Finite Group Databasebeta

← C3×#71898 #11778581

Subgroups — C3×#71898 #11778581

Order 1152.

523 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C32×#544C211yesmaximal, characteristic
2C3×#5043C212yesmaximal
3C3×#4981C211yesmaximal, characteristic
4C3×#4538C212yesmaximal
5C3×#4528C211yesmaximal, characteristic
6C3×#1611—31nomaximal
7#71898C311yesmaximal, characteristic, direct product factor
8C32×C4⋊Q8—22no—
9C32×#125—22no—
10C32×C22⋊Q8—21no—
11C32×C22⋊Q8C2212yes—
12C32×#117—21no—
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊Dic3C2211yescharacteristic
15C2×C3×Dic3⋊C4—21no—
16C3×#835—22no—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic
19C3×#830C2212yes—
20C3×#808—22no—
21C3×#544—21no—
22C3×#544D311yescharacteristic
23C3×#544C2×C311yescharacteristic
24C3×#483—32no—
25C3×#398—31no—
26#5043C2×C312yes—
27#4981C2×C311yescharacteristic
28C3×#328—32no—
29C3×#324—31no—
30#4538C2×C312yes—
31#4528C2×C311yescharacteristic
32C2×C32×Q8—21no—
33C2×C32×Q8D412yes—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C22×C3×Dic3—21no—
38C22×C3×Dic3—21no—
39C32×C4⋊C4—21no—
40C32×C4⋊C4—21no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4—22no—
43C32×C4⋊C4—22no—
44C32×C4⋊C4—22no—
45C32×C4⋊C4D412yes—
46C32×C22⋊C4—22no—
47C32×C42—22no—
48C3×C4⋊Dic3—22no—
49C3×Dic3⋊C4—42no—
50C2×C3×C3⋊C8—22no—
51#1611—31noSylow
52C3×C4⋊Q8—24no—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—22no—
55C3×#125—24no—
56C3×#125—22no—
57C3×#125—22no—
58C3×C22⋊Q8—22no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8C2×D312yes—
63C3×C22⋊Q8C22×C312yes—
64C3×#117—22no—
65C3×#117—21no—
66C3×#117—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—31no—
69C2×C3×C4⋊C4—61no—
70C2×C3×C4⋊C4C2×D311yescharacteristic
71C2×C3×C4⋊C4C22×C311yescharacteristic
72C2×C4⋊Dic3C22×C311yescharacteristic
73C2×Dic3⋊C4—21no—
74C3×#106—62no—
75C3×#103—62no—
76C3×C22⋊C8—32no—
77C3×#95—31no—
78C3×#95—61no—
79#835—22no—
80C3⋊#95—21no—
81C3⋊#95C22×C311yescharacteristic
82#830C22×C312yes—
83#808—22no—
84C23×C32C2×D411yescharacteristic
85C32×Q8—42no—
86C32×Q8—81no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4SD1612yes—
95C2×C32×C4C2×D412yes—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—41no—
102C3×C3⋊C8—42no—
103#544C3×D311yescharacteristic
104#483—32no—
105#398—31no—
106#328—32no—
107#324—31no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8#6712yes—
113C2×C3×Q8C3×D412yes—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—61no—
118C22×C3×C4—61no—
119C22×C3×C4#6711yescharacteristic
120C22×C3×C4#6711yescharacteristic
121C22×C3×C4C3×D411yescharacteristic
122C22×C3×C4C3×D411yescharacteristic
123C22×C3×C4C22×D311yescharacteristic
124C22×C3×C4C23×C311yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—21no—
127C2×C3×C8—62no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—62no—
148C3×C4⋊C4—122no—
149C3×C4⋊C4#6712yes—
150C3×C4⋊C4C3×D412yes—
151C3×C22⋊C4—24no—
152C3×C22⋊C4—22no—
153C3×C22⋊C4—22no—
154C3×C42—24no—
155C3×C42—22no—
156C3×C42—22no—
157C4⋊Dic3—22no—
158Dic3⋊C4—42no—
159C2×C3⋊C8—22no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32#12011yescharacteristic
163C22×C32C2×SD1611yescharacteristic
164C22×C32#13711yescharacteristic
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—42no—
168C32×C4—41no—
169C32×C4—42no—
170C3×Dic3—42no—
171C3×Dic3—81no—
172C4⋊Q8—22no—
173#125—22no—
174C22⋊Q8—21no—
175C22⋊Q8C2×C3×D312yes—
176#117—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4C2×C3×D311yescharacteristic
180#106—62no—
181#103—62no—
182C22⋊C8—32no—
183#95—31no—
184#95—61no—
185C23×C3—21no—
186C23×C3C2×#6711yescharacteristic
187C23×C3C2×C3×D411yescharacteristic
188C3×Q8—42no—
189C3×Q8—42no—
190C3×Q8—82no—
191C3×Q8—82no—
192C3×Q8—81no—
193C3×Q8—81no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—62no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4—121no—
219C2×C3×C4—121no—
220C2×C3×C4—121no—
221C2×C3×C4—121no—
222C2×C3×C4—121no—
223C2×C3×C4#21512yes—
224C2×C3×C4C3×SD1612yes—
225C2×C3×C4C2×#6712yes—
226C2×C3×C4C2×C3×D412yes—
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C3×C8—122no—
234C3⋊C8—42no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32#44811yescharacteristic
238C2×C32#45611yescharacteristic
239C2×C32#45811yescharacteristic
240C2×Q8—21no—
241C2×Q8C3×#6712yes—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4C3×#6711yescharacteristic
245C22×C4C3×#6711yescharacteristic
246C22×C4C22×C3×D311yescharacteristic
247C2×C8—62no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—62no—
255C4⋊C4—122no—
256C4⋊C4C3×#6712yes—
257C22⋊C4—22no—
258C42—22no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3C2×#21511yescharacteristic
269C22×C3#94211yescharacteristic
270C22×C3#95311yescharacteristic
271C22×C3C3×#12011yescharacteristic
272C22×C3C2×C3×SD1611yescharacteristic
273C22×C3C3×#13711yescharacteristic
274C3×C4—24no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—42no—
280C3×C4—42no—
281C3×C4—44no—
282C3×C4—44no—
283C3×C4—41no—
284C3×C4—42no—
285C3×C4—42no—
286C3×C4—42no—
287C3×C4—41no—
288C3×C4—42no—
289C3×C4—122no—
290C3×C4—241no—
291Dic3—42no—
292Dic3—81no—
293C32#161111yescharacteristic, complemented, Sylow
294C23C2×C3×#6711yescharacteristic
295Q8—42no—
296Q8—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—62no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4C3×#21512yes—
311C2×C4C2×C3×#6712yes—
312C8—122no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3#522711yescharacteristic
323C2×C3#524311yescharacteristic
324C2×C3#524611yescharacteristic
325C2×C3C3×#44811yescharacteristic
326C2×C3C3×#45611yescharacteristic
327C2×C3C3×#45811yescharacteristic
328C22—21no—
329C22—21no—
330C22C2×C3×#21511yescharacteristic
331C22C3×#94211yescharacteristic
332C22C3×#95311yescharacteristic
333C4—22no—
334C4—22no—
335C4—41no—
336C4—42no—
337C4—42no—
338C4—122no—
339C4—241no—
340C3—21no—
341C3#7189811yescharacteristic, direct product factor
342C3C3×#161111yescharacteristic, complemented
343C2—21no—
344C2—21no—
345C2C3×#522711yescharacteristic
346C2C3×#524311yescharacteristic
347C2C3×#524611yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#544C2
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC3×#544D3
hypocenterC3C3×#1611
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#544

Complements

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

NH ≅ G/Ntype
7. #71898C3direct (N × H)
293. C32#1611semidirect
341. C3#71898direct (N × H)
342. C3C3×#1611semidirect

Maximal subgroups

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

subgroupindex
C32×#5442
C3×#50432
C3×#49812
C3×#45382
C3×#45282
C3×#16113
#718983