Finite Group Databasebeta

← C3×#87251 #11904476

Subgroups — C3×#87251 #11904476

Order 1152.

3409 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#1007C211yesmaximal, characteristic, complemented
3C3×#5817C218yesmaximal, complemented
4C3×#5816C216yesmaximal, complemented
5C3×Q8×#67C218yesmaximal, complemented
6C3×#5812C211yesmaximal, characteristic, complemented
7C3×#5791C216yesmaximal, complemented
8C3×#3486—31nomaximal
9#87251C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8—28no—
11C22×C32×Q8C2218yes—
12C22×C32×Q8C2213yes—
13C22×C32×Q8C2211yescharacteristic
14C23×C32×C4C2213yes—
15C3×#1007—28no—
16C2×C3×#239C2214yescomplemented
17C2×C3×D3×Q8C2214yescomplemented
18C2×C3×#235C2216yes—
19C3×#953C2214yes—
20C3×#947C22112yescomplemented
21C3×#946C22124yescomplemented
22C3×Q8×Dic3C2218yescomplemented
23C3×#944C22112yescomplemented
24C3×#930C22112yes—
25C3×#929C22112yes—
26C3×C4×#67C22124yes—
27C3×#926C2216yes—
28C3×#924C22112yes—
29C2×C3×2−1+4—31no—
30C23×C3×Q8—21no—
31C23×C3×Q8D311yescharacteristic, complemented
32C23×C3×Q8C2×C311yescharacteristic, complemented
33C2×#1007C2×C311yescharacteristic, complemented
34C3×D4×Q8—38no—
35C3×#548—38no—
36C3×#536—36no—
37C3×#525—36no—
38C3×#519—31no—
39#5817C2×C318yescomplemented
40#5816C2×C316yescomplemented
41Q8×#67C2×C318yescomplemented
42#5812C2×C311yescharacteristic, complemented
43#5791C2×C316yescomplemented
44C24×C32C2311yescharacteristic
45C2×C32×Q8—232no—
46C2×C32×Q8—212no—
47C2×C32×Q8D418yescomplemented
48C2×C32×Q8C23112yes—
49C2×C32×Q8C2314yes—
50C22×C32×C4—212no—
51C22×C32×C4C23112yes—
52C22×C32×C4C2313yes—
53C22×C32×C4C2313yes—
54C2×C3×#67C2318yes—
55C2×C3×#67C2314yes—
56C3×#239—216no—
57C3×D3×Q8—216no—
58C3×#235—224no—
59C2×C3×D12C2316yes—
60C2×C3×C4×D3C23112yes—
61C2×C3×Dic6C2316yes—
62C3×#217C2318yes—
63C3×#217C2314yes—
64C3×(C2×D3)⋊C4C23124yes—
65C3×C4⋊Dic3C23112yes—
66C3×Dic3⋊C4C23124yes—
67C3×C4×Dic3C23112yes—
68#3486—31noSylow
69C3×2−1+4—68no—
70C2×C3×#41—34no—
71C2×C3×#41—36no—
72C22×C3×Q8—216no—
73C22×C3×Q8—28no—
74C22×C3×Q8—28no—
75C22×C3×Q8—23no—
76C22×C3×Q8—28no—
77C22×C3×Q8—21no—
78C22×C3×Q8—34no—
79C22×C3×Q8C2×D313yes—
80C22×C3×Q8C2×D318yes—
81C22×C3×Q8C2×D311yescharacteristic
82C22×C3×Q8C22×C318yes—
83C22×C3×Q8C22×C313yes—
84C22×C3×Q8C22×C311yescharacteristic
85C23×C3×C4—23no—
86C23×C3×C4C2×D313yes—
87C23×C3×C4C22×C313yes—
88#1007—28no—
89C2×#239C22×C314yescomplemented
90C2×D3×Q8C22×C314yescomplemented
91C2×#235C22×C316yes—
92C3×C4⋊Q8—312no—
93C3×#124—312no—
94C3×#123—312no—
95C3×C22⋊Q8—324no—
96C3×C22⋊Q8—312no—
97C3×#121—312no—
98C3×#120—34no—
99C3×C4×Q8—38no—
100C3×C4×D4—324no—
101C3×#117—36no—
102#953C22×C314yes—
103#947C22×C3112yescomplemented
104#946C22×C3124yescomplemented
105Q8×Dic3C22×C318yescomplemented
106#944C22×C3112yescomplemented
107#930C22×C3112yes—
108#929C22×C3112yes—
109C4×#67C22×C3124yes—
110#926C22×C316yes—
111#924C22×C3112yes—
112C23×C32—22no—
113C23×C32—24no—
114C23×C32C2411yescharacteristic
115C23×C32C2412yes—
116C22×C3×D3C2414yes—
117C32×Q8—224no—
118C32×Q8C2×D4116yes—
119C2×C32×C4—26no—
120C2×C32×C4—224no—
121C2×C32×C4C2×D4112yes—
122C2×C32×C4C2416yes—
123C2×C32×C4C2416yes—
124C3×#67—28no—
125C3×#67—216no—
126C2×C3×Dic3C2418yes—
127C2×C3×Dic3C2414yes—
128C3×D12—212no—
129C3×C4×D3—224no—
130C3×Dic6—212no—
131C2×2−1+4—31no—
132C23×Q8C3×D311yescharacteristic, complemented
133D4×Q8—38no—
134#548—38no—
135#536—36no—
136#525—36no—
137#519—31no—
138C24×C3—21no—
139C24×C3C22×D311yescharacteristic
140C24×C3C23×C311yescharacteristic
141C3×#41—616no—
142C3×#41—624no—
143C2×C3×Q8—264no—
144C2×C3×Q8—224no—
145C2×C3×Q8—232no—
146C2×C3×Q8—232no—
147C2×C3×Q8—212no—
148C2×C3×Q8—212no—
149C2×C3×Q8—28no—
150C2×C3×Q8—212no—
151C2×C3×Q8—24no—
152C2×C3×Q8—36no—
153C2×C3×Q8—616no—
154C2×C3×Q8#6718yescomplemented
155C2×C3×Q8C3×D418yescomplemented
156C2×C3×Q8C22×D3112yes—
157C2×C3×Q8C22×D314yes—
158C2×C3×Q8C23×C3112yes—
159C2×C3×Q8C23×C314yes—
160C2×C3×D4—38no—
161C2×C3×D4—34no—
162C2×C3×D4—36no—
163C22×C3×C4—224no—
164C22×C3×C4—212no—
165C22×C3×C4—212no—
166C22×C3×C4—212no—
167C22×C3×C4—23no—
168C22×C3×C4—23no—
169C22×C3×C4—312no—
170C22×C3×C4C22×D3112yes—
171C22×C3×C4C22×D313yes—
172C22×C3×C4C22×D313yes—
173C22×C3×C4C23×C3112yes—
174C22×C3×C4C23×C313yes—
175C22×C3×C4C23×C313yes—
176C2×#67C23×C318yes—
177C2×#67C23×C314yes—
178#239—216no—
179D3×Q8—216no—
180#235—224no—
181C2×D12C23×C316yes—
182C2×C4×D3C23×C3112yes—
183C2×Dic6C23×C316yes—
184C3×C4⋊C4—324no—
185C3×C4⋊C4—312no—
186C3×C22⋊C4—34no—
187C3×C22⋊C4—38no—
188C3×C22⋊C4—324no—
189C3×C42—312no—
190#217C23×C318yes—
191#217C23×C314yes—
192(C2×D3)⋊C4C23×C3124yes—
193C4⋊Dic3C23×C3112yes—
194Dic3⋊C4C23×C3124yes—
195C4×Dic3C23×C3112yes—
196C22×C32—28no—
197C22×C32—22no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32C22×D412yes—
201C22×C322−1+414yescomplemented
202C22×C32C2511yescharacteristic
203C2×C3×D3—24no—
204C2×C3×D3—28no—
205C32×C4—26no—
206C32×C4C22×D4112yes—
207C3×Dic3—24no—
208C3×Dic3—28no—
2092−1+4—68no—
210C2×#41—34no—
211C2×#41—36no—
212C22×Q8—28no—
213C22×Q8—34no—
214C22×Q8C2×C3×D313yes—
215C22×Q8C2×C3×D318yes—
216C22×Q8C2×C3×D311yescharacteristic
217C23×C4C2×C3×D313yes—
218C4⋊Q8—312no—
219#124—312no—
220#123—312no—
221C22⋊Q8—324no—
222C22⋊Q8—312no—
223#121—312no—
224#120—34no—
225C4×Q8—38no—
226C4×D4—324no—
227#117—36no—
228C23×C3—24no—
229C23×C3—28no—
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—24no—
234C23×C3—22no—
235C23×C3—21no—
236C23×C3—34no—
237C23×C3C23×D312yes—
238C23×C3C23×D311yescharacteristic
239C23×C3C24×C311yescharacteristic
240C23×C3C24×C312yes—
241C22×D3C24×C314yes—
242C3×Q8—248no—
243C3×Q8—224no—
244C3×Q8—224no—
245C3×Q8—216no—
246C3×Q8—612no—
247C3×Q8C2×#67116yes—
248C3×Q8C2×C3×D4116yes—
249C3×D4—616no—
250C3×D4—68no—
251C3×D4—612no—
252C2×C3×C4—212no—
253C2×C3×C4—248no—
254C2×C3×C4—26no—
255C2×C3×C4—26no—
256C2×C3×C4—224no—
257C2×C3×C4—224no—
258C2×C3×C4—26no—
259C2×C3×C4—26no—
260C2×C3×C4—212no—
261C2×C3×C4—38no—
262C2×C3×C4—34no—
263C2×C3×C4—624no—
264C2×C3×C4C2×#67112yes—
265C2×C3×C4C2×C3×D4112yes—
266C2×C3×C4C23×D316yes—
267C2×C3×C4C23×D316yes—
268C2×C3×C4C24×C316yes—
269C2×C3×C4C24×C316yes—
270#67—28no—
271#67—216no—
272C2×Dic3C24×C318yes—
273C2×Dic3C24×C314yes—
274D12—212no—
275C4×D3—224no—
276Dic6—212no—
277C2×C32—22no—
278C2×C32—24no—
279C2×C32C23×D411yescharacteristic
280C2×C32C2×2−1+412yes—
281C3×D3—44no—
282C24C22×C3×D311yescharacteristic
283#41—616no—
284#41—624no—
285C2×Q8—232no—
286C2×Q8—212no—
287C2×Q8—36no—
288C2×Q8—616no—
289C2×Q8C3×#6718yescomplemented
290C2×Q8C22×C3×D3112yes—
291C2×Q8C22×C3×D314yes—
292C2×D4—38no—
293C2×D4—34no—
294C2×D4—36no—
295C22×C4—212no—
296C22×C4—312no—
297C22×C4C22×C3×D3112yes—
298C22×C4C22×C3×D313yes—
299C22×C4C22×C3×D313yes—
300C4⋊C4—312no—
301C4⋊C4—324no—
302C22⋊C4—34no—
303C22⋊C4—38no—
304C22⋊C4—324no—
305C42—312no—
306C22×C3—24no—
307C22×C3—216no—
308C22×C3—28no—
309C22×C3—28no—
310C22×C3—24no—
311C22×C3—24no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—24no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—68no—
322C22×C3—64no—
323C22×C3#100714yescomplemented
324C22×C3C22×#6712yes—
325C22×C3C22×C3×D412yes—
326C22×C3C3×2−1+414yescomplemented
327C22×C3C24×D311yescharacteristic
328C22×C3C25×C311yescharacteristic
329C2×D3—24no—
330C2×D3—28no—
331C3×C4—212no—
332C3×C4—26no—
333C3×C4—26no—
334C3×C4—212no—
335C3×C4—68no—
336C3×C4—64no—
337C3×C4C22×#67112yes—
338C3×C4C22×C3×D4112yes—
339Dic3—24no—
340Dic3—28no—
341C32#348611yescharacteristic, complemented, Sylow
342C23—24no—
343C23—22no—
344C23—34no—
345C23C23×C3×D312yes—
346C23C23×C3×D311yescharacteristic
347Q8—224no—
348Q8—612no—
349Q8C2×C3×#67116yes—
350D4—616no—
351D4—68no—
352D4—612no—
353C2×C4—26no—
354C2×C4—224no—
355C2×C4—38no—
356C2×C4—34no—
357C2×C4—624no—
358C2×C4C2×C3×#67112yes—
359C2×C4C23×C3×D316yes—
360C2×C4C23×C3×D316yes—
361C2×C3—28no—
362C2×C3—24no—
363C2×C3—22no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—22no—
367C2×C3—22no—
368C2×C3—21no—
369C2×C3—124no—
370C2×C3C2×#100712yes—
371C2×C3C23×#6711yescharacteristic
372C2×C3C23×C3×D411yescharacteristic
373C2×C3C2×C3×2−1+412yes—
374D3—44no—
375C22—28no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—68no—
380C22—64no—
381C22C3×#100714yescomplemented
382C22C22×C3×#6712yes—
383C22C24×C3×D311yescharacteristic
384C4—26no—
385C4—68no—
386C4—64no—
387C4C22×C3×#67112yes—
388C3—21no—
389C3#8725111yescharacteristic, direct product factor
390C3C3×#348611yescharacteristic, complemented
391C2—24no—
392C2—22no—
393C2—124no—
394C2C2×C3×#100712yes—
395C2C23×C3×#6711yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC23×C32×Q8C2
socleC22×C32C25
hypercenterC23×C3×Q8D3
hypocenterC3C3×#3486
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C23×C32×Q8C2semidirect
2. C2×C3×#1007C2semidirect
3. C3×#5817C2semidirect
4. C3×#5816C2semidirect
5. C3×Q8×#67C2semidirect
6. C3×#5812C2semidirect
7. C3×#5791C2semidirect
9. #87251C3direct (N × H)
16. C2×C3×#239C22semidirect
17. C2×C3×D3×Q8C22semidirect
20. C3×#947C22semidirect
21. C3×#946C22semidirect
22. C3×Q8×Dic3C22semidirect
23. C3×#944C22semidirect
31. C23×C3×Q8D3semidirect
32. C23×C3×Q8C2×C3semidirect
33. C2×#1007C2×C3semidirect
39. #5817C2×C3semidirect
40. #5816C2×C3semidirect
41. Q8×#67C2×C3semidirect
42. #5812C2×C3semidirect
43. #5791C2×C3semidirect
47. C2×C32×Q8D4semidirect
89. C2×#239C22×C3semidirect
90. C2×D3×Q8C22×C3semidirect
103. #947C22×C3semidirect
104. #946C22×C3semidirect
105. Q8×Dic3C22×C3semidirect
106. #944C22×C3semidirect
132. C23×Q8C3×D3semidirect
154. C2×C3×Q8#67semidirect
155. C2×C3×Q8C3×D4semidirect
201. C22×C322−1+4semidirect
289. C2×Q8C3×#67semidirect
323. C22×C3#1007semidirect
326. C22×C3C3×2−1+4semidirect
341. C32#3486semidirect
381. C22C3×#1007semidirect
389. C3#87251direct (N × H)
390. C3C3×#3486semidirect

Maximal subgroups

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

subgroupindex
C23×C32×Q82
C2×C3×#10072
C3×#58172
C3×#58162
C3×Q8×#672
C3×#58122
C3×#57912
C3×#34863
#872513