Finite Group Databasebeta

← #13137442

Subgroups — #13137442

Order 1728.

502 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721715C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic
3C2×#88439C212yesmaximal, complemented
4C2×#88438C212yesmaximal
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10590376—31nomaximal
7#10590374—31nomaximal
8C2×#6229C2212yes—
9C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
10C2×#634⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084—22no—
13#3752⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88439—24no—
16#88438—24no—
17#64084D311yescharacteristic
18C3×#875—31no—
19C3×#869—31no—
20C3×#869—31no—
21C3×#869—61no—
22C2×#3719—32no—
23C2×#3718—32no—
24C2×#3717—32no—
25C2×#3715—32no—
26#63630D311yescharacteristic, complemented
27C3×#803—31no—
28C22×(C3⋊D3)⋊C3—21no—
29C2×C4×H(3,𝔽3)D412yes—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—22no—
32#6229—22no—
33#6229—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36#634⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3D414yes—
38#5086—91no—
39#5060—91no—
40#4715—91no—
41C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
42C2×#634C2×D311yescharacteristic
43C2×C3×D12—31no—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—31no—
47C2×C3×Dic6—62no—
48C32×C42—21no—
49C32×C42C2×D311yescharacteristic
50C32×C42C2×D311yescharacteristic
51#3752—22no—
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C3×(C2×D3)⋊C4—62no—
54C3×C4⋊Dic3—64no—
55C3×C4⋊Dic3—62no—
56C3×C4⋊Dic3—62no—
57C2×C3×C3⋊C8—32no—
58#3719—64no—
59#3718—64no—
60#3717—64no—
61#3715—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—41no—
64C2×(C3⋊D3)⋊C3—42no—
65C4×H(3,𝔽3)—22no—
66C4×H(3,𝔽3)C2×D412yes—
67#1084—42no—
68(C3⋊Dic3)⋊C3—41no—
69C3×C4⋊Q8—91no—
70C3×#124—91no—
71C2×#216—92no—
72C2×#216—92no—
73C2×#215—92no—
74C2×#214—92no—
75C2×Dic12—92no—
76C2×#204—92no—
77#875—31no—
78#875—91no—
79#869—31no—
80#869—31no—
81#869—61no—
82C3×#97—91no—
83#803—31no—
84#803—91no—
85C22×C3⋊D3—21no—
86C22×C3×D3—61no—
87C2×C32×C4—22no—
88C2×C32×C4—21no—
89C2×C32×C4D1212yes—
90C2×C32×C4#6712yes—
91C2×C32×C4C22×D311yescharacteristic
92C2×C32×C4C22×D311yescharacteristic
93C2×C3⋊Dic3—21no—
94C4⋊(C3⋊D3)—22no—
95#634—22no—
96C2×C3×Dic3—64no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C3×D12—62no—
101C3×Dic6—62no—
102C3×Dic6—62no—
103C3×Dic6—62no—
104C3×Dic6—62no—
105C3×Dic6—62no—
106C3×Dic6—122no—
107C3×Dic6—122no—
108C3⋊(C3⋊C8)D1214yes—
109C3×C3⋊C8—34no—
110#497—271noSylow
111C2×H(3,𝔽3)#12711yescharacteristic
112C2×H(3,𝔽3)#13712yes—
113(C3⋊D3)⋊C3—81no—
114C2×C3×Q8—91no—
115C2×C3×Q8—92no—
116C2×C3×D4—91no—
117C2×D12—31no—
118C2×D12—91no—
119C2×Dic6—32no—
120C2×Dic6—32no—
121C2×Dic6—31no—
122C2×Dic6—62no—
123C2×Dic6—91no—
124C2×C3×C8—92no—
125C3×C4⋊C4—182no—
126C3×C22⋊C4—182no—
127C3×C42—31no—
128C3×C42—31no—
129C3×C42—61no—
130C3×C42D3211yescharacteristic
131#216—184no—
132#216—184no—
133#215—184no—
134#214—184no—
135(C2×D3)⋊C4—62no—
136(C2×D3)⋊C4—182no—
137C4⋊Dic3—62no—
138C4⋊Dic3—62no—
139C4⋊Dic3—64no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—92no—
142Dic12—184no—
143#204—184no—
144C22×C32—21no—
145C22×C32C2×D1211yescharacteristic
146C22×C32C2×#6711yescharacteristic
147C2×C3⋊D3—41no—
148C2×C3⋊D3—42no—
149C2×C3×D3—121no—
150C2×C3×D3—122no—
151C32×C4—24no—
152C32×C4—22no—
153C32×C4—22no—
154C32×C4—22no—
155C32×C4C2×D1212yes—
156C32×C4D3×D412yes—
157C3⋊Dic3—41no—
158C3×Dic3—124no—
159C3×Dic3—121no—
160C3×Dic3—122no—
161C3×Dic3—122no—
162C2×Q16—272no—
163C2×SD16—272no—
164C4⋊Q8—91no—
165#124—91no—
166#97—91no—
167H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
168C23×C3—181no—
169C22×D3—61no—
170C22×D3—181no—
171C3×Q8—182no—
172C3×Q8—182no—
173C3×Q8—182no—
174C3×D4—182no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—62no—
180C2×C3×C4—61no—
181C2×C3×C4—181no—
182C2×C3×C4—182no—
183C2×C3×C4#62612yes—
184C2×C3×C4C2×D3211yescharacteristic
185C2×Dic3—62no—
186C2×Dic3—62no—
187C2×Dic3—64no—
188C2×Dic3—61no—
189C2×Dic3—181no—
190D12—62no—
191D12—182no—
192Dic6—62no—
193Dic6—62no—
194Dic6—62no—
195Dic6—62no—
196Dic6—62no—
197Dic6—122no—
198Dic6—122no—
199Dic6—182no—
200C3×C8—94no—
201C3⋊C8—34no—
202C3⋊C8—94no—
203C2×C32—22no—
204C2×C32—21no—
205C2×C32#87411yescharacteristic
206C2×C32#90912yes—
207C2×C32#94011yescharacteristic
208C2×C32#95112yes—
209C3⋊D3—81no—
210C3×D3—241no—
211C2×Q8—91no—
212C2×Q8—92no—
213C2×D4—91no—
214Q16—544no—
215SD16—544no—
216C2×C8—92no—
217C4⋊C4—182no—
218C22⋊C4—182no—
219C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—61no—
223C22×C3—361no—
224C22×C3—362no—
225C22×C3C2×#62611yescharacteristic
226C2×D3—122no—
227C2×D3—121no—
228C2×D3—361no—
229C2×D3—362no—
230C3×C4—22no—
231C3×C4—32no—
232C3×C4—32no—
233C3×C4—62no—
234C3×C4—62no—
235C3×C4—64no—
236C3×C4—62no—
237C3×C4—361no—
238C3×C4—362no—
239C3×C4D3×D1212yes—
240Dic3—122no—
241Dic3—122no—
242Dic3—124no—
243Dic3—121no—
244Dic3—361no—
245C32—21no—
246C32#471511yescharacteristic, complemented
247C32#506011yescharacteristic, complemented
248C23—181no—
249Q8—182no—
250Q8—182no—
251Q8—182no—
252D4—182no—
253C2×C4—181no—
254C2×C4—182no—
255C2×C4#620012yes—
256C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
257C8—94no—
258C2×C3—32no—
259C2×C3—32no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—62no—
263C2×C3—61no—
264C2×C3—721no—
265C2×C3#6383112yes—
266C2×C3#6391111yescharacteristic
267D3—241no—
268D3—721no—
269C22—362no—
270C22—361no—
271C22C2×#620011yescharacteristic
272C4—22no—
273C4—362no—
274C4—361no—
275C4#8865412yes—
276C3—31no—
277C3—31no—
278C3—61no—
279C3#1059037211yescharacteristic
280C2—721no—
281C2#1172107412yes—
282C2#1172115811yescharacteristic
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)#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
1. #11721715C2semidirect
3. C2×#88439C2semidirect
5. #63630⋊C3C2semidirect
26. #63630D3semidirect
167. H(3,𝔽3)#497semidirect
246. C32#4715semidirect
247. C32#5060semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64084⋊C32
C2×#884392
C2×#884382
#63630⋊C32
#105903763
#105903743