Finite Group Databasebeta

← C3×#10590913 #13157879

Subgroups — C3×#10590913 #13157879

Order 1728.

582 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 148 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×C8×C3⋊D3C211yesmaximal, characteristic
2C3×#63818C212yesmaximal
3C32×#819C212yesmaximal, complemented
4C3×#63578C212yesmaximal, complemented
5C3×D3×#108—32nomaximal
6#10590913C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C33×C8C2211yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10C3×C8×C3⋊D3C412yes—
11C3×C8×C3⋊D3C412yes—
12C32×#208C2214yes—
13C3×#3712—22no—
14C3×#3710—22no—
15C2×C8×C3⋊D3C2×C311yescharacteristic
16C3×D3×M16—34no—
17C2×C3×D3×C8—32no—
18C2×C3×D3×C8—61no—
19#63818C2×C312yes—
20C32×#108—32no—
21C3×#822—34no—
22C3×#819—22no—
23C3×#819D312yescomplemented
24C3×#819C2×C312yescomplemented
25#63578C2×C312yescomplemented
26C22×C3×C3⋊D3Q811yescharacteristic
27C2×C33×C4C2311yescharacteristic
28C2×C3×C3⋊Dic3Q811yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3D411yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C33×C8C2×C412yes—
34C3×C3⋊(C3⋊C8)C2×C412yes—
35C32×C3⋊C8—24no—
36C2×C3×#108—91no—
37D3×#108—32no—
38C2×C4×C3⋊D3C22×C311yescharacteristic
39C2×C3×C4×D3—32no—
40C2×C3×C4×D3—61no—
41C32×M16—34no—
42C2×C32×C8—21no—
43C2×C32×C8—22no—
44C2×C32×C8—41no—
45C2×C32×C8C2×D312yes—
46C2×C32×C8C22×C311yescharacteristic
47C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
48C8×C3⋊D3C3×C412yes—
49C8×C3⋊D3C3×C412yes—
50C3×#208—24no—
51C3×#208C2×D314yes—
52C3×#208C22×C314yes—
53C2×C3×C3⋊C8—32no—
54C2×C3×C3⋊C8—61no—
55C3×ΓL(1,𝔽25)—64no—
56C3×D3×C8—34no—
57C3×D3×C8—34no—
58C3×D3×C8—64no—
59C3×D3×C8—62no—
60C3×D3×C8—62no—
61#3712—22no—
62#3710—22no—
63C22×C33C2×Q811yescharacteristic
64C2×C3×C3⋊D3—22no—
65C2×C3×C3⋊D3C4⋊C411yescharacteristic
66C2×C3×C3⋊D3C4⋊C411yescharacteristic
67C33×C4C22×C411yescharacteristic
68C33×C4C2×D411yescharacteristic
69C3×C3⋊Dic3C4⋊C411yescharacteristic
70C3×C3⋊Dic3C4⋊C411yescharacteristic
71C2×C3×M16—92no—
72C22×C3×C8—91no—
73D3×M16—34no—
74C2×D3×C8—32no—
75C2×D3×C8—61no—
76C3×#108—32no—
77C3×#108—32no—
78C3×#108—62no—
79C3×#108—92no—
80#822—34no—
81#819C3×D312yescomplemented
82C22×C3⋊D3C3×Q811yescharacteristic
83C22×C3×D3—32no—
84C22×C3×D3—61no—
85C2×C32×C4—21no—
86C2×C32×C4—22no—
87C2×C32×C4—41no—
88C2×C32×C4C22×D312yes—
89C2×C32×C4C23×C311yescharacteristic
90C2×C3⋊Dic3C3×Q811yescharacteristic
91C4×C3⋊D3C2×C3×C411yescharacteristic
92C4×C3⋊D3C2×C3×C411yescharacteristic
93C4×C3⋊D3C3×D411yescharacteristic
94C4×C3⋊D3C3×D411yescharacteristic
95C2×C3×Dic3—32no—
96C2×C3×Dic3—61no—
97C3×C4×D3—32no—
98C3×C4×D3—32no—
99C3×C4×D3—32no—
100C3×C4×D3—32no—
101C3×C4×D3—61no—
102C3×C4×D3—61no—
103C3×C4×D3—61no—
104C3×C4×D3—61no—
105C32×C8—22no—
106C32×C8—24no—
107C32×C8—42no—
108C32×C8—64no—
109C32×C8C4×D314yes—
110C32×C8C2×C3×C412yes—
111C3⋊(C3⋊C8)C2×C3×C412yes—
112C3×C3⋊C8—24no—
113C3×C3⋊C8—24no—
114C3×C3⋊C8—34no—
115C3×C3⋊C8—44no—
116C3×C3⋊C8—62no—
117C2×#108—91noSylow
118C2×C33—21no—
119C2×C33C2×C4⋊C411yescharacteristic
120C3×C3⋊D3—21no—
121C3×C3⋊D3#10812yescomplemented
122C22×C3×C4—91no—
123C2×C4×D3—32no—
124C2×C4×D3—61no—
125C3×M16—34no—
126C3×M16—34no—
127C3×M16—64no—
128C3×M16—182no—
129C2×C3×C8—22no—
130C2×C3×C8—21no—
131C2×C3×C8—41no—
132C2×C3×C8—91no—
133C2×C3×C8—92no—
134C2×C3×C8—92no—
135C2×C3×C8—182no—
136C2×C3×C8D3211yescharacteristic
137C2×C3×C8C2×C3×D312yes—
138#208C2×C3×D314yes—
139C2×C3⋊C8—32no—
140C2×C3⋊C8—61no—
141ΓL(1,𝔽25)—64no—
142D3×C8—34no—
143D3×C8—34no—
144D3×C8—62no—
145D3×C8—62no—
146D3×C8—64no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—41no—
150C22×C32D3×Q812yes—
151C22×C32C2×C3×Q811yescharacteristic
152C2×C3⋊D3—22no—
153C2×C3⋊D3C3×C4⋊C411yescharacteristic
154C2×C3⋊D3C3×C4⋊C411yescharacteristic
155C2×C3×D3—32no—
156C2×C3×D3—32no—
157C2×C3×D3—61no—
158C2×C3×D3—61no—
159C2×C3×D3—64no—
160C2×C3×D3—64no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—41no—
166C32×C4—41no—
167C32×C4C2×C4×D312yes—
168C32×C4D3×D412yes—
169C32×C4C22×C3×C411yescharacteristic
170C32×C4C2×C3×D411yescharacteristic
171C3⋊Dic3C3×C4⋊C411yescharacteristic
172C3⋊Dic3C3×C4⋊C411yescharacteristic
173C3×Dic3—32no—
174C3×Dic3—32no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C2×M16—92no—
178C22×C8—91no—
179#108—32no—
180#108—92no—
181C33C2×#10811yescharacteristic, complemented, Sylow
182C23×C3—91no—
183C22×D3—32no—
184C22×D3—61no—
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—41no—
188C2×C3×C4—91no—
189C2×C3×C4—91no—
190C2×C3×C4—91no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4C2×D3211yescharacteristic
194C2×C3×C4C22×C3×D312yes—
195C2×Dic3—32no—
196C2×Dic3—61no—
197C4×D3—32no—
198C4×D3—32no—
199C4×D3—32no—
200C4×D3—32no—
201C4×D3—61no—
202C4×D3—61no—
203C4×D3—61no—
204C4×D3—61no—
205C3×C8—24no—
206C3×C8—22no—
207C3×C8—42no—
208C3×C8—64no—
209C3×C8—64no—
210C3×C8—92no—
211C3×C8—124no—
212C3×C8#62412yes—
213C3×C8C3×C4×D314yes—
214C3⋊C8—24no—
215C3⋊C8—34no—
216C3⋊C8—62no—
217C2×C32—21no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32—42no—
224C2×C32—42no—
225C2×C32D3×C4⋊C412yes—
226C2×C32C2×C3×C4⋊C411yescharacteristic
227C3⋊D3—21no—
228C3⋊D3C3×#10812yescomplemented
229C3×D3—34no—
230C3×D3—62no—
231C3×D3—62no—
232C3×D3—62no—
233C22×C4—91no—
234M16—34no—
235M16—182no—
236C2×C8—91no—
237C2×C8—92no—
238C2×C8—92no—
239C2×C8—182no—
240C2×C8C3×D3211yescharacteristic
241C22×C3—21no—
242C22×C3—22no—
243C22×C3—41no—
244C22×C3—91no—
245C22×C3—91no—
246C22×C3—182no—
247C22×C3#379711yescharacteristic
248C22×C3C3×D3×Q812yes—
249C2×D3—32no—
250C2×D3—32no—
251C2×D3—61no—
252C2×D3—61no—
253C2×D3—64no—
254C2×D3—64no—
255C3×C4—22no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—91no—
262C3×C4—91no—
263C3×C4#380211yescharacteristic
264C3×C4C2×#62411yescharacteristic
265C3×C4C2×C3×C4×D312yes—
266C3×C4C3×D3×D412yes—
267Dic3—32no—
268Dic3—32no—
269Dic3—61no—
270Dic3—61no—
271C32—21no—
272C32—22no—
273C32—41no—
274C32D3×#10812yescomplemented
275C32C2×C3×#10811yescharacteristic, complemented
276C23—91no—
277C2×C4—91no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4C2×C3×D3211yescharacteristic
283C8—64no—
284C8—92no—
285C8C3×#62412yes—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—42no—
292C2×C3—42no—
293C2×C3—41no—
294C2×C3—92no—
295C2×C3—181no—
296C2×C3#6390011yescharacteristic
297C2×C3C3×D3×C4⋊C412yes—
298D3—34no—
299D3—62no—
300D3—62no—
301D3—62no—
302C22—91no—
303C22—91no—
304C22—182no—
305C22C3×#379711yescharacteristic
306C4—91no—
307C4—91no—
308C4C3×#380211yescharacteristic
309C4C2×C3×#62411yescharacteristic
310C3—21no—
311C3—22no—
312C3—41no—
313C3#1059091311yescharacteristic, direct product factor
314C3C3×D3×#10812yescomplemented
315C2—21no—
316C2—92no—
317C2—181no—
318C2C3×#6390011yescharacteristic
subgroupG/H
centerC3×C4#3802
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C2×C3×#108
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C32×#819C2semidirect
4. C3×#63578C2semidirect
6. #10590913C3direct (N × H)
23. C3×#819D3semidirect
24. C3×#819C2×C3semidirect
25. #63578C2×C3semidirect
81. #819C3×D3semidirect
121. C3×C3⋊D3#108semidirect
181. C33C2×#108semidirect
228. C3⋊D3C3×#108semidirect
274. C32D3×#108semidirect
275. C32C2×C3×#108semidirect
313. C3#10590913direct (N × H)
314. C3C3×D3×#108semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C3×#638182
C32×#8192
C3×#635782
C3×D3×#1083
#105909133