Finite Group Databasebeta

← C3×#10593436 #13166800

Subgroups — C3×#10593436 #13166800

Order 1728.

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

normal subgroups only

This group has 638 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, complemented
2C2×C3×#3775C212yesmaximal, complemented
3C3×#63912C211yesmaximal, characteristic
4C3×#63778C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593436C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6—21no—
9C2×C3×#627C2212yes—
10C2×C3×#626—21no—
11C2×C3×#625C2211yescharacteristic
12C2×C3×#624—21no—
13C3×#3797—24no—
14C2×C3×#622C2212yes—
15C3×#3775—24no—
16C32×(C2×D3)⋊C4—21no—
17C3×#3724C2211yescharacteristic
18C2×C3×D3×Q8—61no—
19C2×#3797C2×C311yescharacteristic, complemented
20C2×#3775C2×C312yescomplemented
21C3×#884—61no—
22#63912C2×C311yescharacteristic
23#63778C2×C311yescharacteristic, complemented
24#63740C2×C312yescomplemented
25C22×C3×C3⋊D3D411yescharacteristic
26C22×C32×D3—21no—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3×C4×C3⋊D3—22no—
30C2×C32×Dic3—22no—
31C2×C32×Dic3—21no—
32C32×Dic6—24no—
33C3×#627—22no—
34C3×#627D414yescomplemented
35C3×#626—42no—
36C3×#625—22no—
37C3×#624—24no—
38C3×#622—22no—
39C3×#448—91nomaximal
40C2×C32×Q8—61no—
41C2×C4×C3⋊D3C22×C311yescharacteristic
42C2×C3×#67—61no—
43C2×C3×#67—61no—
44C3×D3×Q8—68no—
45C2×C3×D12—61no—
46C2×C3×C4×D3—62no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—61no—
49C2×C3×Dic6—21no—
50C2×C3×Dic6—21no—
51C2×C3×Dic6—41no—
52C2×C3×Dic6—62no—
53C2×#627C22×C312yes—
54C2×#626—21no—
55C2×#625C22×C311yescharacteristic
56C2×#624—21no—
57#3797—24no—
58C2×#622C22×C312yes—
59#3775—24no—
60C32×C22⋊C4—61no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×(C2×D3)⋊C4—41no—
64C3×Dic3⋊C4—61no—
65#3724C22×C311yescharacteristic
66C22×C33C2×D411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3SD1612yescomplemented
70C2×C32×D3—41no—
71C2×C32×D3—41no—
72C2×C32×D3—41no—
73C33×C4—21no—
74C3×C3⋊Dic3C2×D412yes—
75C32×Dic3—24no—
76C32×Dic3—41no—
77C22×C3×Q8—91no—
78C2×D3×Q8—61no—
79C2×C3×SD16—92no—
80C3×#121—91no—
81#884—61no—
82C3×#103—92no—
83C3×C22⋊C8—91no—
84C23×C32—61no—
85C22×C3⋊D3C3×D411yescharacteristic
86C22×C3×D3—21no—
87C22×C3×D3—21no—
88C22×C3×D3—41no—
89C22×C3×D3—61no—
90C22×C3×D3—61no—
91C32×Q8—64no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—41no—
96C2×C32×C4—62no—
97C2×C32×C4—61no—
98C2×C32×C4C3×D411yescharacteristic
99C2×C3⋊Dic3C23×C311yescharacteristic
100C4×C3⋊D3—22no—
101C3×#67—122no—
102C3×#67—122no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—42no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C3×D12—122no—
112C3×C4×D3—68no—
113C3×C4×D3—64no—
114C3×C4×D3—122no—
115C3×Dic6—24no—
116C3×Dic6—24no—
117C3×Dic6—44no—
118C3×Dic6—64no—
119C3×Dic6—64no—
120#627—22no—
121#627C3×D414yescomplemented
122#626—42no—
123#625—22no—
124#624—24no—
125#622—22no—
126#448—91noSylow
127C2×C33#12011yescharacteristic
128C2×C33C2×SD1611yescharacteristic
129C2×C33#13711yescharacteristic
130C3×C3⋊D3—22no—
131C32×D3—81no—
132C2×C3×Q8—61no—
133C2×C3×Q8—61no—
134C2×C3×Q8—92no—
135C2×C3×Q8—121no—
136C2×C3×Q8—184no—
137C2×C3×D4—91no—
138C2×C3×D4—181no—
139C22×C3×C4—91no—
140C22×C3×C4—181no—
141C2×#67—61no—
142C2×#67—61no—
143D3×Q8—68no—
144C2×D12—61no—
145C2×C4×D3—61no—
146C2×C4×D3—61no—
147C2×C4×D3—62no—
148C2×Dic6—21no—
149C2×Dic6—62no—
150C3×SD16—184no—
151C2×C3×C8—92no—
152C3×C4⋊C4—91no—
153C3×C22⋊C4—61no—
154C3×C22⋊C4—61no—
155C3×C22⋊C4—121no—
156(C2×D3)⋊C4—21no—
157Dic3⋊C4—61no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—121no—
163C22×C32—121no—
164C22×C32—121no—
165C22×C32C2×C3×D411yescharacteristic
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3⋊D3C3×SD1612yescomplemented
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—41no—
172C2×C3×D3—42no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C2×C3×D3—41no—
176C2×C3×D3—62no—
177C2×C3×D3—62no—
178C2×C3×D3—62no—
179C2×C3×D3—62no—
180C2×C3×D3—81no—
181C2×C3×D3—81no—
182C2×C3×D3—121no—
183C2×C3×D3—121no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4—42no—
188C32×C4—64no—
189C32×C4—81no—
190C32×C4—121no—
191C3⋊Dic3C2×C3×D412yes—
192C3×Dic3—24no—
193C3×Dic3—24no—
194C3×Dic3—44no—
195C3×Dic3—41no—
196C3×Dic3—41no—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C3×Dic3—81no—
200C22×Q8—91no—
201C2×SD16—92no—
202#121—91no—
203#103—92no—
204C22⋊C8—91no—
205C33#44811yescharacteristic, complemented, Sylow
206C23×C3—61no—
207C23×C3—61no—
208C23×C3—91no—
209C23×C3—121no—
210C22×D3—21no—
211C22×D3—61no—
212C22×D3—61no—
213C3×Q8—64no—
214C3×Q8—64no—
215C3×Q8—94no—
216C3×Q8—124no—
217C3×Q8—182no—
218C3×D4—182no—
219C3×D4—362no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—91no—
229C2×C3×C4—122no—
230C2×C3×C4—121no—
231C2×C3×C4—182no—
232C2×C3×C4—184no—
233C2×C3×C4GO+(4,𝔽2)11yescharacteristic
234#67—122no—
235#67—122no—
236C2×Dic3—22no—
237C2×Dic3—21no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240D12—122no—
241C4×D3—68no—
242C4×D3—64no—
243C4×D3—122no—
244Dic6—24no—
245Dic6—64no—
246Dic6—64no—
247C3×C8—182no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—241no—
261C2×C32C3×#12011yescharacteristic
262C2×C32C2×C3×SD1611yescharacteristic
263C2×C32C3×#13711yescharacteristic
264C3⋊D3—22no—
265C3×D3—64no—
266C3×D3—81no—
267C3×D3—82no—
268C3×D3—81no—
269C3×D3—122no—
270C2×Q8—61no—
271C2×Q8—92no—
272C2×Q8—184no—
273C2×D4—91no—
274C2×D4—181no—
275C22×C4—91no—
276C22×C4—181no—
277SD16—184no—
278C2×C8—92no—
279C4⋊C4—91no—
280C22⋊C4—61no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—92no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3—122no—
290C22×C3—121no—
291C22×C3—121no—
292C22×C3—121no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—241no—
296C22×C3—241no—
297C22×C3C2×GO+(4,𝔽2)11yescharacteristic
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—62no—
302C2×D3—62no—
303C2×D3—62no—
304C2×D3—62no—
305C2×D3—121no—
306C2×D3—121no—
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—41no—
310C3×C4—42no—
311C3×C4—64no—
312C3×C4—64no—
313C3×C4—81no—
314C3×C4—92no—
315C3×C4—124no—
316C3×C4—121no—
317C3×C4—121no—
318C3×C4—241no—
319Dic3—24no—
320Dic3—41no—
321Dic3—62no—
322Dic3—62no—
323C32—21no—
324C32—21no—
325C32—41no—
326C32—41no—
327C32C3×#44811yescharacteristic, complemented
328C23—61no—
329C23—91no—
330Q8—64no—
331Q8—94no—
332Q8—182no—
333D4—182no—
334D4—362no—
335C2×C4—62no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—182no—
339C2×C4—184no—
340C2×C4C3×GO+(4,𝔽2)11yescharacteristic
341C8—182no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—182no—
355C2×C3—241no—
356C2×C3—242no—
357C2×C3—241no—
358C2×C3#6422611yescharacteristic
359C2×C3#6422711yescharacteristic
360C2×C3#6424111yescharacteristic
361D3—64no—
362D3—81no—
363D3—122no—
364C22—92no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—181no—
369C22—181no—
370C22C2×C3×GO+(4,𝔽2)11yescharacteristic
371C4—21no—
372C4—64no—
373C4—92no—
374C4—121no—
375C3—21no—
376C3—21no—
377C3—41no—
378C3—41no—
379C3#1059343611yescharacteristic, direct product factor
380C2—182no—
381C2—241no—
382C2C3×#6422611yescharacteristic
383C2C3×#6422711yescharacteristic
384C2C3×#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 radicalC3×#63912C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#448
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×#3797C2semidirect
2. C2×C3×#3775C2semidirect
4. C3×#63778C2semidirect
5. C3×#63740C2semidirect
6. #10593436C3direct (N × H)
19. C2×#3797C2×C3semidirect
20. C2×#3775C2×C3semidirect
23. #63778C2×C3semidirect
24. #63740C2×C3semidirect
34. C3×#627D4semidirect
69. C2×C3×C3⋊D3SD16semidirect
121. #627C3×D4semidirect
168. C2×C3⋊D3C3×SD16semidirect
205. C33#448semidirect
327. C32C3×#448semidirect
379. C3#10593436direct (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×#37752
C3×#639122
C3×#637782
C3×#637402
#105934363
C3×#4489