Finite Group Databasebeta

← #11904576

Subgroups — #11904576

Order 1152.

3517 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3832C211yesmaximal, characteristic, complemented
2C2×C3×#1007C211yesmaximal, characteristic, complemented
3C3×#5812C211yesmaximal, characteristic, complemented
4#10595410C218yesmaximal, complemented
5#10595342C218yesmaximal, complemented
6#10595045C216yesmaximal, complemented
7#10595034C216yesmaximal, complemented
8#87290—31nomaximal
9#85874—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×#3832—28no—
12C2×#3832C2218yes—
13C22×C4⋊(C3⋊D3)C2213yes—
14C22×C4×C3⋊D3C2213yes—
15C3×#1007—28no—
16C2×C3×#239C2214yescomplemented
17C2×C3×D3×Q8C2214yescomplemented
18C2×C3×#235C2216yes—
19C3×Q8×Dic3C2218yescomplemented
20C3×#926C2216yes—
21#63981C22112yes—
22#63962C2214yes—
23#63916C22112yescomplemented
24#63911C22112yescomplemented
25#63909C22124yescomplemented
26C4×#626C22124yescomplemented
27#63871C22112yescomplemented
28#63864C22112yescomplemented
29C2×C3×2−1+4—31no—
30C2×#1007D311yescharacteristic, complemented
31C22×#239—31no—
32C22×#239—31no—
33C22×#239—61no—
34C3×#519—31no—
35#5834—38no—
36#5830—36no—
37#5828—36no—
38#5817—38no—
39#5812D311yescharacteristic, complemented
40#5577—38no—
41#5576—38no—
42#5525—36no—
43#5524—36no—
44C23×C3⋊D3C2313yes—
45C2×C32×Q8D418yescomplemented
46C2×C32×Q8C2314yes—
47C22×C32×C4C2313yes—
48C22×C3⋊Dic3C2311yescharacteristic
49#3832—232no—
50C2×C4⋊(C3⋊D3)—212no—
51C2×C4⋊(C3⋊D3)C23112yes—
52C2×C4×C3⋊D3—212no—
53C2×C4×C3⋊D3C23112yes—
54C2×C3×#67C2314yes—
55C3×#239—216no—
56C3×D3×Q8—216no—
57C3×#235—224no—
58C2×C3×D12C2316yes—
59C2×C3×C4×D3C23112yes—
60C2×C3×Dic6C2316yes—
61C2×#626C23124yes—
62C3×#217C2314yes—
63C3×C4⋊Dic3C23112yes—
64C3×C4×Dic3C23112yes—
65Dic3⋊Dic3C2318yes—
66#3722C23124yes—
67(C2×D3)⋊Dic3C2318yes—
68#3487—91noSylow
69C3×2−1+4—68no—
70C2×C3×#41—34no—
71C2×C3×#41—36no—
72C22×C3×Q8—21no—
73C22×C3×Q8—34no—
74C22×C3×Q8C2×D311yescharacteristic
75C22×C3×Q8C2×D311yescharacteristic
76#1007—28no—
77C2×#239—38no—
78C2×#239—38no—
79C2×#239—68no—
80C2×#239—616no—
81C2×#239—68no—
82C2×#239—68no—
83C2×#239C2×D314yescomplemented
84C2×D3×Q8C2×D314yescomplemented
85C2×#235C2×D316yes—
86C22×D12—33no—
87C22×D12—33no—
88C22×D12—63no—
89C22×C4×D3—33no—
90C22×C4×D3—33no—
91C22×C4×D3—63no—
92C3×C4×Q8—38no—
93C3×#117—36no—
94#947—312no—
95#946—312no—
96Q8×Dic3C2×D318yescomplemented
97#940—312no—
98#938—312no—
99#937—312no—
100#935—34no—
101#930—324no—
102C4×#67—324no—
103#926C2×D316yes—
104#897—312no—
105#895—312no—
106#895—324no—
107#886—34no—
108#882—312no—
109#875—312no—
110#874—312no—
111C4×D12—324no—
112C23×C32C2411yescharacteristic
113C22×C3⋊D3—26no—
114C22×C3⋊D3—46no—
115C22×C3⋊D3C2416yes—
116C22×C3×D3C2414yes—
117C32×Q8C2×D4116yes—
118C2×C32×C4C2×D4112yes—
119C2×C32×C4C2416yes—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3C2412yes—
122C4⋊(C3⋊D3)—224no—
123C4×C3⋊D3—224no—
124C3×#67—28no—
125C2×C3×Dic3C2418yes—
126C2×C3×Dic3C2414yes—
127C3×D12—212no—
128C3×C4×D3—224no—
129C3×Dic6—212no—
130#626—424no—
131C2×2−1+4—31no—
132C22×#41—91no—
133#550—98no—
134#548—98no—
135#535—96no—
136#525—96no—
137#519—31no—
138C23×D3—33no—
139C23×D3—33no—
140C23×D3—63no—
141C3×#41—616no—
142C3×#41—624no—
143C2×C3×Q8—28no—
144C2×C3×Q8—24no—
145C2×C3×Q8—36no—
146C2×C3×Q8—616no—
147C2×C3×Q8D1218yescomplemented
148C2×C3×Q8#6718yescomplemented
149C2×C3×Q8C22×D314yes—
150C2×C3×Q8C22×D314yes—
151C2×C3×D4—34no—
152C2×C3×D4—36no—
153C22×C3×C4—23no—
154C22×C3×C4—312no—
155C22×C3×C4C22×D313yes—
156C22×C3×C4C22×D313yes—
157C2×#67—324no—
158C2×#67C22×D314yes—
159C22×Dic3—31no—
160C22×Dic3—31no—
161C22×Dic3—61no—
162#239—216no—
163#239—664no—
164#239—632no—
165#239—632no—
166D3×Q8—216no—
167#235—224no—
168C2×D12—312no—
169C2×D12—312no—
170C2×D12—324no—
171C2×D12—612no—
172C2×D12—624no—
173C2×D12—612no—
174C2×D12—612no—
175C2×D12C22×D316yes—
176C2×C4×D3—312no—
177C2×C4×D3—312no—
178C2×C4×D3—612no—
179C2×C4×D3—624no—
180C2×C4×D3—612no—
181C2×C4×D3—612no—
182C2×C4×D3C22×D3112yes—
183C2×Dic6C22×D316yes—
184C3×C4⋊C4—312no—
185C3×C22⋊C4—34no—
186C3×C42—312no—
187#217—38no—
188#217C22×D314yes—
189(C2×D3)⋊C4—38no—
190(C2×D3)⋊C4—324no—
191(C2×D3)⋊C4—324no—
192C4⋊Dic3—38no—
193C4⋊Dic3C22×D3112yes—
194Dic3⋊C4—38no—
195C4×Dic3C22×D3112yes—
196C22×C32—22no—
197C22×C32C22×D412yes—
198C22×C32C2511yescharacteristic
199C2×C3⋊D3—26no—
200C2×C3⋊D3—212no—
201C2×C3⋊D3—412no—
202C2×C3×D3—24no—
203C2×C3×D3—28no—
204C32×C4C22×D4112yes—
205C3⋊Dic3—22no—
206C3×Dic3—24no—
207C3×Dic3—28no—
2082−1+4—68no—
209C2×#41—34no—
210C2×#41—36no—
211C2×#41—98no—
212C2×#41—188no—
213C22×Q8—34no—
214C22×Q8D3211yescharacteristic
215C22×D4—93no—
216C23×C4—93no—
217#127—912no—
218#124—912no—
219#123—94no—
220C22⋊Q8—912no—
221#121—924no—
222#121—912no—
223#120—912no—
224C4×Q8—38no—
225C4×D4—924no—
226#117—36no—
227C23×C3—21no—
228C23×C3—34no—
229C23×C3C23×D311yescharacteristic
230C23×C3C23×D311yescharacteristic
231C22×D3—36no—
232C22×D3—36no—
233C22×D3—66no—
234C22×D3—612no—
235C22×D3—66no—
236C22×D3—66no—
237C22×D3—126no—
238C22×D3—126no—
239C22×D3—1212no—
240C22×D3C23×D314yes—
241C3×Q8—216no—
242C3×Q8—612no—
243C3×Q8C2×D12116yes—
244C3×Q8C2×#67116yes—
245C3×D4—68no—
246C3×D4—612no—
247C2×C3×C4—26no—
248C2×C3×C4—212no—
249C2×C3×C4—34no—
250C2×C3×C4—38no—
251C2×C3×C4—624no—
252C2×C3×C4C2×D12112yes—
253C2×C3×C4C2×#67112yes—
254C2×C3×C4C23×D316yes—
255C2×C3×C4C23×D316yes—
256#67—28no—
257#67—1224no—
258C2×Dic3—32no—
259C2×Dic3—32no—
260C2×Dic3—62no—
261C2×Dic3—64no—
262C2×Dic3—62no—
263C2×Dic3—62no—
264C2×Dic3C23×D318yes—
265C2×Dic3C23×D314yes—
266D12—212no—
267D12—648no—
268D12—624no—
269D12—624no—
270D12—1224no—
271C4×D3—224no—
272C4×D3—648no—
273C4×D3—624no—
274C4×D3—624no—
275Dic6—212no—
276C2×C32—22no—
277C2×C32C23×D411yescharacteristic
278C2×C32#58512yes—
279C3⋊D3—46no—
280C3×D3—44no—
281C24—93no—
282#41—616no—
283#41—624no—
284#41—1832no—
285C2×Q8—36no—
286C2×Q8—616no—
287C2×Q8#62618yescomplemented
288C2×Q8C2×D3214yes—
289C2×D4—34no—
290C2×D4—36no—
291C2×D4—924no—
292C2×D4—912no—
293C2×D4—1812no—
294C22×C4—312no—
295C22×C4—91no—
296C22×C4—912no—
297C22×C4—1812no—
298C22×C4C2×D3213yes—
299C4⋊C4—312no—
300C4⋊C4—98no—
301C22⋊C4—34no—
302C22⋊C4—924no—
303C22⋊C4—98no—
304C42—312no—
305C22×C3—24no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—21no—
310C22×C3—64no—
311C22×C3—68no—
312C22×C3C22×D1212yes—
313C22×C3C22×#6712yes—
314C22×C3C24×D311yescharacteristic
315C22×C3C24×D311yescharacteristic
316C2×D3—28no—
317C2×D3—24no—
318C2×D3—612no—
319C2×D3—66no—
320C2×D3—612no—
321C2×D3—612no—
322C2×D3—66no—
323C2×D3—1212no—
324C2×D3—1224no—
325C2×D3—1212no—
326C2×D3—1212no—
327C3×C4—212no—
328C3×C4—64no—
329C3×C4—68no—
330C3×C4C22×D12112yes—
331C3×C4C22×#67112yes—
332Dic3—28no—
333Dic3—24no—
334Dic3—64no—
335Dic3—62no—
336Dic3—62no—
337C32#348711yescharacteristic, complemented, Sylow
338C23—34no—
339C23—96no—
340C23—186no—
341C23—366no—
342C23C22×D3211yescharacteristic
343Q8—612no—
344Q8C2×#626116yes—
345D4—68no—
346D4—612no—
347D4—1824no—
348D4—3624no—
349C2×C4—38no—
350C2×C4—34no—
351C2×C4—624no—
352C2×C4—92no—
353C2×C4—182no—
354C2×C4—1824no—
355C2×C4C2×#626112yes—
356C2×C4C22×D3216yes—
357C2×C3—24no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—21no—
362C2×C3—124no—
363C2×C3C23×D1211yescharacteristic
364C2×C3#596812yes—
365C2×C3#596812yes—
366C2×C3C23×#6711yescharacteristic
367D3—44no—
368D3—1212no—
369D3—126no—
370D3—126no—
371C22—22no—
372C22—68no—
373C22—64no—
374C22—1812no—
375C22—186no—
376C22—3612no—
377C22C22×#62612yes—
378C22C23×D3211yescharacteristic
379C4—68no—
380C4—64no—
381C4—182no—
382C4C22×#626112yes—
383C3—21no—
384C3#8587411yescharacteristic, complemented
385C3#8729011yescharacteristic, complemented
386C2—22no—
387C2—124no—
388C2—366no—
389C2#1059668912yes—
390C2C23×#62611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC22×C32×Q8C22
hypercenterC22×Q8D32
hypocenterC32#3487
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#3832C2semidirect
2. C2×C3×#1007C2semidirect
3. C3×#5812C2semidirect
4. #10595410C2semidirect
5. #10595342C2semidirect
6. #10595045C2semidirect
7. #10595034C2semidirect
16. C2×C3×#239C22semidirect
17. C2×C3×D3×Q8C22semidirect
19. C3×Q8×Dic3C22semidirect
23. #63916C22semidirect
24. #63911C22semidirect
25. #63909C22semidirect
26. C4×#626C22semidirect
27. #63871C22semidirect
28. #63864C22semidirect
30. C2×#1007D3semidirect
39. #5812D3semidirect
45. C2×C32×Q8D4semidirect
83. C2×#239C2×D3semidirect
84. C2×D3×Q8C2×D3semidirect
96. Q8×Dic3C2×D3semidirect
147. C2×C3×Q8D12semidirect
148. C2×C3×Q8#67semidirect
287. C2×Q8#626semidirect
337. C32#3487semidirect
384. C3#85874semidirect
385. C3#87290semidirect

Maximal subgroups

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

subgroupindex
C22×#38322
C2×C3×#10072
C3×#58122
#105954102
#105953422
#105950452
#105950342
#872903
#858743