Finite Group Databasebeta

← #13161324

Subgroups — #13161324

Order 1728.

1460 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 144 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#11724682C211yesmaximal, characteristic, complemented
2C32×#947C211yesmaximal, characteristic, complemented
3#11723378C212yesmaximal, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5#11722048C212yesmaximal, complemented
6#10592761—31nomaximal
7#10591647—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9#89080—22no—
10Q8×C3⋊(C3⋊D3)—22no—
11#89076—22no—
12#89076C2212yes—
13C2×C32×D12C2211yescharacteristic
14C3×#3748C2212yes—
15C32×(C2×D3)⋊C4—22no—
16C32×C4×Dic3C2211yescharacteristic
17#88794—22no—
18#88790—22no—
19#64364—31no—
20#64364—34no—
21#64364—64no—
22C32×#124—31no—
23#64141—32no—
24C3×#947D314yescomplemented
25#63830—38no—
26#63663D311yescharacteristic, complemented
27#63632—32no—
28C3×#836—34no—
29#63570—38no—
30C22×C32×D3—21no—
31C33×Q8—22no—
32C2×C33×C4D412yes—
33C2×C33×C4C2311yescharacteristic
34#6312—22no—
35C4⋊(C3⋊(C3⋊D3))—22no—
36C4⋊(C3⋊(C3⋊D3))D412yes—
37C4×C3⋊(C3⋊D3)—24no—
38C4×C3⋊(C3⋊D3)—22no—
39#6308—22no—
40#6308D412yes—
41C2×C32×Dic3—21no—
42C32×D12—22no—
43C3×C3⋊(C3⋊C8)—22no—
44#5062—94no—
45#4753—91no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D311yescharacteristic
48C2×C32×Q8C2×D314yes—
49C2×C32×D4—31no—
50#3832—62no—
51#3832—68no—
52#3832—616no—
53Q8×C3⋊D3—62no—
54Q8×C3⋊D3—68no—
55Q8×C3⋊D3—616no—
56#3828—32no—
57#3828—38no—
58#3828—68no—
59#3828—616no—
60#3828—62no—
61#3828—68no—
62C2×C3×D12C2×D314yes—
63C32×C22⋊C4—62no—
64C32×C42—31no—
65#3754—62no—
66#3753—62no—
67#3748C2×D312yes—
68C3×(C2×D3)⋊C4—28no—
69C3×C4×Dic3C2×D314yes—
70C3×#208—38no—
71#3716—68no—
72#3714—68no—
73C22×C33C2×D411yescharacteristic
74C2×C3⋊(C3⋊D3)—22no—
75C2×C32×D3—42no—
76C2×C32×D3—41no—
77C33×C4—22no—
78C33×C4C2×D412yes—
79C3⋊(C3⋊Dic3)—22no—
80C32×Dic3—41no—
81#1007—94no—
82#1007—91no—
83#1007—184no—
84C3×#124—34no—
85#947C3⋊D311yescharacteristic, complemented
86#932—98no—
87#908—92no—
88C3×#101—91no—
89#836—34no—
90#825—92no—
91#805—98no—
92C23×C32—61no—
93C22×C3×D3—24no—
94C32×Q8—216no—
95C32×Q8—28no—
96C32×Q8—22no—
97C32×D4—62no—
98C2×C32×C4—28no—
99C2×C32×C4—24no—
100C2×C32×C4—61no—
101C2×C32×C4D1212yes—
102C2×C32×C4#6718yes—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D311yescharacteristic
105#638—616no—
106#638—62no—
107#638—68no—
108C4⋊(C3⋊D3)—32no—
109C4⋊(C3⋊D3)—38no—
110C4⋊(C3⋊D3)—68no—
111C4⋊(C3⋊D3)—616no—
112C4⋊(C3⋊D3)—62no—
113C4⋊(C3⋊D3)—68no—
114C4×C3⋊D3—632no—
115C4×C3⋊D3—64no—
116C4×C3⋊D3—616no—
117C4×C3⋊D3—616no—
118C4×C3⋊D3—68no—
119C4×C3⋊D3—62no—
120#634—32no—
121#634—38no—
122#634—68no—
123#634—616no—
124#634—62no—
125#634—68no—
126C2×C3×Dic3—24no—
127C3×D12—28no—
128C3⋊(C3⋊C8)—22no—
129C3×C3⋊C8—68no—
130#454—271noSylow
131C2×C33—21no—
132C2×C33#12011yescharacteristic
133C3⋊(C3⋊D3)—42no—
134C32×D3—81no—
135C2×C3×Q8—24no—
136C2×C3×Q8D3214yes—
137C2×C3×Q8C2×C3⋊D311yescharacteristic
138C2×C3×D4—34no—
139#239—188no—
140#239—182no—
141#239—1816no—
142D3×Q8—188no—
143D3×Q8—182no—
144D3×Q8—1816no—
145#235—98no—
146#235—92no—
147#235—188no—
148#235—1816no—
149#235—188no—
150#235—182no—
151C2×D12C2×C3⋊D311yescharacteristic
152C3×M16—92no—
153C3×C22⋊C4—68no—
154C3×C42—34no—
155#214—188no—
156#213—188no—
157(C2×D3)⋊C4—22no—
158C4×Dic3C2×C3⋊D311yescharacteristic
159#208—38no—
160D24—182no—
161#204—182no—
162C22×C32—24no—
163C22×C32—122no—
164C22×C32—121no—
165C22×C32C2×D1211yescharacteristic
166C22×C32C2×#6714yes—
167C2×C3⋊D3—616no—
168C2×C3⋊D3—62no—
169C2×C3⋊D3—68no—
170C2×C3×D3—48no—
171C2×C3×D3—44no—
172C32×C4—216no—
173C32×C4—22no—
174C32×C4—28no—
175C32×C4—28no—
176C32×C4—121no—
177C32×C4D3×D418yes—
178C32×C4D3×D412yes—
179C3⋊Dic3—616no—
180C3⋊Dic3—62no—
181C3⋊Dic3—68no—
182C3×Dic3—44no—
1832−1+4—271no—
184AGL(1,ℤ/8)—272no—
185#124—31no—
186#104—272no—
187#101—91no—
188C33#45411yescharacteristic, complemented, Sylow
189C23×C3—64no—
190C22×D3—21no—
191C3×Q8—216no—
192C3×Q8—28no—
193C3×Q8—22no—
194C3×D4—68no—
195C2×C3×C4—28no—
196C2×C3×C4—24no—
197C2×C3×C4—64no—
198C2×C3×C4#62618yes—
199C2×C3×C4#63812yes—
200C2×C3×C4C2×D3214yes—
201C2×C3×C4C22×C3⋊D311yescharacteristic
202#67—1816no—
203#67—188no—
204#67—182no—
205C2×Dic3—21no—
206D12—22no—
207D12—98no—
208D12—92no—
209D12—188no—
210D12—1816no—
211D12—182no—
212D12—188no—
213C4×D3—1832no—
214C4×D3—184no—
215C4×D3—1816no—
216C4×D3—1816no—
217C4×D3—188no—
218C4×D3—182no—
219Dic6—98no—
220Dic6—92no—
221Dic6—188no—
222Dic6—1816no—
223Dic6—182no—
224Dic6—188no—
225C3×C8—182no—
226C3⋊C8—68no—
227C2×C32—21no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—44no—
231C2×C32—241no—
232C2×C32#88211yescharacteristic
233C2×C32#93714yes—
234C3⋊D3—1216no—
235C3⋊D3—122no—
236C3⋊D3—128no—
237C3×D3—84no—
238#41—272no—
239#41—542no—
240#41—542no—
241C2×Q8—542no—
242C2×Q8D3×C3⋊D311yescharacteristic
243C2×D4—31no—
244SD16—542no—
245D8—542no—
246M16—92no—
247C22⋊C4—62no—
248C42—31no—
249C22×C3—24no—
250C22×C3—128no—
251C22×C3—124no—
252C22×C3C2×#62614yes—
253C22×C3C2×#63811yescharacteristic
254C2×D3—42no—
255C2×D3—41no—
256C2×D3—1816no—
257C2×D3—182no—
258C2×D3—188no—
259C3×C4—216no—
260C3×C4—22no—
261C3×C4—28no—
262C3×C4—28no—
263C3×C4—124no—
264C3×C4#381118yes—
265C3×C4D4×C3⋊D312yes—
266Dic3—41no—
267Dic3—1816no—
268Dic3—182no—
269Dic3—188no—
270C32—24no—
271C32#475311yescharacteristic, complemented
272C32#506214yescomplemented
273C23—61no—
274Q8—22no—
275Q8—272no—
276Q8—542no—
277D4—62no—
278D4—272no—
279D4—542no—
280D4—542no—
281C2×C4—61no—
282C2×C4—544no—
283C2×C4—542no—
284C2×C4#629112yes—
285C2×C4C2×D3×C3⋊D311yescharacteristic
286C8—182no—
287C2×C3—24no—
288C2×C3—21no—
289C2×C3—24no—
290C2×C3—44no—
291C2×C3—244no—
292C2×C3#6398114yes—
293C2×C3#6414611yescharacteristic
294D3—81no—
295D3—3616no—
296D3—368no—
297D3—362no—
298C22—122no—
299C22—121no—
300C22—542no—
301C22C2×#629111yescharacteristic
302C4—22no—
303C4—121no—
304C4—542no—
305C4#8903912yes—
306C3—24no—
307C3#1059164714yescomplemented
308C3#1059276111yescharacteristic, complemented
309C2—21no—
310C2—241no—
311C2—1082no—
312C2#1172363811yescharacteristic
subgroupG/H
centerC2#11723638
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. #11724682C2semidirect
2. C32×#947C2semidirect
3. #11723378C2semidirect
4. C3×#63663C2semidirect
5. #11722048C2semidirect
24. C3×#947D3semidirect
26. #63663D3semidirect
85. #947C3⋊D3semidirect
188. C33#454semidirect
271. C32#4753semidirect
272. C32#5062semidirect
307. C3#10591647semidirect
308. C3#10592761semidirect

Maximal subgroups

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

subgroupindex
#117246822
C32×#9472
#117233782
C3×#636632
#117220482
#105927613
#105916473