Finite Group Databasebeta

← C3×#10592941 #13163281

Subgroups — C3×#10592941 #13163281

Order 1728.

1576 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C33×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#64160C212yesmaximal, complemented
3C3×#64145C211yesmaximal, characteristic, complemented
4C3×#63664C212yesmaximal, complemented
5C3×#63660C211yesmaximal, characteristic, complemented
6#10592941C311yesmaximal, characteristic, direct product factor
7C3×#5242—34nomaximal
8C33×#41—22no—
9C33×#41C2212yes—
10C2×C33×D4—22no—
11C2×C33×D4—22no—
12C2×C33×D4C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic, complemented
14C3×#3757—22no—
15C3×#3756—22no—
16C3×#3754—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
18C3×#3748C2212yescomplemented
19C32×H(5,𝔽2)—24no—
20C32×H(5,𝔽2)D314yes—
21C32×H(5,𝔽2)C2×C311yescharacteristic
22#64160C2×C312yescomplemented
23#64145C2×C311yescharacteristic, complemented
24C3×#951—38no—
25C3×#936—34no—
26#63664C2×C312yescomplemented
27#63660C2×C311yescharacteristic, complemented
28C3×#837—38no—
29C3×#833—34no—
30C23×C33—22no—
31C23×C33D412yes—
32C33×Q8D412yes—
33C33×D4—24no—
34C33×D4—22no—
35C33×D4—22no—
36C33×D4D412yes—
37C2×C33×C4—22no—
38C2×C33×C4—22no—
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×#634—22no—
43C3×C3⋊(C3⋊C8)—22no—
44C3×#455—91no—
45#5242—34no—
46C32×#41—216no—
47C32×#41—28no—
48C32×#41—28no—
49C32×#41—22no—
50C32×#41C2×D318yes—
51C32×#41C22×C312yes—
52C2×C32×D4—216no—
53C2×C32×D4—216no—
54C2×C32×D4—24no—
55C2×C32×D4—28no—
56C2×C32×D4—22no—
57C2×C32×D4—28no—
58C2×C32×D4—22no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C22×C311yescharacteristic
61C2×#634C22×C311yescharacteristic, complemented
62C2×C3×Dic6—34no—
63#3757—22no—
64#3756—22no—
65#3754—22no—
66C4×C3⋊Dic3C22×C311yescharacteristic, complemented
67#3748C22×C312yescomplemented
68C3×#217—68no—
69C3×#216—68no—
70C3×#214—68no—
71C3×C4×Dic3—34no—
72C3×#208—38no—
73C22×C33—22no—
74C22×C33—22no—
75C22×C33—42no—
76C22×C33—82no—
77C22×C33C2×D411yescharacteristic
78C33×C4—22no—
79C33×C4C2×D412yes—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3—41no—
82C3×H(5,𝔽2)—24no—
83C3×H(5,𝔽2)C3×D314yes—
84C3×H(5,𝔽2)C3⋊D311yescharacteristic
85C3×#137—92no—
86C3×#124—91no—
87#951—38no—
88#936—34no—
89C3×#104—92no—
90C3×#100—91no—
91#837—38no—
92#833—34no—
93C23×C32—216no—
94C23×C32—28no—
95C23×C32—22no—
96C23×C32—28no—
97C23×C32#6718yes—
98C23×C32C3×D412yes—
99C32×Q8—28no—
100C32×Q8#6718yes—
101C32×Q8C3×D412yes—
102C32×D4—232no—
103C32×D4—216no—
104C32×D4—24no—
105C32×D4—216no—
106C32×D4—216no—
107C32×D4—28no—
108C32×D4—22no—
109C32×D4—28no—
110C32×D4—22no—
111C32×D4—28no—
112C32×D4#6718yes—
113C32×D4C3×D412yes—
114C2×C32×C4—216no—
115C2×C32×C4—28no—
116C2×C32×C4—22no—
117C2×C32×C4—216no—
118C2×C32×C4—28no—
119C2×C32×C4—22no—
120C2×C32×C4—24no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C23×C311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125#634—22no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C3×Dic6—68no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—68no—
131#455—91noSylow
132C2×C33—21no—
133C2×C33—42no—
134C2×C33—42no—
135C2×C33#12011yescharacteristic
136C3×#41—216no—
137C3×#41—28no—
138C3×#41—22no—
139C3×#41—28no—
140C3×#41C2×C3×D318yes—
141C3×#41C2×C3⋊D312yes—
142C2×C3×Q8—91no—
143C2×C3×D4—216no—
144C2×C3×D4—216no—
145C2×C3×D4—28no—
146C2×C3×D4—22no—
147C2×C3×D4—28no—
148C2×C3×D4—22no—
149C2×C3×D4—24no—
150C2×C3×D4C2×C3×D314yes—
151C2×C3×D4C2×C3⋊D311yescharacteristic
152C2×Dic6—34no—
153C3×Q16—182no—
154C3×SD16—182no—
155C3×M16—92no—
156C3×C22⋊C4—182no—
157C3×C42—91no—
158#217—68no—
159#216—68no—
160#214—68no—
161C4×Dic3—34no—
162#208—38no—
163C22×C32—216no—
164C22×C32—216no—
165C22×C32—28no—
166C22×C32—28no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—24no—
170C22×C32—42no—
171C22×C32—48no—
172C22×C32—88no—
173C22×C32—816no—
174C22×C32—88no—
175C22×C32—82no—
176C22×C32C2×#6714yes—
177C22×C32C2×C3×D411yescharacteristic
178C32×C4—216no—
179C32×C4—28no—
180C32×C4—22no—
181C32×C4—28no—
182C32×C4C2×#6718yes—
183C32×C4C2×C3×D412yes—
184C3⋊Dic3—41no—
185C3⋊Dic3—41no—
186C3×Dic3—124no—
187C3×Dic3—124no—
188H(5,𝔽2)C3×C3⋊D311yescharacteristic
189#137—92no—
190#124—91no—
191#104—92no—
192#100—91no—
193C33#45511yescharacteristic, complemented, Sylow
194C23×C3—216no—
195C23×C3—28no—
196C23×C3—22no—
197C23×C3—28no—
198C23×C3C3×#6718yes—
199C23×C3#63812yes—
200C3×Q8—28no—
201C3×Q8—182no—
202C3×Q8C3×#6718yes—
203C3×Q8#63812yes—
204C3×D4—232no—
205C3×D4—216no—
206C3×D4—24no—
207C3×D4—216no—
208C3×D4—216no—
209C3×D4—28no—
210C3×D4—22no—
211C3×D4—28no—
212C3×D4—22no—
213C3×D4—28no—
214C3×D4C3×#6718yes—
215C3×D4#63812yes—
216C2×C3×C4—216no—
217C2×C3×C4—28no—
218C2×C3×C4—22no—
219C2×C3×C4—216no—
220C2×C3×C4—28no—
221C2×C3×C4—22no—
222C2×C3×C4—24no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C22×C3×D314yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—64no—
228C2×Dic3—64no—
229Dic6—68no—
230C3×C8—182no—
231C3⋊C8—68no—
232C2×C32—21no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—44no—
236C2×C32—416no—
237C2×C32—416no—
238C2×C32—48no—
239C2×C32—48no—
240C2×C32—42no—
241C2×C32—42no—
242C2×C32#95314yes—
243C2×C32C3×#12011yescharacteristic
244#41—22no—
245#41C2×C3×C3⋊D312yes—
246C2×Q8—91no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4C2×C3×C3⋊D311yescharacteristic
250Q16—182no—
251SD16—182no—
252M16—92no—
253C22⋊C4—182no—
254C42—91no—
255C22×C3—216no—
256C22×C3—216no—
257C22×C3—28no—
258C22×C3—28no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—24no—
262C22×C3—48no—
263C22×C3—42no—
264C22×C3—88no—
265C22×C3—816no—
266C22×C3—88no—
267C22×C3—82no—
268C22×C3C2×C3×#6714yes—
269C22×C3C2×#63811yescharacteristic
270C3×C4—216no—
271C3×C4—28no—
272C3×C4—22no—
273C3×C4—28no—
274C3×C4—361no—
275C3×C4—361no—
276C3×C4C2×C3×#6718yes—
277C3×C4C2×#63812yes—
278Dic3—124no—
279Dic3—124no—
280C32—24no—
281C32#524214yescomplemented
282C32C3×#45511yescharacteristic, complemented
283C23—22no—
284C23C3×#63812yes—
285Q8—182no—
286Q8C3×#63812yes—
287D4—24no—
288D4—22no—
289D4—22no—
290D4C3×#63812yes—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C4C22×C3×C3⋊D311yescharacteristic
296C8—182no—
297C2×C3—24no—
298C2×C3—21no—
299C2×C3—24no—
300C2×C3—44no—
301C2×C3—416no—
302C2×C3—416no—
303C2×C3—48no—
304C2×C3—48no—
305C2×C3—42no—
306C2×C3—42no—
307C2×C3C3×#95314yes—
308C2×C3#6416211yescharacteristic
309C22—22no—
310C22—22no—
311C22—42no—
312C22—82no—
313C22C2×C3×#63811yescharacteristic
314C4—22no—
315C4—361no—
316C4—361no—
317C4C2×C3×#63812yes—
318C3—24no—
319C3C3×#524214yescomplemented
320C3#1059294111yescharacteristic, direct product factor
321C2—21no—
322C2—42no—
323C2—42no—
324C2C3×#6416211yescharacteristic
subgroupG/H
centerC2×C3#64162
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
Fitting, prime radicalC33×H(5,𝔽2)C2
socleC2×C33#120
hypercenterC3×H(5,𝔽2)C3⋊D3
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C3×#64160C2semidirect
3. C3×#64145C2semidirect
4. C3×#63664C2semidirect
5. C3×#63660C2semidirect
6. #10592941C3direct (N × H)
13. C2×C3×#634C22semidirect
17. C3×C4×C3⋊Dic3C22semidirect
18. C3×#3748C22semidirect
22. #64160C2×C3semidirect
23. #64145C2×C3semidirect
26. #63664C2×C3semidirect
27. #63660C2×C3semidirect
61. C2×#634C22×C3semidirect
66. C4×C3⋊Dic3C22×C3semidirect
67. #3748C22×C3semidirect
193. C33#455semidirect
281. C32#5242semidirect
282. C32C3×#455semidirect
319. C3C3×#5242semidirect
320. C3#10592941direct (N × H)

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
C3×#641602
C3×#641452
C3×#636642
C3×#636602
#105929413
C3×#52423