Finite Group Databasebeta

← C3×#10593443 #13166810

Subgroups — C3×#10593443 #13166810

Order 1728.

452 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3776C212yesmaximal, complemented
2C2×C3×#3775C212yesmaximal
3C3×#63919C211yesmaximal, characteristic, complemented
4C3×#63915C211yesmaximal, characteristic
5C3×#63777C211yesmaximal, characteristic, complemented
6#10593443C311yesmaximal, characteristic, direct product factor
7C2×C3×#627C2212yes—
8C2×C3×#627C2211yescharacteristic
9C2×C3×#625C2211yescharacteristic
10C2×C3×#622C2212yes—
11C3×#3776—24no—
12C3×#3775—24no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×(C2×D3)⋊C4—21no—
15C32×C4⋊Dic3—21no—
16C3×#3724—21no—
17C3×(C2×D3)⋊Dic3—21no—
18C2×#3776C2×C312yescomplemented
19C2×#3775C2×C312yes—
20C3×#888—61no—
21C3×#885—61no—
22#63919C2×C311yescharacteristic, complemented
23#63915C2×C311yescharacteristic
24#63777C2×C311yescharacteristic, complemented
25C22×C32×D3—21no—
26C2×C33×C4D411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3—21no—
30C2×C32×Dic3—22no—
31C3×#627—22no—
32C3×#627—22no—
33C3×#627—22no—
34C3×#625—22no—
35C3×#622D414yes—
36C3×#497—91nomaximal
37C2×C3×#67—61no—
38C2×C3×Dic6—61no—
39C2×C3×Dic6—62no—
40C2×#627C22×C311yescharacteristic
41C2×#627C22×C312yes—
42C2×#625C22×C311yescharacteristic
43C2×#622C22×C312yes—
44#3776—24no—
45#3775—24no—
46C32×C4⋊C4—61no—
47C32×C22⋊C4—61no—
48C4×C3⋊Dic3C22×C311yescharacteristic
49C3×#217—61no—
50C3×(C2×D3)⋊C4—21no—
51C3×(C2×D3)⋊C4—21no—
52C3×(C2×D3)⋊C4—41no—
53C3×(C2×D3)⋊C4—61no—
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—21no—
56C3×C4⋊Dic3—41no—
57C3×Dic3⋊C4—62no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60#3724—21no—
61(C2×D3)⋊Dic3—21no—
62C22×C33C2×D411yescharacteristic
63C2×C32×D3—41no—
64C2×C32×D3—41no—
65C2×C32×D3—41no—
66C33×C4—21no—
67C3×C3⋊Dic3—21no—
68C3×C3⋊Dic3C2×D412yes—
69C32×Dic3—42no—
70C32×Dic3—41no—
71C2×C3×Q16—92no—
72C2×C3×SD16—92no—
73C3×C4⋊Q8—91no—
74C3×#124—91no—
75#888—61no—
76#885—61no—
77C3×#97—91no—
78C23×C32—61no—
79C22×C3×D3—21no—
80C22×C3×D3—21no—
81C22×C3×D3—41no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4—62no—
87C2×C32×C4—61no—
88C2×C32×C4C3×D411yescharacteristic
89C2×C3⋊Dic3C3×D411yescharacteristic
90C2×C3⋊Dic3C23×C311yescharacteristic
91C3×#67—122no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—21no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—42no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C3×Dic6—122no—
103C3×Dic6—122no—
104C3×Dic6—122no—
105#627—22no—
106#627—22no—
107#627—22no—
108#625—22no—
109#622C3×D414yes—
110#497—91noSylow
111C2×C33#12711yescharacteristic
112C2×C33#13711yescharacteristic
113C2×C33#13711yescharacteristic
114C32×D3—81no—
115C2×C3×Q8—91no—
116C2×C3×Q8—92no—
117C2×C3×D4—91no—
118C2×#67—61no—
119C2×Dic6—61no—
120C2×Dic6—62no—
121C3×Q16—184no—
122C3×SD16—184no—
123C2×C3×C8—92no—
124C3×C4⋊C4—61no—
125C3×C4⋊C4—61no—
126C3×C4⋊C4—121no—
127C3×C4⋊C4—181no—
128C3×C22⋊C4—61no—
129C3×C22⋊C4—61no—
130C3×C22⋊C4—121no—
131C3×C22⋊C4—181no—
132C3×C42—91no—
133#217—61no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—61no—
136C4⋊Dic3—21no—
137Dic3⋊C4—62no—
138C4×Dic3—61no—
139C4×Dic3—61no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32—121no—
145C22×C32—121no—
146C22×C32—121no—
147C22×C32C2×C3×D411yescharacteristic
148C2×C3×D3—41no—
149C2×C3×D3—41no—
150C2×C3×D3—41no—
151C2×C3×D3—42no—
152C2×C3×D3—41no—
153C2×C3×D3—41no—
154C2×C3×D3—41no—
155C2×C3×D3—81no—
156C2×C3×D3—81no—
157C32×C4—22no—
158C32×C4—21no—
159C32×C4—41no—
160C32×C4—42no—
161C32×C4—81no—
162C32×C4—122no—
163C32×C4—121no—
164C3⋊Dic3—21no—
165C3⋊Dic3C2×C3×D412yes—
166C3×Dic3—42no—
167C3×Dic3—41no—
168C3×Dic3—42no—
169C3×Dic3—41no—
170C3×Dic3—62no—
171C3×Dic3—62no—
172C3×Dic3—62no—
173C3×Dic3—82no—
174C3×Dic3—81no—
175C3×Dic3—121no—
176C2×Q16—92no—
177C2×SD16—92no—
178C4⋊Q8—91no—
179#124—91no—
180#97—91no—
181C33#49711yescharacteristic, complemented, Sylow
182C23×C3—61no—
183C23×C3—61no—
184C23×C3—121no—
185C22×D3—21no—
186C3×Q8—182no—
187C3×Q8—182no—
188C3×Q8—182no—
189C3×D4—182no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—62no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—62no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—121no—
201C2×C3×C4—122no—
202C2×C3×C4GO+(4,𝔽2)11yescharacteristic
203#67—122no—
204C2×Dic3—22no—
205C2×Dic3—21no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210Dic6—122no—
211Dic6—122no—
212Dic6—122no—
213C3×C8—94no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—241no—
227C2×C32C3×#12711yescharacteristic
228C2×C32C3×#13711yescharacteristic
229C2×C32C3×#13711yescharacteristic
230C3×D3—81no—
231C3×D3—82no—
232C3×D3—81no—
233C2×Q8—91no—
234C2×Q8—92no—
235C2×D4—91no—
236Q16—184no—
237SD16—184no—
238C2×C8—92no—
239C4⋊C4—61no—
240C4⋊C4—181no—
241C22⋊C4—61no—
242C22⋊C4—181no—
243C42—91no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—121no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3—122no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3—121no—
255C22×C3—241no—
256C22×C3—241no—
257C22×C3C2×GO+(4,𝔽2)11yescharacteristic
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—41no—
264C3×C4—42no—
265C3×C4—81no—
266C3×C4—92no—
267C3×C4—122no—
268C3×C4—121no—
269C3×C4—121no—
270C3×C4—122no—
271C3×C4—181no—
272C3×C4—242no—
273C3×C4—241no—
274Dic3—42no—
275Dic3—41no—
276Dic3—62no—
277Dic3—62no—
278Dic3—62no—
279Dic3—121no—
280C32—21no—
281C32—21no—
282C32—41no—
283C32—41no—
284C32C3×#49711yescharacteristic, complemented
285C23—61no—
286Q8—182no—
287Q8—182no—
288Q8—182no—
289D4—182no—
290C2×C4—61no—
291C2×C4—62no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4C3×GO+(4,𝔽2)11yescharacteristic
295C8—94no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—241no—
309C2×C3—242no—
310C2×C3—241no—
311C2×C3#6422611yescharacteristic
312C2×C3#6423111yescharacteristic
313C2×C3#6423911yescharacteristic
314D3—81no—
315C22—121no—
316C22—121no—
317C22—121no—
318C22C2×C3×GO+(4,𝔽2)11yescharacteristic
319C4—21no—
320C4—92no—
321C4—121no—
322C4—122no—
323C4—181no—
324C3—21no—
325C3—21no—
326C3—41no—
327C3—41no—
328C3#1059344311yescharacteristic, direct product factor
329C2—241no—
330C2C3×#6422611yescharacteristic
331C2C3×#6423111yescharacteristic
332C2C3×#6423911yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#497
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
1. C2×C3×#3776C2semidirect
3. C3×#63919C2semidirect
5. C3×#63777C2semidirect
6. #10593443C3direct (N × H)
18. C2×#3776C2×C3semidirect
22. #63919C2×C3semidirect
24. #63777C2×C3semidirect
181. C33#497semidirect
284. C32C3×#497semidirect
328. C3#10593443direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C2×C3×#37752
C3×#639192
C3×#639152
C3×#637772
#105934433
C3×#4979