Finite Group Databasebeta

← C3×#10593334 #13166754

Subgroups — C3×#10593334 #13166754

Order 1728.

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

normal subgroups only

This group has 412 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
1C3×#63822C211yesmaximal, characteristic, complemented
2C2×C3×#3713C211yesmaximal, characteristic, complemented
3C3×#63773C211yesmaximal, characteristic, complemented
4C3×#63735C212yesmaximal, complemented
5C3×#63734C212yesmaximal, complemented
6#10593334C311yesmaximal, characteristic, direct product factor
7C32×#235—21no—
8C2×C32×D12—21no—
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10C3×#3718C2211yescharacteristic, complemented
11C3×#3715—21no—
12C3×#3713—21no—
13C3×#3713C2212yes—
14C3×#3713C2211yescharacteristic, complemented
15C3×#3708C2212yescomplemented
16C3×#950—61no—
17C2×C3×#213—61no—
18#63822C2×C311yescharacteristic, complemented
19C2×#3713C2×C311yescharacteristic, complemented
20#63773C2×C311yescharacteristic, complemented
21#63735C2×C312yescomplemented
22#63734C2×C312yescomplemented
23C22×C32×D3—41no—
24C2×C33×C4D411yescharacteristic
25C32×#67—41no—
26C32×D12—22no—
27C32×D12—21no—
28C32×D12—41no—
29C32×C4×D3—41no—
30C32×Dic6—21no—
31C3×C3⋊(C3⋊C8)D411yescharacteristic
32C3×C3⋊(C3⋊C8)C2311yescharacteristic
33C3×#509—91nomaximal
34C32×#41—61no—
35C2×C32×D4—61no—
36C3×#235—21no—
37C3×#235—21no—
38C3×#235—41no—
39C2×C3×D12—21no—
40C2×C3×D12—21no—
41C2×C3×D12—41no—
42C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
43C3×#216—61no—
44C3×#215—61no—
45C3×#214—61no—
46C3×#213—62no—
47C3×#213—61no—
48C3×#213—62no—
49C2×C3×C3⋊C8—61no—
50C2×C3×C3⋊C8—61no—
51#3718C22×C311yescharacteristic, complemented
52#3715—21no—
53#3713—21no—
54#3713C22×C312yes—
55#3713C22×C311yescharacteristic, complemented
56#3708C22×C312yescomplemented
57C22×C33D811yescharacteristic
58C2×C32×D3—41no—
59C2×C32×D3—42no—
60C2×C32×D3—82no—
61C33×C4D811yescharacteristic
62C33×C4C2×D411yescharacteristic
63C32×Dic3—41no—
64C3×#135—91no—
65C2×C3×D8—91no—
66#950—61no—
67C2×#213—61no—
68C3×SD32—92no—
69C3×D16—92no—
70C3×M32—91no—
71C23×C32—121no—
72C22×C3×D3—41no—
73C22×C3×D3—42no—
74C22×C3×D3—41no—
75C32×Q8—61no—
76C32×D4—62no—
77C32×D4—61no—
78C32×D4—121no—
79C32×D4—121no—
80C2×C32×C4—21no—
81C2×C32×C4—21no—
82C2×C32×C4—41no—
83C2×C32×C4—41no—
84C2×C32×C4—121no—
85C2×C32×C4C3×D411yescharacteristic
86C3×#67—41no—
87C3×#67—42no—
88C3×#67—41no—
89C3×D12—22no—
90C3×D12—21no—
91C3×D12—22no—
92C3×D12—21no—
93C3×D12—42no—
94C3×D12—41no—
95C3×D12—41no—
96C3×D12—42no—
97C3×D12—41no—
98C3×C4×D3—41no—
99C3×C4×D3—42no—
100C3×C4×D3—41no—
101C3×Dic6—21no—
102C3×Dic6—21no—
103C3×Dic6—41no—
104C3⋊(C3⋊C8)C3×D411yescharacteristic
105C3⋊(C3⋊C8)C23×C311yescharacteristic
106C3×C3⋊C8—61no—
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109C3×C3⋊C8—61no—
110#509—91noSylow
111C2×C33—21no—
112C2×C33C2×D811yescharacteristic
113C32×D3—81no—
114C32×D3—82no—
115C3×#41—61no—
116C3×#41—61no—
117C3×#41—121no—
118C2×C3×D4—61no—
119C2×C3×D4—61no—
120C2×C3×D4—121no—
121#235—21no—
122C2×D12—21no—
123C3×Q16—91no—
124C3×SD16—181no—
125C3×D8—91no—
126C3×D8—92no—
127C3×D8—181no—
128C2×C3×C8—91no—
129#216—61no—
130#215—61no—
131#214—61no—
132#213—62no—
133#213—61no—
134#213—62no—
135C2×C3⋊C8—61no—
136C2×C3⋊C8—61no—
137C3×C16—92no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—41no—
141C22×C32—41no—
142C22×C32—121no—
143C22×C32—122no—
144C22×C32—242no—
145C22×C32C3×D811yescharacteristic
146C2×C3×D3—41no—
147C2×C3×D3—42no—
148C2×C3×D3—42no—
149C2×C3×D3—44no—
150C2×C3×D3—41no—
151C2×C3×D3—42no—
152C2×C3×D3—82no—
153C2×C3×D3—82no—
154C2×C3×D3—84no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—41no—
160C32×C4—41no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4—121no—
164C32×C4C3×D811yescharacteristic
165C32×C4C2×C3×D411yescharacteristic
166C3×Dic3—41no—
167C3×Dic3—42no—
168C3×Dic3—41no—
169#135—91no—
170C2×D8—91no—
171SD32—92no—
172D16—92no—
173M32—91no—
174C33#50911yescharacteristic, complemented, Sylow
175C23×C3—121no—
176C23×C3—122no—
177C23×C3—121no—
178C22×D3—41no—
179C3×Q8—61no—
180C3×Q8—61no—
181C3×Q8—121no—
182C3×D4—62no—
183C3×D4—61no—
184C3×D4—62no—
185C3×D4—61no—
186C3×D4—122no—
187C3×D4—121no—
188C3×D4—121no—
189C3×D4—122no—
190C3×D4—121no—
191C3×D4—122no—
192C3×D4—121no—
193C3×D4—121no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—121no—
199C2×C3×C4—122no—
200C2×C3×C4—121no—
201C2×C3×C4GO+(4,𝔽2)11yescharacteristic
202#67—41no—
203D12—22no—
204D12—21no—
205D12—41no—
206C4×D3—41no—
207Dic6—21no—
208C3×C8—91no—
209C3×C8—91no—
210C3⋊C8—61no—
211C3⋊C8—61no—
212C3⋊C8—61no—
213C3⋊C8—61no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—42no—
222C2×C32—81no—
223C2×C32—241no—
224C2×C32—242no—
225C2×C32C2×C3×D811yescharacteristic
226C3×D3—81no—
227C3×D3—82no—
228C3×D3—82no—
229C3×D3—84no—
230C3×D3—81no—
231C3×D3—82no—
232#41—61no—
233C2×D4—61no—
234Q16—91no—
235SD16—181no—
236D8—91no—
237D8—92no—
238D8—181no—
239C2×C8—91no—
240C16—92no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—121no—
246C22×C3—122no—
247C22×C3—122no—
248C22×C3—124no—
249C22×C3—121no—
250C22×C3—122no—
251C22×C3—242no—
252C22×C3—242no—
253C22×C3—244no—
254C22×C3#377411yescharacteristic
255C2×D3—41no—
256C2×D3—42no—
257C2×D3—82no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—121no—
267C3×C4—122no—
268C3×C4—121no—
269C3×C4#377411yescharacteristic
270C3×C4C2×GO+(4,𝔽2)11yescharacteristic
271Dic3—41no—
272C32—21no—
273C32—21no—
274C32—41no—
275C32—41no—
276C32C3×#50911yescharacteristic, complemented
277C23—121no—
278Q8—61no—
279D4—62no—
280D4—61no—
281D4—121no—
282D4—121no—
283C2×C4—121no—
284C2×C4C3×GO+(4,𝔽2)11yescharacteristic
285C8—91no—
286C8—91no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—41no—
292C2×C3—42no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—81no—
296C2×C3—242no—
297C2×C3—241no—
298C2×C3—244no—
299C2×C3—242no—
300C2×C3—242no—
301C2×C3—241no—
302C2×C3C2×#377411yescharacteristic
303D3—82no—
304D3—81no—
305C22—121no—
306C22—122no—
307C22—242no—
308C22C3×#377411yescharacteristic
309C4—121no—
310C4C3×#377411yescharacteristic
311C4C2×C3×GO+(4,𝔽2)11yescharacteristic
312C3—21no—
313C3—21no—
314C3—41no—
315C3—41no—
316C3#1059333411yescharacteristic, direct product factor
317C2—21no—
318C2—242no—
319C2—241no—
320C2C2×C3×#377411yescharacteristic
subgroupG/H
centerC2×C3C2×#3774
commutator, cosocleC3⋊(C3⋊C8)C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C2×D8
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#509
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. C3×#63822C2semidirect
2. C2×C3×#3713C2semidirect
3. C3×#63773C2semidirect
4. C3×#63735C2semidirect
5. C3×#63734C2semidirect
6. #10593334C3direct (N × H)
10. C3×#3718C22semidirect
14. C3×#3713C22semidirect
15. C3×#3708C22semidirect
18. #63822C2×C3semidirect
19. C2×#3713C2×C3semidirect
20. #63773C2×C3semidirect
21. #63735C2×C3semidirect
22. #63734C2×C3semidirect
51. #3718C22×C3semidirect
55. #3713C22×C3semidirect
56. #3708C22×C3semidirect
174. C33#509semidirect
276. C32C3×#509semidirect
316. C3#10593334direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638222
C2×C3×#37132
C3×#637732
C3×#637352
C3×#637342
#105933343
C3×#5099