Finite Group Databasebeta

← #13173353

Subgroups — #13173353

Order 1728.

396 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C3×#64101C211yesmaximal, characteristic
2#11723104C211yesmaximal, characteristic, complemented
3#11723103C211yesmaximal, characteristic
4#11723103C211yesmaximal, characteristic
5#11723102C211yesmaximal, characteristic, complemented
6C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
7C4×C3⋊#622C211yesmaximal, characteristic, complemented
8#10593753—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C33×C4⋊C4C411yescharacteristic
15C3×#3752C411yescharacteristic
16C3×#3752C411yescharacteristic
17C3×C4⋊(C3⋊Dic3)C411yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20#64101D311yescharacteristic
21C3×#892—62no—
22#63779—31no—
23#63778—31no—
24#63778—31no—
25#63777—31no—
26C4⋊#622—31no—
27C4×#622—31no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×C4×C3⋊D3—21no—
36C3×C4×C3⋊D3—21no—
37C3⋊#622—21no—
38C3⋊#622—21no—
39C3⋊#622—21no—
40C3⋊#622—21no—
41#5000—91nomaximal
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×C3×C4×D3—62no—
44C2×#622—31no—
45C2×#622—31no—
46C2×#622—31no—
47C2×#622—31no—
48C32×C4⋊C4—22no—
49C32×C4⋊C4—42no—
50C32×C4⋊C4Dic311yescharacteristic
51#3752Dic311yescharacteristic
52#3752Dic311yescharacteristic
53C4⋊(C3⋊Dic3)Dic311yescharacteristic
54C4×C3⋊Dic3C2×D311yescharacteristic
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×(C2×D3)⋊C4—62no—
57C3×(C2×D3)⋊C4—62no—
58C3×C4⋊Dic3—62no—
59C3×C4×Dic3—62no—
60C3×C4×Dic3—62no—
61C22×C33C22×C411yescharacteristic
62C2×C3×C3⋊D3—21no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—21no—
65C33×C4—21no—
66C33×C4—21no—
67C33×C4—21no—
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3#4111yescharacteristic
71C3×C3⋊Dic3#4111yescharacteristic
72C3×#117—91no—
73#892—62no—
74#830—91no—
75#830—91no—
76C4⋊(C3⋊C8)—91no—
77C4⋊(C3⋊C8)—91no—
78#803—91no—
79C4×C3⋊C8—91no—
80C22×C3⋊D3C2×Dic311yescharacteristic
81C22×C3×D3—62no—
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—42no—
86C2×C32×C4—42no—
87C2×C32×C4—42no—
88C2×C32×C4C2×Dic311yescharacteristic
89C2×C32×C4C2×Dic311yescharacteristic
90C2×C32×C4C2×Dic311yescharacteristic
91C2×C3⋊Dic3C2×Dic311yescharacteristic
92C2×C3⋊Dic3C2×Dic311yescharacteristic
93C2×C3⋊Dic3C22×D311yescharacteristic
94C4×C3⋊D3—21no—
95C4×C3⋊D3—21no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C3×C4×D3—64no—
100C3×C4×D3—64no—
101#622—61no—
102#622—61no—
103#622—61no—
104#622—61no—
105#433—271noSylow
106C2×C33#11711yescharacteristic
107C2×C33#13111yescharacteristic
108C2×C33#13111yescharacteristic
109C3×C3⋊D3—41no—
110C22×C3×C4—91no—
111C2×C4×D3—62no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—91no—
115C3×C4⋊C4#15711yescharacteristic
116C3×C22⋊C4—91no—
117C3×C22⋊C4—91no—
118C3×C42—91no—
119C3×C42—91no—
120(C2×D3)⋊C4—62no—
121(C2×D3)⋊C4—62no—
122C4⋊Dic3—62no—
123C4×Dic3—62no—
124C4×Dic3—62no—
125C2×C3⋊C8—91no—
126C2×C3⋊C8—91no—
127C2×C3⋊C8—91no—
128C2×C3⋊C8—91no—
129C22×C32—22no—
130C22×C32—42no—
131C22×C32C22×Dic311yescharacteristic
132C2×C3⋊D3—21no—
133C2×C3⋊D3—21no—
134C2×C3⋊D3—21no—
135C2×C3×D3—64no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C32×C4—21no—
139C32×C4—24no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—44no—
143C32×C4—42no—
144C32×C4—42no—
145C32×C4—82no—
146C32×C4—82no—
147C3⋊Dic3—21no—
148C3⋊Dic3—21no—
149C3⋊Dic3#23511yescharacteristic
150C3⋊Dic3#23511yescharacteristic
151C3×Dic3—62no—
152C3×Dic3—62no—
153C3×Dic3—122no—
154C3×Dic3—122no—
155#117—91no—
156C4⋊C8—271no—
157C4⋊C8—271no—
158C22⋊C8—271no—
159C22⋊C8—271no—
160#97—271no—
161C4×C8—271no—
162C33#43311yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C22×D3—62no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—91no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4C2×#15711yescharacteristic
177C2×C3×C4C2×#15711yescharacteristic
178C2×C3×C4C2×#15711yescharacteristic
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C4×D3—64no—
183C4×D3—64no—
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#92611yescharacteristic
195C2×C32#94811yescharacteristic
196C2×C32#94811yescharacteristic
197C3⋊D3—41no—
198C3×D3—124no—
199C22×C4—91no—
200C2×C8—271no—
201C2×C8—271no—
202C2×C8—271no—
203C2×C8—271no—
204C4⋊C4—91no—
205C4⋊C4C3⋊#15711yescharacteristic
206C22⋊C4—91no—
207C22⋊C4—91no—
208C42—91no—
209C42—91no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—181no—
213C22×C3—181no—
214C22×C3—181no—
215C22×C3C22×#15711yescharacteristic
216C2×D3—64no—
217C2×D3—122no—
218C2×D3—122no—
219C3×C4—24no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4—44no—
226C3×C4—82no—
227C3×C4—82no—
228C3×C4—91no—
229C3×C4—91no—
230C3×C4—181no—
231C3×C4—181no—
232Dic3—62no—
233Dic3—62no—
234Dic3—122no—
235Dic3—122no—
236C32—22no—
237C32—42no—
238C32#500011yescharacteristic, complemented
239C23—91no—
240C2×C4—91no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4C2×C3⋊#15711yescharacteristic
246C2×C4C2×C3⋊#15711yescharacteristic
247C2×C4C2×C3⋊#15711yescharacteristic
248C8—541no—
249C8—541no—
250C8—541no—
251C8—541no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—361no—
259C2×C3#6428611yescharacteristic
260C2×C3#6428711yescharacteristic
261C2×C3#6428911yescharacteristic
262D3—124no—
263C22—181no—
264C22—181no—
265C22—181no—
266C22C22×C3⋊#15711yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—91no—
271C4—91no—
272C4—181no—
273C4—181no—
274C3—22no—
275C3—42no—
276C3#1059375311yescharacteristic, complemented
277C2—361no—
278C2#1172451411yescharacteristic
279C2#1172451511yescharacteristic
280C2#1172451711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C4⋊C4C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC4⋊C4C3⋊#157
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723104C2semidirect
5. #11723102C2semidirect
6. C4⋊(C3⋊#622)C2semidirect
7. C4×C3⋊#622C2semidirect
162. C33#433semidirect
238. C32#5000semidirect
276. C3#10593753semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
#117231042
#117231032
#117231022
C4⋊(C3⋊#622)2
C4×C3⋊#6222
#105937533
#50009