Finite Group Databasebeta

← C3×#10593438 #13166802

Subgroups — C3×#10593438 #13166802

Order 1728.

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

normal subgroups only

This group has 462 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×#3775C212yesmaximal
2C3×#63919C211yesmaximal, characteristic, complemented
3C3×#63902C211yesmaximal, characteristic
4C3×#63779C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593438C311yesmaximal, characteristic, direct product factor
7C2×C3×#627C2212yes—
8C2×C3×#625C2211yescharacteristic
9C2×C3×D3×Dic3—21no—
10C2×C3×#622C2212yes—
11C3×#3775—24no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×(C2×D3)⋊C4—21no—
14C32×C4⋊Dic3—21no—
15C3×#3724—21no—
16C3×#3724C2211yescharacteristic
17C2×#3775C2×C312yes—
18C3×#888—61no—
19C3×#67⋊C4—61no—
20#63919C2×C311yescharacteristic, complemented
21#63902C2×C311yescharacteristic
22#63779C2×C311yescharacteristic, complemented
23#63740C2×C312yescomplemented
24C22×C32×D3—21no—
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—21no—
29C2×C32×Dic3—22no—
30C3×#627—22no—
31C3×#627—22no—
32C3×#625—21no—
33C3×#625D412yes—
34C3×D3×Dic3—42no—
35C3×#622—22no—
36C3×#461—91nomaximal
37C2×C3×#67—61no—
38C22×C3×Dic3—61no—
39C2×C3×C4×D3—61no—
40C2×C3×Dic6—62no—
41C2×#627C22×C312yes—
42C2×#625C22×C311yescharacteristic
43C2×D3×Dic3—21no—
44C2×#622C22×C312yes—
45#3775—24no—
46C32×C4⋊C4—61no—
47C32×C22⋊C4—61no—
48C4×C3⋊Dic3C22×C311yescharacteristic
49C3×(C2×D3)⋊C4—21no—
50C3×(C2×D3)⋊C4—21no—
51C3×(C2×D3)⋊C4—41no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3—41no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—62no—
57C3×C4×Dic3—61no—
58C3×C4×Dic3—61no—
59#3724—21no—
60#3724C22×C311yescharacteristic
61C22×C33C2×D411yescharacteristic
62C2×C32×D3—22no—
63C2×C32×D3—41no—
64C2×C32×D3—41no—
65C33×C4—21no—
66C3×C3⋊Dic3SD1612yescomplemented
67C3×C3⋊Dic3C2×D411yescharacteristic
68C3×C3⋊Dic3#4111yescharacteristic
69C32×Dic3—42no—
70C32×Dic3—41no—
71C2×C3×SD16—92no—
72C3×C4⋊Q8—91no—
73C3×C4×D4—91no—
74#888—61no—
75#67⋊C4—61no—
76C3×C4⋊C8—91no—
77C3×#103—92no—
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—61no—
87C2×C32×C4—62no—
88C2×C32×C4C3×D411yescharacteristic
89C2×C3⋊Dic3C3×D411yescharacteristic
90C2×C3⋊Dic3C23×C311yescharacteristic
91C3×#67—62no—
92C3×#67—62no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—21no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—42no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—122no—
104C3×C4×D3—122no—
105C3×Dic6—122no—
106C3×Dic6—122no—
107#627—22no—
108#627—22no—
109#625—21no—
110#625C3×D412yes—
111D3×Dic3—42no—
112#622—22no—
113#461—91noSylow
114C2×C33#12111yescharacteristic
115C2×C33C2×SD1611yescharacteristic
116C2×C33#13711yescharacteristic
117C32×D3—42no—
118C2×C3×Q8—92no—
119C2×C3×D4—91no—
120C22×C3×C4—181no—
121C2×#67—61no—
122C22×Dic3—61no—
123C2×C4×D3—61no—
124C2×Dic6—62no—
125C3×SD16—184no—
126C2×C3×C8—92no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—61no—
129C3×C4⋊C4—91no—
130C3×C4⋊C4—121no—
131C3×C4⋊C4—181no—
132C3×C22⋊C4—61no—
133C3×C22⋊C4—61no—
134C3×C22⋊C4—121no—
135C3×C42—91no—
136(C2×D3)⋊C4—21no—
137C4⋊Dic3—21no—
138Dic3⋊C4—61no—
139Dic3⋊C4—62no—
140C4×Dic3—61no—
141C4×Dic3—61no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—41no—
145C22×C32—41no—
146C22×C32—62no—
147C22×C32—121no—
148C22×C32—121no—
149C22×C32C2×C3×D411yescharacteristic
150C2×C3×D3—22no—
151C2×C3×D3—22no—
152C2×C3×D3—42no—
153C2×C3×D3—41no—
154C2×C3×D3—41no—
155C2×C3×D3—41no—
156C2×C3×D3—41no—
157C2×C3×D3—81no—
158C2×C3×D3—81no—
159C32×C4—22no—
160C32×C4—21no—
161C32×C4—41no—
162C32×C4—42no—
163C32×C4—81no—
164C32×C4—122no—
165C32×C4—121no—
166C3⋊Dic3C3×SD1612yescomplemented
167C3⋊Dic3C2×C3×D411yescharacteristic
168C3⋊Dic3C3×#4111yescharacteristic
169C3×Dic3—42no—
170C3×Dic3—41no—
171C3×Dic3—42no—
172C3×Dic3—41no—
173C3×Dic3—62no—
174C3×Dic3—62no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—82no—
180C3×Dic3—81no—
181C2×SD16—92no—
182C4⋊Q8—91no—
183C4×D4—91no—
184C4⋊C8—91no—
185#103—92no—
186C33#46111yescharacteristic, complemented, Sylow
187C23×C3—61no—
188C23×C3—61no—
189C23×C3—121no—
190C22×D3—21no—
191C3×Q8—182no—
192C3×Q8—182no—
193C3×D4—92no—
194C3×D4—181no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—61no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—121no—
206C2×C3×C4—122no—
207C2×C3×C4—362no—
208C2×C3×C4GO+(4,𝔽2)11yescharacteristic
209#67—62no—
210#67—62no—
211C2×Dic3—22no—
212C2×Dic3—21no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—122no—
218C4×D3—122no—
219Dic6—122no—
220Dic6—122no—
221C3×C8—182no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—122no—
235C2×C32C3×#12111yescharacteristic
236C2×C32C2×C3×SD1611yescharacteristic
237C2×C32C3×#13711yescharacteristic
238C3×D3—42no—
239C3×D3—42no—
240C3×D3—82no—
241C2×Q8—92no—
242C2×D4—91no—
243C22×C4—181no—
244SD16—184no—
245C2×C8—92no—
246C4⋊C4—61no—
247C4⋊C4—91no—
248C4⋊C4—181no—
249C22⋊C4—61no—
250C42—91no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—122no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3—121no—
261C22×C3—121no—
262C22×C3—241no—
263C22×C3—241no—
264C22×C3C2×GO+(4,𝔽2)11yescharacteristic
265C2×D3—22no—
266C2×D3—41no—
267C2×D3—41no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—42no—
272C3×C4—81no—
273C3×C4—92no—
274C3×C4—91no—
275C3×C4—91no—
276C3×C4—122no—
277C3×C4—121no—
278C3×C4—121no—
279C3×C4—122no—
280C3×C4—241no—
281C3×C4—242no—
282Dic3—42no—
283Dic3—41no—
284Dic3—61no—
285Dic3—61no—
286Dic3—62no—
287Dic3—61no—
288Dic3—61no—
289Dic3—62no—
290C32—21no—
291C32—21no—
292C32—41no—
293C32—41no—
294C32C3×#46111yescharacteristic, complemented
295C23—61no—
296Q8—182no—
297Q8—182no—
298D4—92no—
299D4—181no—
300C2×C4—61no—
301C2×C4—62no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—362no—
305C2×C4C3×GO+(4,𝔽2)11yescharacteristic
306C8—182no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—122no—
320C2×C3—122no—
321C2×C3—242no—
322C2×C3#6422711yescharacteristic
323C2×C3#6423011yescharacteristic
324C2×C3#6423911yescharacteristic
325D3—42no—
326C22—62no—
327C22—121no—
328C22—121no—
329C22C2×C3×GO+(4,𝔽2)11yescharacteristic
330C4—21no—
331C4—92no—
332C4—91no—
333C4—91no—
334C4—121no—
335C4—122no—
336C3—21no—
337C3—21no—
338C3—41no—
339C3—41no—
340C3#1059343811yescharacteristic, direct product factor
341C2—122no—
342C2C3×#6422711yescharacteristic
343C2C3×#6423011yescharacteristic
344C2C3×#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×#461
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. C3×#63919C2semidirect
4. C3×#63779C2semidirect
5. C3×#63740C2semidirect
6. #10593438C3direct (N × H)
20. #63919C2×C3semidirect
22. #63779C2×C3semidirect
23. #63740C2×C3semidirect
66. C3×C3⋊Dic3SD16semidirect
166. C3⋊Dic3C3×SD16semidirect
186. C33#461semidirect
294. C32C3×#461semidirect
340. C3#10593438direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37752
C3×#639192
C3×#639022
C3×#637792
C3×#637402
#105934383
C3×#4619