Finite Group Databasebeta

← #11788788

Subgroups — #11788788

Order 1152.

771 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5521C211yesmaximal, characteristic, complemented
2#10595040C211yesmaximal, characteristic
3C3×#5005C211yesmaximal, characteristic, complemented
4#10591111C211yesmaximal, characteristic, complemented
5C2×#63558C211yesmaximal, characteristic, complemented
6#10589397C211yesmaximal, characteristic, complemented
7#10589158C211yesmaximal, characteristic, complemented
8#71652—31nomaximal
9#70408—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C32×#117C2211yescharacteristic
13#64135C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C3×#926C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#893—22no—
18C3×D3×C22⋊C4—22no—
19C3×C4×D12—22no—
20#63976—22no—
21#63914C412yescomplemented
22#63908—22no—
23#63863C412yescomplemented
24C3×#805—22no—
25#63569—22no—
26#63558—22no—
27C3×#518—31no—
28#5830—31no—
29#5805—31no—
30#5521D311yescharacteristic, complemented
31C3×#421—31no—
32#5236—31no—
33C2×#833—31no—
34C2×#823—31no—
35#5005—31no—
36#5005D311yescharacteristic, complemented
37#4557—31no—
38#4531—31no—
39#4531—31no—
40#4531—31no—
41#4531—61no—
42C23×C3×D3D412yes—
43C22×C32×C4C2311yescharacteristic
44C2×C3×#67—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235—21no—
48C3×#235—21no—
49C2×C3×D12—21no—
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12C2×C412yes—
54C2×C3×D12C2×C411yescharacteristic, complemented
55C2×C3×D12D412yes—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—24no—
58C2×C3×Dic6C2×C411yescharacteristic, complemented
59C2×#625—24no—
60C32×C4⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C42D412yescomplemented
63#3757—21no—
64C4⋊(C3⋊Dic3)—21no—
65C4×C3⋊Dic3C2×C412yescomplemented
66#3748—21no—
67#3748—21no—
68C2×C3⋊(C3⋊C8)—21no—
69C3×#217—21no—
70C3×(C2×D3)⋊C4—24no—
71C3×C4⋊Dic3—21no—
72C3×C4×Dic3D412yescomplemented
73C3×#208—21no—
74C3×#208—21no—
75C2×C3×C3⋊C8—21no—
76(C2×D3)⋊Dic3—24no—
77#1911—91noSylow
78C2×C3×#41—31no—
79C22×C3×D4—31no—
80C2×#235C2×D311yescharacteristic
81C22×D12C2×D311yescharacteristic
82C2×C3×M16—31no—
83C3×C4×D4—62no—
84C3×C4×D4—62no—
85C3×#117—21no—
86C3×#117—31no—
87C3×#117C2×D311yescharacteristic
88C3×#117C2×D311yescharacteristic
89C2×C3×C22⋊C4—62no—
90#953—62no—
91#947—32no—
92#940—32no—
93#940—32no—
94#938—62no—
95#937—62no—
96#936—32no—
97#930—62no—
98#926—31no—
99#926—31no—
100#926—61no—
101#926C2×D311yescharacteristic
102C2×#208—31no—
103C2×#208—31no—
104C2×#208—61no—
105C2×#208C2×D311yescharacteristic
106#893—22no—
107D3×C22⋊C4—22no—
108C4×D12—22no—
109C3×#104—62no—
110#837—62no—
111#833—62no—
112#823—62no—
113#805—22no—
114#805—62no—
115C23×C32C2×D411yescharacteristic
116C22×C3×D3—22no—
117C22×C3×D3—22no—
118C22×C3×D3—21no—
119C22×C3×D3—82no—
120C22×C3×D3C22⋊C414yes—
121C2×C32×C4—22no—
122C2×C32×C4C22×C411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4#4111yescharacteristic
127C2×C32×C4#4111yescharacteristic
128C2×C3⋊Dic3—22no—
129C3×#67—41no—
130C3×#67—41no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—22no—
133C3×D12—22no—
134C3×D12—22no—
135C3×D12—21no—
136C3×D12—21no—
137C3×D12—22no—
138C3×D12—22no—
139C3×C4×D3—41no—
140C3×C4×D3—41no—
141C3×C4×D3—84no—
142C3×Dic6—21no—
143C3×Dic6—21no—
144#625—84no—
145C3⋊(C3⋊C8)—41no—
146C3×C3⋊C8—41no—
147#535—91no—
148#518—31no—
149#421—31no—
150#421—91no—
151C2×#100—91no—
152H(3,ℤ/4)—91no—
153H(3,ℤ/4)—91no—
154C24×C3—32no—
155C23×D3#6712yes—
156C3×#41—61no—
157C3×#41—61no—
158C3×#41—61no—
159C3×#41—61no—
160C2×C3×Q8—31no—
161C2×C3×D4—32no—
162C2×C3×D4—31no—
163C2×C3×D4—32no—
164C2×C3×D4—61no—
165C2×C3×D4—61no—
166C2×C3×D4—61no—
167C2×C3×D4—61no—
168C2×C3×D4—61no—
169C22×C3×C4—21no—
170C22×C3×C4—64no—
171C22×C3×C4—61no—
172C22×C3×C4C22×D311yescharacteristic
173C22×C3×C4C22×D311yescharacteristic
174C2×#67—21no—
175C2×#67—64no—
176C2×#67—64no—
177#235—21no—
178#235—21no—
179#235—21no—
180#235—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12C4×D312yes—
186C2×D12C4×D311yescharacteristic, complemented
187C2×D12D1212yes—
188C2×C4×D3—21no—
189C2×C4×D3—24no—
190C2×Dic6C4×D311yescharacteristic, complemented
191C3×M16—61no—
192C3×M16—61no—
193C2×C3×C8—61no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—61no—
198C3×C22⋊C4—21no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—64no—
202C3×C22⋊C4—61no—
203C3×C42—22no—
204C3×C42—32no—
205C3×C42#6712yescomplemented
206C3×C42#6712yescomplemented
207#217—21no—
208#217—64no—
209#217—61no—
210#217—61no—
211#217—121no—
212(C2×D3)⋊C4—24no—
213(C2×D3)⋊C4—64no—
214C4⋊Dic3—21no—
215C4⋊Dic3—61no—
216C4⋊Dic3—61no—
217C4⋊Dic3—121no—
218C4×Dic3—32no—
219C4×Dic3—32no—
220C4×Dic3—62no—
221C4×Dic3#6712yescomplemented
222#208—21no—
223#208—21no—
224#208—61no—
225#208—61no—
226#208—61no—
227#208—61no—
228#208—121no—
229#208—121no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—62no—
232C2×C3⋊C8—61no—
233C2×C3⋊C8—61no—
234C22×C32—41no—
235C22×C32C2×C22⋊C411yescharacteristic
236C22×C32#12011yescharacteristic
237C22×C32#12311yescharacteristic
238C2×C3×D3—24no—
239C2×C3×D3—44no—
240C2×C3×D3—44no—
241C2×C3×D3—42no—
242C2×C3×D3—41no—
243C2×C3×D3—82no—
244C32×C4—42no—
245C32×C4C4×D411yescharacteristic
246C32×C4C4×D411yescharacteristic
247C32×C4#12111yescharacteristic
248C32×C4#12111yescharacteristic
249C3⋊Dic3—42no—
250C3×Dic3—42no—
251C3×Dic3—41no—
252C2×#41—31no—
253C22×D4—31no—
254C2×M16—31no—
255C2×M16—91no—
256#127—92no—
257#124—92no—
258#121—182no—
259#120—182no—
260C4×D4—62no—
261C4×D4—62no—
262#117—31no—
263#117—91no—
264#117D3211yescharacteristic
265C2×C22⋊C4—62no—
266#104—62no—
267#104—182no—
268#100—182no—
269C23×C3—21no—
270C23×C3—34no—
271C23×C3—62no—
272C23×C3—62no—
273C23×C3—61no—
274C23×C3—242no—
275C23×C3D3×D411yescharacteristic
276C23×C3C2×#6711yescharacteristic
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—21no—
280C22×D3—82no—
281C22×D3(C2×D3)⋊C414yes—
282C3×Q8—61no—
283C3×Q8—61no—
284C3×D4—62no—
285C3×D4—62no—
286C3×D4—62no—
287C3×D4—61no—
288C3×D4—61no—
289C3×D4—62no—
290C3×D4—121no—
291C3×D4—121no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—42no—
301C2×C3×C4—61no—
302C2×C3×C4—62no—
303C2×C3×C4—121no—
304C2×C3×C4—121no—
305C2×C3×C4—244no—
306C2×C3×C4C2×C4×D311yescharacteristic
307C2×C3×C4C2×C4×D311yescharacteristic
308C2×C3×C4C2×D1211yescharacteristic
309C2×C3×C4#23511yescharacteristic
310C2×C3×C4#23511yescharacteristic
311C2×C3×C4#23511yescharacteristic
312C2×C3×C4D3×D411yescharacteristic
313C2×C3×C4D3×D411yescharacteristic
314C2×C3×C4#23711yescharacteristic
315C2×C3×C4C2×#6711yescharacteristic
316C2×C3×C4C2×#6711yescharacteristic
317C2×C3×C4C2×#6711yescharacteristic
318#67—41no—
319#67—41no—
320#67—244no—
321#67—244no—
322C2×Dic3—21no—
323C2×Dic3—22no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—122no—
327D12—21no—
328D12—22no—
329D12—22no—
330D12—21no—
331D12—22no—
332D12—22no—
333C4×D3—41no—
334C4×D3—41no—
335C4×D3—84no—
336Dic6—21no—
337Dic6—21no—
338C3×C8—121no—
339C3⋊C8—41no—
340C3⋊C8—122no—
341C3⋊C8—121no—
342C3⋊C8—121no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32#38611yescharacteristic
347C3×D3—44no—
348C3×D3—81no—
349C24—32no—
350#41—61no—
351#41—61no—
352#41—61no—
353#41—61no—
354C2×Q8—31no—
355C2×D4—32no—
356C2×D4—31no—
357C2×D4—32no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—184no—
364C22×C4—64no—
365C22×C4—61no—
366C22×C4C2×D3211yescharacteristic
367M16—61no—
368M16—61no—
369M16—181no—
370M16—181no—
371C2×C8—61no—
372C2×C8—181no—
373C4⋊C4—21no—
374C4⋊C4—61no—
375C4⋊C4—181no—
376C22⋊C4—21no—
377C22⋊C4—64no—
378C22⋊C4—61no—
379C22⋊C4—184no—
380C22⋊C4—181no—
381C42—32no—
382C42—92no—
383C42#62512yescomplemented
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—64no—
390C22×C3—81no—
391C22×C3—124no—
392C22×C3—124no—
393C22×C3—122no—
394C22×C3—121no—
395C22×C3—242no—
396C22×C3D3×C22⋊C411yescharacteristic
397C22×C3#88311yescharacteristic
398C22×C3C2×(C2×D3)⋊C411yescharacteristic
399C22×C3#92911yescharacteristic
400C22×C3#93711yescharacteristic
401C22×C3#95311yescharacteristic
402C2×D3—24no—
403C2×D3—44no—
404C2×D3—44no—
405C2×D3—42no—
406C2×D3—41no—
407C2×D3—82no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—82no—
415C3×C4—122no—
416C3×C4—121no—
417C3×C4#67⋊C411yescharacteristic
418C3×C4#88411yescharacteristic
419C3×C4C4×#6711yescharacteristic
420C3×C4C4×#6711yescharacteristic
421C3×C4C4×#6711yescharacteristic
422C3×C4#93011yescharacteristic
423C3×C4#93011yescharacteristic
424C3×C4#93811yescharacteristic
425Dic3—42no—
426Dic3—41no—
427Dic3—122no—
428Dic3—122no—
429Dic3—242no—
430C32#191111yescharacteristic, complemented, Sylow
431C23—34no—
432C23—62no—
433C23—62no—
434C23—61no—
435C23—242no—
436C23D3×#6711yescharacteristic
437Q8—61no—
438Q8—61no—
439D4—62no—
440D4—61no—
441D4—62no—
442D4—61no—
443D4—62no—
444D4—62no—
445D4—121no—
446D4—121no—
447D4—724no—
448C2×C4—22no—
449C2×C4—61no—
450C2×C4—62no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—182no—
454C2×C4—244no—
455C2×C4#379511yescharacteristic
456C2×C4#379811yescharacteristic
457C2×C4C4×D3211yescharacteristic
458C2×C4D3×D1211yescharacteristic
459C2×C4C2×#62511yescharacteristic
460C2×C4D3×#6711yescharacteristic
461C8—121no—
462C8—361no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—124no—
474C2×C3—241no—
475C2×C3#495611yescharacteristic
476C2×C3#521311yescharacteristic
477D3—44no—
478D3—81no—
479C22—41no—
480C22—64no—
481C22—124no—
482C22—124no—
483C22—122no—
484C22—121no—
485C22—242no—
486C22#6390511yescharacteristic
487C22D3×(C2×D3)⋊C411yescharacteristic
488C22#6397611yescharacteristic
489C4—42no—
490C4—122no—
491C4—121no—
492C4—362no—
493C4#6387911yescharacteristic
494C4C4×#62511yescharacteristic
495C4#6390611yescharacteristic
496C4#6390811yescharacteristic
497C3—21no—
498C3#7040811yescharacteristic, complemented
499C3#7165211yescharacteristic, complemented
500C2—21no—
501C2—21no—
502C2—21no—
503C2—124no—
504C2—241no—
505C2#1059138111yescharacteristic
subgroupG/H
centerC2#10591381
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC2×C32#386
prime radical#10595040C2
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1911
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C3×#5521C2semidirect
3. C3×#5005C2semidirect
4. #10591111C2semidirect
5. C2×#63558C2semidirect
6. #10589397C2semidirect
7. #10589158C2semidirect
21. #63914C4semidirect
23. #63863C4semidirect
30. #5521D3semidirect
36. #5005D3semidirect
54. C2×C3×D12C2×C4semidirect
58. C2×C3×Dic6C2×C4semidirect
62. C32×C42D4semidirect
65. C4×C3⋊Dic3C2×C4semidirect
72. C3×C4×Dic3D4semidirect
186. C2×D12C4×D3semidirect
190. C2×Dic6C4×D3semidirect
205. C3×C42#67semidirect
206. C3×C42#67semidirect
221. C4×Dic3#67semidirect
383. C42#625semidirect
430. C32#1911semidirect
498. C3#70408semidirect
499. C3#71652semidirect

Maximal subgroups

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

subgroupindex
C3×#55212
#105950402
C3×#50052
#105911112
C2×#635582
#105893972
#105891582
#716523
#704083