Finite Group Databasebeta

← C3×#67449 #11771923

Subgroups — C3×#67449 #11771923

Order 1152.

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

normal subgroups only

This group has 373 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
1C3×#4947C211yesmaximal, characteristic
2C32×#323C211yesmaximal, characteristic
3C3×#4526C211yesmaximal, characteristic
4C3×#1351—31nomaximal
5#67449C311yesmaximal, characteristic, direct product factor
6C22×C3×Dic6C411yescharacteristic, complemented
7C2×C32×C22⋊C4C2211yescharacteristic
8C2×C3×#217—21no—
9C2×C3×#217—21no—
10C32×C22⋊C8—21no—
11C3×C3⋊#95—21no—
12C3×C3⋊#95C411yescharacteristic, complemented
13C3×#393—31no—
14#4947C2×C311yescharacteristic
15C3×#342—31no—
16C3×#323—21no—
17C3×#323D311yescharacteristic
18C3×#323C2×C311yescharacteristic
19#4526C2×C311yescharacteristic
20C24×C32D411yescharacteristic
21C22×C32×C4C2×C411yescharacteristic
22C22×C32×C4D411yescharacteristic
23C22×C3×Dic3—21no—
24C22×C3×Dic3—21no—
25C22×C3×Dic3—41no—
26C2×C3×Dic6—41no—
27C2×C3×Dic6—42no—
28C2×C32×C8—41no—
29C32×C22⋊C4—41no—
30C3×#217—22no—
31C3×#217—42no—
32C3×#217—41no—
33#1351—31noSylow
34C22×C3×Q8—31no—
35C22×Dic6C3×C411yescharacteristic, complemented
36C2×C3×C22⋊C4—21no—
37C2×C3×C22⋊C4—61no—
38C2×C3×C22⋊C4—61no—
39C2×C3×C22⋊C4C2×D311yescharacteristic
40C2×C3×C22⋊C4C22×C311yescharacteristic
41C2×#217—21no—
42C2×#217—21no—
43C3×C22⋊C8—22no—
44C3×C22⋊C8—21no—
45C3×C22⋊C8—21no—
46C3×#95—31no—
47C3×#95—61no—
48C3⋊#95—21no—
49C3⋊#95C3×C411yescharacteristic, complemented
50C23×C32—21no—
51C23×C32—21no—
52C23×C32—41no—
53C23×C32—41no—
54C23×C32C22⋊C411yescharacteristic
55C23×C32SD1611yescharacteristic
56C23×C32Q1611yescharacteristic
57C2×C32×C4—21no—
58C2×C32×C4—21no—
59C2×C32×C4—41no—
60C2×C32×C4—41no—
61C2×C3×Dic3—22no—
62C2×C3×Dic3—41no—
63C2×C3×Dic3—41no—
64C2×C3×Dic3—42no—
65C2×C3×Dic3—41no—
66C2×C3×Dic3—41no—
67C2×C3×Dic3—41no—
68C2×C3×Dic3—81no—
69C2×C3×Dic3—81no—
70C3×Dic6—42no—
71C3×Dic6—42no—
72C32×C8—81no—
73#393—31no—
74#342—31no—
75#323C3×D311yescharacteristic
76C24×C3—21no—
77C24×C3D1211yescharacteristic
78C24×C3C3×D411yescharacteristic
79C2×C3×Q8—121no—
80C2×C3×Q8—122no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—61no—
84C22×C3×C4—61no—
85C22×C3×C4—121no—
86C22×C3×C4C4×D311yescharacteristic
87C22×C3×C4#6711yescharacteristic
88C22×C3×C4C2×C3×C411yescharacteristic
89C22×C3×C4C3×D411yescharacteristic
90C22×Dic3—21no—
91C22×Dic3—21no—
92C22×Dic3—41no—
93C2×Dic6—41no—
94C2×Dic6—42no—
95C2×C3×C8—42no—
96C2×C3×C8—41no—
97C2×C3×C8—41no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—42no—
101C3×C22⋊C4—62no—
102C3×C22⋊C4—122no—
103C3×C22⋊C4—121no—
104#217—22no—
105#217—42no—
106#217—41no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32—41no—
114C22×C32—41no—
115C22×C32—81no—
116C22×C32#9911yescharacteristic
117C22×C32#10311yescharacteristic
118C22×C32#10411yescharacteristic
119C32×C4—41no—
120C32×C4—41no—
121C3×Dic3—42no—
122C3×Dic3—81no—
123C3×Dic3—82no—
124C22×Q8—31no—
125C2×C22⋊C4—61no—
126C2×C22⋊C4—61no—
127C2×C22⋊C4C2×C3×D311yescharacteristic
128C22⋊C8—21no—
129#95—31no—
130#95—61no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—21no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3—81no—
145C23×C3—81no—
146C23×C3#20411yescharacteristic
147C23×C3Dic1211yescharacteristic
148C23×C3(C2×D3)⋊C411yescharacteristic
149C23×C3C3×C22⋊C411yescharacteristic
150C23×C3C3×SD1611yescharacteristic
151C23×C3C3×Q1611yescharacteristic
152C3×Q8—122no—
153C3×Q8—122no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—62no—
165C2×C3×C4—81no—
166C2×C3×C4—81no—
167C2×C3×C4—121no—
168C2×C3×C4—121no—
169C2×C3×C4—122no—
170C2×C3×C4—121no—
171C2×C3×C4—121no—
172C2×C3×C4—121no—
173C2×C3×C4—241no—
174C2×C3×C4—241no—
175C2×Dic3—22no—
176C2×Dic3—41no—
177C2×Dic3—41no—
178C2×Dic3—42no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—81no—
183C2×Dic3—81no—
184Dic6—42no—
185Dic6—42no—
186C3×C8—82no—
187C3×C8—81no—
188C3×C8—81no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32#32811yescharacteristic
194C2×C32AΣL(1,𝔽16)11yescharacteristic
195C2×C32#35411yescharacteristic
196C24C3×D1211yescharacteristic
197C2×Q8—121no—
198C2×Q8—122no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4—121no—
202C22×C4C3×C4×D311yescharacteristic
203C22×C4C3×#6711yescharacteristic
204C2×C8—41no—
205C22⋊C4—41no—
206C22⋊C4—62no—
207C22⋊C4—122no—
208C22⋊C4—121no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3—81no—
235C22×C3—81no—
236C22×C3—81no—
237C22×C3—81no—
238C22×C3—82no—
239C22×C3#80611yescharacteristic
240C22×C3#81611yescharacteristic
241C22×C3#82511yescharacteristic
242C22×C3C3×#9911yescharacteristic
243C22×C3C3×#10311yescharacteristic
244C22×C3C3×#10411yescharacteristic
245C3×C4—42no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—81no—
251C3×C4—122no—
252C3×C4—241no—
253C3×C4—242no—
254Dic3—42no—
255Dic3—81no—
256Dic3—82no—
257C32#135111yescharacteristic, complemented, Sylow
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C3×#20411yescharacteristic
263C23C3×Dic1211yescharacteristic
264C23C3×(C2×D3)⋊C411yescharacteristic
265Q8—122no—
266Q8—122no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—62no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—122no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—241no—
279C2×C4—241no—
280C8—81no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3#446711yescharacteristic
297C2×C3#447311yescharacteristic
298C2×C3#447511yescharacteristic
299C2×C3C3×#32811yescharacteristic
300C2×C3C3×AΣL(1,𝔽16)11yescharacteristic
301C2×C3C3×#35411yescharacteristic
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C3×#80611yescharacteristic
312C22C3×#81611yescharacteristic
313C22C3×#82511yescharacteristic
314C4—41no—
315C4—41no—
316C4—122no—
317C4—241no—
318C4—242no—
319C3—21no—
320C3#6744911yescharacteristic, direct product factor
321C3C3×#135111yescharacteristic, complemented
322C2—21no—
323C2—21no—
324C2—41no—
325C2—41no—
326C2C3×#446711yescharacteristic
327C2C3×#447311yescharacteristic
328C2C3×#447511yescharacteristic
subgroupG/H
centerC22×C3#806
commutatorC22×C3×C4C2×C3×C4
FrattiniC2×C22⋊C4C2×C3×D3
FittingC32×#323C2
socleC22×C32#99
prime radicalC24×C32D4
cosocleC2×C3×C22⋊C4C22×C3
hypercenterC3×#323D3
hypocenterC3C3×#1351
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C2×C3×C22⋊C4 ◁ C3×#323

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #67449C3direct (N × H)
6. C22×C3×Dic6C4semidirect
12. C3×C3⋊#95C4semidirect
35. C22×Dic6C3×C4semidirect
49. C3⋊#95C3×C4semidirect
257. C32#1351semidirect
320. C3#67449direct (N × H)
321. C3C3×#1351semidirect

Maximal subgroups

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

subgroupindex
C3×#49472
C32×#3232
C3×#45262
C3×#13513
#674493