Finite Group Databasebeta

← #13161318

Subgroups — #13161318

Order 1728.

1180 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1180 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, complemented
2#11723988C211yesmaximal, characteristic, complemented
3#11723371C212yesmaximal, complemented
4C32×#836C211yesmaximal, characteristic, complemented
5#11722045C212yesmaximal, complemented
6#10592452—31nomaximal
7#10591646—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
10C3×#3832—22no—
11C3×Q8×C3⋊D3—22no—
12C3×#3828—22no—
13C3×#3828C2212yes—
14#88859—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C32×#208C2212yes—
17#88797—22no—
18#88791—22no—
19#64364D311yescharacteristic, complemented
20C3×#1007—34no—
21#64156—31no—
22#64156—34no—
23#64156—64no—
24#64117—32no—
25#63825—38no—
26C32×#101—31no—
27#63652—32no—
28C3×#836D314yescomplemented
29#63568—38no—
30C22×C3⋊(C3⋊D3)—21no—
31C33×Q8—22no—
32C2×C33×C4D412yes—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C4⋊(C3⋊(C3⋊D3))—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#5245—91no—
45#4753—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×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C3×#239—68no—
57C3×D3×Q8—68no—
58C3×#235—38no—
59C3×#235—68no—
60C32×M16—32no—
61#3752—616no—
62#3752—68no—
63#3752—62no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—64no—
67#3745—62no—
68#3744—62no—
69C3×#208C2×D318yes—
70#3717—68no—
71#3714—68no—
72C22×C33C2×D411yescharacteristic
73C2×C3⋊(C3⋊D3)—42no—
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3×C3⋊D3—22no—
76C33×C4—22no—
77C33×C4C2×D412yes—
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—22no—
80C3×2−1+4—91no—
81#1007—34no—
82#949—92no—
83#947—94no—
84#947—91no—
85#947—184no—
86#908—98no—
87C3×#101—34no—
88#837—92no—
89#836C3⋊D311yescharacteristic, complemented
90#825—98no—
91C22×C3⋊D3—68no—
92C22×C3⋊D3—64no—
93C22×C3⋊D3—61no—
94C32×Q8—22no—
95C32×Q8—28no—
96C32×Q8—48no—
97C2×C32×C4—28no—
98C2×C32×C4—24no—
99C2×C32×C4D1218yes—
100C2×C32×C4#6712yes—
101C2×C32×C4C22×D314yes—
102C2×C32×C4C22×D311yescharacteristic
103#638—22no—
104C2×C3⋊Dic3—68no—
105C2×C3⋊Dic3—64no—
106C2×C3⋊Dic3—61no—
107C4⋊(C3⋊D3)—22no—
108C4⋊(C3⋊D3)—62no—
109C4⋊(C3⋊D3)—68no—
110C4⋊(C3⋊D3)—128no—
111C4⋊(C3⋊D3)#6712yes—
112C4×C3⋊D3—24no—
113C4×C3⋊D3—22no—
114#634—22no—
115#634#6712yes—
116C3×#67—68no—
117C3×D12—38no—
118C3×D12—68no—
119C3×C4×D3—616no—
120C3×C4×D3—68no—
121C3×Dic6—38no—
122C3×Dic6—68no—
123C32×C8—62no—
124C3×C3⋊C8—28no—
125#454—271noSylow
126C2×C33—21no—
127C2×C33#12011yescharacteristic
128C3⋊(C3⋊D3)—81no—
129C3×C3⋊D3—42no—
130C3×#41—92no—
131C3×#41—182no—
132C3×#41—182no—
133C2×C3×Q8—24no—
134C2×C3×Q8—182no—
135C2×C3×Q8D3214yes—
136C2×C3×Q8C2×C3⋊D311yescharacteristic
137#239—68no—
138D3×Q8—68no—
139#235—38no—
140#235—68no—
141C2×D12—94no—
142C2×D12—91no—
143C2×D12—184no—
144C3×M16—38no—
145#215—182no—
146#213—182no—
147(C2×D3)⋊C4—1816no—
148(C2×D3)⋊C4—188no—
149(C2×D3)⋊C4—182no—
150C4×Dic3—94no—
151C4×Dic3—91no—
152C4×Dic3—184no—
153#208C2×C3⋊D312yes—
154D24—188no—
155#204—188no—
156C22×C32—24no—
157C22×C32C2×D1214yes—
158C22×C32C2×#6711yescharacteristic
159C2×C3⋊D3—22no—
160C2×C3⋊D3—128no—
161C2×C3⋊D3—1216no—
162C2×C3⋊D3—122no—
163C2×C3⋊D3—128no—
164C2×C3⋊D3—124no—
165C2×C3⋊D3—121no—
166C2×C3×D3—68no—
167C32×C4—22no—
168C32×C4—28no—
169C32×C4—28no—
170C32×C4—48no—
171C32×C4D3×D418yes—
172C32×C4C2×#6712yes—
173C3⋊Dic3—22no—
174C3⋊Dic3—128no—
175C3⋊Dic3—121no—
176C3⋊Dic3—124no—
177C3×Dic3—68no—
1782−1+4—91no—
179AGL(1,ℤ/8)—272no—
180#124—271no—
181#104—272no—
182#101—31no—
183C33#45411yescharacteristic, complemented, Sylow
184C22×D3—188no—
185C22×D3—184no—
186C22×D3—181no—
187C3×Q8—28no—
188C3×Q8—22no—
189C3×Q8—48no—
190C3×Q8—92no—
191C3×Q8—182no—
192C3×D4—92no—
193C3×D4—182no—
194C3×D4—182no—
195C2×C3×C4—28no—
196C2×C3×C4—24no—
197C2×C3×C4—184no—
198C2×C3×C4—182no—
199C2×C3×C4#62618yes—
200C2×C3×C4C4⋊(C3⋊D3)12yes—
201C2×C3×C4C2×D3214yes—
202C2×C3×C4C22×C3⋊D311yescharacteristic
203#67—68no—
204C2×Dic3—188no—
205C2×Dic3—184no—
206C2×Dic3—181no—
207D12—38no—
208D12—68no—
209D12—188no—
210D12—182no—
211D12—368no—
212C4×D3—616no—
213C4×D3—68no—
214Dic6—38no—
215Dic6—68no—
216C3×C8—68no—
217C3⋊C8—22no—
218C2×C32—28no—
219C2×C32—21no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32#88214yes—
223C2×C32#95311yescharacteristic
224C3⋊D3—42no—
225C3⋊D3—248no—
226C3⋊D3—241no—
227C3⋊D3—244no—
228C3×D3—128no—
229#41—92no—
230#41—182no—
231#41—182no—
232C2×Q8—182no—
233C2×Q8D3×C3⋊D311yescharacteristic
234C2×D4—271no—
235SD16—542no—
236D8—542no—
237M16—32no—
238C22⋊C4—542no—
239C42—271no—
240C22×C3—24no—
241C22×C3—182no—
242C22×C3C2×#62614yes—
243C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
244C2×D3—68no—
245C2×D3—368no—
246C2×D3—3616no—
247C2×D3—368no—
248C2×D3—362no—
249C2×D3—361no—
250C2×D3—364no—
251C3×C4—22no—
252C3×C4—28no—
253C3×C4—28no—
254C3×C4—48no—
255C3×C4—182no—
256C3×C4D3×#6718yes—
257C3×C4D4×C3⋊D312yes—
258Dic3—68no—
259Dic3—368no—
260Dic3—364no—
261Dic3—361no—
262C32—24no—
263C32#475314yescomplemented
264C32#524511yescharacteristic, complemented
265C23—541no—
266Q8—22no—
267Q8—92no—
268Q8—182no—
269D4—92no—
270D4—182no—
271D4—182no—
272D4—542no—
273C2×C4—184no—
274C2×C4—182no—
275C2×C4—541no—
276C2×C4#629212yes—
277C2×C4C2×D3×C3⋊D311yescharacteristic
278C8—62no—
279C2×C3—28no—
280C2×C3—24no—
281C2×C3—21no—
282C2×C3—24no—
283C2×C3—362no—
284C2×C3#6397714yes—
285C2×C3#6409111yescharacteristic
286D3—128no—
287D3—728no—
288D3—724no—
289D3—721no—
290C22—182no—
291C22—1082no—
292C22—1081no—
293C22C2×#629211yescharacteristic
294C4—22no—
295C4—182no—
296C4—1081no—
297C4C3⋊D3×#6712yes—
298C3—24no—
299C3#1059164614yescomplemented
300C3#1059245211yescharacteristic, complemented
301C2—21no—
302C2—362no—
303C2—2161no—
304C2#1172363611yescharacteristic
subgroupG/H
centerC2#11723636
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#454
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
1. C3×#64364C2semidirect
2. #11723988C2semidirect
3. #11723371C2semidirect
4. C32×#836C2semidirect
5. #11722045C2semidirect
19. #64364D3semidirect
28. C3×#836D3semidirect
89. #836C3⋊D3semidirect
183. C33#454semidirect
263. C32#4753semidirect
264. C32#5245semidirect
299. C3#10591646semidirect
300. C3#10592452semidirect

Maximal subgroups

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

subgroupindex
C3×#643642
#117239882
#117233712
C32×#8362
#117220452
#105924523
#105916463