Finite Group Databasebeta

← #13139022

Subgroups — #13139022

Order 1728.

412 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721776C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721069C212yesmaximal
4#63646⋊C3C211yesmaximal, characteristic, complemented
5#11720321C212yesmaximal, complemented
6#10590932—31nomaximal
7#10590930—31nomaximal
8#88740C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88538C2211yescharacteristic
12#88527—21no—
13C8×#272—21no—
14#3748⋊C3C2212yes—
15#3746⋊C3—21no—
16#3746⋊C3—21no—
17#88439—22no—
18#88437—22no—
19C2×#3746D311yescharacteristic, complemented
20C2×C3×Dic12—31no—
21C3×#901—31no—
22C3×#901—31no—
23C3×#901—61no—
24#63831—32no—
25#63827—32no—
26#63646D311yescharacteristic, complemented
27C3×#819—31no—
28#63561—32no—
29#63560—32no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31#6235—21no—
32C4⋊#272D411yescharacteristic
33C4×#272—21no—
34#6229D411yescharacteristic
35C2×(C3⋊Dic3)⋊C3—22no—
36#634⋊C3—22no—
37#634⋊C3—22no—
38C8×H(3,𝔽3)D411yescharacteristic
39C8×H(3,𝔽3)D411yescharacteristic
40#6180—21no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5192—91no—
43#5143—91no—
44#4896—91no—
45C2×#634C2×D312yes—
46C3×#235—31no—
47C3×#235—31no—
48C3×#235—61no—
49C2×C3×Dic6—32no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D311yescharacteristic
52C2×C32×C8C2×D311yescharacteristic
53#3748C2×D312yes—
54#3746—21no—
55#3746—21no—
56C3×#208—31no—
57C3×#208—31no—
58C3×#208—32no—
59C3×#208—61no—
60C3×Dic12—61no—
61C3×Dic12—61no—
62C3×ΓL(1,𝔽25)—62no—
63C3×ΓL(1,𝔽25)—61no—
64C3×ΓL(1,𝔽25)—61no—
65C3×D3×C8—62no—
66C3×D3×C8—61no—
67C3×D3×C8—61no—
68#3719—62no—
69#3718—62no—
70#3716—62no—
71#3715—62no—
72C22×H(3,𝔽3)#4111yescharacteristic
73C2×#272—21no—
74C4×H(3,𝔽3)C2×D411yescharacteristic
75C4×H(3,𝔽3)C2×D411yescharacteristic
76#1084—21no—
77(C3⋊Dic3)⋊C3—42no—
78C2×C3×Q16—91no—
79C3×#131—91no—
80#951—92no—
81#942—92no—
82#909—92no—
83C2×Dic12—31no—
84C2×Dic12—91no—
85#901—31no—
86#901—31no—
87#901—61no—
88C3×#108—91no—
89#836—92no—
90#824—92no—
91#824—92no—
92#819—31no—
93#819—91no—
94C2×C32×C4—21no—
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3—22no—
98#634—22no—
99#634—22no—
100C3×#67—62no—
101C3×#67—61no—
102C3×#67—61no—
103C2×C3×Dic3—62no—
104C3×D12—31no—
105C3×D12—31no—
106C3×D12—61no—
107C3×C4×D3—62no—
108C3×C4×D3—61no—
109C3×C4×D3—61no—
110C3×Dic6—31no—
111C3×Dic6—31no—
112C3×Dic6—61no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115C32×C8—21no—
116C32×C8—21no—
117C32×C8D1211yescharacteristic
118C32×C8D1211yescharacteristic
119C32×C8#6711yescharacteristic
120C32×C8#6711yescharacteristic
121C3⋊(C3⋊C8)—22no—
122C3×C3⋊C8—62no—
123C3×C3⋊C8—61no—
124C3×C3⋊C8—62no—
125C3×C3⋊C8—61no—
126#473—271noSylow
127C2×H(3,𝔽3)—21no—
128C2×H(3,𝔽3)#12111yescharacteristic
129#272—41no—
130C3×#41—91no—
131C2×C3×Q8—92no—
132#235—31no—
133#235—31no—
134#235—61no—
135C2×Dic6—32no—
136C2×Dic6—92no—
137C3×Q16—181no—
138C3×Q16—181no—
139C3×M16—92no—
140C3×M16—91no—
141C3×M16—181no—
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C2×C3×C8—61no—
145C2×C3×C8—181no—
146C2×C3×C8D3211yescharacteristic
147#216—182no—
148#216—182no—
149#215—182no—
150#214—182no—
151#208—31no—
152#208—31no—
153#208—32no—
154#208—61no—
155#208—92no—
156Dic12—61no—
157Dic12—61no—
158Dic12—181no—
159Dic12—181no—
160Dic12—182no—
161#204—182no—
162ΓL(1,𝔽25)—61no—
163ΓL(1,𝔽25)—61no—
164ΓL(1,𝔽25)—62no—
165D3×C8—61no—
166D3×C8—61no—
167D3×C8—62no—
168C22×C32—21no—
169C22×C32#23711yescharacteristic
170C22×C32#23911yescharacteristic
171C2×C3×D3—62no—
172C2×C3×D3—61no—
173C2×C3×D3—61no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4C2×D1211yescharacteristic
177C32×C4D3×D411yescharacteristic
178C32×C4D3×D411yescharacteristic
179C32×C4C2×#6711yescharacteristic
180C3⋊Dic3—42no—
181C3×Dic3—62no—
182C3×Dic3—61no—
183C3×Dic3—61no—
184C3×Dic3—122no—
185#137—272no—
186C2×Q16—91no—
187#131—91no—
188#108—91no—
189#101—272no—
190H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
191C3×Q8—91no—
192C3×Q8—182no—
193C3×Q8—182no—
194C3×D4—91no—
195C3×D4—181no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—61no—
199C2×C3×C4—181no—
200C2×C3×C4—182no—
201C2×C3×C4C2×D3211yescharacteristic
202#67—61no—
203#67—61no—
204#67—62no—
205C2×Dic3—62no—
206C2×Dic3—182no—
207D12—31no—
208D12—31no—
209D12—61no—
210C4×D3—61no—
211C4×D3—61no—
212C4×D3—62no—
213Dic6—31no—
214Dic6—31no—
215Dic6—61no—
216Dic6—62no—
217Dic6—62no—
218Dic6—182no—
219Dic6—182no—
220C3×C8—31no—
221C3×C8—31no—
222C3×C8—31no—
223C3×C8—31no—
224C3×C8—61no—
225C3×C8—61no—
226C3×C8—182no—
227C3×C8—181no—
228C3×C8#62611yescharacteristic
229C3×C8#62611yescharacteristic
230C3⋊C8—61no—
231C3⋊C8—61no—
232C3⋊C8—62no—
233C3⋊C8—62no—
234C3⋊C8—182no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—41no—
239C2×C32#89511yescharacteristic
240C2×C32#93811yescharacteristic
241C3×D3—122no—
242C3×D3—121no—
243C3×D3—121no—
244#41—91no—
245C2×Q8—92no—
246Q16—181no—
247Q16—181no—
248Q16—542no—
249SD16—542no—
250M16—91no—
251M16—92no—
252M16—181no—
253C2×C8—181no—
254C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—181no—
259C22×C3#379611yescharacteristic
260C2×D3—61no—
261C2×D3—61no—
262C2×D3—62no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—181no—
270C3×C4—362no—
271C3×C4#380211yescharacteristic
272C3×C4C2×#62611yescharacteristic
273Dic3—61no—
274Dic3—61no—
275Dic3—62no—
276Dic3—122no—
277Dic3—362no—
278C32—21no—
279C32#489611yescharacteristic, complemented
280C32#519211yescharacteristic, complemented
281Q8—91no—
282Q8—182no—
283Q8—182no—
284D4—91no—
285D4—181no—
286C2×C4—181no—
287C2×C4—182no—
288C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C8—182no—
290C8—181no—
291C8#620011yescharacteristic
292C8#620011yescharacteristic
293C2×C3—21no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—61no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—121no—
300C2×C3—361no—
301C2×C3#6391611yescharacteristic
302D3—121no—
303D3—121no—
304D3—122no—
305C22—181no—
306C22#8864911yescharacteristic
307C4—181no—
308C4—362no—
309C4#8865511yescharacteristic
310C4C2×#620011yescharacteristic
311C3—31no—
312C3—31no—
313C3—61no—
314C3#1059093311yescharacteristic
315C2—21no—
316C2—361no—
317C2#1172115911yescharacteristic
subgroupG/H
centerC2#11721159
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63916
prime radical#88740C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#473
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3746⋊C3C2semidirect
4. #63646⋊C3C2semidirect
5. #11720321C2semidirect
19. C2×#3746D3semidirect
26. #63646D3semidirect
190. H(3,𝔽3)#473semidirect
279. C32#4896semidirect
280. C32#5192semidirect

Maximal subgroups

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

subgroupindex
#117217762
C2×#3746⋊C32
#117210692
#63646⋊C32
#117203212
#105909323
#105909303