Finite Group Databasebeta

← C3×#10593369 #13166812

Subgroups — C3×#10593369 #13166812

Order 1728.

412 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3776C211yesmaximal, characteristic
2C3×C4×#627C211yesmaximal, characteristic
3C3×#63838C211yesmaximal, characteristic
4C3×#63777C211yesmaximal, characteristic
5C3×#63742C211yesmaximal, characteristic
6C3×#63740C211yesmaximal, characteristic
7C3×#63740C211yesmaximal, characteristic
8#10593369C311yesmaximal, characteristic, direct product factor
9C2×C3×#627C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C3×#3776C414yescomplemented
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×Dic3⋊C4—21no—
16C32×C4×Dic3—21no—
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×Dic32—21no—
21C2×#3776C2×C311yescharacteristic
22C3×#887—61no—
23C3×C4×Dic6—61no—
24C4×#627C2×C311yescharacteristic
25#63838C2×C311yescharacteristic
26#63777C2×C311yescharacteristic
27#63742C2×C311yescharacteristic
28#63740C2×C311yescharacteristic
29#63740C2×C311yescharacteristic
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3D411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C3×#627—21no—
38C3×#627—21no—
39C3×#627C2×C412yes—
40C3×#627C2×C412yes—
41C3×#622—21no—
42C3×#622C2×C412yes—
43C3×#441—91nomaximal
44C2×C3×Dic6—61no—
45C2×C3×Dic6—61no—
46C2×#627C22×C311yescharacteristic
47C2×#627C22×C311yescharacteristic
48C2×#622C22×C311yescharacteristic
49C2×#622C22×C311yescharacteristic
50#3776C3×C414yescomplemented
51C32×C4⋊C4—61no—
52C32×C42—61no—
53C4×C3⋊Dic3C22×C311yescharacteristic
54C3×C4⋊Dic3—61no—
55C3×Dic3⋊C4—21no—
56C3×Dic3⋊C4—21no—
57C3×Dic3⋊C4—41no—
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—21no—
62C3×C4×Dic3—21no—
63C3×C4×Dic3—41no—
64C3×C4×Dic3—61no—
65C3×C4×Dic3—61no—
66C3×C4×Dic3—61no—
67C3×C4×Dic3—61no—
68#3724C22×C311yescharacteristic
69#3724C22×C311yescharacteristic
70Dic3⋊Dic3—21no—
71Dic32—21no—
72C22×C33C2×D411yescharacteristic
73C33×C4—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3C22×C411yescharacteristic
76C3×C3⋊Dic3#4111yescharacteristic
77C32×Dic3—22no—
78C32×Dic3—22no—
79C32×Dic3—41no—
80C32×Dic3—41no—
81C2×C3×Q16—91no—
82C3×C4×Q8—91no—
83C3×C4×Q8—91no—
84#887—61no—
85C4×Dic6—61no—
86C3×#106—91no—
87C3×#103—91no—
88C3×#103—91no—
89C3×#97—91no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4—61no—
95C2×C32×C4—61no—
96C2×C32×C4—61no—
97C2×C32×C4—61no—
98C2×C32×C4C3×D411yescharacteristic
99C2×C3⋊Dic3C3×D411yescharacteristic
100C2×C3⋊Dic3C23×C311yescharacteristic
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—41no—
110C2×C3×Dic3—41no—
111C2×C3×Dic3—41no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C3×Dic6—62no—
118C3×Dic6—62no—
119C3×Dic6—62no—
120C3×Dic6—62no—
121#627—21no—
122#627—21no—
123#627C2×C3×C412yes—
124#627C2×C3×C412yes—
125#622—21no—
126#622C2×C3×C412yes—
127#441—91noSylow
128C2×C33C4×D411yescharacteristic
129C2×C33#13711yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—91no—
132C2×C3×Q8—91no—
133C2×Dic6—61no—
134C2×Dic6—61no—
135C3×Q16—94no—
136C2×C3×C8—91no—
137C2×C3×C8—91no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—121no—
143C3×C4⋊C4—181no—
144C3×C42—61no—
145C3×C42—61no—
146C3×C42—91no—
147C3×C42—121no—
148C3×C42—181no—
149C4⋊Dic3—61no—
150Dic3⋊C4—21no—
151Dic3⋊C4—61no—
152Dic3⋊C4—61no—
153Dic3⋊C4—61no—
154C4×Dic3—21no—
155C4×Dic3—61no—
156C4×Dic3—61no—
157C4×Dic3—61no—
158C4×Dic3—61no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32C2×C3×D411yescharacteristic
164C32×C4—22no—
165C32×C4—21no—
166C32×C4—41no—
167C32×C4—42no—
168C32×C4—62no—
169C32×C4—62no—
170C32×C4—81no—
171C32×C4—121no—
172C32×C4—121no—
173C3⋊Dic3—21no—
174C3⋊Dic3C22×C3×C411yescharacteristic
175C3⋊Dic3C3×#4111yescharacteristic
176C3×Dic3—22no—
177C3×Dic3—22no—
178C3×Dic3—22no—
179C3×Dic3—22no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—41no—
183C3×Dic3—41no—
184C3×Dic3—42no—
185C3×Dic3—41no—
186C3×Dic3—41no—
187C3×Dic3—61no—
188C3×Dic3—61no—
189C3×Dic3—61no—
190C3×Dic3—61no—
191C3×Dic3—62no—
192C3×Dic3—81no—
193C3×Dic3—121no—
194C2×Q16—91no—
195C4×Q8—91no—
196C4×Q8—91no—
197#106—91no—
198#103—91no—
199#103—91no—
200#97—91no—
201C33#44111yescharacteristic, complemented, Sylow
202C3×Q8—92no—
203C3×Q8—92no—
204C3×Q8—181no—
205C3×Q8—181no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—91no—
219C2×C3×C4—91no—
220C2×C3×C4—121no—
221C2×C3×C4—121no—
222C2×C3×C4—121no—
223C2×C3×C4—121no—
224C2×C3×C4GO+(4,𝔽2)11yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233Dic6—62no—
234Dic6—62no—
235Dic6—62no—
236Dic6—62no—
237C3×C8—92no—
238C3×C8—181no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32C3×C4×D411yescharacteristic
252C2×C32C3×#13711yescharacteristic
253C2×C32C3×#13711yescharacteristic
254C2×Q8—91no—
255C2×Q8—91no—
256Q16—94no—
257C2×C8—91no—
258C2×C8—91no—
259C4⋊C4—61no—
260C4⋊C4—91no—
261C4⋊C4—91no—
262C4⋊C4—181no—
263C42—61no—
264C42—91no—
265C42—181no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3C2×GO+(4,𝔽2)11yescharacteristic
271C3×C4—22no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—42no—
275C3×C4—62no—
276C3×C4—62no—
277C3×C4—62no—
278C3×C4—62no—
279C3×C4—81no—
280C3×C4—91no—
281C3×C4—91no—
282C3×C4—122no—
283C3×C4—122no—
284C3×C4—121no—
285C3×C4—121no—
286C3×C4—122no—
287C3×C4—121no—
288C3×C4—121no—
289C3×C4—181no—
290C3×C4—241no—
291Dic3—22no—
292Dic3—22no—
293Dic3—41no—
294Dic3—41no—
295Dic3—61no—
296Dic3—61no—
297Dic3—61no—
298Dic3—61no—
299Dic3—62no—
300Dic3—121no—
301C32—21no—
302C32—21no—
303C32—41no—
304C32—41no—
305C32C3×#44111yescharacteristic, complemented
306Q8—92no—
307Q8—92no—
308Q8—181no—
309Q8—181no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—91no—
315C2×C4—91no—
316C2×C4C3×GO+(4,𝔽2)11yescharacteristic
317C8—92no—
318C8—181no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3#6422611yescharacteristic
332C2×C3C4×GO+(4,𝔽2)11yescharacteristic
333C2×C3#6423911yescharacteristic
334C22C2×C3×GO+(4,𝔽2)11yescharacteristic
335C4—21no—
336C4—62no—
337C4—62no—
338C4—91no—
339C4—91no—
340C4—121no—
341C4—121no—
342C4—181no—
343C3—21no—
344C3—21no—
345C3—41no—
346C3—41no—
347C3#1059336911yescharacteristic, direct product factor
348C2C3×#6422611yescharacteristic
349C2C3×C4×GO+(4,𝔽2)11yescharacteristic
350C2C3×#6423911yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#441
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
8. #10593369C3direct (N × H)
13. C3×#3776C4semidirect
50. #3776C3×C4semidirect
201. C33#441semidirect
305. C32C3×#441semidirect
347. C3#10593369direct (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
C3×C4×#6272
C3×#638382
C3×#637772
C3×#637422
C3×#637402
#105933693
C3×#4419