Finite Group Databasebeta

← #11870515

Subgroups — #11870515

Order 1152.

955 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C3×#5470C211yesmaximal, characteristic, complemented
3#10592624C211yesmaximal, characteristic, complemented
4#10591169C212yesmaximal
5#10591101C212yesmaximal, complemented
6#10590363C212yesmaximal, complemented
7C4×#3715C214yesmaximal, complemented
8#10590266C212yesmaximal, complemented
9#83351—31nomaximal
10#82314—31nomaximal
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3—21no—
13C2×C32×C42C2211yescharacteristic
14C2×C3×(C2×D3)⋊C4—21no—
15C3×#926—21no—
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C3×#924—21no—
18C3×#924C2212yes—
19C3×C4×D12—21no—
20C3×C4×D12—21no—
21C3×C4×D12C2212yes—
22C3×C4×D12C2212yes—
23C3×#870—21no—
24C3×#869C2211yescharacteristic, complemented
25C3×#869C2211yescharacteristic, complemented
26C3×C4×Dic6—21no—
27C3×C4×Dic6C2212yescomplemented
28C2×#3715C2214yes—
29#63657C2212yes—
30C4⋊(C3⋊(C3⋊C8))C2212yescomplemented
31C4×C3⋊(C3⋊C8)C2212yescomplemented
32#63574C2214yescomplemented
33#63565C2214yescomplemented
34#63562C2214yes—
35C3×#533—31no—
36C2×C3×C4×D4—31no—
37C2×C4×D12D311yescharacteristic
38#5470D311yescharacteristic, complemented
39#5218—32no—
40#5084—32no—
41#5083—32no—
42#5046—32no—
43#5043—32no—
44C4×#215—34no—
45#5019—32no—
46C4×#214—34no—
47#5008—32no—
48#4971—32no—
49#4925—31no—
50#4925—31no—
51#4925—61no—
52C23×C3×D3—21no—
53C22×C32×C4D411yescharacteristic
54C22×C32×C4D411yescharacteristic
55C22×C32×C4C2311yescharacteristic
56C22×C3×Dic3—21no—
57C2×C3×D12—21no—
58C2×C3×D12—24no—
59C2×C3×D12C2312yes—
60C2×C3×C4×D3—22no—
61C2×C3×C4×D3—22no—
62C2×C3×C4×D3—44no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2312yes—
65C32×C42D411yescharacteristic
66C32×C42D411yescharacteristic
67C32×C42C2311yescharacteristic
68C32×C42C2311yescharacteristic
69C2×C3⋊(C3⋊C8)C2314yes—
70C3×#217—22no—
71C3×(C2×D3)⋊C4—22no—
72C3×(C2×D3)⋊C4—22no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3C2312yes—
78C3×C4⋊Dic3C2312yes—
79C3×Dic3⋊C4—22no—
80C3×Dic3⋊C4—22no—
81C3×C4×Dic3—22no—
82#3715—28no—
83#3101—91noSylow
84C22×C3×D4—31no—
85C23×C3×C4—61no—
86C22×D12C2×D311yescharacteristic
87C22×C4×D3—21no—
88C3×C4⋊Q8—31no—
89C3×C4⋊Q8—31no—
90C3×#125—61no—
91C3×C22⋊Q8—32no—
92C3×C22⋊Q8—61no—
93C3×C4×Q8—32no—
94C3×C4×Q8—61no—
95C3×C4×D4—32no—
96C3×C4×D4—32no—
97C3×C4×D4—61no—
98C3×C4×D4—61no—
99C3×#117—61no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C22⋊C4—61no—
102C2×C3×C42—21no—
103C2×C3×C42C2×D311yescharacteristic
104C2×C3×C42C2×D311yescharacteristic
105C2×#215—34no—
106C2×#214—34no—
107C2×(C2×D3)⋊C4—21no—
108#926—21no—
109C2×C4⋊Dic3C2×D311yescharacteristic
110#924—21no—
111#924C2×D312yes—
112C4×D12—21no—
113C4×D12—21no—
114C4×D12C2×D312yes—
115C4×D12C2×D312yes—
116#870—21no—
117#869C2×D311yescharacteristic, complemented
118#869C2×D311yescharacteristic, complemented
119C4×Dic6—21no—
120C4×Dic6C2×D312yescomplemented
121#835—34no—
122#832—34no—
123#830—32no—
124#830—32no—
125#830—62no—
126#810—34no—
127#809—34no—
128#808—34no—
129#808—34no—
130C4⋊(C3⋊C8)—32no—
131C4⋊(C3⋊C8)—32no—
132C4⋊(C3⋊C8)—62no—
133C4×C3⋊C8—32no—
134C4×C3⋊C8—32no—
135C4×C3⋊C8—62no—
136C23×C32C2×D411yescharacteristic
137C22×C3×D3—24no—
138C22×C3×D3—22no—
139C22×C3×D3—42no—
140C22×C3×D3—42no—
141C2×C32×C4—22no—
142C2×C32×C4—41no—
143C2×C32×C4SD1614yescomplemented
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2411yescharacteristic
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—42no—
154C3×D12—24no—
155C3×D12—22no—
156C3×D12#4114yes—
157C3×C4×D3—44no—
158C3×C4×D3—44no—
159C3×Dic6—24no—
160C3×Dic6—42no—
161C3⋊(C3⋊C8)—24no—
162#533—31no—
163C2×C4×D4—31no—
164#483—92no—
165#476—92no—
166#461—92no—
167#450—92no—
168C4×SD16—94no—
169#431—91no—
170C24×C3—61no—
171C23×D3—21no—
172C2×C3×Q8—32no—
173C2×C3×Q8—61no—
174C2×C3×D4—32no—
175C2×C3×D4—64no—
176C2×C3×D4—61no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—61no—
181C22×C3×C4—62no—
182C22×C3×C4—62no—
183C22×C3×C4—124no—
184C22×C3×C4#6711yescharacteristic
185C22×C3×C4#6711yescharacteristic
186C22×C3×C4#6711yescharacteristic
187C22×C3×C4#6711yescharacteristic
188C22×C3×C4C22×D311yescharacteristic
189C22×C3×C4C22×D311yescharacteristic
190C22×Dic3—21no—
191C2×D12—21no—
192C2×D12—24no—
193C2×D12C22×D312yes—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—44no—
197C2×Dic6—21no—
198C2×Dic6C22×D312yes—
199C3×C4⋊C4—32no—
200C3×C4⋊C4—32no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—62no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C22⋊C4—62no—
208C3×C22⋊C4—62no—
209C3×C22⋊C4—62no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—62no—
215C3×C42#6711yescharacteristic
216C3×C42#6711yescharacteristic
217C3×C42#6711yescharacteristic
218C3×C42#6711yescharacteristic
219C3×C42C22×D311yescharacteristic
220C3×C42C22×D311yescharacteristic
221C3×C42C22×D311yescharacteristic
222C3×C42C22×D311yescharacteristic
223#217—22no—
224#215—68no—
225#214—68no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—22no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3C22×D312yes—
233C4⋊Dic3C22×D312yes—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236C4×Dic3—22no—
237C2×C3⋊C8—34no—
238C2×C3⋊C8—34no—
239C2×C3⋊C8—64no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32C2×SD1612yes—
243C22×C32C22×D411yescharacteristic
244C2×C3×D3—24no—
245C2×C3×D3—48no—
246C2×C3×D3—42no—
247C2×C3×D3—42no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4C2×SD1612yes—
252C32×C4C2×SD1612yes—
253C32×C4C2×#4112yes—
254C3×Dic3—42no—
255C3×Dic3—42no—
256C3×Dic3—42no—
257C22×D4—31no—
258C23×C4—61no—
259C2×SD16—94no—
260C4⋊Q8—31no—
261C4⋊Q8—31no—
262#125—61no—
263C22⋊Q8—32no—
264C22⋊Q8—61no—
265C4×Q8—32no—
266C4×Q8—61no—
267C4×D4—32no—
268C4×D4—32no—
269C4×D4—61no—
270C4×D4—61no—
271#117—61no—
272C2×C4⋊C4—31no—
273C2×C22⋊C4—61no—
274C2×C42D3211yescharacteristic
275#106—94no—
276C4⋊C8—92no—
277#103—94no—
278#102—94no—
279C22⋊C8—92no—
280C4×C8—92no—
281C23×C3—21no—
282C23×C3—64no—
283C23×C3—62no—
284C23×C3—122no—
285C23×C3—122no—
286C23×C3C2×#6711yescharacteristic
287C23×C3C2×#6711yescharacteristic
288C22×D3—24no—
289C22×D3—22no—
290C22×D3—42no—
291C22×D3—42no—
292C3×Q8—64no—
293C3×Q8—122no—
294C3×D4—34no—
295C3×D4—64no—
296C3×D4—62no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—24no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—122no—
314C2×C3×C4—124no—
315C2×C3×C4—124no—
316C2×C3×C4#21414yescomplemented
317C2×C3×C4#21514yescomplemented
318C2×C3×C4C2×#6711yescharacteristic
319C2×C3×C4C2×#6711yescharacteristic
320C2×C3×C4C2×#6711yescharacteristic
321C2×C3×C4C2×#6711yescharacteristic
322C2×C3×C4C2×#6711yescharacteristic
323C2×C3×C4C2×#6711yescharacteristic
324C2×C3×C4C2×#6711yescharacteristic
325C2×C3×C4C2×#6711yescharacteristic
326C2×C3×C4C2×#6711yescharacteristic
327C2×C3×C4C2×#6711yescharacteristic
328C2×C3×C4C23×D311yescharacteristic
329C2×C3×C4C23×D311yescharacteristic
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—42no—
334D12—24no—
335D12—22no—
336D12#23514yes—
337C4×D3—44no—
338C4×D3—44no—
339Dic6—24no—
340Dic6—42no—
341C3⋊C8—64no—
342C3⋊C8—64no—
343C3⋊C8—124no—
344C2×C32—22no—
345C2×C32#52511yescharacteristic
346C2×C32C22×SD1611yescharacteristic
347C2×C32#57511yescharacteristic
348C3×D3—44no—
349C24—61no—
350C2×Q8—32no—
351C2×Q8—61no—
352C2×D4—32no—
353C2×D4—64no—
354C2×D4—61no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—124no—
359C22×C4#62511yescharacteristic
360C22×C4#62511yescharacteristic
361C22×C4C2×D3211yescharacteristic
362SD16—188no—
363C2×C8—94no—
364C4⋊C4—32no—
365C4⋊C4—32no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—61no—
372C22⋊C4—62no—
373C22⋊C4—62no—
374C22⋊C4—62no—
375C42—62no—
376C42#62511yescharacteristic
377C42#62511yescharacteristic
378C42C2×D3211yescharacteristic
379C42C2×D3211yescharacteristic
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—64no—
389C22×C3—128no—
390C22×C3—122no—
391C22×C3—122no—
392C22×C3C2×#21412yes—
393C22×C3C2×#21512yes—
394C22×C3C22×#6711yescharacteristic
395C22×C3C22×#6711yescharacteristic
396C2×D3—24no—
397C2×D3—48no—
398C2×D3—42no—
399C2×D3—42no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—122no—
413C3×C4—122no—
414C3×C4—122no—
415C3×C4C2×#21412yes—
416C3×C4C2×#21412yes—
417C3×C4C2×#21512yes—
418C3×C4C2×#21512yes—
419C3×C4C2×#23512yes—
420C3×C4D3×#4112yes—
421Dic3—42no—
422Dic3—42no—
423Dic3—42no—
424C32#310111yescharacteristic, complemented, Sylow
425C23—64no—
426C23—62no—
427C23—122no—
428C23—122no—
429C23C2×#62511yescharacteristic
430Q8—64no—
431Q8—122no—
432D4—34no—
433D4—64no—
434D4—62no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—122no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4#371514yescomplemented
444C2×C4C2×#62511yescharacteristic
445C2×C4C2×#62511yescharacteristic
446C2×C4C2×#62511yescharacteristic
447C2×C4C2×#62511yescharacteristic
448C2×C4C2×#62511yescharacteristic
449C2×C4C22×D3211yescharacteristic
450C8—184no—
451C2×C3—22no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—42no—
457C2×C3—124no—
458C2×C3#579111yescharacteristic
459C2×C3C22×#21411yescharacteristic
460C2×C3C22×#21511yescharacteristic
461C2×C3#582211yescharacteristic
462C2×C3#582211yescharacteristic
463C2×C3#582811yescharacteristic
464D3—44no—
465C22—21no—
466C22—21no—
467C22—64no—
468C22—128no—
469C22—122no—
470C22—122no—
471C22C2×#371512yes—
472C22C22×#62511yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—122no—
477C4—122no—
478C4—122no—
479C4C2×#371512yes—
480C4C2×#371512yes—
481C4D3×#23512yes—
482C3—21no—
483C3#8231411yescharacteristic, complemented
484C3#8335111yescharacteristic, complemented
485C2—22no—
486C2—124no—
487C2#1059478611yescharacteristic
488C2C22×#371511yescharacteristic
489C2#1059502211yescharacteristic
subgroupG/H
centerC2×C4C2×#625
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×C42C22
socleC22×C32C22×D4
prime radicalC22×C3×D12C22
hypercenterC2×C42D32
hypocenterC32#3101
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C3×#5470C2semidirect
3. #10592624C2semidirect
5. #10591101C2semidirect
6. #10590363C2semidirect
7. C4×#3715C2semidirect
8. #10590266C2semidirect
24. C3×#869C22semidirect
25. C3×#869C22semidirect
27. C3×C4×Dic6C22semidirect
30. C4⋊(C3⋊(C3⋊C8))C22semidirect
31. C4×C3⋊(C3⋊C8)C22semidirect
32. #63574C22semidirect
33. #63565C22semidirect
38. #5470D3semidirect
117. #869C2×D3semidirect
118. #869C2×D3semidirect
120. C4×Dic6C2×D3semidirect
143. C2×C32×C4SD16semidirect
316. C2×C3×C4#214semidirect
317. C2×C3×C4#215semidirect
424. C32#3101semidirect
443. C2×C4#3715semidirect
483. C3#82314semidirect
484. C3#83351semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D122
C3×#54702
#105926242
#105911692
#105911012
#105903632
C4×#37152
#105902662
#833513
#823143