Finite Group Databasebeta

← #11802904

Subgroups — #11802904

Order 1152.

1969 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×C3⋊D3C211yesmaximal, characteristic
2C2×#63974C211yesmaximal, characteristic
3C22×#3722C211yesmaximal, characteristic
4C3×#4950C212yesmaximal, complemented
5#10591693C212yesmaximal, complemented
6#10591378C214yesmaximal, complemented
7#10591355C212yesmaximal, complemented
8#10591325C212yesmaximal, complemented
9#74591—32nomaximal
10C24×C3⋊D3C2211yescharacteristic
11C22×C4×C3⋊D3C2212yes—
12C23×C3×Dic3C2212yes—
13C22×#624C2212yes—
14C2×C32×C22⋊C4C2211yescharacteristic
15C2×#3757C2211yescharacteristic
16C2×#3752C2212yes—
17C22⋊C4×C3⋊D3C418yescomplemented
18C2×C3×#217C2212yes—
19C2×C3×C4⋊Dic3C2214yescomplemented
20#63974—24no—
21C2×#3724C2214yescomplemented
22C2×#3722—24no—
23C2×#3722C2211yescharacteristic
24C2×#3722C2212yes—
25C2×#3722C2211yescharacteristic
26C2×#3722C2212yes—
27C3×C3⋊#95C2214yescomplemented
28#63579C2214yescomplemented
29C22×(C2×D3)⋊C4—32no—
30C2×D3×C22⋊C4—32no—
31C2×D3×C22⋊C4—32no—
32C2×D3×C22⋊C4—61no—
33C3×#385—32no—
34#5210—32no—
35#4989—34no—
36#4987—34no—
37#4950D312yescomplemented
38#4946—32no—
39#4668—34no—
40#4667—34no—
41C24×C32C2311yescharacteristic
42C23×C3⋊D3—44no—
43C23×C3⋊D3C2×C414yes—
44C23×C3⋊D3D414yes—
45C23×C3⋊D3Q814yes—
46C23×C3⋊D3C2311yescharacteristic
47C23×C3⋊D3C2311yescharacteristic
48C22×C32×C4C2312yes—
49C22×C3⋊Dic3C2312yes—
50C2×C4×C3⋊D3—44no—
51C2×C4×C3⋊D3C2×C418yescomplemented
52C22×C3×Dic3—24no—
53C22×C3×Dic3—44no—
54C22×C3×Dic3C2312yes—
55C22×C3×Dic3C2314yes—
56C22×C3×Dic3C2312yes—
57C2×#624—24no—
58C2×#624—44no—
59C32×C22⋊C4C2×C414yescomplemented
60#3757C2×C414yescomplemented
61#3752C2×C418yescomplemented
62C3×#217—24no—
63C3×C4⋊Dic3—44no—
64#3724—44no—
65#3722—24no—
66#3722—24no—
67#3722—22no—
68#3722—22no—
69#2315—91noSylow
70C24×D3—32no—
71C24×D3—61no—
72C23×C3×C4—32no—
73C23×Dic3C2×D312yes—
74C22×C4×D3—34no—
75C22×C4×D3—32no—
76C22×C4×D3—32no—
77C22×C4×D3—62no—
78C2×C3×C4⋊C4—34no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—32no—
81C2×C3×C22⋊C4C2×D312yes—
82C2×#217—32no—
83C2×#217—61no—
84C2×#217C2×D312yes—
85C2×(C2×D3)⋊C4—34no—
86C2×(C2×D3)⋊C4—32no—
87C2×(C2×D3)⋊C4—34no—
88C2×(C2×D3)⋊C4—34no—
89C2×(C2×D3)⋊C4—32no—
90C2×(C2×D3)⋊C4—62no—
91C2×(C2×D3)⋊C4—68no—
92C2×C4⋊Dic3C2×D314yescomplemented
93C2×Dic3⋊C4—34no—
94C2×Dic3⋊C4—34no—
95D3×C22⋊C4—316no—
96D3×C22⋊C4—68no—
97D3×C22⋊C4—68no—
98C3×#95—34no—
99C3⋊#95—34no—
100C3⋊#95—34no—
101C3⋊#95C2×D314yescomplemented
102C23×C32—24no—
103C23×C32C22×C412yes—
104C23×C32C2×D412yes—
105C23×C32C2×Q812yes—
106C23×C32C2411yescharacteristic
107C22×C3⋊D3—24no—
108C22×C3⋊D3—24no—
109C22×C3⋊D3—48no—
110C22×C3⋊D3—48no—
111C22×C3⋊D3—48no—
112C22×C3⋊D3C4⋊C4116yescomplemented
113C22×C3⋊D3C22×C412yes—
114C22×C3⋊D3C22×C412yes—
115C22×C3⋊D3C2×D412yes—
116C22×C3⋊D3C2×D412yes—
117C22×C3⋊D3C2×Q812yes—
118C22×C3⋊D3C2×Q812yes—
119C2×C32×C4—24no—
120C2×C32×C4C22×C414yes—
121C2×C3⋊Dic3—24no—
122C2×C3⋊Dic3C22×C414yes—
123C4×C3⋊D3—48no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—48no—
132#624—48no—
133C22×C22⋊C4—91no—
134C22×C22⋊C4—91no—
135C22×C22⋊C4—91no—
136#385—32no—
137#385—94no—
138#385—92no—
139#380—92no—
140#380—92no—
141C24×C3—21no—
142C24×C3C22×D312yes—
143C23×D3—32no—
144C23×D3—32no—
145C23×D3—38no—
146C23×D3—38no—
147C23×D3—38no—
148C23×D3—61no—
149C23×D3—64no—
150C23×D3—64no—
151C23×D3—64no—
152C23×D3—61no—
153C23×D3—124no—
154C23×D3—124no—
155C23×D3—128no—
156C22×C3×C4—22no—
157C22×C3×C4—32no—
158C22×C3×C4—34no—
159C22×C3×C4—32no—
160C22×C3×C4—64no—
161C22×C3×C4—124no—
162C22×C3×C4C22×D314yes—
163C22×Dic3—24no—
164C22×Dic3—34no—
165C22×Dic3—44no—
166C22×Dic3—62no—
167C22×Dic3C22×D312yes—
168C22×Dic3C22×D314yes—
169C22×Dic3C22×D312yes—
170C2×C4×D3—316no—
171C2×C4×D3—68no—
172C2×C4×D3—64no—
173C2×C4×D3—64no—
174C2×C4×D3—124no—
175C2×C4×D3—124no—
176C2×C4×D3—124no—
177C2×C4×D3—124no—
178C2×C4×D3—128no—
179C3×C4⋊C4—124no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—64no—
182C3×C22⋊C4C4×D318yescomplemented
183#217—24no—
184#217—38no—
185#217—64no—
186(C2×D3)⋊C4—316no—
187(C2×D3)⋊C4—68no—
188(C2×D3)⋊C4—64no—
189(C2×D3)⋊C4—64no—
190(C2×D3)⋊C4—68no—
191(C2×D3)⋊C4—68no—
192C4⋊Dic3—44no—
193Dic3⋊C4—124no—
194Dic3⋊C4—124no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32—24no—
198C22×C32C2×C4⋊C414yes—
199C22×C32C23×C411yescharacteristic
200C22×C32C22×D411yescharacteristic
201C22×C32C22×Q811yescharacteristic
202C22×C32H(5,𝔽2)14yes—
203C2×C3⋊D3—24no—
204C2×C3⋊D3—24no—
205C2×C3⋊D3—24no—
206C2×C3⋊D3—24no—
207C2×C3⋊D3—24no—
208C2×C3⋊D3—24no—
209C2×C3⋊D3—416no—
210C2×C3⋊D3C2×C4⋊C414yes—
211C2×C3⋊D3C2×C4⋊C414yes—
212C32×C4—24no—
213C3⋊Dic3—24no—
214C3×Dic3—42no—
215C3×Dic3—42no—
216C3×Dic3—44no—
217C25—91no—
218C23×C4—32no—
219C23×C4—92no—
220C23×C4—92no—
221C2×C4⋊C4—34no—
222C2×C4⋊C4—94no—
223C2×C22⋊C4—32no—
224C2×C22⋊C4—91no—
225C2×C22⋊C4—92no—
226C2×C22⋊C4—98no—
227C2×C22⋊C4—91no—
228C2×C22⋊C4—92no—
229C2×C22⋊C4—91no—
230C2×C22⋊C4—92no—
231C2×C22⋊C4—184no—
232C2×C22⋊C4—184no—
233C2×C22⋊C4D3211yescharacteristic
234#95—34no—
235#95—94no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—21no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—44no—
243C23×C3C2×C4×D314yes—
244C23×C3D3×D414yes—
245C23×C3D3×Q814yes—
246C23×C3C23×D312yes—
247C22×D3—34no—
248C22×D3—34no—
249C22×D3—34no—
250C22×D3—34no—
251C22×D3—34no—
252C22×D3—34no—
253C22×D3—332no—
254C22×D3—62no—
255C22×D3—62no—
256C22×D3—62no—
257C22×D3—616no—
258C22×D3—62no—
259C22×D3—62no—
260C22×D3—62no—
261C22×D3—68no—
262C22×D3—64no—
263C22×D3—64no—
264C22×D3—64no—
265C22×D3—64no—
266C22×D3—124no—
267C22×D3—1216no—
268C22×D3—128no—
269C22×D3—1216no—
270C22×D3—128no—
271C22×D3—1216no—
272C22×D3—1216no—
273C22×D3—1216no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—44no—
278C2×C3×C4—64no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—64no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—64no—
285C2×C3×C4—128no—
286C2×C3×C4C2×C4×D318yes—
287C2×Dic3—24no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—24no—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—24no—
294C2×Dic3—38no—
295C2×Dic3—48no—
296C2×Dic3—64no—
297C2×Dic3—64no—
298C2×Dic3—64no—
299C2×Dic3—124no—
300C4×D3—128no—
301C4×D3—128no—
302C4×D3—1216no—
303C4×D3—1216no—
304C2×C32—24no—
305C2×C32C22×C4⋊C411yescharacteristic
306C2×C32#51812yes—
307C2×C32#53412yes—
308C2×C32#54312yes—
309C3⋊D3—24no—
310C3⋊D3—24no—
311C24—91no—
312C24—91no—
313C24—94no—
314C24—94no—
315C24—94no—
316C24—364no—
317C24C2×D3211yescharacteristic
318C22×C4—32no—
319C22×C4—34no—
320C22×C4—32no—
321C22×C4—64no—
322C22×C4—92no—
323C22×C4—98no—
324C22×C4—124no—
325C22×C4—184no—
326C22×C4—364no—
327C22×C4—364no—
328C22×C4C2×D3212yes—
329C4⋊C4—124no—
330C4⋊C4—364no—
331C22⋊C4—64no—
332C22⋊C4—94no—
333C22⋊C4—98no—
334C22⋊C4—182no—
335C22⋊C4—182no—
336C22⋊C4—184no—
337C22⋊C4—184no—
338C22⋊C4#62414yescomplemented
339C22×C3—28no—
340C22×C3—24no—
341C22×C3—28no—
342C22×C3—24no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—24no—
346C22×C3—21no—
347C22×C3—24no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3D3×C4⋊C418yes—
352C22×C3C22×C4×D312yes—
353C22×C3C2×D3×D412yes—
354C22×C3#100414yes—
355C22×C3C2×D3×Q812yes—
356C22×C3#100914yes—
357C2×D3—38no—
358C2×D3—38no—
359C2×D3—64no—
360C2×D3—64no—
361C2×D3—68no—
362C2×D3—68no—
363C2×D3—68no—
364C2×D3—68no—
365C2×D3—68no—
366C2×D3—64no—
367C2×D3—64no—
368C2×D3—68no—
369C2×D3—68no—
370C2×D3—64no—
371C2×D3—64no—
372C2×D3—124no—
373C2×D3—124no—
374C2×D3—124no—
375C2×D3—1232no—
376C2×D3—1232no—
377C3×C4—28no—
378C3×C4—44no—
379C3×C4—122no—
380C3×C4—124no—
381C3×C4—122no—
382Dic3—42no—
383Dic3—42no—
384Dic3—44no—
385Dic3—68no—
386Dic3—124no—
387C32#231511yescharacteristic, complemented, Sylow
388C23—24no—
389C23—92no—
390C23—92no—
391C23—92no—
392C23—92no—
393C23—92no—
394C23—92no—
395C23—916no—
396C23—184no—
397C23—184no—
398C23—368no—
399C23—368no—
400C23—368no—
401C23#379712yes—
402C23#380212yes—
403C23C2×#62412yes—
404C23C22×D3211yescharacteristic
405C2×C4—24no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—64no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—64no—
412C2×C4—64no—
413C2×C4—94no—
414C2×C4—128no—
415C2×C4—184no—
416C2×C4—368no—
417C2×C4—368no—
418C2×C4C2×#62414yes—
419C2×C3—28no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—44no—
425C2×C3C2×D3×C4⋊C412yes—
426C2×C3#554514yes—
427C2×C3#558912yes—
428C2×C3#563414yes—
429C2×C3#565312yes—
430D3—68no—
431D3—68no—
432D3—68no—
433D3—124no—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—94no—
438C22—94no—
439C22—184no—
440C22—184no—
441C22—184no—
442C22—184no—
443C22—184no—
444C22—184no—
445C22—3616no—
446C22#6390014yes—
447C22C2×#379711yescharacteristic
448C22C2×#380211yescharacteristic
449C22#6431414yes—
450C22C22×#62411yescharacteristic
451C4—24no—
452C4—124no—
453C4—122no—
454C4—122no—
455C4—184no—
456C3—21no—
457C3#7459112yescomplemented
458C2—24no—
459C2—184no—
460C2—184no—
461C2#1059418012yes—
462C2#1059429612yes—
463C2C2×#6390011yescharacteristic
464C2#1059522312yes—
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C22⋊C4C22
socle, cosocleC23×C32C24
prime radicalC24×C3⋊D3C22
hypercenterC2×C22⋊C4D32
hypocenterC32#2315
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#4950C2semidirect
5. #10591693C2semidirect
6. #10591378C2semidirect
7. #10591355C2semidirect
8. #10591325C2semidirect
17. C22⋊C4×C3⋊D3C4semidirect
19. C2×C3×C4⋊Dic3C22semidirect
21. C2×#3724C22semidirect
27. C3×C3⋊#95C22semidirect
28. #63579C22semidirect
37. #4950D3semidirect
51. C2×C4×C3⋊D3C2×C4semidirect
59. C32×C22⋊C4C2×C4semidirect
60. #3757C2×C4semidirect
61. #3752C2×C4semidirect
92. C2×C4⋊Dic3C2×D3semidirect
101. C3⋊#95C2×D3semidirect
112. C22×C3⋊D3C4⋊C4semidirect
182. C3×C22⋊C4C4×D3semidirect
338. C22⋊C4#624semidirect
387. C32#2315semidirect
457. C3#74591semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×C3⋊D32
C2×#639742
C22×#37222
C3×#49502
#105916932
#105913782
#105913552
#105913252
#745913