Finite Group Databasebeta

← #13146675

Subgroups — #13146675

Order 1728.

357 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 43 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#11721318C211yesmaximal, characteristic
2C2×#88449C211yesmaximal, characteristic, complemented
3C22⋊#6188C211yesmaximal, characteristic, complemented
4#11720705C212yesmaximal, complemented
5#11720704C212yesmaximal, complemented
6C22×#1084C2211yescharacteristic
7C2×#638⋊C3—21no—
8C22×(C3⋊Dic3)⋊C3—21no—
9C2×#6198C2211yescharacteristic
10C2×#6188C2212yes—
11#88450—21no—
12#88449—21no—
13#88449C2211yescharacteristic
14#88449C2212yes—
15#63978—61no—
16C2×Dic3⋊Dic3—61no—
17C23×H(3,𝔽3)D411yescharacteristic
18C22×(C3⋊D3)⋊C3—21no—
19C2×#1084—22no—
20C2×#1084D411yescharacteristic
21C2×#1084C2311yescharacteristic
22#638⋊C3—42no—
23C2×(C3⋊Dic3)⋊C3—22no—
24C2×(C3⋊Dic3)⋊C3—21no—
25C2×(C3⋊Dic3)⋊C3—42no—
26#6198—22no—
27#6188—22no—
28#4736—91nomaximal
29C2×#638—21no—
30C22×C3⋊Dic3—21no—
31C2×C3×#67—61no—
32C22×C3×Dic3—61no—
33C22×C3×Dic3—61no—
34C22×C3×Dic3—61no—
35C2×#626—61no—
36Dic3⋊Dic3—62no—
37Dic3⋊Dic3—61no—
38Dic3⋊Dic3—62no—
39#3722—61no—
40(C2×D3)⋊Dic3—61no—
41C22×H(3,𝔽3)—21no—
42C22×H(3,𝔽3)—21no—
43C22×H(3,𝔽3)D812yes—
44C22×H(3,𝔽3)C2×D411yescharacteristic
45C2×(C3⋊D3)⋊C3—41no—
46C2×(C3⋊D3)⋊C3—41no—
47C2×(C3⋊D3)⋊C3—41no—
48#1084—21no—
49#1084#4112yes—
50(C3⋊Dic3)⋊C3—42no—
51(C3⋊Dic3)⋊C3—41no—
52#939—181no—
53#930—91no—
54C2×C4⋊Dic3—91no—
55C2×Dic3⋊C4—181no—
56C3×C22⋊C8—91no—
57#821—92no—
58#818—92no—
59C23×C32—21no—
60C23×C32—21no—
61C22×C3⋊D3—21no—
62C22×C3×D3—61no—
63#638—42no—
64C2×C3⋊Dic3—21no—
65C2×C3⋊Dic3—22no—
66C2×C3⋊Dic3—42no—
67C3×#67—122no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3—61no—
70C2×C3×Dic3—61no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—64no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—122no—
76C2×C3×Dic3—122no—
77#626—122no—
78#480—271noSylow
79C2×H(3,𝔽3)—22no—
80C2×H(3,𝔽3)#12311yescharacteristic
81C2×H(3,𝔽3)C2×D811yescharacteristic
82C2×H(3,𝔽3)#13711yescharacteristic
83(C3⋊D3)⋊C3—81no—
84C2×C3×D4—181no—
85C22×C3×C4—91no—
86C22×C3×C4—181no—
87C2×#67—61no—
88C2×#67—181no—
89C2×#67—181no—
90C22×Dic3—61no—
91C22×Dic3—61no—
92C22×Dic3—61no—
93C22×Dic3—181no—
94C2×D12—91no—
95C2×C3×C8—92no—
96#217—181no—
97(C2×D3)⋊C4—181no—
98(C2×D3)⋊C4—181no—
99C4⋊Dic3—91no—
100C4⋊Dic3—92no—
101C4⋊Dic3—181no—
102Dic3⋊C4—182no—
103Dic3⋊C4—181no—
104Dic3⋊C4—182no—
105C22×C32—22no—
106C22×C32—22no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32—22no—
110C22×C32—22no—
111C22×C32—41no—
112C22×C32—41no—
113C2×C3⋊D3—41no—
114C2×C3⋊D3—41no—
115C2×C3⋊D3—41no—
116C2×C3×D3—121no—
117C2×C3×D3—121no—
118C2×C3×D3—121no—
119C3⋊Dic3—42no—
120C3⋊Dic3—41no—
121C3×Dic3—62no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—121no—
125C3×Dic3—122no—
126C3×Dic3—121no—
127#121—271no—
128C2×C4⋊C4—271no—
129#107—272no—
130#102—272no—
131C22⋊C8—91no—
132H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
133C23×C3—61no—
134C23×C3—61no—
135C23×C3—181no—
136C23×C3GO+(4,𝔽2)11yescharacteristic
137C22×D3—61no—
138C22×D3—181no—
139C3×D4—362no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4—181no—
145C2×C3×C4—362no—
146#67—122no—
147#67—362no—
148#67—362no—
149C2×Dic3—61no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—64no—
154C2×Dic3—62no—
155C2×Dic3—61no—
156C2×Dic3—122no—
157C2×Dic3—122no—
158C2×Dic3—182no—
159C2×Dic3—181no—
160C2×Dic3—362no—
161D12—182no—
162C3×C8—182no—
163C2×C32—24no—
164C2×C32—21no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32—21no—
170C2×C32—42no—
171C3⋊D3—81no—
172C3×D3—241no—
173C2×D4—181no—
174C2×D4—271no—
175C22×C4—91no—
176C22×C4—181no—
177C2×C8—92no—
178C4⋊C4—272no—
179C4⋊C4—271no—
180C4⋊C4—541no—
181C22⋊C4—541no—
182C22×C3—21no—
183C22×C3—21no—
184C22×C3—62no—
185C22×C3—62no—
186C22×C3—62no—
187C22×C3—62no—
188C22×C3—61no—
189C22×C3—61no—
190C22×C3—121no—
191C22×C3—121no—
192C22×C3—361no—
193C22×C3—361no—
194C22×C3—361no—
195C22×C3#377412yes—
196C22×C3C2×GO+(4,𝔽2)11yescharacteristic
197C2×D3—121no—
198C2×D3—121no—
199C2×D3—121no—
200C2×D3—361no—
201C2×D3—361no—
202C2×D3—361no—
203C3×C4—92no—
204C3×C4—181no—
205C3×C4—362no—
206C3×C4—361no—
207Dic3—62no—
208Dic3—62no—
209Dic3—62no—
210Dic3—121no—
211Dic3—122no—
212Dic3—121no—
213Dic3—362no—
214Dic3—361no—
215C32—21no—
216C32—21no—
217C23—181no—
218C23#1088⋊C211yescharacteristic
219D4—362no—
220D4—542no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—182no—
224C2×C4—181no—
225C2×C4—182no—
226C2×C4—362no—
227C8—182no—
228C2×C3—22no—
229C2×C3—64no—
230C2×C3—61no—
231C2×C3—61no—
232C2×C3—61no—
233C2×C3—61no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—122no—
237C2×C3—721no—
238C2×C3#6423311yescharacteristic
239C2×C3C2×#377411yescharacteristic
240C2×C3#6423911yescharacteristic
241D3—241no—
242D3—721no—
243C22—21no—
244C22—21no—
245C22—361no—
246C22—361no—
247C22—361no—
248C22#8858812yes—
249C22C2×#1088⋊C211yescharacteristic
250C4—92no—
251C4—181no—
252C4—362no—
253C4—361no—
254C3—61no—
255C3—61no—
256C3#1059349211yescharacteristic
257C2—22no—
258C2—721no—
259C2#1172230711yescharacteristic
260C2C2×#8858811yescharacteristic
261C2#1172231411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC23×H(3,𝔽3)D4
prime radical#11721318C2
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)#480
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88449C2semidirect
3. C22⋊#6188C2semidirect
4. #11720705C2semidirect
5. #11720704C2semidirect
132. H(3,𝔽3)#480semidirect

Maximal subgroups

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

subgroupindex
#117213182
C2×#884492
C22⋊#61882
#117207052
#117207042
#47369