Finite Group Databasebeta

← #13137387

Subgroups — #13137387

Order 1728.

474 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 79 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#64084⋊C3C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C2×#88433C212yesmaximal, complemented
4C2×#88431C212yesmaximal
5#11720520C211yesmaximal, characteristic, complemented
6#10590374—31nomaximal
7#10590372—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×#634⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2212yes—
13#3752⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15#88433—24no—
16#88431—24no—
17#64084D311yescharacteristic
18#64078D311yescharacteristic, complemented
19C3×#875—31no—
20C3×#869—31no—
21C2×#3719—32no—
22C2×#3719—32no—
23C2×#3717—32no—
24C2×#3716—32no—
25C3×#803—31no—
26C3×#803—31no—
27C3×#803—61no—
28C22×(C3⋊D3)⋊C3—21no—
29C2×C4×H(3,𝔽3)D412yes—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3—21no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#634⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37#6180D414yes—
38#5086—91no—
39#5060—91no—
40#4715—91no—
41C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
42C2×#634C2×D312yes—
43C2×#634C2×D311yescharacteristic
44C2×C3×D12—31no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—31no—
47C32×C42—21no—
48C32×C42C2×D311yescharacteristic
49C32×C42C2×D311yescharacteristic
50#3752—22no—
51C4⋊(C3⋊Dic3)—22no—
52C3×(C2×D3)⋊C4—62no—
53C3×C4⋊Dic3—62no—
54C2×C3×C3⋊C8—32no—
55C2×C3×C3⋊C8—32no—
56C2×C3×C3⋊C8—62no—
57#3719—64no—
58#3719—64no—
59#3717—64no—
60#3716—64no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C2×(C3⋊D3)⋊C3—41no—
63C2×(C3⋊D3)⋊C3—42no—
64C4×H(3,𝔽3)—22no—
65C4×H(3,𝔽3)C2×D412yes—
66(C3⋊Dic3)⋊C3—42no—
67(C3⋊Dic3)⋊C3—41no—
68C3×C4⋊Q8—91no—
69C3×#124—91no—
70C2×#216—92no—
71C2×#216—92no—
72C2×#215—92no—
73C2×#214—92no—
74C2×Dic12—92no—
75C2×#204—92no—
76#875—31no—
77#875—91no—
78#869—31no—
79#869—91no—
80C3×#97—91no—
81#803—31no—
82#803—31no—
83#803—61no—
84C22×C3⋊D3—21no—
85C22×C3×D3—61no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4#6712yes—
89C2×C32×C4#6712yes—
90C2×C32×C4C22×D311yescharacteristic
91C2×C32×C4C22×D311yescharacteristic
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—21no—
94C4⋊(C3⋊D3)—22no—
95#634—22no—
96#634—22no—
97#634—22no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—61no—
100C3×D12—62no—
101C3×Dic6—62no—
102C3×Dic6—62no—
103C3×Dic6—62no—
104C3×C3⋊C8—34no—
105C3×C3⋊C8—34no—
106C3×C3⋊C8—64no—
107#497—271noSylow
108C2×H(3,𝔽3)#12711yescharacteristic
109C2×H(3,𝔽3)#13712yes—
110(C3⋊D3)⋊C3—81no—
111C2×C3×Q8—92no—
112C2×C3×Q8—91no—
113C2×C3×D4—91no—
114C2×D12—31no—
115C2×D12—91no—
116C2×Dic6—32no—
117C2×Dic6—31no—
118C2×Dic6—92no—
119C2×Dic6—91no—
120C2×C3×C8—92no—
121C3×C4⋊C4—182no—
122C3×C22⋊C4—182no—
123C3×C42—31no—
124C3×C42—31no—
125C3×C42—61no—
126C3×C42D3211yescharacteristic
127#216—184no—
128#216—184no—
129#215—184no—
130#214—184no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4—182no—
133C4⋊Dic3—62no—
134C4⋊Dic3—182no—
135C2×C3⋊C8—32no—
136C2×C3⋊C8—32no—
137C2×C3⋊C8—62no—
138Dic12—184no—
139#204—184no—
140C22×C32—21no—
141C22×C32C2×#6711yescharacteristic
142C22×C32C2×#6711yescharacteristic
143C2×C3⋊D3—41no—
144C2×C3⋊D3—42no—
145C2×C3×D3—121no—
146C2×C3×D3—122no—
147C32×C4—22no—
148C32×C4—22no—
149C32×C4—22no—
150C32×C4—42no—
151C32×C4D3×D412yes—
152C32×C4D3×D412yes—
153C3⋊Dic3—41no—
154C3⋊Dic3—42no—
155C3×Dic3—122no—
156C3×Dic3—121no—
157C2×Q16—272no—
158C2×SD16—272no—
159C4⋊Q8—91no—
160#124—91no—
161#97—91no—
162H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
163C23×C3—181no—
164C22×D3—61no—
165C22×D3—181no—
166C3×Q8—182no—
167C3×Q8—182no—
168C3×Q8—182no—
169C3×D4—182no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—62no—
175C2×C3×C4—61no—
176C2×C3×C4—181no—
177C2×C3×C4—182no—
178C2×C3×C4#62512yes—
179C2×C3×C4C2×D3211yescharacteristic
180C2×Dic3—62no—
181C2×Dic3—61no—
182C2×Dic3—182no—
183C2×Dic3—181no—
184D12—62no—
185D12—182no—
186Dic6—62no—
187Dic6—62no—
188Dic6—62no—
189Dic6—182no—
190Dic6—182no—
191Dic6—182no—
192C3×C8—94no—
193C3⋊C8—34no—
194C3⋊C8—34no—
195C3⋊C8—64no—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32#94011yescharacteristic
199C2×C32#94011yescharacteristic
200C2×C32#94212yes—
201C2×C32#95112yes—
202C3⋊D3—81no—
203C3×D3—241no—
204C2×Q8—91no—
205C2×Q8—92no—
206C2×D4—91no—
207Q16—544no—
208SD16—544no—
209C2×C8—92no—
210C4⋊C4—182no—
211C22⋊C4—182no—
212C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
213C22×C3—31no—
214C22×C3—31no—
215C22×C3—61no—
216C22×C3—361no—
217C22×C3—362no—
218C22×C3C2×#62511yescharacteristic
219C2×D3—122no—
220C2×D3—121no—
221C2×D3—361no—
222C2×D3—362no—
223C3×C4—22no—
224C3×C4—32no—
225C3×C4—32no—
226C3×C4—62no—
227C3×C4—62no—
228C3×C4—62no—
229C3×C4—122no—
230C3×C4—361no—
231C3×C4—362no—
232C3×C4#380212yes—
233Dic3—122no—
234Dic3—121no—
235Dic3—362no—
236Dic3—361no—
237C32—21no—
238C32#506011yescharacteristic, complemented
239C32#508611yescharacteristic, complemented
240C23—181no—
241Q8—182no—
242Q8—182no—
243Q8—182no—
244D4—182no—
245C2×C4—181no—
246C2×C4—182no—
247C2×C4#619812yes—
248C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
249C8—94no—
250C2×C3—32no—
251C2×C3—32no—
252C2×C3—31no—
253C2×C3—31no—
254C2×C3—62no—
255C2×C3—61no—
256C2×C3—721no—
257C2×C3#6382712yes—
258C2×C3#6391411yescharacteristic
259D3—241no—
260D3—721no—
261C22—362no—
262C22—361no—
263C22C2×#619811yescharacteristic
264C4—22no—
265C4—361no—
266C4—362no—
267C4#8865512yes—
268C3—31no—
269C3—31no—
270C3—61no—
271C3#1059037611yescharacteristic
272C2—721no—
273C2#1172105812yes—
274C2#1172114411yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#497
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
2. #64078⋊C3C2semidirect
3. C2×#88433C2semidirect
5. #11720520C2semidirect
18. #64078D3semidirect
162. H(3,𝔽3)#497semidirect
238. C32#5060semidirect
239. C32#5086semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
#64078⋊C32
C2×#884332
C2×#884312
#117205202
#105903743
#105903723