Finite Group Databasebeta

← C3×#10590888 #13157850

Subgroups — C3×#10590888 #13157850

Order 1728.

602 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 116 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×#63914C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic, complemented
3C3×#63642C211yesmaximal, characteristic, complemented
4C32×#821C212yesmaximal, complemented
5C3×#63562C212yesmaximal, complemented
6C3×#4764—32nomaximal
7#10590888C311yesmaximal, characteristic, direct product factor
8C2×C32×D12C2212yes—
9C2×C3×#625—22no—
10C2×C33×C8C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×C4⋊Dic3C2212yes—
14C3×Dic3⋊Dic3—22no—
15C3×#940—32no—
16C3×#890—32no—
17#63914C2×C311yescharacteristic
18#63872C2×C311yescharacteristic, complemented
19C32×#102—32no—
20#63642C2×C311yescharacteristic, complemented
21C3×#832—32no—
22C3×#821—22no—
23C3×#821D312yescomplemented
24C3×#821C2×C312yescomplemented
25C3×#815—32no—
26C3×#815—61no—
27C3×#809—32no—
28#63562C2×C312yescomplemented
29C22×C32×D3—22no—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3D412yes—
32C2×C32×Dic3—22no—
33C32×D12—22no—
34C32×D12—22no—
35C3×#625—44no—
36C33×C8—21no—
37C3×C3⋊(C3⋊C8)—21no—
38C3×#490—91no—
39#4764—32no—
40C2×C32×D4—32no—
41C2×C3×#67—62no—
42C2×C3×#67—62no—
43C2×C3×D12—22no—
44C2×C3×D12C2×D312yes—
45C2×C3×D12C22×C312yes—
46C2×#625—22no—
47C2×C32×C8—21no—
48C2×C32×C8—22no—
49C2×C32×C8—41no—
50C2×C32×C8C2×D312yes—
51C2×C32×C8C22×C311yescharacteristic
52C32×C4⋊C4—32no—
53C4×C3⋊Dic3C22×C311yescharacteristic
54C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3C2×D312yes—
58C3×C4⋊Dic3C22×C312yes—
59C3×Dic3⋊C4—62no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—61no—
64Dic3⋊Dic3—22no—
65C22×C33C2×D411yescharacteristic
66C2×C32×D3—42no—
67C2×C32×D3—42no—
68C2×C32×D3—42no—
69C33×C4#4111yescharacteristic
70C33×C4#4111yescharacteristic
71C3×C3⋊Dic3—22no—
72C32×Dic3—42no—
73C3×#127—91no—
74C3×#125—91no—
75#940—32no—
76#890—32no—
77C3×#102—32no—
78C3×#102—32no—
79C3×#102—62no—
80C3×#102—92no—
81C3×#97—91no—
82#832—32no—
83#821C3×D312yescomplemented
84#815—32no—
85#815—61no—
86#809—32no—
87C23×C32—62no—
88C22×C3×D3—22no—
89C22×C3×D3—22no—
90C22×C3×D3—42no—
91C32×D4—62no—
92C32×D4—62no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—41no—
96C2×C32×C4—62no—
97C2×C32×C4C22×D312yes—
98C2×C32×C4C23×C311yescharacteristic
99C2×C3⋊Dic3C3×D412yes—
100C3×#67—124no—
101C3×#67—124no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—42no—
107C2×C3×Dic3—62no—
108C3×D12—24no—
109C3×D12—22no—
110C3×D12—22no—
111C3×D12—22no—
112C3×D12—22no—
113C3×D12—42no—
114#625—44no—
115C32×C8—21no—
116C32×C8—22no—
117C32×C8—22no—
118C32×C8—42no—
119C32×C8—42no—
120C3⋊(C3⋊C8)—21no—
121C3×C3⋊C8—62no—
122C3×C3⋊C8—62no—
123#490—91noSylow
124C2×C33#12411yescharacteristic
125C2×C33AGL(1,ℤ/8)12yes—
126C32×D3—82no—
127C2×C3×D4—32no—
128C2×C3×D4—32no—
129C2×C3×D4—62no—
130C2×C3×D4—182no—
131C2×#67—62no—
132C2×#67—62no—
133C2×D12C2×C3×D312yes—
134C2×C3×C8—21no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—91no—
138C2×C3×C8D3211yescharacteristic
139C2×C3×C8C2×C3×D312yes—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4—62no—
143C3×C4⋊C4—182no—
144C3×C42—91no—
145C4⋊Dic3—62no—
146C4⋊Dic3C2×C3×D312yes—
147Dic3⋊C4—62no—
148C4×Dic3—32no—
149C4×Dic3—61no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—61no—
152C22×C32—21no—
153C22×C32—22no—
154C22×C32—41no—
155C22×C32—122no—
156C22×C32—122no—
157C22×C32—122no—
158C22×C32D3×D412yes—
159C22×C32C2×C3×D411yescharacteristic
160C2×C3×D3—44no—
161C2×C3×D3—42no—
162C2×C3×D3—42no—
163C2×C3×D3—42no—
164C2×C3×D3—42no—
165C2×C3×D3—42no—
166C2×C3×D3—42no—
167C2×C3×D3—82no—
168C2×C3×D3—82no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4—122no—
176C32×C4#23512yes—
177C32×C4#23712yes—
178C32×C4C3×#4111yescharacteristic
179C32×C4C3×#4111yescharacteristic
180C3⋊Dic3—22no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—82no—
186C3×Dic3—122no—
187#127—91no—
188#125—91no—
189#102—32no—
190#102—92no—
191#97—91no—
192C33#49011yescharacteristic, complemented, Sylow
193C23×C3—62no—
194C23×C3—62no—
195C23×C3—122no—
196C22×D3—22no—
197C3×D4—64no—
198C3×D4—62no—
199C3×D4—62no—
200C3×D4—62no—
201C3×D4—62no—
202C3×D4—122no—
203C3×D4—364no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—92no—
210C2×C3×C4—122no—
211C2×C3×C4C2×D3211yescharacteristic
212C2×C3×C4C22×C3×D312yes—
213#67—124no—
214#67—124no—
215C2×Dic3—22no—
216C2×Dic3—32no—
217C2×Dic3—32no—
218C2×Dic3—62no—
219D12—22no—
220D12—22no—
221C3×C8—22no—
222C3×C8—21no—
223C3×C8—22no—
224C3×C8—42no—
225C3×C8—42no—
226C3×C8—181no—
227C3⋊C8—62no—
228C3⋊C8—62no—
229C2×C32—22no—
230C2×C32—22no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—21no—
234C2×C32—41no—
235C2×C32—42no—
236C2×C32—242no—
237C2×C32#88512yes—
238C2×C32#91112yes—
239C2×C32#91412yes—
240C2×C32C3×#12411yescharacteristic
241C2×C32C3×AGL(1,ℤ/8)12yes—
242C3×D3—84no—
243C3×D3—82no—
244C3×D3—82no—
245C2×D4—32no—
246C2×D4—182no—
247C2×C8—91no—
248C2×C8C3×D3211yescharacteristic
249C4⋊C4—32no—
250C4⋊C4—182no—
251C42—91no—
252C22×C3—21no—
253C22×C3—22no—
254C22×C3—41no—
255C22×C3—124no—
256C22×C3—122no—
257C22×C3—122no—
258C22×C3—122no—
259C22×C3—122no—
260C22×C3—122no—
261C22×C3—122no—
262C22×C3—242no—
263C22×C3—242no—
264C22×C3#380211yescharacteristic
265C22×C3C3×D3×D412yes—
266C2×D3—42no—
267C2×D3—42no—
268C2×D3—42no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—22no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—122no—
276C3×C4—122no—
277C3×C4—182no—
278C3×C4—242no—
279C3×C4#379811yescharacteristic
280C3×C4#380511yescharacteristic
281C3×C4C3×#23512yes—
282C3×C4C3×#23712yes—
283Dic3—42no—
284Dic3—62no—
285Dic3—62no—
286Dic3—122no—
287C32—22no—
288C32—21no—
289C32—41no—
290C32#476412yescomplemented
291C32C3×#49011yescharacteristic, complemented
292C23—62no—
293D4—62no—
294D4—62no—
295D4—364no—
296C2×C4—62no—
297C2×C4—92no—
298C2×C4C2×C3×D3211yescharacteristic
299C8—21no—
300C8—181no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—21no—
306C2×C3—42no—
307C2×C3—41no—
308C2×C3—244no—
309C2×C3—242no—
310C2×C3—242no—
311C2×C3#6379812yes—
312C2×C3#6391511yescharacteristic
313C2×C3C3×#88512yes—
314C2×C3C3×#91112yes—
315C2×C3C3×#91412yes—
316D3—82no—
317C22—122no—
318C22—122no—
319C22—122no—
320C22C3×#380211yescharacteristic
321C4—122no—
322C4—182no—
323C4C3×#379811yescharacteristic
324C4C3×#380511yescharacteristic
325C3—22no—
326C3—21no—
327C3—41no—
328C3#1059088811yescharacteristic, direct product factor
329C3C3×#476412yescomplemented
330C2—242no—
331C2C3×#6379812yes—
332C2C3×#6391511yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63872C2semidirect
3. C3×#63642C2semidirect
4. C32×#821C2semidirect
5. C3×#63562C2semidirect
7. #10590888C3direct (N × H)
18. #63872C2×C3semidirect
20. #63642C2×C3semidirect
23. C3×#821D3semidirect
24. C3×#821C2×C3semidirect
28. #63562C2×C3semidirect
83. #821C3×D3semidirect
192. C33#490semidirect
290. C32#4764semidirect
291. C32C3×#490semidirect
328. C3#10590888direct (N × H)
329. C3C3×#4764semidirect

Maximal subgroups

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

subgroupindex
C3×#639142
C3×#638722
C3×#636422
C32×#8212
C3×#635622
C3×#47643
#105908883