Finite Group Databasebeta

← C3×#68141 #11778814

Subgroups — C3×#68141 #11778814

Order 1152.

443 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 106 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×#5500C211yesmaximal, characteristic
2C3×#4986C211yesmaximal, characteristic
3C3×#4722C212yesmaximal
4C32×#324C211yesmaximal, characteristic
5C3×#4467C212yesmaximal
6C3×#1617—31nomaximal
7#68141C311yesmaximal, characteristic, direct product factor
8C2×C32×C4⋊C4C2211yescharacteristic
9C3×#926—21no—
10C2×C3×C4⋊Dic3C2211yescharacteristic
11C3×#924—21no—
12C3×#924C2212yes—
13C3×#890—22no—
14C3×#888—22no—
15C32×C22⋊C8C2212yes—
16C3×C3⋊#95—21no—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic
19C3×#818—22no—
20C3×#816—22no—
21C3×#544—31no—
22#5500C2×C311yescharacteristic
23C3×#484—32no—
24C3×#400—31no—
25#4986C2×C311yescharacteristic
26#4722C2×C312yes—
27C3×#328—32no—
28C3×#324—21no—
29C3×#324D311yescharacteristic
30C3×#324C2×C311yescharacteristic
31#4467C2×C312yes—
32C22×C32×C4D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C2×C3×Dic6—21no—
38C2×C3×Dic6D412yes—
39C2×C32×C8—22no—
40C32×C4⋊C4—22no—
41C3×#217—22no—
42C3×C4⋊Dic3—21no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—22no—
46C3×C4⋊Dic3D412yes—
47C3×Dic3⋊C4—22no—
48C3×Dic3⋊C4—22no—
49C3×C4×Dic3—22no—
50#1617—31noSylow
51C3×C4⋊Q8—62no—
52C3×#125—62no—
53C3×C22⋊Q8—32no—
54C3×C22⋊Q8—61no—
55C3×#117—61no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—31no—
58C2×C3×C4⋊C4C2×D311yescharacteristic
59C2×C3×C4⋊C4C22×C311yescharacteristic
60#926—21no—
61C2×C4⋊Dic3C22×C311yescharacteristic
62#924—21no—
63#924C22×C312yes—
64#890—22no—
65#888—22no—
66C3×#107—62no—
67C3×#103—62no—
68C3×C22⋊C8—22no—
69C3×C22⋊C8C2×D312yes—
70C3×C22⋊C8C22×C312yes—
71C3×#95—31no—
72C3×#95—61no—
73C3×#95—61no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95C22×C311yescharacteristic
77#818—22no—
78#816—22no—
79C23×C32C2×D411yescharacteristic
80C2×C32×C4—21no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4Q1612yes—
86C2×C32×C4C2×D412yes—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C3×Dic6—42no—
95C3×Dic6—81no—
96C32×C8—42no—
97#544—31no—
98#484—32no—
99#400—31no—
100#328—32no—
101#324C3×D311yescharacteristic
102C2×C3×Q8—32no—
103C2×C3×Q8—61no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—61no—
108C22×C3×C4—61no—
109C22×C3×C4D1211yescharacteristic
110C22×C3×C4D1211yescharacteristic
111C22×C3×C4C3×D411yescharacteristic
112C22×C3×C4C3×D411yescharacteristic
113C22×C3×C4C22×D311yescharacteristic
114C22×C3×C4C23×C311yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C2×Dic6—21no—
118C2×Dic6C3×D412yes—
119C2×C3×C8—22no—
120C2×C3×C8—22no—
121C2×C3×C8—42no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—62no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—61no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—62no—
131C3×C4⋊C4—62no—
132C3×C22⋊C4—62no—
133C3×C42—62no—
134#217—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3C3×D412yes—
140Dic3⋊C4—22no—
141Dic3⋊C4—22no—
142C4×Dic3—22no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32#12011yescharacteristic
146C22×C32C2×Q1611yescharacteristic
147C22×C32#13711yescharacteristic
148C32×C4—22no—
149C32×C4—22no—
150C32×C4—41no—
151C3×Dic3—42no—
152C3×Dic3—42no—
153C3×Dic3—81no—
154C3×Dic3—81no—
155C4⋊Q8—62no—
156#125—62no—
157C22⋊Q8—32no—
158C22⋊Q8—61no—
159#117—61no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4C2×C3×D311yescharacteristic
162#107—62no—
163#103—62no—
164C22⋊C8C2×C3×D312yes—
165#95—31no—
166#95—61no—
167#95—61no—
168C23×C3—21no—
169C23×C3C2×D1211yescharacteristic
170C23×C3C2×C3×D411yescharacteristic
171C3×Q8—122no—
172C3×Q8—241no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4—121no—
193C2×C3×C4—121no—
194C2×C3×C4—122no—
195C2×C3×C4—121no—
196C2×C3×C4—121no—
197C2×C3×C4Dic1212yes—
198C2×C3×C4C3×Q1612yes—
199C2×C3×C4D3×D412yes—
200C2×C3×C4C2×C3×D412yes—
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208Dic6—42no—
209Dic6—81no—
210C3×C8—42no—
211C3×C8—42no—
212C3×C8—82no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32#45111yescharacteristic
216C2×C32#45411yescharacteristic
217C2×C32#45811yescharacteristic
218C2×Q8—32no—
219C2×Q8—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C3×D1211yescharacteristic
223C22×C4C3×D1211yescharacteristic
224C22×C4C22×C3×D311yescharacteristic
225C2×C8—22no—
226C4⋊C4—22no—
227C4⋊C4—32no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C22⋊C4—62no—
235C42—62no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3#88211yescharacteristic
246C22×C3C2×Dic1211yescharacteristic
247C22×C3#90911yescharacteristic
248C22×C3C3×#12011yescharacteristic
249C22×C3C2×C3×Q1611yescharacteristic
250C22×C3C3×#13711yescharacteristic
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—42no—
256C3×C4—42no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—81no—
260C3×C4—122no—
261C3×C4—122no—
262C3×C4—241no—
263C3×C4—241no—
264Dic3—42no—
265Dic3—42no—
266Dic3—81no—
267Dic3—81no—
268C32#161711yescharacteristic, complemented, Sylow
269C23C2×C3×D1211yescharacteristic
270Q8—122no—
271Q8—241no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—122no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4C3×Dic1212yes—
285C2×C4C3×D3×D412yes—
286C8—42no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3#473911yescharacteristic
297C2×C3#474211yescharacteristic
298C2×C3#475311yescharacteristic
299C2×C3C3×#45111yescharacteristic
300C2×C3C3×#45411yescharacteristic
301C2×C3C3×#45811yescharacteristic
302C22—21no—
303C22—21no—
304C22C3×#88211yescharacteristic
305C22C2×C3×Dic1211yescharacteristic
306C22C3×#90911yescharacteristic
307C4—22no—
308C4—22no—
309C4—41no—
310C4—122no—
311C4—122no—
312C4—241no—
313C4—241no—
314C3—21no—
315C3#6814111yescharacteristic, direct product factor
316C3C3×#161711yescharacteristic, complemented
317C2—21no—
318C2—21no—
319C2C3×#473911yescharacteristic
320C2C3×#474211yescharacteristic
321C2C3×#475311yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#324C2
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC3×#324D3
hypocenterC3C3×#1617
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#324

Complements

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

NH ≅ G/Ntype
7. #68141C3direct (N × H)
268. C32#1617semidirect
315. C3#68141direct (N × H)
316. C3C3×#1617semidirect

Maximal subgroups

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

subgroupindex
C3×#55002
C3×#49862
C3×#47222
C32×#3242
C3×#44672
C3×#16173
#681413