Finite Group Databasebeta

← #13173341

Subgroups — #13173341

Order 1728.

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

normal subgroups only

This group has 412 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×#64086C211yesmaximal, characteristic
2#11723112C211yesmaximal, characteristic
3#11723112C211yesmaximal, characteristic
4#11723104C211yesmaximal, characteristic, complemented
5#11723102C211yesmaximal, characteristic, complemented
6C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
7C4×C3⋊#622C211yesmaximal, characteristic, complemented
8#10593744—31nomaximal
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C33×C22⋊C4C411yescharacteristic
15C3×#3757C411yescharacteristic
16C3×#3750C411yescharacteristic
17C3×#3750C411yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20#64086D311yescharacteristic
21C3×#877—62no—
22C22⋊#622—31no—
23C22⋊#622—31no—
24#63779—31no—
25#63777—31no—
26C4⋊#622—31no—
27C4×#622—31no—
28C23×C33C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2×C411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3⋊#622—21no—
38C3⋊#622—21no—
39C3⋊#622—21no—
40C3⋊#622—21no—
41#5000—91nomaximal
42C22×C3⋊Dic3C2×D311yescharacteristic
43C22×C3×Dic3—62no—
44C2×#622—31no—
45C2×#622—31no—
46C2×#622—31no—
47C2×#622—31no—
48C32×C22⋊C4—22no—
49C32×C22⋊C4—42no—
50C32×C22⋊C4Dic311yescharacteristic
51#3757Dic311yescharacteristic
52#3750Dic311yescharacteristic
53#3750Dic311yescharacteristic
54C4×C3⋊Dic3C2×D311yescharacteristic
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×#217—62no—
57C3×Dic3⋊C4—62no—
58C3×Dic3⋊C4—62no—
59C3×C4×Dic3—62no—
60C3×C4×Dic3—62no—
61C22×C33—21no—
62C22×C33—21no—
63C22×C33—21no—
64C22×C33C22×C411yescharacteristic
65C33×C4—21no—
66C33×C4—21no—
67C3×C3⋊Dic3—21no—
68C3×C3⋊Dic3—21no—
69C3×C3⋊Dic3—21no—
70C3×C3⋊Dic3#4111yescharacteristic
71C3×C3⋊Dic3#4111yescharacteristic
72C3×#117—91no—
73#877—62no—
74#830—91no—
75#830—91no—
76C4⋊(C3⋊C8)—91no—
77C4⋊(C3⋊C8)—91no—
78#803—91no—
79C4×C3⋊C8—91no—
80C23×C32—22no—
81C23×C32—42no—
82C23×C32C2×Dic311yescharacteristic
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—42no—
86C2×C32×C4—42no—
87C2×C32×C4C2×Dic311yescharacteristic
88C2×C32×C4C2×Dic311yescharacteristic
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3C2×Dic311yescharacteristic
92C2×C3⋊Dic3C2×Dic311yescharacteristic
93C2×C3⋊Dic3C2×Dic311yescharacteristic
94C2×C3⋊Dic3C22×D311yescharacteristic
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—64no—
101#622—61no—
102#622—61no—
103#622—61no—
104#622—61no—
105#433—271noSylow
106C2×C33—41no—
107C2×C33#11711yescharacteristic
108C2×C33#13111yescharacteristic
109C2×C33#13111yescharacteristic
110C22×C3×C4—91no—
111C22×Dic3—62no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—91no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—91no—
117C3×C22⋊C4#15711yescharacteristic
118C3×C42—91no—
119C3×C42—91no—
120#217—62no—
121Dic3⋊C4—62no—
122Dic3⋊C4—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—21no—
130C22×C32—24no—
131C22×C32—22no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—44no—
135C22×C32—42no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—82no—
139C22×C32—82no—
140C22×C32C22×Dic311yescharacteristic
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—42no—
144C32×C4—42no—
145C32×C4—82no—
146C32×C4—82no—
147C3⋊Dic3—21no—
148C3⋊Dic3—21no—
149C3⋊Dic3—21no—
150C3⋊Dic3#23511yescharacteristic
151C3⋊Dic3#23511yescharacteristic
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—64no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157#117—91no—
158C4⋊C8—271no—
159C4⋊C8—271no—
160C22⋊C8—271no—
161C22⋊C8—271no—
162#97—271no—
163C4×C8—271no—
164C33#43311yescharacteristic, complemented, Sylow
165C23×C3—22no—
166C23×C3—42no—
167C23×C3C2×#15711yescharacteristic
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4C2×#15711yescharacteristic
179C2×C3×C4C2×#15711yescharacteristic
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—64no—
185C2×Dic3—64no—
186C3⋊C8—181no—
187C3⋊C8—181no—
188C3⋊C8—181no—
189C3⋊C8—181no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—42no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—41no—
197C2×C32—44no—
198C2×C32—84no—
199C2×C32#92611yescharacteristic
200C2×C32#94811yescharacteristic
201C2×C32#94811yescharacteristic
202C22×C4—91no—
203C2×C8—271no—
204C2×C8—271no—
205C2×C8—271no—
206C2×C8—271no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C22⋊C4—91no—
210C22⋊C4C3⋊#15711yescharacteristic
211C42—91no—
212C42—91no—
213C22×C3—24no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—22no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—44no—
221C22×C3—42no—
222C22×C3—82no—
223C22×C3—82no—
224C22×C3C22×#15711yescharacteristic
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—42no—
228C3×C4—42no—
229C3×C4—82no—
230C3×C4—82no—
231C3×C4—91no—
232C3×C4—91no—
233C3×C4—181no—
234C3×C4—181no—
235C3×C4—181no—
236Dic3—62no—
237Dic3—62no—
238Dic3—64no—
239Dic3—122no—
240Dic3—122no—
241C32—22no—
242C32—42no—
243C32#500011yescharacteristic, complemented
244C23C2×C3⋊#15711yescharacteristic
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—181no—
250C2×C4—181no—
251C2×C4C2×C3⋊#15711yescharacteristic
252C2×C4C2×C3⋊#15711yescharacteristic
253C8—541no—
254C8—541no—
255C8—541no—
256C8—541no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—44no—
261C2×C3—41no—
262C2×C3—42no—
263C2×C3—42no—
264C2×C3—42no—
265C2×C3—84no—
266C2×C3#6428611yescharacteristic
267C2×C3#6428711yescharacteristic
268C2×C3#6428711yescharacteristic
269C22—21no—
270C22—21no—
271C22—21no—
272C22C22×C3⋊#15711yescharacteristic
273C4—21no—
274C4—21no—
275C4—91no—
276C4—91no—
277C4—181no—
278C4—181no—
279C4—181no—
280C3—22no—
281C3—42no—
282C3#1059374411yescharacteristic, complemented
283C2—41no—
284C2#1172451411yescharacteristic
285C2#1172451511yescharacteristic
286C2#1172451511yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C22⋊C4C4
prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22⋊C4C3⋊#157
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #11723104C2semidirect
5. #11723102C2semidirect
6. C4⋊(C3⋊#622)C2semidirect
7. C4×C3⋊#622C2semidirect
164. C33#433semidirect
243. C32#5000semidirect
282. C3#10593744semidirect

Maximal subgroups

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

subgroupindex
C3×#640862
#117231122
#117231042
#117231022
C4⋊(C3⋊#622)2
C4×C3⋊#6222
#105937443
#50009