Finite Group Databasebeta

← C3×#10593373 #13166746

Subgroups — C3×#10593373 #13166746

Order 1728.

358 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C3×#63870C211yesmaximal, characteristic
2C3×#63838C211yesmaximal, characteristic
3C3×C4⋊#622C211yesmaximal, characteristic
4C3×#63743C211yesmaximal, characteristic
5C3×#63742C211yesmaximal, characteristic
6C3×#63740C211yesmaximal, characteristic
7C3×#63740C211yesmaximal, characteristic
8#10593373C311yesmaximal, characteristic, direct product factor
9C2×C3×#627C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×Dic3⋊C4—21no—
14C32×Dic3⋊C4—21no—
15C3×#3724C2211yescharacteristic
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×Dic32—21no—
20C3×#889—61no—
21C3×#887—61no—
22#63870C2×C311yescharacteristic
23#63838C2×C311yescharacteristic
24C4⋊#622C2×C311yescharacteristic
25#63743C2×C311yescharacteristic
26#63742C2×C311yescharacteristic
27#63740C2×C311yescharacteristic
28#63740C2×C311yescharacteristic
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3D411yescharacteristic
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C2×C32×Dic3—21no—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C3×#627—21no—
37C3×#627Q812yes—
38C3×#622—21no—
39C3×#622—21no—
40C3×#479—91nomaximal
41C2×C3×Dic6—61no—
42C2×#627C22×C311yescharacteristic
43C2×#622C22×C311yescharacteristic
44C2×#622C22×C311yescharacteristic
45C32×C4⋊C4—61no—
46C32×C4⋊C4—61no—
47C4×C3⋊Dic3C22×C311yescharacteristic
48C3×C4⋊Dic3—61no—
49C3×Dic3⋊C4—21no—
50C3×Dic3⋊C4—21no—
51C3×Dic3⋊C4—21no—
52C3×Dic3⋊C4—21no—
53C3×Dic3⋊C4—41no—
54C3×Dic3⋊C4—41no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58C3×Dic3⋊C4—61no—
59C3×C4×Dic3—61no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—61no—
63#3724C22×C311yescharacteristic
64#3724C22×C311yescharacteristic
65#3724C22×C311yescharacteristic
66Dic3⋊Dic3—21no—
67Dic32—21no—
68C22×C33C2×D411yescharacteristic
69C33×C4—21no—
70C3×C3⋊Dic3—21no—
71C3×C3⋊Dic3C2×Q811yescharacteristic
72C3×C3⋊Dic3#4111yescharacteristic
73C32×Dic3—22no—
74C32×Dic3—41no—
75C32×Dic3—41no—
76C32×Dic3—41no—
77C3×#125—91no—
78C3×C4×Q8—91no—
79#889—61no—
80#887—61no—
81C3×#107—91no—
82C3×#106—91no—
83C3×C4⋊C8—91no—
84C3×#103—91no—
85C3×#103—91no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4—61no—
91C2×C32×C4—61no—
92C2×C32×C4—61no—
93C2×C32×C4—61no—
94C2×C32×C4C3×D411yescharacteristic
95C2×C3⋊Dic3C3×D411yescharacteristic
96C2×C3⋊Dic3C23×C311yescharacteristic
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115#627—21no—
116#627C3×Q812yes—
117#622—21no—
118#622—21no—
119#479—91noSylow
120C2×C33C22⋊Q811yescharacteristic
121C2×C33#13511yescharacteristic
122C2×C33#13711yescharacteristic
123C2×C3×Q8—91no—
124C2×Dic6—61no—
125C2×C3×C8—91no—
126C2×C3×C8—91no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—61no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—91no—
132C3×C4⋊C4—91no—
133C3×C4⋊C4—91no—
134C3×C4⋊C4—121no—
135C3×C4⋊C4—121no—
136C3×C4⋊C4—181no—
137C3×C42—91no—
138C3×C42—181no—
139C4⋊Dic3—61no—
140Dic3⋊C4—21no—
141Dic3⋊C4—21no—
142Dic3⋊C4—61no—
143Dic3⋊C4—61no—
144Dic3⋊C4—61no—
145Dic3⋊C4—61no—
146C4×Dic3—61no—
147C4×Dic3—61no—
148C4×Dic3—61no—
149C4×Dic3—61no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—41no—
153C22×C32—41no—
154C22×C32C2×C3×D411yescharacteristic
155C32×C4—21no—
156C32×C4—41no—
157C32×C4—41no—
158C32×C4—62no—
159C32×C4—81no—
160C32×C4—81no—
161C32×C4—121no—
162C32×C4—121no—
163C32×C4—121no—
164C3⋊Dic3—21no—
165C3⋊Dic3C2×C3×Q811yescharacteristic
166C3⋊Dic3C3×#4111yescharacteristic
167C3×Dic3—22no—
168C3×Dic3—22no—
169C3×Dic3—42no—
170C3×Dic3—41no—
171C3×Dic3—41no—
172C3×Dic3—41no—
173C3×Dic3—42no—
174C3×Dic3—41no—
175C3×Dic3—41no—
176C3×Dic3—41no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—61no—
180C3×Dic3—61no—
181C3×Dic3—62no—
182C3×Dic3—81no—
183C3×Dic3—81no—
184C3×Dic3—121no—
185#125—91no—
186C4×Q8—91no—
187#107—91no—
188#106—91no—
189C4⋊C8—91no—
190#103—91no—
191#103—91no—
192C33#47911yescharacteristic, complemented, Sylow
193C3×Q8—92no—
194C3×Q8—181no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4—121no—
210C2×C3×C4—121no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4GO+(4,𝔽2)11yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222Dic6—62no—
223Dic6—62no—
224C3×C8—181no—
225C3×C8—181no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32C3×C22⋊Q811yescharacteristic
239C2×C32C3×#13511yescharacteristic
240C2×C32C3×#13711yescharacteristic
241C2×Q8—91no—
242C2×C8—91no—
243C2×C8—91no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—91no—
247C4⋊C4—91no—
248C4⋊C4—91no—
249C4⋊C4—181no—
250C42—91no—
251C42—181no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C2×GO+(4,𝔽2)11yescharacteristic
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—62no—
261C3×C4—62no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—91no—
265C3×C4—91no—
266C3×C4—122no—
267C3×C4—121no—
268C3×C4—121no—
269C3×C4—121no—
270C3×C4—122no—
271C3×C4—121no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—181no—
275C3×C4—241no—
276C3×C4—241no—
277Dic3—22no—
278Dic3—41no—
279Dic3—41no—
280Dic3—41no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—61no—
285Dic3—62no—
286Dic3—121no—
287C32—21no—
288C32—21no—
289C32—41no—
290C32—41no—
291C32C3×#47911yescharacteristic, complemented
292Q8—92no—
293Q8—181no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C4C3×GO+(4,𝔽2)11yescharacteristic
301C8—181no—
302C8—181no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3D32⋊Q811yescharacteristic
316C2×C3#6422611yescharacteristic
317C2×C3#6423711yescharacteristic
318C22C2×C3×GO+(4,𝔽2)11yescharacteristic
319C4—21no—
320C4—62no—
321C4—91no—
322C4—91no—
323C4—121no—
324C4—121no—
325C4—121no—
326C4—181no—
327C3—21no—
328C3—21no—
329C3—41no—
330C3—41no—
331C3#1059337311yescharacteristic, direct product factor
332C2C3×D32⋊Q811yescharacteristic
333C2C3×#6422611yescharacteristic
334C2C3×#6423711yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#479
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
8. #10593373C3direct (N × H)
192. C33#479semidirect
291. C32C3×#479semidirect
331. C3#10593373direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638702
C3×#638382
C3×C4⋊#6222
C3×#637432
C3×#637422
C3×#637402
#105933733
C3×#4799