Finite Group Databasebeta

← #13146553

Subgroups — #13146553

Order 1728.

353 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 45 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
1#11721144C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic, complemented
3C4×#6188C211yesmaximal, characteristic, complemented
4#11720705C214yesmaximal, complemented
5C2×C4⋊((C3⋊D3)⋊C3)—21no—
6C2×#6198—21no—
7C2×#6198C2212yes—
8C2×#6196—21no—
9C2×#6188C2212yes—
10C4×#1084C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)—21no—
12#88449C2212yes—
13#63918—61no—
14#63911—61no—
15C22×(C3⋊D3)⋊C3—22no—
16C2×C4×H(3,𝔽3)D411yescharacteristic
17C2×#1084D411yescharacteristic
18C2×#1084C2311yescharacteristic
19C2×(C3⋊Dic3)⋊C3—22no—
20C4⋊((C3⋊D3)⋊C3)—42no—
21#6198—24no—
22#6198—42no—
23#6196—42no—
24#6188—22no—
25#4694—91nomaximal
26C2×C4⋊(C3⋊D3)—21no—
27C2×C3×D12—61no—
28C2×#627—62no—
29C2×#626—62no—
30C2×#626—62no—
31C4⋊(C3⋊Dic3)—21no—
32C3×C4⋊Dic3—61no—
33C3×C4×Dic3—61no—
34C3×C4×Dic3—61no—
35Dic3⋊Dic3—62no—
36C22×H(3,𝔽3)C2×D411yescharacteristic
37C2×(C3⋊D3)⋊C3—42no—
38C2×(C3⋊D3)⋊C3—42no—
39C2×(C3⋊D3)⋊C3—42no—
40C4×H(3,𝔽3)SD1612yes—
41#1084D812yes—
42#1084#4112yes—
43(C3⋊Dic3)⋊C3—42no—
44#940—181no—
45#888—181no—
46#874—91no—
47#869—91no—
48C3×C4×C8—91no—
49#821—94no—
50C22×C3⋊D3—22no—
51C22×C3×D3—62no—
52C2×C32×C4—21no—
53C2×C32×C4—21no—
54C2×C3⋊Dic3—22no—
55C4⋊(C3⋊D3)—42no—
56C2×C3×Dic3—61no—
57C2×C3×Dic3—61no—
58C2×C3×Dic3—61no—
59C2×C3×Dic3—61no—
60C2×C3×Dic3—62no—
61C3×D12—122no—
62#627—124no—
63#626—124no—
64#626—124no—
65#486—271noSylow
66C2×H(3,𝔽3)#12411yescharacteristic
67C2×H(3,𝔽3)C2×D811yescharacteristic
68C2×H(3,𝔽3)C2×SD1611yescharacteristic
69(C3⋊D3)⋊C3—82no—
70C2×C3×D4—181no—
71C2×#67—182no—
72C2×#67—182no—
73C2×D12—61no—
74C2×D12—92no—
75C2×D12—181no—
76C2×D12—181no—
77C2×Dic6—181no—
78C2×Dic6—182no—
79C2×C3×C8—92no—
80C3×C4⋊C4—181no—
81C3×C42—91no—
82C4⋊Dic3—61no—
83C4⋊Dic3—92no—
84C4⋊Dic3—181no—
85Dic3⋊C4—182no—
86C4×Dic3—61no—
87C4×Dic3—61no—
88C22×C32—21no—
89C22×C32—21no—
90C2×C3⋊D3—42no—
91C2×C3⋊D3—42no—
92C2×C3⋊D3—42no—
93C2×C3×D3—122no—
94C2×C3×D3—122no—
95C2×C3×D3—122no—
96C32×C4—22no—
97C32×C4—22no—
98C3⋊Dic3—42no—
99C3×Dic3—62no—
100C3×Dic3—62no—
101C3×Dic3—62no—
102C3×Dic3—62no—
103C3×Dic3—122no—
104C4⋊Q8—271no—
105#127—271no—
106#102—274no—
107C4×C8—91no—
108H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
109C23×C3—182no—
110C22×D3—62no—
111C22×D3—182no—
112C3×D4—362no—
113C2×C3×C4—61no—
114C2×C3×C4—61no—
115C2×C3×C4—91no—
116C2×C3×C4—91no—
117C2×C3×C4—182no—
118C2×C3×C4GO+(4,𝔽2)11yescharacteristic
119#67—364no—
120#67—364no—
121C2×Dic3—61no—
122C2×Dic3—61no—
123C2×Dic3—61no—
124C2×Dic3—61no—
125C2×Dic3—62no—
126C2×Dic3—182no—
127D12—122no—
128D12—184no—
129D12—362no—
130D12—362no—
131Dic6—364no—
132Dic6—362no—
133C3×C8—182no—
134C2×C32—21no—
135C2×C32—21no—
136C2×C32—21no—
137C2×C32—21no—
138C2×C32—21no—
139C2×C32—21no—
140C3⋊D3—82no—
141C3×D3—242no—
142C2×Q8—541no—
143C2×D4—181no—
144C2×D4—272no—
145C2×D4—541no—
146C2×C8—92no—
147C4⋊C4—181no—
148C4⋊C4—272no—
149C42—91no—
150C22×C3—61no—
151C22×C3—61no—
152C22×C3—362no—
153C22×C3—362no—
154C22×C3—362no—
155C22×C3C2×GO+(4,𝔽2)11yescharacteristic
156C2×D3—122no—
157C2×D3—122no—
158C2×D3—122no—
159C2×D3—362no—
160C2×D3—362no—
161C2×D3—362no—
162C3×C4—62no—
163C3×C4—62no—
164C3×C4—92no—
165C3×C4—92no—
166C3×C4—362no—
167C3×C4#377512yes—
168Dic3—62no—
169Dic3—62no—
170Dic3—62no—
171Dic3—62no—
172Dic3—122no—
173Dic3—362no—
174C32—21no—
175C32—21no—
176C23—182no—
177Q8—1082no—
178D4—362no—
179D4—544no—
180D4—1082no—
181C2×C4—91no—
182C2×C4—91no—
183C2×C4—182no—
184C2×C4#1088⋊C211yescharacteristic
185C8—182no—
186C2×C3—61no—
187C2×C3—61no—
188C2×C3—61no—
189C2×C3—61no—
190C2×C3—61no—
191C2×C3—61no—
192C2×C3—722no—
193C2×C3#6422111yescharacteristic
194C2×C3#6422511yescharacteristic
195C2×C3C2×#377511yescharacteristic
196D3—242no—
197D3—722no—
198C22—362no—
199C22—362no—
200C22—362no—
201C22C2×#1088⋊C211yescharacteristic
202C4—92no—
203C4—92no—
204C4—362no—
205C4#8858712yes—
206C3—61no—
207C3—61no—
208C3#1059336611yescharacteristic
209C2—722no—
210C2#1172229811yescharacteristic
211C2#1172230211yescharacteristic
212C2C2×#8858711yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radical#11721144C2
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#486
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11721117C2semidirect
3. C4×#6188C2semidirect
4. #11720705C2semidirect
108. H(3,𝔽3)#486semidirect

Maximal subgroups

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

subgroupindex
#117211442
#117211172
C4×#61882
#117207052
#46949