Finite Group Databasebeta

← #13167064

Subgroups — #13167064

Order 1728.

396 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 67 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⋊#6296C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3C4×C3⋊#622C211yesmaximal, characteristic
4#11722778C214yesmaximal
5#10593384—31nomaximal
6C2×C32×Dic6—21no—
7C2×#6296—21no—
8C2×C3⋊#622C2212yes—
9C3×C4⋊(C3⋊Dic3)—21no—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11#88815C2212yes—
12C3×#3724—21no—
13C3×#3724C2212yes—
14C3×#944—61no—
15#63919—31no—
16#63918—61no—
17#63873D311yescharacteristic
18C4×#622—31no—
19#63742—34no—
20C2×C33×C4D411yescharacteristic
21C2×C3×C3⋊Dic3—22no—
22C2×C3×C3⋊Dic3D411yescharacteristic
23C2×C3×C3⋊Dic3C2311yescharacteristic
24C2×C32×Dic3—22no—
25C32×Dic6—42no—
26#6296—42no—
27C3⋊#622Q814yescomplemented
28#5080—91nomaximal
29C2×C32×Q8—61no—
30C2×C3×Dic6—21no—
31C2×C3×Dic6—21no—
32C2×C3×Dic6—41no—
33C2×#627—62no—
34C2×#627—61no—
35C2×#622—32no—
36C4⋊(C3⋊Dic3)—21no—
37C4×C3⋊Dic3C2×D311yescharacteristic
38C3×C4⋊Dic3—61no—
39C3×C4⋊Dic3—61no—
40C3×C4⋊Dic3—121no—
41C3×Dic3⋊C4—62no—
42C3×Dic3⋊C4—62no—
43C3×C4×Dic3—61no—
44C3×C4×Dic3—61no—
45#3724—21no—
46#3724—32no—
47#3724C2×D312yes—
48Dic3⋊Dic3—62no—
49C22×C33C2×D411yescharacteristic
50C33×C4SD1612yes—
51C3×C3⋊Dic3—42no—
52C3×C3⋊Dic3SD1612yes—
53C3×C3⋊Dic3C2×Q812yes—
54C32×Dic3—42no—
55C3×C4⋊Q8—91no—
56#944—61no—
57#888—181no—
58#869—91no—
59#808—94no—
60C4×C3⋊C8—91no—
61C32×Q8—122no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—41no—
65C2×C32×C4—41no—
66C2×C32×C4—62no—
67C2×C32×C4#6711yescharacteristic
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3#6711yescharacteristic
70C2×C3⋊Dic3C22×D311yescharacteristic
71C2×C3×Dic3—22no—
72C2×C3×Dic3—22no—
73C2×C3×Dic3—42no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—122no—
81C3×Dic6—42no—
82C3×Dic6—44no—
83C3×Dic6—42no—
84#627—124no—
85#627—122no—
86#622—34no—
87#498—271noSylow
88C2×C33C4⋊Q811yescharacteristic
89C2×C33C2×SD1611yescharacteristic
90C2×C33C2×SD1611yescharacteristic
91C2×C3×Q8—61no—
92C2×C3×Q8—61no—
93C2×C3×Q8—121no—
94C2×Dic6—21no—
95C2×Dic6—181no—
96C2×Dic6—182no—
97C3×C4⋊C4—92no—
98C3×C4⋊C4—181no—
99C3×C4⋊C4—181no—
100C3×C42—91no—
101C4⋊Dic3—61no—
102C4⋊Dic3—61no—
103C4⋊Dic3—92no—
104C4⋊Dic3—121no—
105Dic3⋊C4—62no—
106Dic3⋊C4—62no—
107Dic3⋊C4—182no—
108C4×Dic3—61no—
109C4×Dic3—61no—
110C2×C3⋊C8—92no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—41no—
114C22×C32—41no—
115C22×C32C2×#6711yescharacteristic
116C32×C4—22no—
117C32×C4—22no—
118C32×C4—42no—
119C32×C4—42no—
120C32×C4—122no—
121C32×C4#21412yes—
122C3⋊Dic3—42no—
123C3⋊Dic3#21512yes—
124C3⋊Dic3D3×Q812yes—
125C3×Dic3—42no—
126C3×Dic3—44no—
127C3×Dic3—42no—
128C3×Dic3—62no—
129C3×Dic3—62no—
130C3×Dic3—62no—
131C3×Dic3—62no—
132C3×Dic3—122no—
133C3×Dic3—122no—
134C3×Dic3—242no—
135C4⋊Q8—91no—
136C4⋊Q8—271no—
137#106—274no—
138C4×C8—271no—
139C33#49811yescharacteristic, complemented, Sylow
140C3×Q8—122no—
141C3×Q8—124no—
142C3×Q8—122no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—41no—
146C2×C3×C4—41no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—122no—
152C2×C3×C4—182no—
153C2×C3×C4GO+(4,𝔽2)11yescharacteristic
154C2×Dic3—22no—
155C2×Dic3—61no—
156C2×Dic3—61no—
157C2×Dic3—61no—
158C2×Dic3—61no—
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—122no—
162Dic6—42no—
163Dic6—364no—
164Dic6—362no—
165C3⋊C8—94no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—41no—
173C2×C32—41no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32C2×#21411yescharacteristic
179C2×C32C2×#21511yescharacteristic
180C2×C32#94411yescharacteristic
181C2×Q8—61no—
182C2×Q8—541no—
183C2×C8—272no—
184C4⋊C4—92no—
185C4⋊C4—181no—
186C4⋊C4—181no—
187C4⋊C4—272no—
188C42—91no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3C2×GO+(4,𝔽2)11yescharacteristic
194C3×C4—22no—
195C3×C4—22no—
196C3×C4—42no—
197C3×C4—42no—
198C3×C4—92no—
199C3×C4—92no—
200C3×C4—122no—
201C3×C4—124no—
202C3×C4—122no—
203C3×C4—362no—
204C3×C4#377512yes—
205Dic3—42no—
206Dic3—62no—
207Dic3—62no—
208Dic3—62no—
209Dic3—62no—
210Dic3—122no—
211Dic3—122no—
212Dic3—242no—
213C32—21no—
214C32—21no—
215C32—41no—
216C32—41no—
217C32#508011yescharacteristic, complemented
218Q8—122no—
219Q8—1082no—
220C2×C4—62no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—182no—
224C2×C4D32⋊D311yescharacteristic
225C8—274no—
226C2×C3—21no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3#6422211yescharacteristic
239C2×C3#6422711yescharacteristic
240C2×C3C2×#377511yescharacteristic
241C22C2×D32⋊D311yescharacteristic
242C4—92no—
243C4—92no—
244C4—122no—
245C4—362no—
246C4#8893712yes—
247C3—21no—
248C3—21no—
249C3—41no—
250C3—41no—
251C3#1059338411yescharacteristic, complemented
252C2#1172430211yescharacteristic
253C2#1172430811yescharacteristic
254C2C2×#8893711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#498
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
27. C3⋊#622Q8semidirect
139. C33#498semidirect
217. C32#5080semidirect
251. C3#10593384semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#638732
C4×C3⋊#6222
#117227782
#105933843
#50809