Finite Group Databasebeta

← #13166328

Subgroups — #13166328

Order 1728.

496 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C4×Dic3×C3⋊D3C211yesmaximal, characteristic
2#11723103C211yesmaximal, characteristic
3C3×#63778C211yesmaximal, characteristic
4Dic3⋊#622C212yesmaximal, complemented
5#11722724C212yesmaximal, complemented
6#10593730—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×Dic3×C3⋊D3C411yescharacteristic
9C2×Dic3×C3⋊D3C411yescharacteristic
10C2×C3⋊#622C2212yes—
11C2×C3×#622C2212yes—
12C4×C3⋊(C3⋊Dic3)C411yescharacteristic
13C32×C4×Dic3C411yescharacteristic
14Dic3×C3⋊Dic3C2211yescharacteristic
15Dic3×C3⋊Dic3C2211yescharacteristic
16C42×C3⋊D3—31no—
17C4×D3×Dic3—62no—
18#63778—31no—
19#63778D311yescharacteristic
20#63777—32no—
21C4⋊#622—32no—
22C22×C3×C3⋊D3C2×C411yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
25C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3—22no—
28C2×C32×Dic3C2×C411yescharacteristic
29C2×C32×Dic3C2×C411yescharacteristic
30Dic3×C3⋊D3—22no—
31Dic3×C3⋊D3—22no—
32C3⋊#622—22no—
33C3×#622—22no—
34#4719—91nomaximal
35C2×C4×C3⋊D3—31no—
36C2×C4×C3⋊D3—31no—
37C2×C4×C3⋊D3C2×D311yescharacteristic
38C2×C3×C4×D3—62no—
39C2×D3×Dic3—62no—
40C2×D3×Dic3—62no—
41C2×#622—32no—
42C2×#622C2×D312yes—
43C32×C42—31no—
44C4×C3⋊Dic3—31no—
45C4×C3⋊Dic3—31no—
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—62no—
48C4×C3⋊Dic3—122no—
49C3×C4×Dic3—22no—
50Dic32—62no—
51Dic32—62no—
52C22×C33C22×C411yescharacteristic
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—21no—
55C2×C3×C3⋊D3M1612yes—
56C33×C4—21no—
57C3⋊(C3⋊Dic3)—21no—
58C3⋊(C3⋊Dic3)—21no—
59C3×C3⋊Dic3#4112yes—
60C32×Dic3—21no—
61C32×Dic3M1612yes—
62C2×C4×Dic3—91no—
63C42×D3—182no—
64C3×C22⋊C8—91no—
65#830—91no—
66#815—92no—
67Dic3⋊C8—92no—
68C22×C3⋊D3C4×D311yescharacteristic
69C22×C3×D3—62no—
70C2×C32×C4—22no—
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—42no—
74C2×C32×C4C4×D311yescharacteristic
75C2×C3⋊Dic3—31no—
76C2×C3⋊Dic3—31no—
77C2×C3⋊Dic3—62no—
78C2×C3⋊Dic3—62no—
79C2×C3⋊Dic3—122no—
80C2×C3⋊Dic3—122no—
81C2×C3⋊Dic3C22×D311yescharacteristic
82C4×C3⋊D3—22no—
83C4×C3⋊D3—62no—
84C4×C3⋊D3—62no—
85C2×C3×Dic3—22no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—62no—
88C3×C4×D3—68no—
89D3×Dic3—68no—
90D3×Dic3—68no—
91#622—22no—
92#622—62no—
93#432—271noSylow
94C2×C33#11711yescharacteristic
95C2×C33C2×M1611yescharacteristic
96C2×C33C2×M1611yescharacteristic
97C3×C3⋊D3—22no—
98C22×C3×C4—91no—
99C22×Dic3—91no—
100C22×Dic3—91no—
101C2×C4×D3—62no—
102C2×C4×D3—182no—
103C2×C4×D3—182no—
104C2×C3×C8—92no—
105C3×C42—62no—
106C4×Dic3—91no—
107C4×Dic3—91no—
108C4×Dic3—91no—
109C4×Dic3—182no—
110C4×Dic3—182no—
111C4×Dic3—182no—
112C4×Dic3—362no—
113C4×Dic3#15711yescharacteristic
114C2×C3⋊C8—92no—
115C22×C32—22no—
116C22×C32—42no—
117C22×C32C2×C4×D311yescharacteristic
118C2×C3⋊D3—21no—
119C2×C3⋊D3—21no—
120C2×C3⋊D3ΓL(1,𝔽25)12yes—
121C2×C3×D3—64no—
122C2×C3×D3—64no—
123C2×C3×D3—64no—
124C32×C4—21no—
125C32×C4—24no—
126C32×C4—32no—
127C32×C4—44no—
128C32×C4—61no—
129C3⋊Dic3—61no—
130C3⋊Dic3—61no—
131C3⋊Dic3—64no—
132C3⋊Dic3—64no—
133C3⋊Dic3—124no—
134C3⋊Dic3—124no—
135C3⋊Dic3#23712yes—
136C3×Dic3—24no—
137C3×Dic3—24no—
138C3×Dic3—64no—
139C2×C42—271no—
140C4⋊C8—272no—
141C22⋊C8—91no—
142C22⋊C8—271no—
143#97—272no—
144C33#43211yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C22×D3—62no—
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—62no—
150C2×C3×C4—62no—
151C2×C3×C4—91no—
152C2×C3×C4—182no—
153C2×C3×C4C2×#15711yescharacteristic
154C2×Dic3—62no—
155C2×Dic3—91no—
156C2×Dic3—91no—
157C2×Dic3—182no—
158C2×Dic3—182no—
159C2×Dic3—182no—
160C2×Dic3—182no—
161C2×Dic3—362no—
162C2×Dic3—362no—
163C2×Dic3C2×#15711yescharacteristic
164C2×Dic3C2×#15711yescharacteristic
165C4×D3—68no—
166C4×D3—188no—
167C4×D3—188no—
168C3×C8—182no—
169C3⋊C8—182no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—22no—
173C2×C32—42no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32#87711yescharacteristic
177C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
178C2×C32D3×M1611yescharacteristic
179C3⋊D3—22no—
180C3×D3—68no—
181C22×C4—91no—
182C22×C4—271no—
183C22×C4—271no—
184C2×C8—92no—
185C2×C8—272no—
186C42—31no—
187C42—271no—
188C42—271no—
189C42—271no—
190C22×C3—22no—
191C22×C3—42no—
192C22×C3—92no—
193C22×C3—181no—
194C22×C3—181no—
195C22×C3C22×#15711yescharacteristic
196C2×D3—64no—
197C2×D3—64no—
198C2×D3—64no—
199C3×C4—24no—
200C3×C4—21no—
201C3×C4—44no—
202C3×C4—64no—
203C3×C4—64no—
204C3×C4—92no—
205Dic3—21no—
206Dic3—64no—
207Dic3—184no—
208Dic3—184no—
209Dic3—181no—
210Dic3—181no—
211Dic3—364no—
212Dic3—364no—
213Dic3#378812yes—
214C32—22no—
215C32—42no—
216C32#471911yescharacteristic, complemented
217C23—91no—
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—91no—
221C2×C4—182no—
222C2×C4—271no—
223C2×C4—271no—
224C2×C4—542no—
225C2×C4—542no—
226C2×C4D3×#15711yescharacteristic
227C8—182no—
228C8—542no—
229C2×C3—22no—
230C2×C3—22no—
231C2×C3—22no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—182no—
236C2×C3C2×#378811yescharacteristic
237C2×C3#6428311yescharacteristic
238C2×C3#6428611yescharacteristic
239D3—68no—
240C22—92no—
241C22—181no—
242C22—181no—
243C22C2×D3×#15711yescharacteristic
244C4—21no—
245C4—32no—
246C4—61no—
247C4—92no—
248C4—541no—
249C4—541no—
250C3—22no—
251C3—42no—
252C3#1059373011yescharacteristic, complemented
253C2—182no—
254C2D3×#378811yescharacteristic
255C2#1172425411yescharacteristic
256C2#1172426111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. Dic3⋊#622C2semidirect
5. #11722724C2semidirect
144. C33#432semidirect
216. C32#4719semidirect
252. C3#10593730semidirect

Maximal subgroups

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

subgroupindex
C4×Dic3×C3⋊D32
#117231032
C3×#637782
Dic3⋊#6222
#117227242
#105937303
#47199