Finite Group Databasebeta

← #13137327

Subgroups — #13137327

Order 1728.

392 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721716C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic
3C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#11720309C212yesmaximal, complemented
5#11720308C212yesmaximal
6#10590293—31nomaximal
7#10590278—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12#88540—22no—
13#3752⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#64084D311yescharacteristic
16C3×#875—31no—
17C3×#870—31no—
18C3×#870—31no—
19C3×#870—61no—
20C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
21C3×C4×C3⋊C8—31no—
22#63567—32no—
23#63565—32no—
24#63564—32no—
25#63562—32no—
26C22×(C3⋊D3)⋊C3—21no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×(C3⋊Dic3)⋊C3—21no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32#634⋊C3—22no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5068—91no—
35#5056—91no—
36#4697—91no—
37C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
38C2×#634C2×D311yescharacteristic
39C2×C3×D12—31no—
40C2×C3×Dic6—31no—
41C32×C42—21no—
42C32×C42C2×D311yescharacteristic
43C32×C42C2×D311yescharacteristic
44#3752—22no—
45C2×C3⋊(C3⋊C8)C2×D312yes—
46C3×(C2×D3)⋊C4—62no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—62no—
50C3×Dic3⋊C4—64no—
51C3×Dic3⋊C4—62no—
52C3×Dic3⋊C4—62no—
53C2×C3×C3⋊C8—32no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×(C3⋊D3)⋊C3—41no—
56C2×(C3⋊D3)⋊C3—42no—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)#4112yes—
59#1084—42no—
60(C3⋊Dic3)⋊C3—41no—
61C3×#125—91no—
62C3×#124—91no—
63#875—31no—
64#875—91no—
65#870—31no—
66#870—31no—
67#870—61no—
68C3×C4×C8—91no—
69#835—92no—
70#832—92no—
71#821—92no—
72#816—92no—
73#810—92no—
74#809—92no—
75C4×C3⋊C8—31no—
76C4×C3⋊C8—91no—
77C22×C3⋊D3—21no—
78C22×C3×D3—61no—
79C2×C32×C4—22no—
80C2×C32×C4—21no—
81C2×C32×C4D1212yes—
82C2×C32×C4#6712yes—
83C2×C32×C4C22×D311yescharacteristic
84C2×C32×C4C22×D311yescharacteristic
85C2×C3⋊Dic3—21no—
86C4⋊(C3⋊D3)—22no—
87#634—22no—
88C2×C3×Dic3—64no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C3×D12—62no—
93C3×Dic6—62no—
94C3⋊(C3⋊C8)—22no—
95C3×C3⋊C8—62no—
96#488—271noSylow
97C2×H(3,𝔽3)#12411yescharacteristic
98C2×H(3,𝔽3)#13512yes—
99(C3⋊D3)⋊C3—81no—
100C2×C3×Q8—91no—
101C2×C3×D4—91no—
102C2×D12—31no—
103C2×D12—91no—
104C2×Dic6—31no—
105C2×Dic6—91no—
106C2×C3×C8—92no—
107C3×C4⋊C4—92no—
108C3×C4⋊C4—182no—
109C3×C22⋊C4—182no—
110C3×C42—31no—
111C3×C42—31no—
112C3×C42—61no—
113C3×C42D3211yescharacteristic
114(C2×D3)⋊C4—62no—
115(C2×D3)⋊C4—182no—
116C4⋊Dic3—32no—
117C4⋊Dic3—32no—
118C4⋊Dic3—62no—
119Dic3⋊C4—62no—
120Dic3⋊C4—62no—
121Dic3⋊C4—64no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—92no—
124C22×C32—21no—
125C22×C32C2×D1211yescharacteristic
126C22×C32C2×#6711yescharacteristic
127C2×C3⋊D3—41no—
128C2×C3⋊D3—42no—
129C2×C3×D3—121no—
130C2×C3×D3—122no—
131C32×C4—22no—
132C32×C4—22no—
133C32×C4—22no—
134C32×C4—42no—
135C32×C4#23512yes—
136C32×C4#23712yes—
137C3⋊Dic3—41no—
138C3×Dic3—124no—
139C3×Dic3—121no—
140C3×Dic3—122no—
141C3×Dic3—122no—
142#125—91no—
143#124—91no—
144#103—272no—
145#102—272no—
146C4×C8—91no—
147H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
148C23×C3—181no—
149C22×D3—61no—
150C22×D3—181no—
151C3×Q8—182no—
152C3×D4—182no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—62no—
158C2×C3×C4—61no—
159C2×C3×C4—181no—
160C2×C3×C4—182no—
161C2×C3×C4#62612yes—
162C2×C3×C4C2×D3211yescharacteristic
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—64no—
166C2×Dic3—61no—
167C2×Dic3—181no—
168D12—62no—
169D12—182no—
170Dic6—62no—
171Dic6—182no—
172C3×C8—182no—
173C3⋊C8—62no—
174C3⋊C8—182no—
175C2×C32—22no—
176C2×C32—21no—
177C2×C32#87511yescharacteristic
178C2×C32#90412yes—
179C2×C32#93611yescharacteristic
180C2×C32#95012yes—
181C3⋊D3—81no—
182C3×D3—241no—
183C2×Q8—91no—
184C2×D4—91no—
185C2×C8—92no—
186C4⋊C4—92no—
187C4⋊C4—182no—
188C22⋊C4—182no—
189C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
190C22×C3—31no—
191C22×C3—31no—
192C22×C3—61no—
193C22×C3—361no—
194C22×C3—362no—
195C22×C3C2×#62611yescharacteristic
196C2×D3—122no—
197C2×D3—121no—
198C2×D3—361no—
199C2×D3—362no—
200C3×C4—22no—
201C3×C4—32no—
202C3×C4—32no—
203C3×C4—62no—
204C3×C4—62no—
205C3×C4—62no—
206C3×C4—122no—
207C3×C4—361no—
208C3×C4—362no—
209C3×C4#379512yes—
210Dic3—122no—
211Dic3—122no—
212Dic3—124no—
213Dic3—121no—
214Dic3—361no—
215C32—21no—
216C32#469711yescharacteristic, complemented
217C32#505611yescharacteristic, complemented
218C23—181no—
219Q8—182no—
220D4—182no—
221C2×C4—181no—
222C2×C4—182no—
223C2×C4#620012yes—
224C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C8—182no—
226C2×C3—32no—
227C2×C3—32no—
228C2×C3—31no—
229C2×C3—31no—
230C2×C3—62no—
231C2×C3—61no—
232C2×C3—721no—
233C2×C3#6382912yes—
234C2×C3#6386011yescharacteristic
235D3—241no—
236D3—721no—
237C22—362no—
238C22—361no—
239C22C2×#620011yescharacteristic
240C4—22no—
241C4—362no—
242C4—361no—
243C4#8864812yes—
244C3—31no—
245C3—31no—
246C3—61no—
247C3#1059027411yescharacteristic
248C2—721no—
249C2#1172107312yes—
250C2#1172111011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721716C2semidirect
3. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #11720309C2semidirect
20. C4×C3⋊(C3⋊C8)D3semidirect
147. H(3,𝔽3)#488semidirect
216. C32#4697semidirect
217. C32#5056semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64084⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#117203092
#117203082
#105902933
#105902783