Finite Group Databasebeta

← C3×#82612 #11856156

Subgroups — C3×#82612 #11856156

Order 1152.

7175 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 7175 conjugacy classes of subgroups — too many to draw. Only its 490 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
1C25×C3×D3C211yesmaximal, characteristic, complemented
2C2×C32×#120C211yesmaximal, characteristic, complemented
3C2×C3×#937C216yesmaximal, complemented
4C3×#5247C211yesmaximal, characteristic, complemented
5C3×#4955C216yesmaximal, complemented
6C3×#2846—31nomaximal
7#82612C311yesmaximal, characteristic, direct product factor
8C25×C32C2211yescharacteristic
9C24×C3×D3—24no—
10C24×C3×D3—224no—
11C24×C3×D3C2216yes—
12C22×C32×D4C2213yescomplemented
13C22×C3×#67C2214yescomplemented
14C32×#120—24no—
15C2×C32×C22⋊C4C2213yescomplemented
16C2×C3×#217C2216yescomplemented
17C3×#937—224no—
18C2×C3×(C2×D3)⋊C4C22112yescomplemented
19C26×C3—31no—
20C25×D3C2×C311yescharacteristic, complemented
21C2×C3×#120—21no—
22C2×C3×#120—36no—
23C2×C3×#120D311yescharacteristic, complemented
24C2×C3×#120C2×C311yescharacteristic, complemented
25C2×#937C2×C316yescomplemented
26C3×#379—36no—
27C3×#379—31no—
28#5247C2×C311yescharacteristic, complemented
29#4955C2×C316yescomplemented
30C24×C32—212no—
31C24×C32D414yescomplemented
32C24×C32C2313yes—
33C24×C32C2311yescharacteristic, complemented
34C23×C3×D3—296no—
35C23×C3×D3—2128no—
36C23×C3×D3—212no—
37C23×C3×D3—212no—
38C23×C3×D3—248no—
39C23×C3×D3D4124yescomplemented
40C23×C3×D3C2314yes—
41C2×C32×D4—26no—
42C2×C32×D4—212no—
43C22×C32×C4C2313yescomplemented
44C2×C3×#67—224no—
45C22×C3×Dic3C2314yescomplemented
46C32×C22⋊C4—26no—
47C3×#217—212no—
48C3×(C2×D3)⋊C4—224no—
49#2846—31noSylow
50C25×C3—21no—
51C25×C3—36no—
52C25×C3—624no—
53C25×C3—64no—
54C25×C3C2×D311yescharacteristic
55C25×C3C22×C311yescharacteristic
56C24×D3—24no—
57C24×D3—224no—
58C24×D3C22×C316yes—
59C22×C3×D4—23no—
60C22×C3×D4—34no—
61C22×C3×D4C2×D313yescomplemented
62C22×C3×D4C22×C313yescomplemented
63C22×#67C22×C314yescomplemented
64C3×#120—28no—
65C3×#120—24no—
66C3×#120—24no—
67C3×#120—624no—
68C2×C3×C22⋊C4—23no—
69C2×C3×C22⋊C4—36no—
70C2×C3×C22⋊C4—312no—
71C2×C3×C22⋊C4C2×D313yescomplemented
72C2×C3×C22⋊C4C22×C313yescomplemented
73C2×#217C22×C316yescomplemented
74#937—224no—
75C2×(C2×D3)⋊C4C22×C3112yescomplemented
76C23×C32—21no—
77C23×C32—232no—
78C23×C32—224no—
79C23×C32—212no—
80C23×C32—26no—
81C23×C32C2×D4112yes—
82C23×C32C2×D416yes—
83C23×C32C2411yescharacteristic
84C22×C3×D3—2256no—
85C22×C3×D3—296no—
86C22×C3×D3—296no—
87C22×C3×D3—248no—
88C22×C3×D3—212no—
89C22×C3×D3—24no—
90C22×C3×D3—296no—
91C22×C3×D3C2×D4124yes—
92C32×D4—412no—
93C2×C32×C4—23no—
94C2×C32×C4—26no—
95C3×#67—416no—
96C2×C3×Dic3—212no—
97C26—31no—
98C2×#120—36no—
99C2×#120C3×D311yescharacteristic, complemented
100#379—31no—
101#379—36no—
102C24×C3—212no—
103C24×C3—212no—
104C24×C3—24no—
105C24×C3—23no—
106C24×C3—21no—
107C24×C3—34no—
108C24×C3—324no—
109C24×C3—412no—
110C24×C3—648no—
111C24×C3—6128no—
112C24×C3—696no—
113C24×C3—612no—
114C24×C3—612no—
115C24×C3#6714yescomplemented
116C24×C3C3×D414yescomplemented
117C24×C3C22×D313yes—
118C24×C3C22×D311yescharacteristic, complemented
119C24×C3C23×C313yes—
120C24×C3C23×C311yescharacteristic, complemented
121C23×D3—296no—
122C23×D3—2128no—
123C23×D3—212no—
124C23×D3—212no—
125C23×D3—248no—
126C23×D3C3×D4124yescomplemented
127C23×D3C23×C314yes—
128C2×C3×D4—212no—
129C2×C3×D4—212no—
130C2×C3×D4—26no—
131C2×C3×D4—212no—
132C2×C3×D4—26no—
133C2×C3×D4—412no—
134C2×C3×D4—624no—
135C22×C3×C4—23no—
136C22×C3×C4—34no—
137C22×C3×C4C22×D313yescomplemented
138C22×C3×C4C23×C313yescomplemented
139C2×#67—224no—
140C22×Dic3C23×C314yescomplemented
141C3×C22⋊C4—212no—
142C3×C22⋊C4—26no—
143C3×C22⋊C4—26no—
144C3×C22⋊C4—624no—
145C3×C22⋊C4—612no—
146#217—212no—
147(C2×D3)⋊C4—224no—
148C22×C32—232no—
149C22×C32—212no—
150C22×C32—224no—
151C22×C32—43no—
152C22×C32—44no—
153C22×C32#120112yescomplemented
154C22×C32C22×D416yes—
155C22×C32C22×D411yescharacteristic
156C2×C3×D3—2192no—
157C2×C3×D3—224no—
158C2×C3×D3—224no—
159C2×C3×D3#120116yescomplemented
160C32×C4—43no—
161C3×Dic3—44no—
162C25—36no—
163C25—624no—
164C25—64no—
165C25C2×C3×D311yescharacteristic
166C22×D4—34no—
167C22×D4C2×C3×D313yescomplemented
168#120—24no—
169#120—624no—
170C2×C22⋊C4—36no—
171C2×C22⋊C4—312no—
172C2×C22⋊C4C2×C3×D313yescomplemented
173C23×C3—232no—
174C23×C3—232no—
175C23×C3—224no—
176C23×C3—224no—
177C23×C3—212no—
178C23×C3—212no—
179C23×C3—22no—
180C23×C3—26no—
181C23×C3—21no—
182C23×C3—26no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—212no—
186C23×C3—26no—
187C23×C3—324no—
188C23×C3—424no—
189C23×C3—412no—
190C23×C3—46no—
191C23×C3—432no—
192C23×C3—696no—
193C23×C3—6256no—
194C23×C3—696no—
195C23×C3—696no—
196C23×C3—648no—
197C23×C3—612no—
198C23×C3—64no—
199C23×C3D3×D4112yes—
200C23×C3C2×#6716yes—
201C23×C3C2×C3×D4112yes—
202C23×C3C2×C3×D416yes—
203C23×C3C23×D311yescharacteristic
204C23×C3C24×C311yescharacteristic
205C22×D3—2256no—
206C22×D3—296no—
207C22×D3—296no—
208C22×D3—248no—
209C22×D3—212no—
210C22×D3—24no—
211C22×D3—296no—
212C22×D3C2×C3×D4124yes—
213C3×D4—424no—
214C3×D4—412no—
215C3×D4—412no—
216C3×D4—1216no—
217C2×C3×C4—26no—
218C2×C3×C4—26no—
219C2×C3×C4—23no—
220C2×C3×C4—26no—
221C2×C3×C4—23no—
222C2×C3×C4—46no—
223C2×C3×C4—612no—
224#67—416no—
225C2×Dic3—212no—
226C2×C32—212no—
227C2×C32—81no—
228C2×C32C2×#12013yes—
229C2×C32C2×#12014yes—
230C3×D3—216no—
231C24—212no—
232C24—34no—
233C24—324no—
234C24—648no—
235C24—6128no—
236C24—696no—
237C24—612no—
238C24—612no—
239C24C3×#6714yescomplemented
240C24C22×C3×D313yes—
241C24C22×C3×D311yescharacteristic, complemented
242C2×D4—212no—
243C2×D4—26no—
244C2×D4—624no—
245C22×C4—34no—
246C22×C4C22×C3×D313yescomplemented
247C22⋊C4—26no—
248C22⋊C4—624no—
249C22⋊C4—612no—
250C22×C3—232no—
251C22×C3—232no—
252C22×C3—212no—
253C22×C3—224no—
254C22×C3—212no—
255C22×C3—224no—
256C22×C3—212no—
257C22×C3—26no—
258C22×C3—21no—
259C22×C3—316no—
260C22×C3—412no—
261C22×C3—424no—
262C22×C3—432no—
263C22×C3—46no—
264C22×C3—43no—
265C22×C3—44no—
266C22×C3—44no—
267C22×C3—43no—
268C22×C3—6192no—
269C22×C3—624no—
270C22×C3—624no—
271C22×C3—84no—
272C22×C3#937112yescomplemented
273C22×C3C3×#120112yescomplemented
274C22×C3C2×D3×D416yes—
275C22×C3C22×#6711yescharacteristic
276C22×C3C22×C3×D416yes—
277C22×C3C22×C3×D411yescharacteristic
278C2×D3—2192no—
279C2×D3—224no—
280C2×D3—224no—
281C2×D3C3×#120116yescomplemented
282C3×C4—46no—
283C3×C4—43no—
284C3×C4—43no—
285C3×C4—124no—
286Dic3—44no—
287C32#284611yescharacteristic, complemented, Sylow
288C23—232no—
289C23—224no—
290C23—212no—
291C23—26no—
292C23—21no—
293C23—324no—
294C23—696no—
295C23—6256no—
296C23—696no—
297C23—696no—
298C23—648no—
299C23—612no—
300C23—64no—
301C23C3×D3×D4112yes—
302C23C2×C3×#6716yes—
303C23C23×C3×D311yescharacteristic
304D4—412no—
305D4—1216no—
306C2×C4—26no—
307C2×C4—23no—
308C2×C4—612no—
309C2×C3—212no—
310C2×C3—212no—
311C2×C3—23no—
312C2×C3—24no—
313C2×C3—412no—
314C2×C3—616no—
315C2×C3—82no—
316C2×C3—81no—
317C2×C3—81no—
318C2×C3D3×#12014yes—
319C2×C3C2×#93713yes—
320C2×C3C2×C3×#12013yes—
321C2×C3C2×C3×#12014yes—
322D3—216no—
323C22—232no—
324C22—212no—
325C22—224no—
326C22—316no—
327C22—43no—
328C22—44no—
329C22—6192no—
330C22—624no—
331C22—624no—
332C22C3×#937112yescomplemented
333C22C2×C3×D3×D416yes—
334C22C22×C3×#6711yescharacteristic
335C4—43no—
336C4—124no—
337C3—21no—
338C3#8261211yescharacteristic, direct product factor
339C3C3×#284611yescharacteristic, complemented
340C2—212no—
341C2—616no—
342C2—81no—
343C2C3×D3×#12014yes—
344C2C2×C3×#93713yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC2×C32×#120C2
socleC23×C32C24
hypercenterC2×C3×#120D3
hypocenterC3C3×#2846
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#120

Complements

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

NH ≅ G/Ntype
1. C25×C3×D3C2semidirect
2. C2×C32×#120C2semidirect
3. C2×C3×#937C2semidirect
4. C3×#5247C2semidirect
5. C3×#4955C2semidirect
7. #82612C3direct (N × H)
12. C22×C32×D4C22semidirect
13. C22×C3×#67C22semidirect
15. C2×C32×C22⋊C4C22semidirect
16. C2×C3×#217C22semidirect
18. C2×C3×(C2×D3)⋊C4C22semidirect
20. C25×D3C2×C3semidirect
23. C2×C3×#120D3semidirect
24. C2×C3×#120C2×C3semidirect
25. C2×#937C2×C3semidirect
28. #5247C2×C3semidirect
29. #4955C2×C3semidirect
31. C24×C32D4semidirect
33. C24×C32C23semidirect
39. C23×C3×D3D4semidirect
43. C22×C32×C4C23semidirect
45. C22×C3×Dic3C23semidirect
61. C22×C3×D4C2×D3semidirect
62. C22×C3×D4C22×C3semidirect
63. C22×#67C22×C3semidirect
71. C2×C3×C22⋊C4C2×D3semidirect
72. C2×C3×C22⋊C4C22×C3semidirect
73. C2×#217C22×C3semidirect
75. C2×(C2×D3)⋊C4C22×C3semidirect
99. C2×#120C3×D3semidirect
115. C24×C3#67semidirect
116. C24×C3C3×D4semidirect
118. C24×C3C22×D3semidirect
120. C24×C3C23×C3semidirect
126. C23×D3C3×D4semidirect
137. C22×C3×C4C22×D3semidirect
138. C22×C3×C4C23×C3semidirect
140. C22×Dic3C23×C3semidirect
153. C22×C32#120semidirect
159. C2×C3×D3#120semidirect
167. C22×D4C2×C3×D3semidirect
172. C2×C22⋊C4C2×C3×D3semidirect
239. C24C3×#67semidirect
241. C24C22×C3×D3semidirect
246. C22×C4C22×C3×D3semidirect
272. C22×C3#937semidirect
273. C22×C3C3×#120semidirect
281. C2×D3C3×#120semidirect
287. C32#2846semidirect
332. C22C3×#937semidirect
338. C3#82612direct (N × H)
339. C3C3×#2846semidirect

Maximal subgroups

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

subgroupindex
C25×C3×D32
C2×C32×#1202
C2×C3×#9372
C3×#52472
C3×#49552
C3×#28463
#826123