Finite Group Databasebeta

← #13167463

Subgroups — #13167463

Order 1728.

364 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C2×#88812C211yesmaximal, characteristic, complemented
2#11723660C211yesmaximal, characteristic
3C3×#63773C211yesmaximal, characteristic, complemented
4#11722790C212yesmaximal, complemented
5#11722789C212yesmaximal, complemented
6#10593336—31nomaximal
7C3×#3828—21no—
8C2×C3×#634—21no—
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10#88812—21no—
11#88812C2212yes—
12#88812C2211yescharacteristic
13#88811—21no—
14#88810C2211yescharacteristic
15C3×#3708C2212yes—
16C2×#3719—61no—
17C2×#3718—31no—
18#63829—61no—
19#63822—31no—
20#63773D311yescharacteristic, complemented
21#63736—32no—
22#63735—32no—
23C2×C33×C4D411yescharacteristic
24C3×#638—41no—
25C2×C3×C3⋊Dic3—41no—
26C3×C4⋊(C3⋊D3)—21no—
27C3×C4×C3⋊D3—41no—
28C3×#634—22no—
29C3×#634—21no—
30C3×#634—41no—
31C3×C3⋊(C3⋊C8)D411yescharacteristic
32C3×C3⋊(C3⋊C8)C2311yescharacteristic
33#4909—91nomaximal
34#3828—21no—
35C2×#634—21no—
36C3×#235—61no—
37C3×#235—61no—
38C3×#235—121no—
39C2×C3×Dic6—61no—
40C2×C3×Dic6—61no—
41C2×C3×Dic6—121no—
42C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
43C2×C3×C3⋊C8—61no—
44C2×C3×C3⋊C8—61no—
45#3719—61no—
46#3719—62no—
47#3719—62no—
48#3718—31no—
49#3718—32no—
50#3718—61no—
51#3717—61no—
52#3716—61no—
53#3715—61no—
54#3714—61no—
55#3713—31no—
56#3708C2×D312yes—
57C22×C33D811yescharacteristic
58C2×C3×C3⋊D3—41no—
59C33×C4D811yescharacteristic
60C33×C4C2×D411yescharacteristic
61C3×C3⋊Dic3—42no—
62C3×C3⋊Dic3—41no—
63#950—181no—
64C2×#216—181no—
65C2×Dic12—91no—
66#904—91no—
67C3×M32—91no—
68Dic24—92no—
69#800—92no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4D1211yescharacteristic
75#638—41no—
76C2×C3⋊Dic3—41no—
77C4⋊(C3⋊D3)—21no—
78C4×C3⋊D3—41no—
79#634—21no—
80#634—22no—
81#634—41no—
82C3×#67—121no—
83C3×#67—122no—
84C3×#67—121no—
85C2×C3×Dic3—121no—
86C2×C3×Dic3—122no—
87C2×C3×Dic3—121no—
88C3×D12—61no—
89C3×D12—61no—
90C3×D12—121no—
91C3×C4×D3—121no—
92C3×C4×D3—122no—
93C3×C4×D3—121no—
94C3×Dic6—61no—
95C3×Dic6—62no—
96C3×Dic6—61no—
97C3×Dic6—62no—
98C3×Dic6—122no—
99C3×Dic6—121no—
100C3×Dic6—121no—
101C3×Dic6—122no—
102C3×Dic6—121no—
103C3⋊(C3⋊C8)D1211yescharacteristic
104C3⋊(C3⋊C8)C22×D311yescharacteristic
105C3×C3⋊C8—61no—
106C3×C3⋊C8—61no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109#510—271noSylow
110C2×C33—21no—
111C2×C33C2×D811yescharacteristic
112C3×C3⋊D3—81no—
113C3×#41—181no—
114C2×C3×Q8—181no—
115#235—61no—
116#235—61no—
117#235—121no—
118C2×Dic6—61no—
119C2×Dic6—61no—
120C2×Dic6—121no—
121C2×C3×C8—91no—
122#216—182no—
123#216—181no—
124#216—182no—
125#215—181no—
126#214—181no—
127#213—181no—
128C2×C3⋊C8—61no—
129C2×C3⋊C8—61no—
130Dic12—92no—
131Dic12—91no—
132Dic12—181no—
133D24—91no—
134#204—181no—
135C3×C16—92no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32D2411yescharacteristic
141C2×C3⋊D3—41no—
142C2×C3×D3—121no—
143C2×C3×D3—122no—
144C2×C3×D3—121no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—41no—
152C32×C4—41no—
153C32×C4D2411yescharacteristic
154C32×C4C2×D1211yescharacteristic
155C3⋊Dic3—42no—
156C3⋊Dic3—41no—
157C3×Dic3—122no—
158C3×Dic3—121no—
159C3×Dic3—124no—
160C3×Dic3—122no—
161C3×Dic3—122no—
162C3×Dic3—121no—
163#135—271no—
164C2×Q16—271no—
165Q32—272no—
166SD32—272no—
167M32—91no—
168C33#51011yescharacteristic, complemented, Sylow
169C3×Q8—181no—
170C3×Q8—182no—
171C3×Q8—361no—
172C3×D4—181no—
173C3×D4—361no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—361no—
179C2×C3×C4—361no—
180C2×C3×C4GO+(4,𝔽2)11yescharacteristic
181#67—121no—
182#67—122no—
183#67—121no—
184C2×Dic3—121no—
185C2×Dic3—122no—
186C2×Dic3—121no—
187D12—61no—
188D12—61no—
189D12—121no—
190C4×D3—121no—
191C4×D3—122no—
192C4×D3—121no—
193Dic6—61no—
194Dic6—62no—
195Dic6—61no—
196Dic6—62no—
197Dic6—122no—
198Dic6—121no—
199Dic6—121no—
200Dic6—122no—
201Dic6—121no—
202C3×C8—91no—
203C3×C8—91no—
204C3⋊C8—61no—
205C3⋊C8—61no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C2×C32—22no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—42no—
216C2×C32—81no—
217C2×C32C2×D2411yescharacteristic
218C3⋊D3—81no—
219C3×D3—241no—
220C3×D3—242no—
221C3×D3—241no—
222#41—181no—
223C2×Q8—181no—
224Q16—272no—
225Q16—271no—
226Q16—541no—
227SD16—541no—
228D8—271no—
229C2×C8—91no—
230C16—92no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—361no—
236C22×C3#377411yescharacteristic
237C2×D3—121no—
238C2×D3—122no—
239C2×D3—121no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—361no—
249C3×C4—362no—
250C3×C4#377411yescharacteristic
251C3×C4C2×GO+(4,𝔽2)11yescharacteristic
252Dic3—122no—
253Dic3—121no—
254Dic3—124no—
255Dic3—122no—
256Dic3—122no—
257Dic3—121no—
258C32—21no—
259C32—21no—
260C32—41no—
261C32—41no—
262C32#490911yescharacteristic, complemented
263Q8—181no—
264Q8—182no—
265Q8—361no—
266D4—181no—
267D4—361no—
268C2×C4—361no—
269C2×C4—361no—
270C2×C4#157⋊D311yescharacteristic
271C8—91no—
272C8—91no—
273C2×C3—22no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—41no—
278C2×C3—42no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—81no—
282C2×C3—721no—
283C2×C3C2×#377411yescharacteristic
284D3—241no—
285D3—242no—
286D3—241no—
287C22—361no—
288C22#8894111yescharacteristic
289C4—361no—
290C4—362no—
291C4#8894111yescharacteristic
292C4C2×#157⋊D311yescharacteristic
293C3—21no—
294C3—21no—
295C3—41no—
296C3—41no—
297C3#1059333611yescharacteristic, complemented
298C2—21no—
299C2—721no—
300C2C2×#8894111yescharacteristic
subgroupG/H
centerC2C2×#88941
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×D8
prime radical#11723660C2
hypercenterC2×C4#157⋊D3
hypocenterC33#510
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88812C2semidirect
3. C3×#63773C2semidirect
4. #11722790C2semidirect
5. #11722789C2semidirect
20. #63773D3semidirect
168. C33#510semidirect
262. C32#4909semidirect
297. C3#10593336semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117236602
C3×#637732
#117227902
#117227892
#105933363
#49099