Finite Group Databasebeta

← C3×#10593371 #13166814

Subgroups — C3×#10593371 #13166814

Order 1728.

590 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3797C211yesmaximal, characteristic
2C2×C3×#3776C212yesmaximal, complemented
3C3×#63865C211yesmaximal, characteristic
4C3×#63778C211yesmaximal, characteristic
5C3×#63740C212yesmaximal, complemented
6#10593371C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6—21no—
9C2×C3×#627C2211yescharacteristic
10C2×C3×#627C2212yes—
11C2×C3×#624—21no—
12C3×#3797—24no—
13C2×C3×#622C2212yes—
14C3×#3776—24no—
15C32×Dic3⋊C4—21no—
16C3×#3724C2211yescharacteristic
17C3×#3722—21no—
18C2×C3×D3×Q8—61no—
19C2×#3797C2×C311yescharacteristic
20C2×#3776C2×C312yescomplemented
21C3×#896—61no—
22#63865C2×C311yescharacteristic
23#63778C2×C311yescharacteristic
24#63740C2×C312yescomplemented
25C22×C3×C3⋊D3D411yescharacteristic
26C2×C33×C4D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4×C3⋊D3—22no—
29C2×C32×Dic3—22no—
30C2×C32×Dic3—21no—
31C2×C32×Dic3—21no—
32C32×Dic6—24no—
33C3×#627—22no—
34C3×#627—22no—
35C3×#627D414yes—
36C3×#624—24no—
37C3×#622—22no—
38C3×#451—91nomaximal
39C2×C32×Q8—61no—
40C2×C4×C3⋊D3C22×C311yescharacteristic
41C3×D3×Q8—68no—
42C2×C3×C4×D3—62no—
43C2×C3×C4×D3—61no—
44C2×C3×C4×D3—61no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6—21no—
47C2×C3×Dic6—41no—
48C2×C3×Dic6—62no—
49C2×C3×Dic6—61no—
50C2×#627C22×C311yescharacteristic
51C2×#627C22×C312yes—
52C2×#624—21no—
53#3797—24no—
54C2×#622C22×C312yes—
55#3776—24no—
56C32×C4⋊C4—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—61no—
59C3×Dic3⋊C4—21no—
60C3×Dic3⋊C4—21no—
61C3×Dic3⋊C4—41no—
62C3×Dic3⋊C4—61no—
63#3724C22×C311yescharacteristic
64#3722—21no—
65C22×C33C2×D411yescharacteristic
66C2×C3×C3⋊D3—21no—
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3Q1612yescomplemented
69C33×C4—21no—
70C3×C3⋊Dic3C2×D412yes—
71C32×Dic3—24no—
72C32×Dic3—41no—
73C32×Dic3—41no—
74C22×C3×Q8—91no—
75C2×D3×Q8—61no—
76C2×C3×Q16—92no—
77C3×C22⋊Q8—91no—
78#896—61no—
79C3×#103—92no—
80C3×C22⋊C8—91no—
81C22×C3⋊D3C3×D411yescharacteristic
82C22×C3×D3—61no—
83C22×C3×D3—61no—
84C32×Q8—64no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—41no—
88C2×C32×C4—41no—
89C2×C32×C4—62no—
90C2×C32×C4—61no—
91C2×C32×C4—61no—
92C2×C32×C4C3×D411yescharacteristic
93C2×C3⋊Dic3C23×C311yescharacteristic
94C4×C3⋊D3—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—42no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×C4×D3—68no—
107C3×C4×D3—64no—
108C3×C4×D3—122no—
109C3×Dic6—24no—
110C3×Dic6—24no—
111C3×Dic6—44no—
112C3×Dic6—64no—
113C3×Dic6—64no—
114C3×Dic6—122no—
115#627—22no—
116#627—22no—
117#627C3×D414yes—
118#624—24no—
119#622—22no—
120#451—91noSylow
121C2×C33#12011yescharacteristic
122C2×C33C2×Q1611yescharacteristic
123C2×C33#13711yescharacteristic
124C3×C3⋊D3—22no—
125C2×C3×Q8—61no—
126C2×C3×Q8—61no—
127C2×C3×Q8—92no—
128C2×C3×Q8—91no—
129C2×C3×Q8—121no—
130C2×C3×Q8—184no—
131C22×C3×C4—91no—
132C22×C3×C4—181no—
133D3×Q8—68no—
134C2×C4×D3—61no—
135C2×C4×D3—61no—
136C2×C4×D3—62no—
137C2×Dic6—21no—
138C2×Dic6—62no—
139C2×Dic6—61no—
140C3×Q16—184no—
141C2×C3×C8—92no—
142C3×C4⋊C4—61no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—121no—
146C3×C22⋊C4—181no—
147(C2×D3)⋊C4—61no—
148(C2×D3)⋊C4—61no—
149Dic3⋊C4—21no—
150Dic3⋊C4—61no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32C2×C3×D411yescharacteristic
156C2×C3⋊D3—21no—
157C2×C3⋊D3—21no—
158C2×C3⋊D3C3×Q1612yescomplemented
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—121no—
164C2×C3×D3—121no—
165C32×C4—22no—
166C32×C4—21no—
167C32×C4—41no—
168C32×C4—42no—
169C32×C4—64no—
170C32×C4—81no—
171C32×C4—121no—
172C32×C4—121no—
173C3⋊Dic3C2×C3×D412yes—
174C3×Dic3—24no—
175C3×Dic3—24no—
176C3×Dic3—44no—
177C3×Dic3—41no—
178C3×Dic3—41no—
179C3×Dic3—42no—
180C3×Dic3—41no—
181C3×Dic3—41no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—81no—
185C22×Q8—91no—
186C2×Q16—92no—
187C22⋊Q8—91no—
188#103—92no—
189C22⋊C8—91no—
190C33#45111yescharacteristic, complemented, Sylow
191C23×C3—91no—
192C22×D3—61no—
193C22×D3—61no—
194C3×Q8—64no—
195C3×Q8—64no—
196C3×Q8—94no—
197C3×Q8—124no—
198C3×Q8—182no—
199C3×Q8—182no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—91no—
211C2×C3×C4—121no—
212C2×C3×C4—122no—
213C2×C3×C4—121no—
214C2×C3×C4—182no—
215C2×C3×C4—184no—
216C2×C3×C4GO+(4,𝔽2)11yescharacteristic
217C2×Dic3—22no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C4×D3—68no—
223C4×D3—64no—
224C4×D3—122no—
225Dic6—24no—
226Dic6—64no—
227Dic6—64no—
228Dic6—122no—
229C3×C8—182no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32C3×#12011yescharacteristic
243C2×C32C2×C3×Q1611yescharacteristic
244C2×C32C3×#13711yescharacteristic
245C3⋊D3—22no—
246C3×D3—64no—
247C3×D3—122no—
248C2×Q8—61no—
249C2×Q8—92no—
250C2×Q8—91no—
251C2×Q8—184no—
252C22×C4—91no—
253C22×C4—181no—
254Q16—184no—
255C2×C8—92no—
256C4⋊C4—61no—
257C4⋊C4—91no—
258C22⋊C4—181no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—92no—
264C22×C3—181no—
265C22×C3—181no—
266C22×C3C2×GO+(4,𝔽2)11yescharacteristic
267C2×D3—62no—
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—121no—
272C2×D3—121no—
273C3×C4—22no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—42no—
277C3×C4—64no—
278C3×C4—64no—
279C3×C4—81no—
280C3×C4—92no—
281C3×C4—124no—
282C3×C4—121no—
283C3×C4—121no—
284C3×C4—122no—
285C3×C4—121no—
286C3×C4—121no—
287C3×C4—241no—
288Dic3—24no—
289Dic3—41no—
290Dic3—41no—
291Dic3—62no—
292Dic3—62no—
293C32—21no—
294C32—21no—
295C32—41no—
296C32—41no—
297C32C3×#45111yescharacteristic, complemented
298C23—91no—
299Q8—64no—
300Q8—94no—
301Q8—182no—
302Q8—182no—
303C2×C4—62no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—182no—
308C2×C4—184no—
309C2×C4C3×GO+(4,𝔽2)11yescharacteristic
310C8—182no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—182no—
324C2×C3#6422611yescharacteristic
325C2×C3#6422811yescharacteristic
326C2×C3#6424111yescharacteristic
327D3—64no—
328D3—122no—
329C22—92no—
330C22—181no—
331C22—181no—
332C22C2×C3×GO+(4,𝔽2)11yescharacteristic
333C4—21no—
334C4—64no—
335C4—92no—
336C4—121no—
337C4—121no—
338C3—21no—
339C3—21no—
340C3—41no—
341C3—41no—
342C3#1059337111yescharacteristic, direct product factor
343C2—182no—
344C2C3×#6422611yescharacteristic
345C2C3×#6422811yescharacteristic
346C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#451
LCSG ▷ C2×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×#3776C2semidirect
5. C3×#63740C2semidirect
6. #10593371C3direct (N × H)
20. C2×#3776C2×C3semidirect
24. #63740C2×C3semidirect
68. C2×C3×C3⋊D3Q16semidirect
158. C2×C3⋊D3C3×Q16semidirect
190. C33#451semidirect
297. C32C3×#451semidirect
342. C3#10593371direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×C3×#37762
C3×#638652
C3×#637782
C3×#637402
#105933713
C3×#4519