Finite Group Databasebeta

← #13166385

Subgroups — #13166385

Order 1728.

552 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 108 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×#88988C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic
3C4×Dic3×C3⋊D3C211yesmaximal, characteristic
4#11722732C212yesmaximal, complemented
5Dic3⋊#622C212yesmaximal, complemented
6#10593726—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×Dic3×C3⋊D3C411yescharacteristic
9C2×Dic3×C3⋊D3C411yescharacteristic
10C2×C3⋊#622C2212yes—
11#88988—22no—
12C2×C3×#622C2212yes—
13C3×#3787—22no—
14C4×C3⋊(C3⋊Dic3)C411yescharacteristic
15C32×C4×Dic3C411yescharacteristic
16Dic3×C3⋊Dic3C2211yescharacteristic
17Dic3×C3⋊Dic3C2211yescharacteristic
18C2×#3787—31no—
19C2×#3787D311yescharacteristic
20C42×C3⋊D3—31no—
21C4×D3×Dic3—62no—
22#63779—32no—
23C4⋊#622—32no—
24C22×C3×C3⋊D3C2×C411yescharacteristic
25C2×C33×C4C2×C411yescharacteristic
26C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
27C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3×C4×C3⋊D3D412yes—
30C3×C4×C3⋊D3Q812yes—
31C2×C32×Dic3C2×C411yescharacteristic
32C2×C32×Dic3C2×C411yescharacteristic
33Dic3×C3⋊D3—22no—
34Dic3×C3⋊D3C814yescomplemented
35C3⋊#622—22no—
36C3×#622—22no—
37C2×Dic3⋊C8—91nomaximal
38C2×C4×C3⋊D3—31no—
39C2×C4×C3⋊D3—31no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—62no—
42C2×D3×Dic3—62no—
43C2×D3×Dic3—62no—
44C2×#622—32no—
45C2×#622C2×D312yes—
46#3787—22no—
47#3787—62no—
48C32×C42—31no—
49C4×C3⋊Dic3—31no—
50C4×C3⋊Dic3—31no—
51C4×C3⋊Dic3—31no—
52C4×C3⋊Dic3—62no—
53C4×C3⋊Dic3—122no—
54C3×C4×Dic3—22no—
55Dic32—62no—
56Dic32—62no—
57C22×C33C22×C411yescharacteristic
58C2×C3×C3⋊D3C4⋊C412yes—
59C2×C3×C3⋊D3C2×C812yes—
60C2×C3×C3⋊D3M1612yes—
61C33×C4C4⋊C412yes—
62C3⋊(C3⋊Dic3)—21no—
63C3⋊(C3⋊Dic3)C2×C812yescomplemented
64C3×C3⋊Dic3C2×D411yescharacteristic
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C32×Dic3—21no—
67C32×Dic3C2×C812yescomplemented
68C22×C3×C8—91no—
69C2×C4×Dic3—91no—
70C22×C3⋊C8—91no—
71C42×D3—182no—
72Dic3⋊C8—92no—
73Dic3⋊C8—92no—
74C22×C3⋊D3C4×D311yescharacteristic
75C22×C3×D3—62no—
76C2×C32×C4—22no—
77C2×C32×C4—31no—
78C2×C32×C4—31no—
79C2×C32×C4—42no—
80C2×C32×C4C4×D311yescharacteristic
81C2×C3⋊Dic3—31no—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—62no—
84C2×C3⋊Dic3—62no—
85C2×C3⋊Dic3—122no—
86C2×C3⋊Dic3—122no—
87C2×C3⋊Dic3C22×D311yescharacteristic
88C4×C3⋊D3—34no—
89C4×C3⋊D3—62no—
90C4×C3⋊D3Dic612yes—
91C4×C3⋊D3#6712yes—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—62no—
95C3×C4×D3—64no—
96C3×C4×D3—64no—
97D3×Dic3—68no—
98D3×Dic3—68no—
99#622—22no—
100#622—62no—
101C2×C4⋊C8—271noSylow
102C2×C33C2×C4⋊C411yescharacteristic
103C2×C33C22×C811yescharacteristic
104C2×C33C2×M1611yescharacteristic
105C3×C3⋊D3C4⋊C814yescomplemented
106C22×C3×C4—91no—
107C22×Dic3—91no—
108C22×Dic3—91no—
109C2×C4×D3—62no—
110C2×C4×D3—182no—
111C2×C4×D3—182no—
112C2×C3×C8—92no—
113C2×C3×C8—182no—
114C3×C42—62no—
115C4×Dic3—91no—
116C4×Dic3—91no—
117C4×Dic3—91no—
118C4×Dic3—182no—
119C4×Dic3—182no—
120C4×Dic3—182no—
121C4×Dic3—362no—
122C4×Dic3#15711yescharacteristic
123C2×C3⋊C8—92no—
124C2×C3⋊C8—182no—
125C22×C32—22no—
126C22×C32—42no—
127C22×C32C2×C4×D311yescharacteristic
128C2×C3⋊D3D3×C812yes—
129C2×C3⋊D3ΓL(1,𝔽25)12yes—
130C2×C3⋊D3Dic3⋊C412yes—
131C2×C3×D3—64no—
132C2×C3×D3—64no—
133C2×C3×D3—64no—
134C32×C4—24no—
135C32×C4—32no—
136C32×C4—44no—
137C32×C4—61no—
138C32×C4Dic3⋊C412yes—
139C3⋊Dic3—32no—
140C3⋊Dic3—64no—
141C3⋊Dic3—61no—
142C3⋊Dic3—64no—
143C3⋊Dic3—124no—
144C3⋊Dic3—124no—
145C3⋊Dic3C2×Dic611yescharacteristic
146C3⋊Dic3C2×#6711yescharacteristic
147C3×Dic3—24no—
148C3×Dic3—24no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C22×C8—91no—
152C22×C8—271no—
153C2×C42—271no—
154C4⋊C8—272no—
155C4⋊C8—272no—
156C33C2×C4⋊C811yescharacteristic, complemented, Sylow
157C23×C3—91no—
158C22×D3—62no—
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—62no—
162C2×C3×C4—62no—
163C2×C3×C4—91no—
164C2×C3×C4—92no—
165C2×C3×C4—92no—
166C2×C3×C4C2×#15711yescharacteristic
167C2×Dic3—62no—
168C2×Dic3—91no—
169C2×Dic3—94no—
170C2×Dic3—91no—
171C2×Dic3—182no—
172C2×Dic3—182no—
173C2×Dic3—182no—
174C2×Dic3—362no—
175C2×Dic3—362no—
176C2×Dic3C2×#15711yescharacteristic
177C2×Dic3C2×#15711yescharacteristic
178C4×D3—64no—
179C4×D3—64no—
180C4×D3—188no—
181C4×D3—188no—
182C3×C8—182no—
183C3⋊C8—182no—
184C2×C32—22no—
185C2×C32—22no—
186C2×C32—22no—
187C2×C32—42no—
188C2×C32—42no—
189C2×C32—42no—
190C2×C32C2×D3×C811yescharacteristic
191C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
192C2×C32C2×Dic3⋊C411yescharacteristic
193C3⋊D3Dic3⋊C814yescomplemented
194C3×D3—68no—
195C22×C4—91no—
196C22×C4—271no—
197C22×C4—271no—
198C2×C8—92no—
199C2×C8—182no—
200C2×C8—272no—
201C2×C8—542no—
202C42—31no—
203C42—271no—
204C42—271no—
205C42—271no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—92no—
209C22×C3—92no—
210C22×C3—92no—
211C22×C3C22×#15711yescharacteristic
212C2×D3—64no—
213C2×D3—64no—
214C2×D3—64no—
215C3×C4—24no—
216C3×C4—44no—
217C3×C4—64no—
218C3×C4—64no—
219C3×C4—91no—
220C3×C4—91no—
221C3×C4C4⋊#15712yes—
222Dic3—21no—
223Dic3—62no—
224Dic3—62no—
225Dic3—92no—
226Dic3—184no—
227Dic3—184no—
228Dic3—181no—
229Dic3—364no—
230Dic3—364no—
231Dic3#378712yescomplemented
232C32—22no—
233C32—42no—
234C32C2×Dic3⋊C811yescharacteristic, complemented
235C23—91no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—91no—
239C2×C4—92no—
240C2×C4—92no—
241C2×C4—274no—
242C2×C4—271no—
243C2×C4—271no—
244C2×C4—542no—
245C2×C4D3×#15711yescharacteristic
246C8—182no—
247C8—542no—
248C2×C3—22no—
249C2×C3—22no—
250C2×C3—22no—
251C2×C3—42no—
252C2×C3—42no—
253C2×C3—42no—
254C2×C3—94no—
255C2×C3C2×#378711yescharacteristic
256C2×C3#6428311yescharacteristic
257C2×C3C2×C4⋊#15711yescharacteristic
258D3—68no—
259C22—92no—
260C22—92no—
261C22—92no—
262C22C2×D3×#15711yescharacteristic
263C4—32no—
264C4—61no—
265C4—91no—
266C4—91no—
267C4—272no—
268C4—541no—
269C4Dic3⋊#15712yes—
270C3—22no—
271C3—42no—
272C3#1059372611yescharacteristic, complemented
273C2—94no—
274C2D3×#378711yescharacteristic
275C2#1172425411yescharacteristic
276C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutatorC2×C33C22×C8
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×C4⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722732C2semidirect
5. Dic3⋊#622C2semidirect
34. Dic3×C3⋊D3C8semidirect
63. C3⋊(C3⋊Dic3)C2×C8semidirect
67. C32×Dic3C2×C8semidirect
105. C3×C3⋊D3C4⋊C8semidirect
156. C33C2×C4⋊C8semidirect
193. C3⋊D3Dic3⋊C8semidirect
231. Dic3#3787semidirect
234. C32C2×Dic3⋊C8semidirect
272. C3#10593726semidirect

Maximal subgroups

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

subgroupindex
C2×#889882
C2×C3×#37872
C4×Dic3×C3⋊D32
#117227322
Dic3⋊#6222
#105937263
C2×Dic3⋊C89