Finite Group Databasebeta

← #13166248

Subgroups — #13166248

Order 1728.

424 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 68 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#11724247C212yesmaximal
2D3×C8×C3⋊D3C211yesmaximal, characteristic
3#11723094C211yesmaximal, characteristic, complemented
4C3×#63769C211yesmaximal, characteristic, complemented
5#11722714C212yesmaximal, complemented
6#10593768—31nomaximal
7C4×D3×C3⋊D3C2211yescharacteristic
8C4⋊(C3⋊#157)C2212yes—
9C3×C4⋊#157C2212yes—
10(C2×D3)⋊#157—22no—
11Dic3×#157—22no—
12C8×C3⋊(C3⋊D3)C411yescharacteristic, complemented
13C3×C8×C3⋊D3C2211yescharacteristic
14C32×D3×C8C411yescharacteristic, complemented
15#88785C411yescharacteristic, complemented
16C3⋊D3×C3⋊C8C2211yescharacteristic
17D3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
18#64286—32no—
19C2×C8×C3⋊D3—31no—
20D32×C8—62no—
21#63769—31no—
22#63769D311yescharacteristic, complemented
23#63768—32no—
24C2×D3×C3⋊D3D411yescharacteristic
25C2×C3⋊#157—22no—
26C2×C3×#157—22no—
27C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
28C3×C4×C3⋊D3C2311yescharacteristic
29C32×C4×D3C2×C411yescharacteristic
30#6289Q811yescharacteristic
31Dic3×C3⋊D3D411yescharacteristic
32D3×C3⋊Dic3Q811yescharacteristic
33C33×C8C2×C411yescharacteristic
34C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
35C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
36C32×C3⋊C8C2×C411yescharacteristic
37#4880—91nomaximal
38C2×C4×C3⋊D3—31no—
39C4×D32—62no—
40C22⋊#157—62no—
41C4⋊#157—32no—
42C4⋊#157C2×D312yes—
43C4×#157—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×D411yescharacteristic
59C2×C32×D3C4⋊C411yescharacteristic
60D3×C3⋊D3—21no—
61D3×C3⋊D3—21no—
62C3⋊#157—42no—
63C3×#157—42no—
64C33×C4C22×C411yescharacteristic
65C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
66C3×C3⋊Dic3C2×Q811yescharacteristic
67C32×Dic3C4⋊C411yescharacteristic
68C2×D3×C8—91no—
69C2×D3×C8—182no—
70#892—92no—
71C3×#107—91no—
72#818—91no—
73#808—92no—
74C22×C3⋊D3—31no—
75C2×D32—62no—
76C2×#157—22no—
77C2×#157—62no—
78C2×C32×C4—31no—
79C2×C3⋊Dic3—31no—
80C4×C3⋊D3—31no—
81C4×C3⋊D3—31no—
82C4×C3⋊D3—31no—
83C4×C3⋊D3—62no—
84C4×C3⋊D3—122no—
85C4×C3⋊D3C22×D311yescharacteristic
86C3×C4×D3—22no—
87C3×C4×D3—62no—
88#624—62no—
89D3×Dic3—62no—
90D3×Dic3—62no—
91C32×C8—22no—
92C32×C8—31no—
93C32×C8—42no—
94C32×C8C4×D311yescharacteristic
95C3⋊(C3⋊C8)—31no—
96C3⋊(C3⋊C8)—62no—
97C3⋊(C3⋊C8)—122no—
98C3⋊(C3⋊C8)C4×D311yescharacteristic
99C3×C3⋊C8—22no—
100C3×C3⋊C8—62no—
101#427—271noSylow
102C2×C33C2×C4⋊C411yescharacteristic
103C3⋊(C3⋊D3)—21no—
104C3×C3⋊D3#13511yescharacteristic
105C3×C3⋊D3#13511yescharacteristic
106C32×D3—21no—
107C2×C4×D3—91no—
108C2×C4×D3—182no—
109C2×C3×C8—62no—
110C2×C3×C8—91no—
111C3×C4⋊C4—92no—
112(C2×D3)⋊C4—182no—
113C4⋊Dic3—92no—
114C4×Dic3—182no—
115C2×C3⋊C8—91no—
116C2×C3⋊C8—182no—
117D3×C8—62no—
118D3×C8—91no—
119D3×C8—91no—
120D3×C8—91no—
121D3×C8—182no—
122D3×C8—182no—
123D3×C8—182no—
124D3×C8—362no—
125D3×C8#15711yescharacteristic, complemented
126C22×C32—31no—
127C2×C3⋊D3—31no—
128C2×C3⋊D3—62no—
129C2×C3⋊D3—61no—
130C2×C3⋊D3—61no—
131C2×C3⋊D3—122no—
132C2×C3⋊D3D3×D411yescharacteristic
133C2×C3×D3—22no—
134C2×C3×D3—62no—
135D32—64no—
136D32—64no—
137#157—42no—
138#157—122no—
139C32×C4—22no—
140C32×C4—31no—
141C32×C4—42no—
142C32×C4C2×C4×D311yescharacteristic
143C3⋊Dic3—31no—
144C3⋊Dic3—62no—
145C3⋊Dic3—122no—
146C3⋊Dic3D3×Q811yescharacteristic
147C3×Dic3—22no—
148C3×Dic3—62no—
149C22×C8—271no—
150#117—272no—
151#107—91no—
152#107—271no—
153#106—272no—
154C33#42711yescharacteristic, complemented, Sylow
155C22×D3—91no—
156C22×D3—182no—
157C2×C3×C4—62no—
158C2×C3×C4—91no—
159C2×C3×C4—182no—
160C2×Dic3—91no—
161C2×Dic3—182no—
162C2×Dic3—182no—
163C4×D3—62no—
164C4×D3—91no—
165C4×D3—91no—
166C4×D3—91no—
167C4×D3—182no—
168C4×D3—182no—
169C4×D3—182no—
170C4×D3—362no—
171C4×D3C2×#15711yescharacteristic
172C3×C8—22no—
173C3×C8—42no—
174C3×C8—62no—
175C3×C8—91no—
176C3×C8C2×#15711yescharacteristic
177C3⋊C8—62no—
178C3⋊C8—91no—
179C3⋊C8—182no—
180C3⋊C8—362no—
181C3⋊C8C2×#15711yescharacteristic
182C2×C32—22no—
183C2×C32—42no—
184C2×C32—61no—
185C2×C32D3×C4⋊C411yescharacteristic
186C3⋊D3—61no—
187C3⋊D3—64no—
188C3⋊D3—124no—
189C3⋊D3#91211yescharacteristic
190C3⋊D3#91911yescharacteristic
191C3×D3—24no—
192C3×D3—62no—
193C3×D3—62no—
194C22×C4—271no—
195C2×C8—31no—
196C2×C8—91no—
197C2×C8—271no—
198C2×C8—271no—
199C2×C8—271no—
200C2×C8—271no—
201C4⋊C4—92no—
202C4⋊C4—272no—
203C22⋊C4—542no—
204C42—542no—
205C22×C3—62no—
206C22×C3—91no—
207C2×D3—62no—
208C2×D3—91no—
209C2×D3—182no—
210C2×D3—181no—
211C2×D3—181no—
212C2×D3—184no—
213C2×D3—184no—
214C2×D3—362no—
215C2×D3C4⋊#15711yescharacteristic
216C3×C4—22no—
217C3×C4—42no—
218C3×C4—62no—
219C3×C4—91no—
220C3×C4—362no—
221C3×C4C22×#15711yescharacteristic
222Dic3—62no—
223Dic3—91no—
224Dic3—182no—
225Dic3—362no—
226Dic3—362no—
227Dic3C4⋊#15711yescharacteristic
228C32—22no—
229C32—42no—
230C32#488011yescharacteristic, complemented
231C23—271no—
232C2×C4—31no—
233C2×C4—91no—
234C2×C4—182no—
235C2×C4—271no—
236C2×C4—271no—
237C2×C4—271no—
238C2×C4—271no—
239C2×C4—542no—
240C8—31no—
241C8—91no—
242C8—271no—
243C8D3×#15711yescharacteristic
244C2×C3—22no—
245C2×C3—42no—
246C2×C3—64no—
247C2×C3—91no—
248C2×C3—91no—
249C2×C3C2×C4⋊#15711yescharacteristic
250D3—21no—
251D3—62no—
252D3—62no—
253D3—184no—
254D3—181no—
255D3—364no—
256C22—31no—
257C22—91no—
258C22—271no—
259C22—541no—
260C22—541no—
261C4—31no—
262C4—91no—
263C4—271no—
264C4—362no—
265C4—1082no—
266C4C2×D3×#15711yescharacteristic
267C3—22no—
268C3—42no—
269C3#1059376811yescharacteristic, complemented
270C2—61no—
271C2—91no—
272C2—91no—
273C2—541no—
274C2D3×C4⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C4⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C2×C4⋊C4
prime radicalC2×D3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#427
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. #11723094C2semidirect
4. C3×#63769C2semidirect
5. #11722714C2semidirect
12. C8×C3⋊(C3⋊D3)C4semidirect
14. C32×D3×C8C4semidirect
15. #88785C4semidirect
17. D3×C3⋊(C3⋊C8)C4semidirect
22. #63769D3semidirect
125. D3×C8#157semidirect
154. C33#427semidirect
230. C32#4880semidirect
269. C3#10593768semidirect

Maximal subgroups

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

subgroupindex
#117242472
D3×C8×C3⋊D32
#117230942
C3×#637692
#117227142
#105937683
#48809