Finite Group Databasebeta

← #13161327

Subgroups — #13161327

Order 1728.

1124 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C3×#64364C211yesmaximal, characteristic
2#11723985C211yesmaximal, characteristic, complemented
3#11723388C212yesmaximal, complemented
4C32×#836C211yesmaximal, characteristic, complemented
5#11722045C212yesmaximal, complemented
6#10592451—31nomaximal
7#10591649—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×#6308C2211yescharacteristic
10C3×#3832—22no—
11C3×Q8×C3⋊D3—22no—
12C3×#3828—22no—
13C3×#3828C2212yes—
14#88857—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C32×#208C2212yes—
17#88800—22no—
18#88796—22no—
19#64364D311yescharacteristic
20C3×#1007—34no—
21#64153—31no—
22#64153—34no—
23#64153—64no—
24#64118—32no—
25#63831—38no—
26C32×#101—31no—
27#63652—32no—
28C3×#836D314yescomplemented
29#63568—38no—
30C33×Q8—22no—
31C2×C33×C4D412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)—21no—
35#6308—22no—
36C3×#638—22no—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4⋊(C3⋊D3)D412yes—
39C3×C4×C3⋊D3—24no—
40C3×C4×C3⋊D3—22no—
41C3×#634—22no—
42C3×#634D412yes—
43C32×C3⋊C8—22no—
44#5246—91no—
45#4752—94no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C2×D311yescharacteristic
49#3832—22no—
50Q8×C3⋊D3—22no—
51#3828—22no—
52#3828C2×D312yes—
53C2×#634—31no—
54C2×#634—34no—
55C2×#634—64no—
56C3×#239—68no—
57C3×D3×Q8—68no—
58C3×#235—38no—
59C3×#235—68no—
60C32×M16—32no—
61#3750—616no—
62#3750—68no—
63#3750—62no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—64no—
67#3746—62no—
68#3744—62no—
69C3×#208C2×D318yes—
70#3719—68no—
71#3716—68no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—22no—
74C33×C4—22no—
75C33×C4C2×D412yes—
76C3⋊(C3⋊Dic3)—22no—
77C3⋊(C3⋊Dic3)—41no—
78C3×C3⋊Dic3—22no—
79C3×2−1+4—91no—
80#1007—34no—
81#951—92no—
82#944—94no—
83#944—91no—
84#944—184no—
85#909—98no—
86C3×#101—34no—
87#837—92no—
88#836C3⋊D311yescharacteristic, complemented
89#825—98no—
90C32×Q8—22no—
91C32×Q8—28no—
92C32×Q8—48no—
93C2×C32×C4—28no—
94C2×C32×C4—24no—
95C2×C32×C4D1218yes—
96C2×C32×C4#6712yes—
97C2×C32×C4C22×D314yes—
98C2×C32×C4C22×D311yescharacteristic
99#638—22no—
100C2×C3⋊Dic3—68no—
101C2×C3⋊Dic3—64no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—68no—
104C2×C3⋊Dic3—64no—
105C2×C3⋊Dic3—61no—
106C4⋊(C3⋊D3)—22no—
107C4⋊(C3⋊D3)#6712yes—
108C4×C3⋊D3—24no—
109C4×C3⋊D3—22no—
110#634—22no—
111#634—62no—
112#634—68no—
113#634—128no—
114#634#6712yes—
115C3×#67—68no—
116C3×D12—38no—
117C3×D12—68no—
118C3×C4×D3—616no—
119C3×C4×D3—68no—
120C3×Dic6—38no—
121C3×Dic6—68no—
122C32×C8—62no—
123C3×C3⋊C8—28no—
124#456—271noSylow
125C2×C33—21no—
126C2×C33#12011yescharacteristic
127C3×C3⋊D3—42no—
128C3×#41—92no—
129C3×#41—182no—
130C3×#41—182no—
131C2×C3×Q8—24no—
132C2×C3×Q8—182no—
133C2×C3×Q8D3214yes—
134C2×C3×Q8C2×C3⋊D311yescharacteristic
135#239—68no—
136D3×Q8—68no—
137#235—38no—
138#235—68no—
139C2×Dic6—94no—
140C2×Dic6—91no—
141C2×Dic6—184no—
142C3×M16—38no—
143#216—182no—
144#214—182no—
145Dic3⋊C4—1816no—
146Dic3⋊C4—188no—
147Dic3⋊C4—182no—
148C4×Dic3—94no—
149C4×Dic3—91no—
150C4×Dic3—184no—
151#208C2×C3⋊D312yes—
152Dic12—188no—
153#204—188no—
154C22×C32—24no—
155C22×C32C2×D1214yes—
156C22×C32C2×#6711yescharacteristic
157C2×C3⋊D3—22no—
158C2×C3×D3—68no—
159C32×C4—22no—
160C32×C4—28no—
161C32×C4—28no—
162C32×C4—48no—
163C32×C4D3×D418yes—
164C32×C4C2×#6712yes—
165C3⋊Dic3—22no—
166C3⋊Dic3—616no—
167C3⋊Dic3—62no—
168C3⋊Dic3—68no—
169C3⋊Dic3—128no—
170C3⋊Dic3—124no—
171C3⋊Dic3—121no—
172C3×Dic3—68no—
1732−1+4—91no—
174#137—272no—
175C4⋊Q8—271no—
176#104—272no—
177#101—31no—
178C33#45611yescharacteristic, complemented, Sylow
179C3×Q8—28no—
180C3×Q8—22no—
181C3×Q8—48no—
182C3×Q8—92no—
183C3×Q8—182no—
184C3×D4—92no—
185C3×D4—182no—
186C3×D4—182no—
187C2×C3×C4—28no—
188C2×C3×C4—24no—
189C2×C3×C4—184no—
190C2×C3×C4—182no—
191C2×C3×C4#62618yes—
192C2×C3×C4C4⋊(C3⋊D3)12yes—
193C2×C3×C4C2×D3214yes—
194C2×C3×C4C22×C3⋊D311yescharacteristic
195#67—68no—
196C2×Dic3—188no—
197C2×Dic3—184no—
198C2×Dic3—181no—
199C2×Dic3—188no—
200C2×Dic3—184no—
201C2×Dic3—181no—
202D12—38no—
203D12—68no—
204C4×D3—616no—
205C4×D3—68no—
206Dic6—38no—
207Dic6—68no—
208Dic6—188no—
209Dic6—182no—
210Dic6—368no—
211C3×C8—68no—
212C3⋊C8—22no—
213C2×C32—28no—
214C2×C32—21no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32#88214yes—
218C2×C32#95311yescharacteristic
219C3⋊D3—42no—
220C3×D3—128no—
221#41—92no—
222#41—182no—
223#41—182no—
224C2×Q8—182no—
225C2×Q8—271no—
226C2×Q8D3×C3⋊D311yescharacteristic
227Q16—542no—
228SD16—542no—
229M16—32no—
230C4⋊C4—542no—
231C42—271no—
232C22×C3—24no—
233C22×C3—182no—
234C22×C3C2×#62614yes—
235C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
236C2×D3—68no—
237C3×C4—22no—
238C3×C4—28no—
239C3×C4—28no—
240C3×C4—48no—
241C3×C4—182no—
242C3×C4D3×#6718yes—
243C3×C4D4×C3⋊D312yes—
244Dic3—68no—
245Dic3—1816no—
246Dic3—188no—
247Dic3—182no—
248Dic3—368no—
249Dic3—361no—
250Dic3—364no—
251C32—24no—
252C32#475214yescomplemented
253C32#524611yescharacteristic, complemented
254Q8—22no—
255Q8—92no—
256Q8—182no—
257Q8—542no—
258D4—92no—
259D4—182no—
260D4—182no—
261C2×C4—184no—
262C2×C4—182no—
263C2×C4—541no—
264C2×C4—541no—
265C2×C4#629212yes—
266C2×C4C2×D3×C3⋊D311yescharacteristic
267C8—62no—
268C2×C3—28no—
269C2×C3—24no—
270C2×C3—21no—
271C2×C3—24no—
272C2×C3—362no—
273C2×C3#6397714yes—
274C2×C3#6409111yescharacteristic
275D3—128no—
276C22—182no—
277C22C2×#629211yescharacteristic
278C4—22no—
279C4—182no—
280C4—542no—
281C4—1081no—
282C4C3⋊D3×#6712yes—
283C3—24no—
284C3#1059164914yescomplemented
285C3#1059245111yescharacteristic, complemented
286C2—21no—
287C2—362no—
288C2#1172363611yescharacteristic
subgroupG/H
centerC2#11723636
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
prime radicalC3×#64364C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#456
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723985C2semidirect
3. #11723388C2semidirect
4. C32×#836C2semidirect
5. #11722045C2semidirect
28. C3×#836D3semidirect
88. #836C3⋊D3semidirect
178. C33#456semidirect
252. C32#4752semidirect
253. C32#5246semidirect
284. C3#10591649semidirect
285. C3#10592451semidirect

Maximal subgroups

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

subgroupindex
C3×#643642
#117239852
#117233882
C32×#8362
#117220452
#105924513
#105916493