Finite Group Databasebeta

← #13166311

Subgroups — #13166311

Order 1728.

396 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 62 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#11723459C211yesmaximal, characteristic
2#11723104C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4(C2×D3)⋊#622C211yesmaximal, characteristic
5(C2×D3)⋊#622C211yesmaximal, characteristic
6#11722724C211yesmaximal, characteristic, complemented
7Dic3×#622C211yesmaximal, characteristic, complemented
8#10593744—31nomaximal
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88857C411yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×(C2×D3)⋊C4C411yescharacteristic
17#88806C411yescharacteristic
18(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
19Dic3×C3⋊Dic3C2211yescharacteristic
20#64086—31no—
21#63877—62no—
22C22⋊#622—31no—
23C22⋊#622—31no—
24#63779—31no—
25#63777—31no—
26C4⋊#622D311yescharacteristic, complemented
27C4×#622—31no—
28C22×C32×D3C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3C2×C411yescharacteristic
35D3×C3⋊Dic3—21no—
36D3×C3⋊Dic3—21no—
37C3⋊#622—21no—
38C3⋊#622—21no—
39C3×#622—21no—
40C3×#622—21no—
41#4840—91nomaximal
42C22×C3⋊Dic3—31no—
43C2×D3×Dic3—62no—
44C2×#622—31no—
45C2×#622—31no—
46C2×#622C2×D311yescharacteristic
47C2×#622C2×D311yescharacteristic
48C32×C22⋊C4—31no—
49#3757—31no—
50#3750—31no—
51#3750—31no—
52#3750—62no—
53#3750—122no—
54C4×C3⋊Dic3—31no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×(C2×D3)⋊C4—22no—
57C3×C4×Dic3—62no—
58#3724—62no—
59(C2×D3)⋊Dic3—62no—
60Dic32—62no—
61C22×C33C22×C411yescharacteristic
62C2×C32×D3—21no—
63C2×C32×D3—21no—
64C2×C32×D3—21no—
65C33×C4—21no—
66C3⋊(C3⋊Dic3)—21no—
67C3⋊(C3⋊Dic3)—21no—
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3#4111yescharacteristic
70C3×C3⋊Dic3#4111yescharacteristic
71C32×Dic3—21no—
72#877—182no—
73#872—91no—
74C3×C4⋊C8—91no—
75(C2×D3)⋊C8—91no—
76(C2×D3)⋊C8—91no—
77#815—91no—
78C8×Dic3—91no—
79C4⋊(C3⋊C8)—91no—
80C23×C32—31no—
81C22×C3×D3—22no—
82C2×C32×C4—22no—
83C2×C32×C4—31no—
84C2×C32×C4—42no—
85C2×C32×C4C4×D311yescharacteristic
86C2×C3⋊Dic3—31no—
87C2×C3⋊Dic3—31no—
88C2×C3⋊Dic3—62no—
89C2×C3⋊Dic3—62no—
90C2×C3⋊Dic3—61no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3—122no—
93C2×C3⋊Dic3—122no—
94C2×C3⋊Dic3C4×D311yescharacteristic
95C2×C3⋊Dic3C22×D311yescharacteristic
96C2×C3×Dic3—22no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99D3×Dic3—64no—
100D3×Dic3—64no—
101#622—21no—
102#622—21no—
103#622—61no—
104#622—61no—
105#433—271noSylow
106C2×C33#11711yescharacteristic
107C2×C33#13111yescharacteristic
108C2×C33#13111yescharacteristic
109C32×D3—41no—
110C22×Dic3—182no—
111C2×C4×D3—91no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C3×C22⋊C4—62no—
115C3×C42—91no—
116#217—182no—
117(C2×D3)⋊C4—91no—
118(C2×D3)⋊C4#15711yescharacteristic
119Dic3⋊C4—91no—
120Dic3⋊C4—91no—
121Dic3⋊C4—182no—
122Dic3⋊C4—182no—
123Dic3⋊C4—362no—
124C4×Dic3—62no—
125C4×Dic3—91no—
126C4×Dic3—182no—
127C2×C3⋊C8—91no—
128C2×C3⋊C8—91no—
129C22×C32—22no—
130C22×C32—42no—
131C22×C32—61no—
132C22×C32—61no—
133C22×C32—61no—
134C22×C32C2×C4×D311yescharacteristic
135C2×C3×D3—24no—
136C2×C3×D3—42no—
137C2×C3×D3—42no—
138C32×C4—21no—
139C32×C4—42no—
140C32×C4—61no—
141C32×C4—82no—
142C3⋊Dic3—21no—
143C3⋊Dic3—61no—
144C3⋊Dic3—64no—
145C3⋊Dic3—61no—
146C3⋊Dic3—124no—
147C3⋊Dic3—122no—
148C3⋊Dic3—242no—
149C3⋊Dic3#23711yescharacteristic
150C3⋊Dic3#23911yescharacteristic
151C3×Dic3—42no—
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—122no—
155#117—271no—
156C4⋊C8—91no—
157C4⋊C8—271no—
158C22⋊C8—271no—
159C22⋊C8—271no—
160#97—271no—
161C4×C8—271no—
162C33#43311yescharacteristic, complemented, Sylow
163C23×C3—62no—
164C22×D3C2×#15711yescharacteristic
165C2×C3×C4—22no—
166C2×C3×C4—42no—
167C2×C3×C4—62no—
168C2×C3×C4—91no—
169C2×C3×C4—91no—
170C2×C3×C4C2×#15711yescharacteristic
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—91no—
174C2×Dic3—91no—
175C2×Dic3—182no—
176C2×Dic3—182no—
177C2×Dic3—184no—
178C2×Dic3—184no—
179C2×Dic3—362no—
180C2×Dic3—362no—
181C2×Dic3C2×#15711yescharacteristic
182C4×D3—181no—
183C4×D3—181no—
184C3×C8—181no—
185C3×C8—181no—
186C3⋊C8—181no—
187C3⋊C8—181no—
188C2×C32—22no—
189C2×C32—22no—
190C2×C32—22no—
191C2×C32—42no—
192C2×C32—42no—
193C2×C32—42no—
194C2×C32—121no—
195C2×C32#89211yescharacteristic
196C2×C32#90111yescharacteristic
197C2×C32#90711yescharacteristic
198C3×D3—44no—
199C22×C4—271no—
200C2×C8—91no—
201C2×C8—91no—
202C2×C8—271no—
203C2×C8—271no—
204C4⋊C4—271no—
205C4⋊C4—271no—
206C22⋊C4—31no—
207C22⋊C4—271no—
208C42—91no—
209C42—271no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—64no—
213C22×C3—122no—
214C22×C3—122no—
215C22×C3C22×#15711yescharacteristic
216C2×D3—21no—
217C2×D3—21no—
218C2×D3—21no—
219C3×C4—21no—
220C3×C4—42no—
221C3×C4—82no—
222C3×C4—91no—
223C3×C4—91no—
224C3×C4—122no—
225C3×C4—181no—
226Dic3—21no—
227Dic3—62no—
228Dic3—62no—
229Dic3—122no—
230Dic3—184no—
231Dic3—181no—
232Dic3—181no—
233Dic3—362no—
234Dic3—364no—
235Dic3—722no—
236C32—22no—
237C32—42no—
238C32#484011yescharacteristic, complemented
239C23—31no—
240C2×C4—31no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—271no—
244C2×C4—271no—
245C2×C4—541no—
246C2×C4—541no—
247C2×C4D3×#15711yescharacteristic
248C8—181no—
249C8—181no—
250C8—541no—
251C8—541no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—124no—
259C2×C3#6428611yescharacteristic
260C2×C3#6428711yescharacteristic
261C2×C3#6428711yescharacteristic
262D3—41no—
263C22—61no—
264C22—61no—
265C22—61no—
266C22C2×D3×#15711yescharacteristic
267C4—21no—
268C4—61no—
269C4—91no—
270C4—91no—
271C4—181no—
272C4—541no—
273C4—541no—
274C3—22no—
275C3—42no—
276C3#1059374411yescharacteristic, complemented
277C2—121no—
278C2#1172424711yescharacteristic
279C2#1172425111yescharacteristic
280C2#1172426611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723104C2semidirect
3. C3×C4⋊#622C2semidirect
6. #11722724C2semidirect
7. Dic3×#622C2semidirect
26. C4⋊#622D3semidirect
162. C33#433semidirect
238. C32#4840semidirect
276. C3#10593744semidirect

Maximal subgroups

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

subgroupindex
#117234592
#117231042
C3×C4⋊#6222
(C2×D3)⋊#6222
#117227242
Dic3×#6222
#105937443
#48409