Finite Group Databasebeta

← #13173420

Subgroups — #13173420

Order 1728.

752 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 93 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×C3×#3830C211yesmaximal, characteristic
2C2×#88989C211yesmaximal, characteristic, complemented
3C2×#88988C211yesmaximal, characteristic, complemented
4#11723112C214yesmaximal, complemented
5#10593823—31nomaximal
6C2×C33×D4C411yescharacteristic
7C2×C3×#638C412yes—
8C22×C3×C3⋊Dic3C2212yes—
9C3×#3830—22no—
10C3×#3830—22no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C411yescharacteristic
13C2×C3⋊#622C2212yes—
14C2×C3⋊#622C2212yes—
15#88989—22no—
16#88988—22no—
17C2×#3830D311yescharacteristic
18C2×C3×#237—62no—
19C2×#3788—31no—
20C2×#3787—31no—
21C22⋊#622—34no—
22C23×C33C2×C412yes—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C33×D4—22no—
25C2×C33×C4C2×C411yescharacteristic
26C3×#638—24no—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3—22no—
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×C4×C3⋊D3D412yes—
32C3×C4×C3⋊D3D412yes—
33C3×#634—22no—
34C3⋊#622—22no—
35C3⋊#622—22no—
36#5234—91nomaximal
37C2×C32×D4—22no—
38C2×C32×D4—42no—
39C2×C32×D4Dic311yescharacteristic
40C2×#638Dic312yes—
41C22×C3⋊Dic3C2×D312yes—
42#3830—22no—
43#3830—22no—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×#634Dic311yescharacteristic
46C2×C3×#67—64no—
47C22×C3×Dic3—64no—
48C3×#237—68no—
49C3×#237—68no—
50C2×C3×C4×D3—62no—
51C2×C3×Dic6—62no—
52C2×#622—32no—
53C2×#622—32no—
54#3788—62no—
55#3787—62no—
56C22×C33—22no—
57C22×C33—24no—
58C22×C33C22×C411yescharacteristic
59C2×C3×C3⋊D3—22no—
60C2×C3×C3⋊D3C22⋊C412yes—
61C33×C4C22⋊C412yes—
62C3×C3⋊Dic3—22no—
63C3×C3⋊Dic3C2×D411yescharacteristic
64C3×C3⋊Dic3C2×D411yescharacteristic
65C2×C3×#41—91no—
66C2×#237—62no—
67C2×#208—91no—
68C22×C3⋊C8—91no—
69#830—94no—
70C23×C32—24no—
71C23×C32—44no—
72C23×C32C2×Dic312yes—
73C22×C3⋊D3C2×Dic311yescharacteristic
74C22×C3×D3—62no—
75C32×D4—22no—
76C32×D4—28no—
77C32×D4—48no—
78C2×C32×C4—22no—
79C2×C32×C4—42no—
80C2×C32×C4C2×Dic311yescharacteristic
81#638—24no—
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3C2×Dic312yes—
85C2×C3⋊Dic3C22×D311yescharacteristic
86C4×C3⋊D3#6712yes—
87C4×C3⋊D3#6712yes—
88#634—22no—
89C3×#67—616no—
90C2×C3×Dic3—64no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—68no—
93C2×C3×Dic3—68no—
94C3×C4×D3—64no—
95C3×C4×D3—64no—
96C3×Dic6—68no—
97#622—62no—
98#622—62no—
99#408—271noSylow
100C2×C33—42no—
101C2×C33C2×C22⋊C411yescharacteristic
102C2×C33#13112yes—
103C3×C3⋊D3—22no—
104C3×#41—182no—
105C3×#41—182no—
106C2×C3×Q8—91no—
107C2×C3×D4—22no—
108C2×C3×D4—42no—
109C2×C3×D4—92no—
110C2×C3×D4#15711yescharacteristic
111C22×C3×C4—91no—
112C22×C3×C4—92no—
113C2×#67—64no—
114C22×Dic3—64no—
115#237—68no—
116#237—68no—
117C2×C4×D3—62no—
118C2×Dic6—62no—
119#208—182no—
120C2×C3⋊C8—92no—
121C2×C3⋊C8—92no—
122C2×C3⋊C8—182no—
123C22×C32—22no—
124C22×C32—28no—
125C22×C32—22no—
126C22×C32—24no—
127C22×C32—48no—
128C22×C32—42no—
129C22×C32—48no—
130C22×C32—88no—
131C22×C32C22×Dic311yescharacteristic
132C2×C3⋊D3—22no—
133C2×C3⋊D3#21712yes—
134C2×C3×D3—64no—
135C2×C3×D3—124no—
136C32×C4—24no—
137C32×C4—44no—
138C32×C4#21712yes—
139C3⋊Dic3—22no—
140C3⋊Dic3C2×#6711yescharacteristic
141C3⋊Dic3C2×#6711yescharacteristic
142C3×Dic3—62no—
143C3×Dic3—62no—
144C3×Dic3—68no—
145C2×#41—91no—
146C2×M16—271no—
147C22×C8—271no—
148C22⋊C8—274no—
149C33#40811yescharacteristic, complemented, Sylow
150C23×C3—24no—
151C23×C3—44no—
152C23×C3—91no—
153C23×C3C2×#15712yes—
154C22×D3—62no—
155C3×Q8—182no—
156C3×D4—28no—
157C3×D4—22no—
158C3×D4—48no—
159C3×D4—184no—
160C2×C3×C4—22no—
161C2×C3×C4—42no—
162C2×C3×C4—91no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4—92no—
166C2×C3×C4—182no—
167C2×C3×C4—182no—
168C2×C3×C4C2×#15711yescharacteristic
169#67—616no—
170C2×Dic3—64no—
171C2×Dic3—62no—
172C2×Dic3—68no—
173C2×Dic3—68no—
174C4×D3—64no—
175C4×D3—64no—
176Dic6—68no—
177C3⋊C8—182no—
178C3⋊C8—182no—
179C2×C32—24no—
180C2×C32—22no—
181C2×C32—42no—
182C2×C32—44no—
183C2×C32—42no—
184C2×C32—48no—
185C2×C32—88no—
186C2×C32#94812yes—
187C2×C32C2×#21711yescharacteristic
188C3⋊D3—22no—
189C3×D3—124no—
190#41—182no—
191#41—182no—
192C2×Q8—91no—
193C2×D4—92no—
194C2×D4C3⋊#15711yescharacteristic
195C22×C4—91no—
196C22×C4—92no—
197M16—542no—
198C2×C8—272no—
199C2×C8—272no—
200C2×C8—542no—
201C22×C3—28no—
202C22×C3—24no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—48no—
206C22×C3—48no—
207C22×C3—42no—
208C22×C3—88no—
209C22×C3—92no—
210C22×C3—182no—
211C22×C3C22×#15711yescharacteristic
212C2×D3—64no—
213C2×D3—124no—
214C3×C4—24no—
215C3×C4—44no—
216C3×C4—91no—
217C3×C4—91no—
218C3×C4—182no—
219C3×C4C22⋊#15712yes—
220Dic3—62no—
221Dic3—62no—
222Dic3—68no—
223C32—22no—
224C32—42no—
225C32#523411yescharacteristic, complemented
226C23—91no—
227C23C2×C3⋊#15712yes—
228Q8—182no—
229D4—22no—
230D4—184no—
231C2×C4—91no—
232C2×C4—92no—
233C2×C4—92no—
234C2×C4—92no—
235C2×C4—182no—
236C2×C4—182no—
237C2×C4C2×C3⋊#15711yescharacteristic
238C8—542no—
239C8—542no—
240C2×C3—24no—
241C2×C3—22no—
242C2×C3—48no—
243C2×C3—42no—
244C2×C3—44no—
245C2×C3—42no—
246C2×C3—88no—
247C2×C3—182no—
248C2×C3#6428712yes—
249C2×C3C2×C22⋊#15711yescharacteristic
250D3—124no—
251C22—24no—
252C22—22no—
253C22—92no—
254C22—182no—
255C22C22×C3⋊#15711yescharacteristic
256C4—91no—
257C4—91no—
258C4—182no—
259C4#8899412yes—
260C3—22no—
261C3—42no—
262C3#1059382311yescharacteristic, complemented
263C2—42no—
264C2—182no—
265C2#1172451512yes—
266C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC2×C33×D4C4
prime radicalC2×C3×#3830C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88989C2semidirect
3. C2×#88988C2semidirect
4. #11723112C2semidirect
149. C33#408semidirect
225. C32#5234semidirect
262. C3#10593823semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#889892
C2×#889882
#117231122
#105938233
#52349