Finite Group Databasebeta

← C3×#10593336 #13166756

Subgroups — C3×#10593336 #13166756

Order 1728.

364 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 58 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×C3×#3718C211yesmaximal, characteristic, complemented
2C3×#63822C211yesmaximal, characteristic
3C3×#63773C211yesmaximal, characteristic, complemented
4C3×#63736C212yesmaximal, complemented
5C3×#63735C212yesmaximal, complemented
6#10593336C311yesmaximal, characteristic, direct product factor
7C32×#235—21no—
8C2×C32×Dic6—21no—
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10C3×#3718—21no—
11C3×#3718C2211yescharacteristic
12C3×#3718C2212yes—
13C3×#3715—21no—
14C3×#3713C2211yescharacteristic
15C3×#3708C2212yes—
16C3×#950—61no—
17C2×C3×#216—61no—
18C2×#3718C2×C311yescharacteristic, complemented
19#63822C2×C311yescharacteristic
20#63773C2×C311yescharacteristic, complemented
21#63736C2×C312yescomplemented
22#63735C2×C312yescomplemented
23C2×C33×C4D411yescharacteristic
24C32×#67—41no—
25C2×C32×Dic3—41no—
26C32×D12—21no—
27C32×C4×D3—41no—
28C32×Dic6—21no—
29C32×Dic6—22no—
30C32×Dic6—41no—
31C3×C3⋊(C3⋊C8)D411yescharacteristic
32C3×C3⋊(C3⋊C8)C2311yescharacteristic
33C3×#510—91nomaximal
34C32×#41—61no—
35C2×C32×Q8—61no—
36C3×#235—21no—
37C3×#235—21no—
38C3×#235—41no—
39C2×C3×Dic6—21no—
40C2×C3×Dic6—21no—
41C2×C3×Dic6—41no—
42C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
43C3×#216—62no—
44C3×#216—61no—
45C3×#216—62no—
46C3×#215—61no—
47C3×#214—61no—
48C3×#213—61no—
49C2×C3×C3⋊C8—61no—
50C2×C3×C3⋊C8—61no—
51#3718—21no—
52#3718C22×C311yescharacteristic
53#3718C22×C312yes—
54#3715—21no—
55#3713C22×C311yescharacteristic
56#3708C22×C312yes—
57C22×C33D811yescharacteristic
58C2×C32×D3—41no—
59C33×C4D811yescharacteristic
60C33×C4C2×D411yescharacteristic
61C32×Dic3—42no—
62C32×Dic3—41no—
63C3×#135—91no—
64C2×C3×Q16—91no—
65#950—61no—
66C2×#216—61no—
67C3×Q32—92no—
68C3×SD32—92no—
69C3×M32—91no—
70C32×Q8—61no—
71C32×Q8—62no—
72C32×Q8—121no—
73C32×D4—61no—
74C32×D4—121no—
75C2×C32×C4—21no—
76C2×C32×C4—21no—
77C2×C32×C4—41no—
78C2×C32×C4—41no—
79C2×C32×C4—121no—
80C2×C32×C4—121no—
81C2×C32×C4C3×D411yescharacteristic
82C3×#67—41no—
83C3×#67—42no—
84C3×#67—41no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—41no—
88C3×D12—21no—
89C3×D12—21no—
90C3×D12—41no—
91C3×C4×D3—41no—
92C3×C4×D3—42no—
93C3×C4×D3—41no—
94C3×Dic6—21no—
95C3×Dic6—22no—
96C3×Dic6—21no—
97C3×Dic6—22no—
98C3×Dic6—42no—
99C3×Dic6—41no—
100C3×Dic6—41no—
101C3×Dic6—42no—
102C3×Dic6—41no—
103C3⋊(C3⋊C8)C3×D411yescharacteristic
104C3⋊(C3⋊C8)C23×C311yescharacteristic
105C3×C3⋊C8—61no—
106C3×C3⋊C8—61no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109#510—91noSylow
110C2×C33—21no—
111C2×C33C2×D811yescharacteristic
112C32×D3—81no—
113C3×#41—61no—
114C3×#41—61no—
115C3×#41—121no—
116C2×C3×Q8—61no—
117C2×C3×Q8—61no—
118C2×C3×Q8—121no—
119#235—21no—
120C2×Dic6—21no—
121C3×Q16—92no—
122C3×Q16—91no—
123C3×Q16—181no—
124C3×SD16—181no—
125C3×D8—91no—
126C2×C3×C8—91no—
127#216—62no—
128#216—61no—
129#216—62no—
130#215—61no—
131#214—61no—
132#213—61no—
133C2×C3⋊C8—61no—
134C2×C3⋊C8—61no—
135C3×C16—92no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32—121no—
141C22×C32C3×D811yescharacteristic
142C2×C3×D3—41no—
143C2×C3×D3—42no—
144C2×C3×D3—41no—
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×C4—121no—
154C32×C4—122no—
155C32×C4C3×D811yescharacteristic
156C32×C4C2×C3×D411yescharacteristic
157C3×Dic3—42no—
158C3×Dic3—41no—
159C3×Dic3—44no—
160C3×Dic3—42no—
161C3×Dic3—42no—
162C3×Dic3—41no—
163#135—91no—
164C2×Q16—91no—
165Q32—92no—
166SD32—92no—
167M32—91no—
168C33#51011yescharacteristic, complemented, Sylow
169C3×Q8—61no—
170C3×Q8—62no—
171C3×Q8—61no—
172C3×Q8—62no—
173C3×Q8—122no—
174C3×Q8—121no—
175C3×Q8—121no—
176C3×Q8—122no—
177C3×Q8—121no—
178C3×D4—61no—
179C3×D4—61no—
180C3×D4—121no—
181C3×D4—121no—
182C3×D4—122no—
183C3×D4—121no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—121no—
189C2×C3×C4—122no—
190C2×C3×C4—121no—
191C2×C3×C4—122no—
192C2×C3×C4—121no—
193C2×C3×C4—121no—
194C2×C3×C4GO+(4,𝔽2)11yescharacteristic
195#67—41no—
196C2×Dic3—41no—
197D12—21no—
198C4×D3—41no—
199Dic6—21no—
200Dic6—22no—
201Dic6—41no—
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×C32—241no—
218C2×C32C2×C3×D811yescharacteristic
219C3×D3—81no—
220C3×D3—82no—
221C3×D3—81no—
222#41—61no—
223C2×Q8—61no—
224Q16—92no—
225Q16—91no—
226Q16—181no—
227SD16—181no—
228D8—91no—
229C2×C8—91no—
230C16—92no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—121no—
236C22×C3—122no—
237C22×C3—121no—
238C22×C3#377411yescharacteristic
239C2×D3—41no—
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—121no—
249C3×C4—122no—
250C3×C4—122no—
251C3×C4—124no—
252C3×C4—121no—
253C3×C4—122no—
254C3×C4#377411yescharacteristic
255C3×C4C2×GO+(4,𝔽2)11yescharacteristic
256Dic3—42no—
257Dic3—41no—
258C32—21no—
259C32—21no—
260C32—41no—
261C32—41no—
262C32C3×#51011yescharacteristic, complemented
263Q8—61no—
264Q8—62no—
265Q8—121no—
266D4—61no—
267D4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4C3×GO+(4,𝔽2)11yescharacteristic
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—241no—
283C2×C3—242no—
284C2×C3—241no—
285C2×C3C2×#377411yescharacteristic
286D3—81no—
287C22—121no—
288C22C3×#377411yescharacteristic
289C4—121no—
290C4—122no—
291C4C3×#377411yescharacteristic
292C4C2×C3×GO+(4,𝔽2)11yescharacteristic
293C3—21no—
294C3—21no—
295C3—41no—
296C3—41no—
297C3#1059333611yescharacteristic, direct product factor
298C2—21no—
299C2—241no—
300C2C2×C3×#377411yescharacteristic
subgroupG/H
centerC2×C3C2×#3774
commutator, cosocleC3⋊(C3⋊C8)C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C2×D8
prime radicalC3×#63822C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#510
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3718C2semidirect
3. C3×#63773C2semidirect
4. C3×#63736C2semidirect
5. C3×#63735C2semidirect
6. #10593336C3direct (N × H)
18. C2×#3718C2×C3semidirect
20. #63773C2×C3semidirect
21. #63736C2×C3semidirect
22. #63735C2×C3semidirect
168. C33#510semidirect
262. C32C3×#510semidirect
297. C3#10593336direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37182
C3×#638222
C3×#637732
C3×#637362
C3×#637352
#105933363
C3×#5109