Finite Group Databasebeta

← #13161099

Subgroups — #13161099

Order 1728.

1216 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1216 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×#64361C211yesmaximal, characteristic
2#11723977C211yesmaximal, characteristic, complemented
3#11723388C212yesmaximal, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5#11722045C212yesmaximal, complemented
6#10592445—31nomaximal
7#10591513—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×#6308C2211yescharacteristic, complemented
10C2×C3×#638—22no—
11C3×#3830—22no—
12C3×D4×C3⋊D3—22no—
13C3×#3828C2212yes—
14#88864—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
16C32×#208C2212yescomplemented
17#88800—22no—
18#88796—22no—
19#64361D311yescharacteristic
20C3×#1004—34no—
21#64145—31no—
22#64145—34no—
23#64145—64no—
24#64118—32no—
25#63831—38no—
26C32×#100—31no—
27#63652—32no—
28C3×#833D314yescomplemented
29#63568—38no—
30C23×C33D412yes—
31C22×C3×C3⋊D3—22no—
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3⋊(C3⋊Dic3)—21no—
36#6308—22no—
37C3×#638—24no—
38C3×#638—22no—
39C2×C3×C3⋊Dic3—22no—
40C3×C4⋊(C3⋊D3)D412yes—
41C3×C4×C3⋊D3—22no—
42C3×#634D412yes—
43C32×C3⋊C8—22no—
44#5242—91no—
45#4746—94no—
46C2×C32×D4—24no—
47C2×C32×D4C2×D314yes—
48C2×C32×D4C2×D311yescharacteristic
49C2×#638—22no—
50#3830—22no—
51D4×C3⋊D3—22no—
52#3828C2×D312yes—
53C2×#634—31no—
54C2×#634—34no—
55C2×#634—64no—
56C2×C3×#67—68no—
57C3×#237—68no—
58C3×D3×D4—68no—
59C3×#235—38no—
60C32×M16—32no—
61#3757—616no—
62#3757—68no—
63#3757—62no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—64no—
67#3746—62no—
68#3744—62no—
69C3×#208C2×D318yescomplemented
70#3719—68no—
71#3716—68no—
72C22×C33—22no—
73C22×C33—42no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—82no—
77C33×C4C2×D412yes—
78C3⋊(C3⋊Dic3)—41no—
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3—22no—
81C3×H(5,𝔽2)—91no—
82#1004—34no—
83#951—92no—
84#936—94no—
85#936—91no—
86#936—184no—
87#909—98no—
88C3×#100—34no—
89#837—92no—
90#833C3⋊D311yescharacteristic, complemented
91#825—98no—
92C23×C32—28no—
93C23×C32D1218yes—
94C23×C32#6712yes—
95C22×C3⋊D3—22no—
96C22×C3×D3—68no—
97C32×D4—22no—
98C32×D4—28no—
99C32×D4—48no—
100C2×C32×C4—24no—
101C2×C32×C4C22×D314yes—
102C2×C32×C4C22×D311yescharacteristic
103#638—24no—
104#638—22no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—68no—
107C2×C3⋊Dic3—68no—
108C2×C3⋊Dic3—64no—
109C2×C3⋊Dic3—64no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112C4⋊(C3⋊D3)#6712yes—
113C4×C3⋊D3—22no—
114#634—62no—
115#634—68no—
116#634—128no—
117#634#6712yes—
118C3×#67—616no—
119C3×#67—68no—
120C2×C3×Dic3—68no—
121C3×D12—38no—
122C3×C4×D3—68no—
123C3×Dic6—38no—
124C32×C8—62no—
125C3×C3⋊C8—28no—
126#455—271noSylow
127C2×C33—21no—
128C2×C33—42no—
129C2×C33#12011yescharacteristic
130C3×C3⋊D3—42no—
131C3×#41—92no—
132C3×#41—182no—
133C2×C3×D4—24no—
134C2×C3×D4—182no—
135C2×C3×D4—182no—
136C2×C3×D4D3214yes—
137C2×C3×D4C2×C3⋊D311yescharacteristic
138C2×#67—68no—
139#237—68no—
140D3×D4—68no—
141#235—38no—
142C2×Dic6—94no—
143C2×Dic6—91no—
144C2×Dic6—184no—
145C3×M16—38no—
146#217—1816no—
147#217—188no—
148#217—182no—
149#216—182no—
150#214—182no—
151C4×Dic3—94no—
152C4×Dic3—91no—
153C4×Dic3—184no—
154#208C2×C3⋊D312yescomplemented
155Dic12—188no—
156#204—188no—
157C22×C32—22no—
158C22×C32—24no—
159C22×C32—28no—
160C22×C32—48no—
161C22×C32—48no—
162C22×C32—42no—
163C22×C32—416no—
164C22×C32C2×D1214yes—
165C22×C32C2×#6711yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—82no—
168C2×C3×D3—68no—
169C2×C3×D3—248no—
170C32×C4—28no—
171C32×C4D3×D418yes—
172C32×C4C2×#6712yes—
173C3⋊Dic3—22no—
174C3⋊Dic3—128no—
175C3⋊Dic3—128no—
176C3⋊Dic3—121no—
177C3⋊Dic3—124no—
178C3⋊Dic3—124no—
179C3⋊Dic3—121no—
180C3×Dic3—68no—
181H(5,𝔽2)—91no—
182#137—272no—
183#124—271no—
184#104—272no—
185#100—31no—
186C33#45511yescharacteristic, complemented, Sylow
187C23×C3—28no—
188C23×C3—182no—
189C23×C3#62618yes—
190C23×C3C4⋊(C3⋊D3)12yes—
191C22×D3—68no—
192C3×Q8—92no—
193C3×D4—28no—
194C3×D4—22no—
195C3×D4—48no—
196C3×D4—92no—
197C3×D4—184no—
198C3×D4—182no—
199C2×C3×C4—24no—
200C2×C3×C4—182no—
201C2×C3×C4—182no—
202C2×C3×C4C2×D3214yes—
203C2×C3×C4C22×C3⋊D311yescharacteristic
204#67—616no—
205#67—68no—
206C2×Dic3—68no—
207C2×Dic3—188no—
208C2×Dic3—188no—
209C2×Dic3—184no—
210C2×Dic3—184no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213D12—38no—
214C4×D3—68no—
215Dic6—38no—
216Dic6—188no—
217Dic6—182no—
218Dic6—368no—
219C3×C8—68no—
220C3⋊C8—22no—
221C2×C32—28no—
222C2×C32—21no—
223C2×C32—24no—
224C2×C32—24no—
225C2×C32—42no—
226C2×C32—48no—
227C2×C32—88no—
228C2×C32#88214yes—
229C2×C32#95311yescharacteristic
230C3⋊D3—42no—
231C3×D3—128no—
232#41—92no—
233#41—182no—
234C2×Q8—271no—
235C2×D4—182no—
236C2×D4—182no—
237C2×D4D3×C3⋊D311yescharacteristic
238Q16—542no—
239SD16—542no—
240M16—32no—
241C22⋊C4—542no—
242C42—271no—
243C22×C3—22no—
244C22×C3—28no—
245C22×C3—24no—
246C22×C3—48no—
247C22×C3—48no—
248C22×C3—42no—
249C22×C3—416no—
250C22×C3—182no—
251C22×C3—722no—
252C22×C3C2×#62614yes—
253C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
254C2×D3—68no—
255C2×D3—248no—
256C3×C4—28no—
257C3×C4—182no—
258C3×C4D3×#6718yes—
259C3×C4D4×C3⋊D312yes—
260Dic3—68no—
261Dic3—368no—
262Dic3—368no—
263Dic3—364no—
264Dic3—361no—
265Dic3—361no—
266Dic3—364no—
267C32—24no—
268C32#474614yescomplemented
269C32#524211yescharacteristic, complemented
270C23—182no—
271C23#629212yes—
272Q8—92no—
273Q8—542no—
274D4—22no—
275D4—92no—
276D4—184no—
277D4—182no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4C2×D3×C3⋊D311yescharacteristic
283C8—62no—
284C2×C3—28no—
285C2×C3—24no—
286C2×C3—21no—
287C2×C3—24no—
288C2×C3—48no—
289C2×C3—42no—
290C2×C3—88no—
291C2×C3—362no—
292C2×C3#6397714yes—
293C2×C3#6409111yescharacteristic
294D3—128no—
295C22—22no—
296C22—42no—
297C22—182no—
298C22—722no—
299C22C2×#629211yescharacteristic
300C4—182no—
301C4—1081no—
302C4—1081no—
303C4C3⋊D3×#6712yes—
304C3—24no—
305C3#1059151314yescomplemented
306C3#1059244511yescharacteristic, complemented
307C2—21no—
308C2—42no—
309C2—362no—
310C2#1172363611yescharacteristic
subgroupG/H
centerC2#11723636
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723977C2semidirect
3. #11723388C2semidirect
4. C32×#833C2semidirect
5. #11722045C2semidirect
9. C2×#6308C22semidirect
15. C4×C3⋊(C3⋊Dic3)C22semidirect
16. C32×#208C22semidirect
28. C3×#833D3semidirect
69. C3×#208C2×D3semidirect
90. #833C3⋊D3semidirect
154. #208C2×C3⋊D3semidirect
186. C33#455semidirect
268. C32#4746semidirect
269. C32#5242semidirect
305. C3#10591513semidirect
306. C3#10592445semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
#117239772
#117233882
C32×#8332
#117220452
#105924453
#105915133