Finite Group Databasebeta

← C3×#67451 #11771940

Subgroups — C3×#67451 #11771940

Order 1152.

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

normal subgroups only

This group has 357 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×#4949C211yesmaximal, characteristic
2C32×#323C211yesmaximal, characteristic
3C3×#4526C211yesmaximal, characteristic
4C3×#1352—31nomaximal
5#67451C311yesmaximal, characteristic, direct product factor
6C2×C32×C22⋊C4C2211yescharacteristic
7C2×C3×#217—21no—
8C2×C3×#217—21no—
9C2×C3×C4⋊Dic3C411yescharacteristic, complemented
10C2×C3×Dic3⋊C4—21no—
11C32×C22⋊C8—21no—
12C3×C3⋊#95C411yescharacteristic, complemented
13C3×#399—31no—
14#4949C2×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×C32×C8—41no—
27C32×C22⋊C4—41no—
28C3×#217—22no—
29C3×#217—42no—
30C3×#217—41no—
31C3×C4⋊Dic3—41no—
32C3×Dic3⋊C4—42no—
33#1352—31noSylow
34C2×C3×C4⋊C4—31no—
35C2×C3×C4⋊C4—61no—
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—
43C2×C4⋊Dic3C3×C411yescharacteristic, complemented
44C2×Dic3⋊C4—21no—
45C3×C22⋊C8—22no—
46C3×C22⋊C8—21no—
47C3×C22⋊C8—21no—
48C3×#95—31no—
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—41no—
65C2×C3×Dic3—42no—
66C2×C3×Dic3—41no—
67C2×C3×Dic3—41no—
68C2×C3×Dic3—81no—
69C2×C3×Dic3—81no—
70C32×C8—81no—
71#399—31no—
72#342—31no—
73#323C3×D311yescharacteristic
74C24×C3—21no—
75C24×C3D1211yescharacteristic
76C24×C3C3×D411yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—61no—
80C22×C3×C4—61no—
81C22×C3×C4—121no—
82C22×C3×C4C4×D311yescharacteristic
83C22×C3×C4#6711yescharacteristic
84C22×C3×C4C2×C3×C411yescharacteristic
85C22×C3×C4C3×D411yescharacteristic
86C22×Dic3—21no—
87C22×Dic3—21no—
88C22×Dic3—41no—
89C2×C3×C8—42no—
90C2×C3×C8—41no—
91C2×C3×C8—41no—
92C3×C4⋊C4—121no—
93C3×C4⋊C4—122no—
94C3×C22⋊C4—41no—
95C3×C22⋊C4—41no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—62no—
98C3×C22⋊C4—122no—
99C3×C22⋊C4—121no—
100#217—22no—
101#217—42no—
102#217—41no—
103C4⋊Dic3—41no—
104Dic3⋊C4—42no—
105C22×C32—21no—
106C22×C32—21no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32—41no—
110C22×C32—41no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32—81no—
114C22×C32#9911yescharacteristic
115C22×C32#10311yescharacteristic
116C22×C32#10411yescharacteristic
117C32×C4—41no—
118C32×C4—41no—
119C3×Dic3—42no—
120C3×Dic3—82no—
121C3×Dic3—81no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—61no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—61no—
126C2×C22⋊C4C2×C3×D311yescharacteristic
127C22⋊C8—21no—
128#95—31no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—41no—
137C23×C3—41no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3—81no—
143C23×C3—81no—
144C23×C3#20411yescharacteristic
145C23×C3Dic1211yescharacteristic
146C23×C3(C2×D3)⋊C411yescharacteristic
147C23×C3C3×C22⋊C411yescharacteristic
148C23×C3C3×SD1611yescharacteristic
149C23×C3C3×Q1611yescharacteristic
150C2×C3×C4—22no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—62no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×C3×C4—121no—
164C2×C3×C4—121no—
165C2×C3×C4—122no—
166C2×C3×C4—121no—
167C2×C3×C4—121no—
168C2×C3×C4—121no—
169C2×C3×C4—241no—
170C2×C3×C4—241no—
171C2×Dic3—22no—
172C2×Dic3—41no—
173C2×Dic3—41no—
174C2×Dic3—41no—
175C2×Dic3—42no—
176C2×Dic3—41no—
177C2×Dic3—41no—
178C2×Dic3—81no—
179C2×Dic3—81no—
180C3×C8—82no—
181C3×C8—81no—
182C3×C8—81no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32—41no—
186C2×C32—41no—
187C2×C32#32811yescharacteristic
188C2×C32#35211yescharacteristic
189C2×C32#35311yescharacteristic
190C24C3×D1211yescharacteristic
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—121no—
194C22×C4C3×C4×D311yescharacteristic
195C22×C4C3×#6711yescharacteristic
196C2×C8—41no—
197C4⋊C4—121no—
198C4⋊C4—122no—
199C22⋊C4—41no—
200C22⋊C4—62no—
201C22⋊C4—122no—
202C22⋊C4—121no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—81no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3—81no—
230C22×C3—81no—
231C22×C3—81no—
232C22×C3—82no—
233C22×C3#80611yescharacteristic
234C22×C3#81611yescharacteristic
235C22×C3#82511yescharacteristic
236C22×C3C3×#9911yescharacteristic
237C22×C3C3×#10311yescharacteristic
238C22×C3C3×#10411yescharacteristic
239C3×C4—42no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—81no—
245C3×C4—122no—
246C3×C4—242no—
247C3×C4—241no—
248Dic3—42no—
249Dic3—82no—
250Dic3—81no—
251C32#135211yescharacteristic, complemented, Sylow
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C3×#20411yescharacteristic
257C23C3×Dic1211yescharacteristic
258C23C3×(C2×D3)⋊C411yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—62no—
264C2×C4—121no—
265C2×C4—121no—
266C2×C4—122no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4—241no—
271C2×C4—241no—
272C8—81no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—81no—
287C2×C3—81no—
288C2×C3#446711yescharacteristic
289C2×C3#447411yescharacteristic
290C2×C3#447611yescharacteristic
291C2×C3C3×#32811yescharacteristic
292C2×C3C3×#35211yescharacteristic
293C2×C3C3×#35311yescharacteristic
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C3×#80611yescharacteristic
304C22C3×#81611yescharacteristic
305C22C3×#82511yescharacteristic
306C4—41no—
307C4—41no—
308C4—122no—
309C4—242no—
310C4—241no—
311C3—21no—
312C3#6745111yescharacteristic, direct product factor
313C3C3×#135211yescharacteristic, complemented
314C2—21no—
315C2—21no—
316C2—41no—
317C2—41no—
318C2C3×#446711yescharacteristic
319C2C3×#447411yescharacteristic
320C2C3×#447611yescharacteristic
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×#1352
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. #67451C3direct (N × H)
9. C2×C3×C4⋊Dic3C4semidirect
12. C3×C3⋊#95C4semidirect
43. C2×C4⋊Dic3C3×C4semidirect
49. C3⋊#95C3×C4semidirect
251. C32#1352semidirect
312. C3#67451direct (N × H)
313. C3C3×#1352semidirect

Maximal subgroups

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

subgroupindex
C3×#49492
C32×#3232
C3×#45262
C3×#13523
#674513