Finite Group Databasebeta

← C3×#68140 #11778587

Subgroups — C3×#68140 #11778587

Order 1152.

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

normal subgroups only

This group has 459 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×#4981C211yesmaximal, characteristic
3C3×#4721C212yesmaximal
4C32×#324C211yesmaximal, characteristic
5C3×#4467C212yesmaximal
6C3×#1611—31nomaximal
7#68140C311yesmaximal, characteristic, direct product factor
8C2×C32×C4⋊C4C2211yescharacteristic
9C3×#926—21no—
10C2×C3×C4⋊Dic3C2211yescharacteristic
11C3×#924—21no—
12C3×#924C2212yes—
13C2×C3×Dic3⋊C4—21no—
14C3×#890—22no—
15C3×#888—22no—
16C32×C22⋊C8C2212yes—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic
19C3×#817—22no—
20C3×#816—22no—
21C3×#544—31no—
22#5500C2×C311yescharacteristic
23C3×#483—32no—
24C3×#398—31no—
25#4981C2×C311yescharacteristic
26#4721C2×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×Dic3⋊C4—42no—
50C3×C4×Dic3—22no—
51#1611—31noSylow
52C3×C4⋊Q8—62no—
53C3×#125—62no—
54C3×C22⋊Q8—32no—
55C3×C22⋊Q8—61no—
56C3×#117—61no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—31no—
59C2×C3×C4⋊C4—61no—
60C2×C3×C4⋊C4C2×D311yescharacteristic
61C2×C3×C4⋊C4C22×C311yescharacteristic
62#926—21no—
63C2×C4⋊Dic3C22×C311yescharacteristic
64#924—21no—
65#924C22×C312yes—
66C2×Dic3⋊C4—21no—
67#890—22no—
68#888—22no—
69C3×#106—62no—
70C3×#103—62no—
71C3×C22⋊C8—22no—
72C3×C22⋊C8C2×D312yes—
73C3×C22⋊C8C22×C312yes—
74C3×#95—31no—
75C3×#95—61no—
76C3⋊#95—21no—
77C3⋊#95C22×C311yescharacteristic
78#817—22no—
79#816—22no—
80C23×C32C2×D411yescharacteristic
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4SD1612yes—
87C2×C32×C4C2×D412yes—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C3×Dic6—42no—
97C3×Dic6—81no—
98C32×C8—42no—
99#544—31no—
100#483—32no—
101#398—31no—
102#328—32no—
103#324C3×D311yescharacteristic
104C2×C3×Q8—32no—
105C2×C3×Q8—61no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—61no—
110C22×C3×C4—61no—
111C22×C3×C4D1211yescharacteristic
112C22×C3×C4D1211yescharacteristic
113C22×C3×C4C3×D411yescharacteristic
114C22×C3×C4C3×D411yescharacteristic
115C22×C3×C4C22×D311yescharacteristic
116C22×C3×C4C23×C311yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C2×Dic6—21no—
120C2×Dic6C3×D412yes—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—42no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—42no—
128C3×C4⋊C4—62no—
129C3×C4⋊C4—61no—
130C3×C4⋊C4—61no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4—62no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—122no—
135C3×C22⋊C4—62no—
136C3×C42—62no—
137#217—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3C3×D412yes—
143Dic3⋊C4—22no—
144Dic3⋊C4—22no—
145Dic3⋊C4—42no—
146C4×Dic3—22no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32#12011yescharacteristic
150C22×C32C2×SD1611yescharacteristic
151C22×C32#13711yescharacteristic
152C32×C4—22no—
153C32×C4—22no—
154C32×C4—41no—
155C3×Dic3—42no—
156C3×Dic3—42no—
157C3×Dic3—42no—
158C3×Dic3—81no—
159C4⋊Q8—62no—
160#125—62no—
161C22⋊Q8—32no—
162C22⋊Q8—61no—
163#117—61no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4C2×C3×D311yescharacteristic
167#106—62no—
168#103—62no—
169C22⋊C8C2×C3×D312yes—
170#95—31no—
171#95—61no—
172C23×C3—21no—
173C23×C3C2×D1211yescharacteristic
174C23×C3C2×C3×D411yescharacteristic
175C3×Q8—122no—
176C3×Q8—241no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4—121no—
202C2×C3×C4#20412yes—
203C2×C3×C4C3×SD1612yes—
204C2×C3×C4D3×D412yes—
205C2×C3×C4C2×C3×D412yes—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214Dic6—42no—
215Dic6—81no—
216C3×C8—42no—
217C3×C8—42no—
218C3×C8—82no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32#44811yescharacteristic
222C2×C32#45611yescharacteristic
223C2×C32#45811yescharacteristic
224C2×Q8—32no—
225C2×Q8—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C3×D1211yescharacteristic
229C22×C4C3×D1211yescharacteristic
230C22×C4C22×C3×D311yescharacteristic
231C2×C8—22no—
232C4⋊C4—22no—
233C4⋊C4—32no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—122no—
241C22⋊C4—62no—
242C42—62no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3#88211yescharacteristic
253C22×C3C2×#20411yescharacteristic
254C22×C3#90911yescharacteristic
255C22×C3C3×#12011yescharacteristic
256C22×C3C2×C3×SD1611yescharacteristic
257C22×C3C3×#13711yescharacteristic
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—42no—
263C3×C4—42no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—81no—
267C3×C4—122no—
268C3×C4—122no—
269C3×C4—122no—
270C3×C4—241no—
271Dic3—42no—
272Dic3—42no—
273Dic3—42no—
274Dic3—81no—
275C32#161111yescharacteristic, complemented, Sylow
276C23C2×C3×D1211yescharacteristic
277Q8—122no—
278Q8—241no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4C3×#20412yes—
293C2×C4C3×D3×D412yes—
294C8—42no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3#473811yescharacteristic
305C2×C3#474211yescharacteristic
306C2×C3#475211yescharacteristic
307C2×C3C3×#44811yescharacteristic
308C2×C3C3×#45611yescharacteristic
309C2×C3C3×#45811yescharacteristic
310C22—21no—
311C22—21no—
312C22C3×#88211yescharacteristic
313C22C2×C3×#20411yescharacteristic
314C22C3×#90911yescharacteristic
315C4—22no—
316C4—22no—
317C4—41no—
318C4—122no—
319C4—122no—
320C4—122no—
321C4—241no—
322C3—21no—
323C3#6814011yescharacteristic, direct product factor
324C3C3×#161111yescharacteristic, complemented
325C2—21no—
326C2—21no—
327C2C3×#473811yescharacteristic
328C2C3×#474211yescharacteristic
329C2C3×#475211yescharacteristic
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×#1611
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. #68140C3direct (N × H)
275. C32#1611semidirect
323. C3#68140direct (N × H)
324. C3C3×#1611semidirect

Maximal subgroups

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

subgroupindex
C3×#55002
C3×#49812
C3×#47212
C32×#3242
C3×#44672
C3×#16113
#681403