Finite Group Databasebeta

← #13166450

Subgroups — #13166450

Order 1728.

464 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×#88925C211yesmaximal, characteristic
2C2×D3×#622C211yesmaximal, characteristic
3#11723459C211yesmaximal, characteristic
4C4×C3⋊#622C211yesmaximal, characteristic, complemented
5C3×#63777C211yesmaximal, characteristic, complemented
6#11722724C211yesmaximal, characteristic, complemented
7Dic3×#622C211yesmaximal, characteristic, complemented
8#10593738—31nomaximal
9C2×D3×C3⋊Dic3C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14#88925C414yescomplemented
15D3×#622C414yescomplemented
16#88857C411yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×(C2×D3)⋊C4C411yescharacteristic
19#88806C411yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
21Dic3×C3⋊Dic3C2211yescharacteristic
22C2×#3792—31no—
23C22×#622—31no—
24#64086—31no—
25#63877—62no—
26#63777—31no—
27#63777D311yescharacteristic, complemented
28C4×#622—31no—
29C4×#622—31no—
30C22×C32×D3C2×C411yescharacteristic
31C2×C33×C4C2×C411yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3C2×C411yescharacteristic
37D3×C3⋊Dic3C2×C412yes—
38D3×C3⋊Dic3C2×C412yes—
39C3⋊#622C2×C412yes—
40C3⋊#622C2×C412yes—
41C3×#622C2×C412yes—
42C3×#622C2×C412yes—
43#4708—91nomaximal
44C22×C3⋊Dic3—31no—
45C2×D3×Dic3—62no—
46#3792—34no—
47C2×#622—31no—
48C2×#622—31no—
49C2×#622—34no—
50C2×#622C2×D311yescharacteristic
51C2×#622C2×D311yescharacteristic
52C32×C22⋊C4—31no—
53#3757—31no—
54#3750—31no—
55#3750—31no—
56#3750—62no—
57#3750—122no—
58C4×C3⋊Dic3—31no—
59C4×C3⋊Dic3C2×D311yescharacteristic
60C3×(C2×D3)⋊C4—22no—
61C3×C4×Dic3—62no—
62#3724—62no—
63(C2×D3)⋊Dic3—62no—
64Dic32—62no—
65C22×C33C22×C411yescharacteristic
66C2×C32×D3—21no—
67C2×C32×D3—21no—
68C2×C32×D3C4212yes—
69C33×C4—21no—
70C3⋊(C3⋊Dic3)—21no—
71C3⋊(C3⋊Dic3)C4212yes—
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3C22×C411yescharacteristic
74C3×C3⋊Dic3C22×C411yescharacteristic
75C32×Dic3—21no—
76C2×ΓL(1,𝔽25)—91no—
77C2×D3×C8—91no—
78#877—182no—
79#872—91no—
80C3×#97—91no—
81#815—91no—
82C8×Dic3—91no—
83C4×C3⋊C8—91no—
84C23×C32—31no—
85C22×C3×D3—22no—
86C2×C32×C4—22no—
87C2×C32×C4—31no—
88C2×C32×C4—42no—
89C2×C32×C4C4×D311yescharacteristic
90C2×C3⋊Dic3—31no—
91C2×C3⋊Dic3—32no—
92C2×C3⋊Dic3—32no—
93C2×C3⋊Dic3—31no—
94C2×C3⋊Dic3—62no—
95C2×C3⋊Dic3—62no—
96C2×C3⋊Dic3—122no—
97C2×C3⋊Dic3—122no—
98C2×C3⋊Dic3C4×D311yescharacteristic
99C2×C3⋊Dic3C22×D311yescharacteristic
100C2×C3×Dic3—22no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103D3×Dic3—64no—
104D3×Dic3—64no—
105#622—32no—
106#622—32no—
107#622C4×D312yes—
108#622C4×D312yes—
109#405—271noSylow
110C2×C33C2×C4211yescharacteristic
111C2×C33#13111yescharacteristic
112C2×C33#13111yescharacteristic
113C32×D3—22no—
114C22×Dic3—182no—
115C2×C4×D3—91no—
116C2×C3×C8—91no—
117C2×C3×C8—91no—
118C3×C22⋊C4—62no—
119C3×C42—91no—
120#217—182no—
121(C2×D3)⋊C4—91no—
122(C2×D3)⋊C4#15711yescharacteristic
123Dic3⋊C4—91no—
124Dic3⋊C4—91no—
125Dic3⋊C4—182no—
126Dic3⋊C4—182no—
127Dic3⋊C4—362no—
128C4×Dic3—62no—
129C4×Dic3—91no—
130C4×Dic3—182no—
131C2×C3⋊C8—91no—
132C2×C3⋊C8—91no—
133ΓL(1,𝔽25)—94no—
134D3×C8—94no—
135C22×C32—22no—
136C22×C32—32no—
137C22×C32—42no—
138C22×C32—61no—
139C22×C32—61no—
140C22×C32C2×C4×D311yescharacteristic
141C2×C3×D3—24no—
142C2×C3×D3—42no—
143C2×C3×D3—42no—
144C32×C4—21no—
145C32×C4—42no—
146C32×C4—61no—
147C32×C4—82no—
148C3⋊Dic3—21no—
149C3⋊Dic3—32no—
150C3⋊Dic3—64no—
151C3⋊Dic3—61no—
152C3⋊Dic3—124no—
153C3⋊Dic3—122no—
154C3⋊Dic3—242no—
155C3⋊Dic3C2×C4×D311yescharacteristic
156C3⋊Dic3C2×C4×D311yescharacteristic
157C3×Dic3—42no—
158C3×Dic3—62no—
159C3×Dic3—62no—
160C3×Dic3—122no—
161C2×M16—271no—
162C22×C8—271no—
163#117—271no—
164#97—91no—
165#97—271no—
166C4×C8—271no—
167C4×C8—271no—
168C33#40511yescharacteristic, complemented, Sylow
169C23×C3—62no—
170C22×D3C2×#15711yescharacteristic
171C2×C3×C4—22no—
172C2×C3×C4—42no—
173C2×C3×C4—62no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4C2×#15711yescharacteristic
177C2×Dic3—62no—
178C2×Dic3—62no—
179C2×Dic3—91no—
180C2×Dic3—91no—
181C2×Dic3—184no—
182C2×Dic3—184no—
183C2×Dic3—182no—
184C2×Dic3—182no—
185C2×Dic3—362no—
186C2×Dic3—362no—
187C2×Dic3C2×#15711yescharacteristic
188C4×D3—92no—
189C4×D3—92no—
190C3×C8—92no—
191C3×C8—92no—
192C3⋊C8—92no—
193C3⋊C8—92no—
194C2×C32—22no—
195C2×C32—22no—
196C2×C32—22no—
197C2×C32—42no—
198C2×C32—42no—
199C2×C32—42no—
200C2×C32—62no—
201C2×C32C42×D311yescharacteristic
202C2×C32#90711yescharacteristic
203C2×C32#90711yescharacteristic
204C3×D3—44no—
205C22×C4—271no—
206M16—274no—
207C2×C8—91no—
208C2×C8—91no—
209C2×C8—274no—
210C2×C8—271no—
211C2×C8—271no—
212C4⋊C4—271no—
213C4⋊C4—271no—
214C22⋊C4—31no—
215C22⋊C4—271no—
216C42—91no—
217C42—271no—
218C22×C3—22no—
219C22×C3—42no—
220C22×C3—64no—
221C22×C3—122no—
222C22×C3—122no—
223C22×C3C22×#15711yescharacteristic
224C2×D3—21no—
225C2×D3—21no—
226C2×D3C4×#15712yes—
227C3×C4—21no—
228C3×C4—42no—
229C3×C4—82no—
230C3×C4—91no—
231C3×C4—91no—
232C3×C4—122no—
233C3×C4—181no—
234Dic3—21no—
235Dic3—62no—
236Dic3—62no—
237Dic3—92no—
238Dic3—122no—
239Dic3—184no—
240Dic3—181no—
241Dic3—364no—
242Dic3—362no—
243Dic3—722no—
244C32—22no—
245C32—42no—
246C32#470811yescharacteristic, complemented
247C23—31no—
248C2×C4—31no—
249C2×C4—91no—
250C2×C4—91no—
251C2×C4—272no—
252C2×C4—272no—
253C2×C4—271no—
254C2×C4—271no—
255C2×C4D3×#15711yescharacteristic
256C8—92no—
257C8—92no—
258C8—272no—
259C8—272no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—22no—
263C2×C3—42no—
264C2×C3—42no—
265C2×C3—42no—
266C2×C3—124no—
267C2×C3C2×C4×#15711yescharacteristic
268C2×C3#6428711yescharacteristic
269C2×C3#6428711yescharacteristic
270D3—22no—
271C22—32no—
272C22—61no—
273C22—61no—
274C22C2×D3×#15711yescharacteristic
275C4—21no—
276C4—61no—
277C4—91no—
278C4—91no—
279C4—181no—
280C4—272no—
281C4—541no—
282C3—22no—
283C3—42no—
284C3#1059373811yescharacteristic, complemented
285C2—62no—
286C2#1172425111yescharacteristic
287C2C4×D3×#15711yescharacteristic
288C2#1172426711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C2×C42
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C4×C3⋊#622C2semidirect
5. C3×#63777C2semidirect
6. #11722724C2semidirect
7. Dic3×#622C2semidirect
14. #88925C4semidirect
15. D3×#622C4semidirect
27. #63777D3semidirect
168. C33#405semidirect
246. C32#4708semidirect
284. C3#10593738semidirect

Maximal subgroups

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

subgroupindex
C2×#889252
C2×D3×#6222
#117234592
C4×C3⋊#6222
C3×#637772
#117227242
Dic3×#6222
#105937383
#47089