Finite Group Databasebeta

← C3×#71949 #11780564

Subgroups — C3×#71949 #11780564

Order 1152.

586 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 98 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×H(3,𝔽4)C211yesmaximal, characteristic
2C3×#5060C211yesmaximal, characteristic, complemented
3C3×#5055C212yesmaximal, complemented
4C3×#4540C211yesmaximal, characteristic, complemented
5C3×#4540C212yesmaximal, complemented
6C3×#1659—31nomaximal
7#71949C311yesmaximal, characteristic, direct product factor
8C32×#124—21no—
9C32×#124—22no—
10C32×#124C2212yes—
11C32×#124C2211yescharacteristic
12C32×#120—21no—
13C32×#120—22no—
14C2×C3×#216—21no—
15C2×C3×#214—21no—
16C3×#869C2211yescharacteristic, complemented
17C3×#835—22no—
18C3×#832—22no—
19C3×#803C2212yescomplemented
20C3×#803C2211yescharacteristic, complemented
21C3×H(3,𝔽4)—21no—
22C3×H(3,𝔽4)D311yescharacteristic
23C3×H(3,𝔽4)C2×C311yescharacteristic
24C3×#497—31no—
25C3×#489—32no—
26#5060C2×C311yescharacteristic, complemented
27#5055C2×C312yescomplemented
28C3×#329—31no—
29C3×#329—32no—
30#4540C2×C311yescharacteristic, complemented
31#4540C2×C312yescomplemented
32C24×C32—21no—
33C2×C32×Q8D411yescharacteristic
34C2×C32×Q8D412yes—
35C2×C32×D4—21no—
36C2×C32×D4—22no—
37C2×C32×D4D411yescharacteristic
38C2×C32×D4D412yes—
39C2×C3×Dic6—21no—
40C32×C22⋊C4—22no—
41C32×C22⋊C4—21no—
42C32×C22⋊C4—22no—
43C32×C22⋊C4—22no—
44C32×C22⋊C4—22no—
45C32×C42—21no—
46C32×C42C2311yescharacteristic
47C3×#216—42no—
48C3×#214—42no—
49C3×C4⋊Dic3—22no—
50C2×C3×C3⋊C8—21no—
51C2×C3×C3⋊C8—22no—
52#1659—31noSylow
53C2×C3×Q16—61no—
54C2×C3×SD16—61no—
55C3×C4⋊Q8—31no—
56C3×#124—24no—
57C3×#124—22no—
58C3×#124—22no—
59C3×#124—21no—
60C3×#124—22no—
61C3×#124—21no—
62C3×#124—22no—
63C3×#124—21no—
64C3×#124C2×D312yes—
65C3×#124C2×D311yescharacteristic
66C3×#124C22×C312yes—
67C3×#124C22×C311yescharacteristic
68C3×#120—24no—
69C3×#120—22no—
70C3×#120—22no—
71C3×#120—21no—
72C3×#120—22no—
73C3×#120—21no—
74C2×#216—21no—
75C2×#214—21no—
76#869C22×C311yescharacteristic, complemented
77C3×#103—62no—
78C3×#102—62no—
79C3×#97—32no—
80C3×#97—31no—
81#835—22no—
82#832—22no—
83#803C22×C312yescomplemented
84#803C22×C311yescharacteristic, complemented
85C23×C32—21no—
86C23×C32—22no—
87C23×C32—81no—
88C23×C32—82no—
89C32×Q8—41no—
90C32×Q8—42no—
91C32×D4—41no—
92C32×D4—42no—
93C32×D4—81no—
94C32×D4—82no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4C2×D412yes—
98C2×C32×C4C2×D411yescharacteristic
99C2×C3×Dic3—41no—
100C3×Dic6—41no—
101C3×Dic6—41no—
102C3×C3⋊C8—22no—
103C3×C3⋊C8—42no—
104H(3,𝔽4)C3×D311yescharacteristic
105#497—31no—
106#489—32no—
107#329—31no—
108#329—32no—
109C24×C3—22no—
110C24×C3—21no—
111C24×C3—21no—
112C2×C3×Q8—22no—
113C2×C3×Q8—21no—
114C2×C3×Q8—61no—
115C2×C3×Q8#6712yes—
116C2×C3×Q8#6711yescharacteristic
117C2×C3×Q8C3×D412yes—
118C2×C3×Q8C3×D411yescharacteristic
119C2×C3×D4—22no—
120C2×C3×D4—24no—
121C2×C3×D4—21no—
122C2×C3×D4—22no—
123C2×C3×D4—21no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—21no—
127C2×C3×D4#6712yes—
128C2×C3×D4#6711yescharacteristic
129C2×C3×D4C3×D412yes—
130C2×C3×D4C3×D411yescharacteristic
131C2×Dic6—21no—
132C3×Q16—122no—
133C3×SD16—122no—
134C2×C3×C8—61no—
135C2×C3×C8—62no—
136C3×C4⋊C4—62no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—21no—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—21no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C42—22no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C42—21no—
156C3×C42C22×D311yescharacteristic
157C3×C42C23×C311yescharacteristic
158#216—42no—
159#214—42no—
160C4⋊Dic3—22no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—22no—
163C22×C32—41no—
164C22×C32—42no—
165C22×C32—42no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32—84no—
169C22×C32#12011yescharacteristic
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—41no—
173C32×C4—42no—
174C3×Dic3—81no—
175C2×Q16—61no—
176C2×SD16—61no—
177C4⋊Q8—31no—
178#124—22no—
179#124—21no—
180#124C2×C3×D312yes—
181#124C2×C3×D311yescharacteristic
182#120—22no—
183#120—21no—
184#103—62no—
185#102—62no—
186#97—32no—
187#97—31no—
188C23×C3—22no—
189C23×C3—24no—
190C23×C3—21no—
191C23×C3—22no—
192C23×C3—21no—
193C23×C3—22no—
194C23×C3—82no—
195C23×C3—84no—
196C23×C3—81no—
197C23×C3—82no—
198C23×C3—81no—
199C23×C3—82no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—42no—
203C3×Q8—42no—
204C3×Q8—81no—
205C3×Q8—82no—
206C3×Q8—121no—
207C3×Q8—121no—
208C3×D4—41no—
209C3×D4—41no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—81no—
213C3×D4—82no—
214C3×D4—82no—
215C3×D4—84no—
216C3×D4—81no—
217C3×D4—82no—
218C3×D4—81no—
219C3×D4—82no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—121no—
229C2×C3×C4C2×#6712yes—
230C2×C3×C4C2×#6711yescharacteristic
231C2×C3×C4C2×C3×D412yes—
232C2×C3×C4C2×C3×D411yescharacteristic
233C2×Dic3—41no—
234Dic6—41no—
235Dic6—41no—
236C3×C8—62no—
237C3×C8—122no—
238C3⋊C8—22no—
239C3⋊C8—42no—
240C2×C32—81no—
241C2×C32—82no—
242C2×C32#45311yescharacteristic
243C2×C32#45512yes—
244C24—21no—
245C2×Q8—61no—
246C2×Q8C3×#6712yes—
247C2×Q8C3×#6711yescharacteristic
248C2×D4—21no—
249C2×D4—22no—
250C2×D4C3×#6712yes—
251C2×D4C3×#6711yescharacteristic
252Q16—122no—
253SD16—122no—
254C2×C8—61no—
255C2×C8—62no—
256C4⋊C4—62no—
257C22⋊C4—22no—
258C22⋊C4—21no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C42—21no—
263C42C22×C3×D311yescharacteristic
264C22×C3—21no—
265C22×C3—42no—
266C22×C3—44no—
267C22×C3—44no—
268C22×C3—44no—
269C22×C3—44no—
270C22×C3—41no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3—42no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—88no—
281C22×C3—84no—
282C22×C3—84no—
283C22×C3#95311yescharacteristic
284C22×C3C3×#12011yescharacteristic
285C3×C4—21no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—22no—
289C3×C4—41no—
290C3×C4—42no—
291C3×C4—42no—
292C3×C4—44no—
293C3×C4—41no—
294C3×C4—42no—
295C3×C4—41no—
296C3×C4—42no—
297C3×C4—241no—
298Dic3—81no—
299C32#165911yescharacteristic, complemented, Sylow
300C23—21no—
301C23—22no—
302C23—81no—
303C23—82no—
304Q8—41no—
305Q8—42no—
306Q8—121no—
307Q8—121no—
308D4—41no—
309D4—42no—
310D4—81no—
311D4—82no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—121no—
315C2×C4C2×C3×#6712yes—
316C2×C4C2×C3×#6711yescharacteristic
317C8—62no—
318C8—122no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—82no—
322C2×C3—84no—
323C2×C3—81no—
324C2×C3—82no—
325C2×C3—81no—
326C2×C3—82no—
327C2×C3#524111yescharacteristic
328C2×C3#524212yes—
329C2×C3C3×#45311yescharacteristic
330C2×C3C3×#45512yes—
331C22—41no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—84no—
337C22C3×#95311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—42no—
342C4—241no—
343C3—21no—
344C3#7194911yescharacteristic, direct product factor
345C3C3×#165911yescharacteristic, complemented
346C2—81no—
347C2—82no—
348C2C3×#524111yescharacteristic
349C2C3×#524212yes—
subgroupG/H
centerC22×C3#953
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
Fitting, prime radicalC32×H(3,𝔽4)C2
socleC22×C32#120
hypercenterC3×H(3,𝔽4)D3
hypocenterC3C3×#1659
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
2. C3×#5060C2semidirect
3. C3×#5055C2semidirect
4. C3×#4540C2semidirect
5. C3×#4540C2semidirect
7. #71949C3direct (N × H)
16. C3×#869C22semidirect
19. C3×#803C22semidirect
20. C3×#803C22semidirect
26. #5060C2×C3semidirect
27. #5055C2×C3semidirect
30. #4540C2×C3semidirect
31. #4540C2×C3semidirect
76. #869C22×C3semidirect
83. #803C22×C3semidirect
84. #803C22×C3semidirect
299. C32#1659semidirect
344. C3#71949direct (N × H)
345. C3C3×#1659semidirect

Maximal subgroups

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

subgroupindex
C32×H(3,𝔽4)2
C3×#50602
C3×#50552
C3×#45402
C3×#16593
#719493