Finite Group Databasebeta

← #13135213

Subgroups — #13135213

Order 1728.

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

normal subgroups only

This group has 262 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#64156⋊C3C211yesmaximal, characteristic
2#11720584C211yesmaximal, characteristic, complemented
3#63663⋊C3C211yesmaximal, characteristic, complemented
4#10589183—31nomaximal
5#10589182—31nomaximal
6C2×Q8×H(3,𝔽3)C2211yescharacteristic
7C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
8#88538—21no—
9#3752⋊C3—21no—
10#3752⋊C3—21no—
11C4×(C3⋊Dic3)⋊C3C411yescharacteristic
12#3748⋊C3—21no—
13#64156D311yescharacteristic
14C3×#947—31no—
15#63663D311yescharacteristic, complemented
16C3×#836—31no—
17C3×#836—31no—
18C3×#836—31no—
19C3×#836—61no—
20C22×(C3⋊D3)⋊C3—21no—
21Q8×H(3,𝔽3)—41no—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)D411yescharacteristic
25C2×(C3⋊Dic3)⋊C3—21no—
26C4⋊((C3⋊D3)⋊C3)—41no—
27#6180—41no—
28(C3⋊(C3⋊C8))⋊C3—41no—
29#4542—91no—
30#4477—91no—
31#4477—91no—
32C2×C32×Q8—21no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C32×Q8C2×D311yescharacteristic
35C2×C4⋊(C3⋊D3)Dic311yescharacteristic
36C2×C3×D12—31no—
37#3752—21no—
38#3752—21no—
39C4×C3⋊Dic3Dic311yescharacteristic
40#3748—21no—
41C3×(C2×D3)⋊C4—61no—
42C3×(C2×D3)⋊C4—61no—
43C3×C4×Dic3—31no—
44C3×#208—62no—
45C3×#208—61no—
46C3×#208—61no—
47C3×#208—61no—
48C22×H(3,𝔽3)C22⋊C411yescharacteristic
49C2×(C3⋊D3)⋊C3—42no—
50C2×(C3⋊D3)⋊C3—41no—
51C4×H(3,𝔽3)—21no—
52C4×H(3,𝔽3)—21no—
53C4×H(3,𝔽3)—21no—
54(C3⋊Dic3)⋊C3—41no—
55C3×#124—91no—
56#947—31no—
57#947—91no—
58C3×#101—91no—
59C3×#101—91no—
60#836—31no—
61#836—31no—
62#836—31no—
63#836—61no—
64#836—91no—
65C22×C3⋊D3—21no—
66C22×C3×D3—61no—
67C32×Q8—41no—
68C32×Q8—41no—
69C32×Q8—42no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C32×C4C4×D311yescharacteristic
74C2×C32×C4D1211yescharacteristic
75C2×C32×C4C2×Dic311yescharacteristic
76C2×C32×C4#6711yescharacteristic
77C2×C32×C4#6711yescharacteristic
78C2×C32×C4#6711yescharacteristic
79C2×C3⋊Dic3—21no—
80C4⋊(C3⋊D3)—41no—
81C2×C3×Dic3—61no—
82C3×D12—121no—
83C3⋊(C3⋊C8)—41no—
84C3×C3⋊C8—121no—
85C3×C3⋊C8—122no—
86C3×C3⋊C8—121no—
87C3×C3⋊C8—121no—
88#355—271noSylow
89C2×H(3,𝔽3)—21no—
90C2×H(3,𝔽3)#9911yescharacteristic
91(C3⋊D3)⋊C3—81no—
92C2×C3×Q8—31no—
93C2×C3×Q8—31no—
94C2×C3×Q8—61no—
95C2×C3×Q8D3211yescharacteristic
96C2×C3×D4—91no—
97C2×D12—31no—
98C2×D12—91no—
99C3×M16—181no—
100C3×M16—181no—
101C3×C22⋊C4—181no—
102C3×C22⋊C4—181no—
103C3×C42—91no—
104(C2×D3)⋊C4—61no—
105(C2×D3)⋊C4—61no—
106(C2×D3)⋊C4—181no—
107(C2×D3)⋊C4—181no—
108C4×Dic3—31no—
109C4×Dic3—91no—
110#208—61no—
111#208—61no—
112#208—62no—
113#208—61no—
114#208—181no—
115C22×C32—21no—
116C22×C32(C2×D3)⋊C411yescharacteristic
117C22×C32#21711yescharacteristic
118C2×C3⋊D3—42no—
119C2×C3⋊D3—41no—
120C2×C3×D3—122no—
121C2×C3×D3—121no—
122C32×C4—22no—
123C32×C4—21no—
124C32×C4—21no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4—21no—
128C32×C4—21no—
129C32×C4—41no—
130C32×C4—41no—
131C3⋊Dic3—41no—
132C3×Dic3—121no—
133#124—91no—
134#101—91no—
135#101—91no—
136H(3,𝔽3)#35511yescharacteristic, complemented, Sylow
137C23×C3—181no—
138C22×D3—61no—
139C22×D3—181no—
140C3×Q8—41no—
141C3×Q8—121no—
142C3×Q8—121no—
143C3×Q8—122no—
144C3×D4—361no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—61no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—181no—
155C2×C3×C4D3×Dic311yescharacteristic
156C2×C3×C4#62511yescharacteristic
157C2×C3×C4#62611yescharacteristic
158C2×Dic3—61no—
159C2×Dic3—181no—
160D12—121no—
161D12—361no—
162C3×C8—361no—
163C3×C8—361no—
164C3⋊C8—121no—
165C3⋊C8—121no—
166C3⋊C8—121no—
167C3⋊C8—122no—
168C3⋊C8—361no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—41no—
173C2×C32#80611yescharacteristic
174C2×C32#83411yescharacteristic
175C3⋊D3—81no—
176C3×D3—241no—
177C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
178C2×D4—91no—
179M16—181no—
180M16—181no—
181C22⋊C4—181no—
182C22⋊C4—181no—
183C42—91no—
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—61no—
187C22×C3—361no—
188C22×C3—362no—
189C22×C3(C2×D3)⋊Dic311yescharacteristic
190C2×D3—122no—
191C2×D3—121no—
192C2×D3—361no—
193C2×D3—362no—
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—61no—
198C3×C4—61no—
199C3×C4—61no—
200C3×C4—61no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—62no—
204C3×C4—121no—
205C3×C4—121no—
206C3×C4—361no—
207Dic3—121no—
208Dic3—361no—
209C32—21no—
210C32#447711yescharacteristic, complemented
211C32#454211yescharacteristic, complemented
212C23—181no—
213Q8—41no—
214D4—361no—
215C2×C4—181no—
216C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
217C2×C4#619811yescharacteristic
218C2×C4#620011yescharacteristic
219C8—361no—
220C8—361no—
221C2×C3—21no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—61no—
225C2×C3—61no—
226C2×C3—61no—
227C2×C3—121no—
228C2×C3—721no—
229C2×C3#6358011yescharacteristic
230D3—241no—
231D3—721no—
232C22—362no—
233C22—361no—
234C22#8845011yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—361no—
239C3—31no—
240C3—31no—
241C3—61no—
242C3#1058918211yescharacteristic
243C2—21no—
244C2—721no—
245C2#1172033311yescharacteristic
subgroupG/H
centerC2#11720333
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×Q8D32
Fitting, cosocleC2×Q8×H(3,𝔽3)C22
socleC2×C3#63580
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#355
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11720584C2semidirect
3. #63663⋊C3C2semidirect
15. #63663D3semidirect
136. H(3,𝔽3)#355semidirect
210. C32#4477semidirect
211. C32#4542semidirect

Maximal subgroups

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

subgroupindex
#64156⋊C32
#117205842
#63663⋊C32
#105891833
#105891823