Finite Group Databasebeta

← C3×#10593288 #13166657

Subgroups — C3×#10593288 #13166657

Order 1728.

404 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 66 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×D32⋊Q8C212yesmaximal
2C3×#63802C211yesmaximal, characteristic, complemented
3C3×D32×C8C211yesmaximal, characteristic
4C3×#63769C211yesmaximal, characteristic, complemented
5C3×#63730C212yesmaximal, complemented
6#10593288C311yesmaximal, characteristic, direct product factor
7C3×C4×D32C2211yescharacteristic
8C3×#3797C2212yes—
9C3×C4⋊#157C2212yes—
10C3×#3773—22no—
11C3×D32⋊C4—22no—
12C3×C8×C3⋊D3C2211yescharacteristic
13C32×Dic12—21no—
14C32×D3×C8—21no—
15C3×#3718—21no—
16C3×#3710C2211yescharacteristic
17C3×D3×C3⋊C8—21no—
18D32⋊Q8C2×C312yes—
19C3×D3×Q16—61no—
20C2×C3×D3×C8—61no—
21#63802C2×C311yescharacteristic, complemented
22D32×C8C2×C311yescharacteristic
23#63769C2×C311yescharacteristic, complemented
24#63730C2×C312yescomplemented
25C2×C3×D32D411yescharacteristic
26C2×C3×#157—22no—
27C3×C4×C3⋊D3C2311yescharacteristic
28C32×C4×D3—21no—
29C32×Dic6—22no—
30C3×#627—22no—
31C3×#624—22no—
32C3×#624D411yescharacteristic
33C3×D3×Dic3—21no—
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)D411yescharacteristic
36C32×C3⋊C8—21no—
37C3×#467—91nomaximal
38C3×D3×Q8—62no—
39C2×C3×C4×D3—61no—
40C4×D32C22×C311yescharacteristic
41#3797C22×C312yes—
42C4⋊#157C22×C312yes—
43#3773—22no—
44D32⋊C4—22no—
45C32×Q16—61no—
46C2×C32×C8—61no—
47C8×C3⋊D3C22×C311yescharacteristic
48C3×#216—62no—
49C2×C3×C3⋊C8—61no—
50C3×Dic12—21no—
51C3×Dic12—21no—
52C3×Dic12—41no—
53C3×D3×C8—21no—
54C3×D3×C8—21no—
55C3×D3×C8—41no—
56C3×D3×C8—61no—
57C3×D3×C8—61no—
58C3×D3×C8—61no—
59C3×D3×C8—61no—
60#3718—21no—
61#3710C22×C311yescharacteristic
62D3×C3⋊C8—21no—
63C2×C3×C3⋊D3C2×D411yescharacteristic
64C2×C32×D3—21no—
65C3×D32—21no—
66C3×D32Q1612yescomplemented
67C3×#157—42no—
68C33×C4C2×D411yescharacteristic
69C3×C3⋊Dic3#4111yescharacteristic
70C32×Dic3—21no—
71C32×Dic3—42no—
72C2×C3×Q16—91no—
73C22×C3×C8—91no—
74C3×C22⋊Q8—92no—
75D3×Q16—61no—
76C2×D3×C8—61no—
77C3×#107—91no—
78C3×#103—92no—
79C22×C3×D3—61no—
80C2×D32C3×D411yescharacteristic
81C2×#157—22no—
82C32×Q8—62no—
83C2×C32×C4—61no—
84C4×C3⋊D3C23×C311yescharacteristic
85C2×C3×Dic3—61no—
86C3×C4×D3—21no—
87C3×C4×D3—21no—
88C3×C4×D3—41no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C3×C4×D3—122no—
94C3×Dic6—22no—
95C3×Dic6—22no—
96C3×Dic6—42no—
97C3×Dic6—122no—
98#627—22no—
99#624—22no—
100#624C3×D411yescharacteristic
101D3×Dic3—21no—
102C32×C8—21no—
103C32×C8—21no—
104C32×C8—41no—
105C32×C8—41no—
106C32×C8—61no—
107C32×C8C3×D411yescharacteristic
108C3⋊(C3⋊C8)C3×D411yescharacteristic
109C3×C3⋊C8—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8—61no—
113C3×C3⋊C8—61no—
114#467—91noSylow
115C2×C33#12111yescharacteristic
116C3×C3⋊D3C2×Q1611yescharacteristic
117C3×C3⋊D3#13511yescharacteristic
118C32×D3—22no—
119C2×C3×Q8—92no—
120C22×C3×C4—91no—
121D3×Q8—62no—
122C2×C4×D3—61no—
123C3×Q16—61no—
124C3×Q16—61no—
125C3×Q16—121no—
126C3×Q16—181no—
127C2×C3×C8—61no—
128C2×C3×C8—61no—
129C2×C3×C8—91no—
130C2×C3×C8—91no—
131C2×C3×C8—121no—
132C2×C3×C8—181no—
133C3×C4⋊C4—92no—
134C3×C4⋊C4—182no—
135C3×C22⋊C4—182no—
136#216—62no—
137C2×C3⋊C8—61no—
138Dic12—21no—
139D3×C8—21no—
140D3×C8—61no—
141D3×C8—61no—
142D3×C8—61no—
143D3×C8—61no—
144C22×C32—61no—
145C2×C3⋊D3C2×C3×D411yescharacteristic
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—41no—
149C2×C3×D3—62no—
150C2×C3×D3—61no—
151C2×C3×D3—61no—
152C2×C3×D3—62no—
153D32—21no—
154D32C3×Q1612yescomplemented
155#157—42no—
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—41no—
159C32×C4—41no—
160C32×C4—61no—
161C32×C4—122no—
162C32×C4C2×C3×D411yescharacteristic
163C3⋊Dic3C3×#4111yescharacteristic
164C3×Dic3—21no—
165C3×Dic3—21no—
166C3×Dic3—41no—
167C3×Dic3—42no—
168C3×Dic3—42no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—82no—
172C2×Q16—91no—
173C22×C8—91no—
174C22⋊Q8—92no—
175#107—91no—
176#103—92no—
177C33#46711yescharacteristic, complemented, Sylow
178C23×C3—91no—
179C22×D3—61no—
180C3×Q8—62no—
181C3×Q8—62no—
182C3×Q8—122no—
183C3×Q8—182no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—91no—
187C2×C3×C4—91no—
188C2×C3×C4—121no—
189C2×C3×C4—181no—
190C2×C3×C4—182no—
191C2×C3×C4—182no—
192C2×Dic3—61no—
193C4×D3—21no—
194C4×D3—61no—
195C4×D3—61no—
196C4×D3—61no—
197C4×D3—61no—
198C4×D3—122no—
199Dic6—22no—
200Dic6—122no—
201C3×C8—21no—
202C3×C8—21no—
203C3×C8—41no—
204C3×C8—41no—
205C3×C8—61no—
206C3×C8—61no—
207C3×C8—91no—
208C3×C8—121no—
209C3×C8GO+(4,𝔽2)11yescharacteristic
210C3⋊C8—21no—
211C3⋊C8—61no—
212C3⋊C8—61no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—62no—
218C2×C32C3×#12111yescharacteristic
219C3⋊D3C2×C3×Q1611yescharacteristic
220C3⋊D3C3×#13511yescharacteristic
221C3×D3—22no—
222C3×D3—22no—
223C3×D3—42no—
224C3×D3—61no—
225C3×D3—61no—
226C3×D3—61no—
227C3×D3—61no—
228C2×Q8—92no—
229C22×C4—91no—
230Q16—61no—
231Q16—181no—
232C2×C8—61no—
233C2×C8—91no—
234C2×C8—91no—
235C2×C8—181no—
236C4⋊C4—92no—
237C4⋊C4—182no—
238C22⋊C4—182no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—92no—
242C22×C3—91no—
243C22×C3—121no—
244C22×C3—181no—
245C2×D3—21no—
246C2×D3—62no—
247C2×D3—61no—
248C2×D3—61no—
249C2×D3—62no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—91no—
257C3×C4—121no—
258C3×C4—122no—
259C3×C4—122no—
260C3×C4—242no—
261C3×C4—362no—
262C3×C4C2×GO+(4,𝔽2)11yescharacteristic
263Dic3—21no—
264Dic3—42no—
265Dic3—61no—
266Dic3—61no—
267C32—21no—
268C32—21no—
269C32—41no—
270C32—41no—
271C32C3×#46711yescharacteristic, complemented
272C23—91no—
273Q8—62no—
274Q8—182no—
275C2×C4—61no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—181no—
279C2×C4—182no—
280C2×C4—182no—
281C8—61no—
282C8—91no—
283C8C3×GO+(4,𝔽2)11yescharacteristic
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—62no—
289C2×C3—62no—
290C2×C3—91no—
291C2×C3—91no—
292C2×C3—122no—
293C2×C3#6423011yescharacteristic
294D3—22no—
295D3—61no—
296D3—61no—
297D3—61no—
298D3—61no—
299C22—61no—
300C22—92no—
301C22—91no—
302C22—181no—
303C4—61no—
304C4—91no—
305C4—122no—
306C4—362no—
307C4C2×C3×GO+(4,𝔽2)11yescharacteristic
308C3—21no—
309C3—21no—
310C3—41no—
311C3—41no—
312C3#1059328811yescharacteristic, direct product factor
313C2—62no—
314C2—91no—
315C2—91no—
316C2C3×#6423011yescharacteristic
subgroupG/H
centerC2×C3#64230
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#121
prime radicalC2×C3×D32D4
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#467
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63802C2semidirect
4. C3×#63769C2semidirect
5. C3×#63730C2semidirect
6. #10593288C3direct (N × H)
21. #63802C2×C3semidirect
23. #63769C2×C3semidirect
24. #63730C2×C3semidirect
66. C3×D32Q16semidirect
154. D32C3×Q16semidirect
177. C33#467semidirect
271. C32C3×#467semidirect
312. C3#10593288direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D32⋊Q82
C3×#638022
C3×D32×C82
C3×#637692
C3×#637302
#105932883
C3×#4679