×C3⋊D3Q811yescharacteristic
25C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
26C3×C4×C3⋊D3C2311yescharacteristic
27C32×C4×D3C2×C411yescharacteristic
28#6289D411yescharacteristic
29Dic3×C3⋊D3Q811yescharacteristic
30D3×C3⋊Dic3D411yescharacteristic
31C3⋊#622—22no—
32C3×#622—22no—
33C33×C8C2×C411yescharacteristic
34C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
35C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
36C32×C3⋊C8C2×C411yescharacteristic
37D3×#108—91nomaximal
38C2×C4×C3⋊D3—31no—
39C4×D32—62no—
40#3792—62no—
41#3788—32no—
42#3788C2×D312yes—
43#3787—62no—
44C2×C32×C8—31no—
45C2×C3⋊(C3⋊C8)—31no—
46C8×C3⋊D3—31no—
47C8×C3⋊D3—31no—
48C8×C3⋊D3—31no—
49C8×C3⋊D3—62no—
50C8×C3⋊D3—122no—
51C8×C3⋊D3C2×D311yescharacteristic
52C3×D3×C8—22no—
53C3×D3×C8—62no—
54#3710—62no—
55D3×C3⋊C8—62no—
56D3×C3⋊C8—62no—
57C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
58C2×C3×C3⋊D3C2×Q811yescharacteristic
59C2×C32×D3C4⋊C411yescharacteristic
60D3×C3⋊D3—22no—
61C33×C4C22×C411yescharacteristic
62C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
63C3×C3⋊Dic3C2×D411yescharacteristic
64C32×Dic3C4⋊C411yescharacteristic
65D3×M16—92no—
66C2×D3×C8—91no—
67C2×D3×C8—182no—
68C3×#108—91no—
69#822—92no—
70#819—91no—
71C22×C3⋊D3—31no—
72C2×D32—62no—
73C2×C32×C4—31no—
74C2×C3⋊Dic3—31no—
75C4×C3⋊D3—31no—
76C4×C3⋊D3—31no—
77C4×C3⋊D3—31no—
78C4×C3⋊D3—62no—
79C4×C3⋊D3—122no—
80C4×C3⋊D3C22×D311yescharacteristic
81C3×C4×D3—22no—
82C3×C4×D3—62no—
83#624—62no—
84D3×Dic3—62no—
85D3×Dic3—62no—
86#622—22no—
87#622—62no—
88C32×C8—22no—
89C32×C8—31no—
90C32×C8—42no—
91C32×C8C4×D311yescharacteristic
92C3⋊(C3⋊C8)—31no—
93C3⋊(C3⋊C8)—62no—
94C3⋊(C3⋊C8)—122no—
95C3⋊(C3⋊C8)C4×D311yescharacteristic
96C3×C3⋊C8—22no—
97C3×C3⋊C8—62no—
98C2×#108—271noSylow
99C2×C33C2×C4⋊C411yescharacteristic
100C3⋊(C3⋊D3)#10812yescomplemented
101C3×C3⋊D3—21no—
102C32×D3—21no—
103C2×C4×D3—91no—
104C2×C4×D3—182no—
105C3×M16—92no—
106C2×C3×C8—62no—
107C2×C3×C8—91no—
108#208—92no—
109C2×C3⋊C8—91no—
110C2×C3⋊C8—182no—
111ΓL(1,𝔽25)—182no—
112D3×C8—62no—
113D3×C8—91no—
114D3×C8—91no—
115D3×C8—91no—
116D3×C8—182no—
117D3×C8—182no—
118D3×C8—182no—
119D3×C8—182no—
120D3×C8—362no—
121D3×C8#15711yescharacteristic
122C22×C32—31no—
123C2×C3⋊D3—31no—
124C2×C3⋊D3—62no—
125C2×C3⋊D3—62no—
126C2×C3⋊D3—122no—
127C2×C3⋊D3D3×Q811yescharacteristic
128C2×C3×D3—22no—
129C2×C3×D3—62no—
130D32—68no—
131C32×C4—22no—
132C32×C4—31no—
133C32×C4—42no—
134C32×C4C2×C4×D311yescharacteristic
135C3⋊Dic3—31no—
136C3⋊Dic3—62no—
137C3⋊Dic3—122no—
138C3⋊Dic3D3×D411yescharacteristic
139C3×Dic3—22no—
140C3×Dic3—62no—
141C2×M16—272no—
142C22×C8—271no—
143#108—91no—
144#108—271no—
145#108—272no—
146C33C2×#10811yescharacteristic, complemented, Sylow
147C22×D3—91no—
148C22×D3—182no—
149C2×C3×C4—62no—
150C2×C3×C4—91no—
151C2×Dic3—91no—
152C2×Dic3—182no—
153C4×D3—62no—
154C4×D3—91no—
155C4×D3—91no—
156C4×D3—91no—
157C4×D3—182no—
158C4×D3—182no—
159C4×D3—182no—
160C4×D3—362no—
161C4×D3C2×#15711yescharacteristic
162C3×C8—22no—
163C3×C8—42no—
164C3×C8—62no—
165C3×C8—91no—
166C3×C8—182no—
167C3×C8C2×#15711yescharacteristic
168C3⋊C8—62no—
169C3⋊C8—91no—
170C3⋊C8—182no—
171C3⋊C8—182no—
172C3⋊C8—362no—
173C3⋊C8C2×#15711yescharacteristic
174C2×C32—22no—
175C2×C32—42no—
176C2×C32—61no—
177C2×C32D3×C4⋊C411yescharacteristic
178C3⋊D3—21no—
179C3⋊D3—32no—
180C3⋊D3—64no—
181C3⋊D3—124no—
182C3×D3—24no—
183C3×D3—64no—
184C22×C4—271no—
185M16—92no—
186M16—272no—
187M16—542no—
188C2×C8—31no—
189C2×C8—91no—
190C2×C8—271no—
191C2×C8—271no—
192C2×C8—271no—
193C2×C8—271no—
194C2×C8—542no—
195C22×C3—62no—
196C22×C3—91no—
197C2×D3—62no—
198C2×D3—91no—
199C2×D3—182no—
200C2×D3—182no—
201C2×D3—188no—
202C2×D3—362no—
203C2×D3C4⋊#15711yescharacteristic
204C3×C4—22no—
205C3×C4—42no—
206C3×C4—62no—
207C3×C4—91no—
208C3×C4C22×#15711yescharacteristic
209Dic3—62no—
210Dic3—91no—
211Dic3—182no—
212Dic3—362no—
213Dic3C4⋊#15711yescharacteristic
214C32—22no—
215C32—42no—
216C32D3×#10811yescharacteristic, complemented
217C23—271no—
218C2×C4—31no—
219C2×C4—91no—
220C2×C4—271no—
221C2×C4—271no—
222C2×C4—271no—
223C2×C4—271no—
224C8—31no—
225C8—91no—
226C8—182no—
227C8—271no—
228C8—542no—
229C8D3×#15711yescharacteristic
230C2×C3—22no—
231C2×C3—42no—
232C2×C3—64no—
233C2×C3—181no—
234C2×C3C2×C4⋊#15711yescharacteristic
235D3—21no—
236D3—64no—
237D3—92no—
238D3—184no—
239D3—364no—
240C22—31no—
241C22—91no—
242C22—271no—
243C22—542no—
244C4—31no—
245C4—91no—
246C4—271no—
247C4C2×D3×#15711yescharacteristic
248C3—22no—
249C3—42no—
250C3#1059377111yescharacteristic, complemented
251C2—61no—
252C2—181no—
253C2—272no—
254C2D3×C4⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C4⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C2×C4⋊C4
prime radicalC2×D3×C3⋊D3Q8
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33C2×#108
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
3. #11723096C2semidirect
4. C3×#63771C2semidirect
5. #11722717C2semidirect
22. #63771D3semidirect
100. C3⋊(C3⋊D3)#108semidirect
146. C33C2×#108semidirect
216. C32D3×#108semidirect
250. C3#10593771semidirect

Maximal subgroups

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

subgroupindex
#117242552
D3×C8×C3⋊D32
#117230962
C3×#637712
#117227172
#105937713
D3×#1089