Finite Group Databasebeta

← C3×#10593363 #13166704

Subgroups — C3×#10593363 #13166704

Order 1728.

776 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3802C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic, complemented
3C3×#63874C211yesmaximal, characteristic, complemented
4C3×#63739C214yesmaximal, complemented
5#10593363C311yesmaximal, characteristic, direct product factor
6C22×C3×D32—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12—21no—
9C2×C3×#625C2212yes—
10C3×#3802—22no—
11C3×#3802C414yescomplemented
12C2×C3×#622C2212yescomplemented
13C3×#3788—22no—
14C32×C4×Dic3—21no—
15C3×#3724C2212yescomplemented
16C3×#3722—21no—
17C2×C3×D3×D4—61no—
18C2×#3802C2×C311yescharacteristic
19C2×#3788C2×C311yescharacteristic, complemented
20C3×#872—61no—
21#63874C2×C311yescharacteristic, complemented
22#63739C2×C314yescomplemented
23C22×C3×C3⋊D3D411yescharacteristic
24C22×C32×D3—22no—
25C2×C3×D32—24no—
26C2×C3×D32—44no—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3×C4×C3⋊D3C2×C412yes—
30C3×C4×C3⋊D3D412yes—
31C2×C32×Dic3—22no—
32C32×D12—24no—
33C3×#625—22no—
34C3×#625C2×C414yes—
35C3×#622—22no—
36C3×#418—91nomaximal
37C23×C3×D3—62no—
38C22×D32—21no—
39C2×C32×D4—61no—
40C2×C4×C3⋊D3C22×C311yescharacteristic
41C2×C3×#67—62no—
42C3×D3×D4—64no—
43C3×D3×D4—64no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C2×C3×D12—41no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—61no—
49C2×#625C22×C312yes—
50#3802—22no—
51#3802C3×C414yescomplemented
52C2×#622C22×C312yescomplemented
53#3788—22no—
54C32×C42—61no—
55C3×(C2×D3)⋊C4—62no—
56C3×Dic3⋊C4—62no—
57C3×C4×Dic3—21no—
58C3×C4×Dic3—21no—
59C3×C4×Dic3—41no—
60#3724C22×C312yescomplemented
61#3722—21no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—22no—
64C2×C3×C3⋊D3C22⋊C412yes—
65C2×C32×D3—24no—
66C2×C32×D3—44no—
67C3×D32—48no—
68C33×C4C22⋊C412yes—
69C3×C3⋊Dic3C22×C411yescharacteristic
70C3×C3⋊Dic3C2×D411yescharacteristic
71C32×Dic3—42no—
72C22×C3×D4—91no—
73C2×D3×D4—61no—
74C2×C3×M16—91no—
75C3×#117—91no—
76#872—61no—
77C3×#102—94no—
78C23×C32—62no—
79C22×C3⋊D3C3×D411yescharacteristic
80C22×C3×D3—22no—
81C22×C3×D3—22no—
82C22×C3×D3—42no—
83C22×C3×D3—61no—
84C22×C3×D3—61no—
85C22×C3×D3—68no—
86C22×C3×D3—128no—
87C2×D32—24no—
88C2×D32—44no—
89C32×D4—64no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4—62no—
95C2×C32×C4C3×D411yescharacteristic
96C2×C3⋊Dic3C23×C311yescharacteristic
97C4×C3⋊D3C2×C3×C412yes—
98C4×C3⋊D3C3×D412yes—
99C3×#67—64no—
100C3×#67—64no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—42no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×D12—24no—
107C3×D12—24no—
108C3×D12—44no—
109C3×C4×D3—62no—
110C3×C4×D3—62no—
111C3×C4×D3—62no—
112C3×C4×D3—62no—
113#625—22no—
114#625C2×C3×C414yes—
115#622—22no—
116#418—91noSylow
117C2×C33C2×C22⋊C411yescharacteristic
118C2×C33AGL(1,ℤ/8)12yes—
119C3×C3⋊D3—22no—
120C32×D3—44no—
121C24×C3—181no—
122C23×D3—62no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—94no—
126C2×C3×D4—92no—
127C2×C3×D4—121no—
128C2×C3×D4—182no—
129C22×C3×C4—91no—
130C2×#67—62no—
131D3×D4—64no—
132D3×D4—64no—
133C2×D12—21no—
134C2×C4×D3—61no—
135C2×C4×D3—61no—
136C3×M16—182no—
137C2×C3×C8—92no—
138C3×C4⋊C4—92no—
139C3×C22⋊C4—181no—
140C3×C42—61no—
141C3×C42—61no—
142C3×C42—121no—
143(C2×D3)⋊C4—62no—
144Dic3⋊C4—62no—
145C4×Dic3—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—64no—
151C22×C32—124no—
152C22×C32C2×C3×D411yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3C3×C22⋊C412yes—
155C2×C3×D3—24no—
156C2×C3×D3—24no—
157C2×C3×D3—44no—
158C2×C3×D3—44no—
159C2×C3×D3—44no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—64no—
163C2×C3×D3—84no—
164C2×C3×D3—122no—
165C2×C3×D3—1216no—
166D32—48no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—42no—
170C32×C4—42no—
171C32×C4—122no—
172C32×C4C3×C22⋊C412yes—
173C3⋊Dic3C22×C3×C411yescharacteristic
174C3⋊Dic3C2×C3×D411yescharacteristic
175C3×Dic3—42no—
176C3×Dic3—44no—
177C3×Dic3—42no—
178C3×Dic3—61no—
179C3×Dic3—61no—
180C3×Dic3—61no—
181C3×Dic3—61no—
182C22×D4—91no—
183C2×M16—91no—
184#117—91no—
185#102—94no—
186C33#41811yescharacteristic, complemented, Sylow
187C23×C3—62no—
188C23×C3—62no—
189C23×C3—91no—
190C23×C3—122no—
191C23×C3—184no—
192C23×C3—364no—
193C22×D3—22no—
194C22×D3—61no—
195C22×D3—61no—
196C22×D3—68no—
197C22×D3—128no—
198C3×D4—64no—
199C3×D4—64no—
200C3×D4—94no—
201C3×D4—124no—
202C3×D4—182no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—91no—
210C2×C3×C4—92no—
211C2×C3×C4—92no—
212C2×C3×C4—122no—
213C2×C3×C4GO+(4,𝔽2)11yescharacteristic
214#67—64no—
215#67—64no—
216C2×Dic3—22no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219D12—24no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—62no—
224C3×C8—182no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32—42no—
233C2×C32—124no—
234C2×C32C2×C3×C22⋊C411yescharacteristic
235C2×C32C3×AGL(1,ℤ/8)12yes—
236C3⋊D3—22no—
237C3×D3—44no—
238C3×D3—44no—
239C3×D3—64no—
240C3×D3—84no—
241C3×D3—122no—
242C24—181no—
243C2×D4—61no—
244C2×D4—94no—
245C2×D4—92no—
246C2×D4—182no—
247C22×C4—91no—
248M16—182no—
249C2×C8—92no—
250C4⋊C4—92no—
251C22⋊C4—181no—
252C42—61no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—64no—
258C22×C3—64no—
259C22×C3—92no—
260C22×C3—124no—
261C22×C3—124no—
262C22×C3—124no—
263C22×C3—182no—
264C22×C3—244no—
265C22×C3—368no—
266C22×C3C2×GO+(4,𝔽2)11yescharacteristic
267C2×D3—24no—
268C2×D3—44no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—64no—
272C2×D3—122no—
273C2×D3—1216no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—91no—
279C3×C4—91no—
280C3×C4—122no—
281C3×C4—124no—
282C3×C4—122no—
283C3×C4D32⋊C412yes—
284Dic3—42no—
285Dic3—61no—
286Dic3—61no—
287Dic3—61no—
288Dic3—61no—
289C32—21no—
290C32—21no—
291C32—41no—
292C32—41no—
293C32C3×#41811yescharacteristic, complemented
294C23—62no—
295C23—91no—
296C23—184no—
297C23—364no—
298D4—64no—
299D4—94no—
300D4—182no—
301C2×C4—62no—
302C2×C4—91no—
303C2×C4—92no—
304C2×C4—92no—
305C2×C4C3×GO+(4,𝔽2)11yescharacteristic
306C8—182no—
307C2×C3—22no—
308C2×C3—22no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—42no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—124no—
316C2×C3—124no—
317C2×C3—182no—
318C2×C3—244no—
319C2×C3#6422412yes—
320C2×C3C2×D32⋊C411yescharacteristic
321D3—44no—
322D3—64no—
323D3—122no—
324C22—64no—
325C22—92no—
326C22—124no—
327C22—182no—
328C22—368no—
329C22C2×C3×GO+(4,𝔽2)11yescharacteristic
330C4—91no—
331C4—91no—
332C4—122no—
333C4C3×D32⋊C412yes—
334C3—21no—
335C3—21no—
336C3—41no—
337C3—41no—
338C3#1059336311yescharacteristic, direct product factor
339C2—124no—
340C2—182no—
341C2C3×#6422412yes—
342C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#418
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3788C2semidirect
3. C3×#63874C2semidirect
4. C3×#63739C2semidirect
5. #10593363C3direct (N × H)
11. C3×#3802C4semidirect
12. C2×C3×#622C22semidirect
15. C3×#3724C22semidirect
19. C2×#3788C2×C3semidirect
21. #63874C2×C3semidirect
22. #63739C2×C3semidirect
51. #3802C3×C4semidirect
52. C2×#622C22×C3semidirect
60. #3724C22×C3semidirect
186. C33#418semidirect
293. C32C3×#418semidirect
338. C3#10593363direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×C3×#37882
C3×#638742
C3×#637392
#105933633
C3×#4189