Finite Group Databasebeta

← #11895352

Subgroups — #11895352

Order 1152.

842 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596056C211yesmaximal, characteristic, complemented
2C3×#5579C211yesmaximal, characteristic, complemented
3C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4#10590366C212yesmaximal, complemented
5C4×#3716C211yesmaximal, characteristic, complemented
6#10590349C211yesmaximal, characteristic
7C4×#3714C212yesmaximal, complemented
8#10590308C212yesmaximal, complemented
9#10590293C212yesmaximal, complemented
10#10590284C211yesmaximal, characteristic, complemented
11#10590281C211yesmaximal, characteristic, complemented
12#83643—31nomaximal
13#79927—31nomaximal
14C32×C4×Q8C2211yescharacteristic
15#64156—21no—
16#64103—22no—
17#64102—21no—
18#64084C2212yes—
19#64083C2211yescharacteristic
20C4×C4⋊(C3⋊D3)C2212yes—
21C4×#634C2211yescharacteristic
22C3×#946—21no—
23C3×#896—22no—
24C3×D3×C4⋊C4—21no—
25C3×C4×D12C2212yes—
26C3×C4×D12C2211yescharacteristic
27C3×#870C2212yes—
28C3×#869C2211yescharacteristic
29C2×#3716C2211yescharacteristic
30C2×#3714C2212yes—
31C3×C4⋊(C3⋊C8)C2211yescharacteristic
32C3×C4⋊(C3⋊C8)C2212yes—
33C3×C4×C3⋊C8C2211yescharacteristic
34C3×C4×C3⋊C8C2212yes—
35#63575C2212yes—
36#63573C2211yescharacteristic
37#63567C2212yes—
38#63567C2211yescharacteristic
39#63564C2212yes—
40#63564C2212yes—
41#63564C2212yes—
42#63563C2211yescharacteristic
43C3×#554—31no—
44#5585—31no—
45#5585—31no—
46#5585—61no—
47#5579D311yescharacteristic, complemented
48C3×C8×Q8—31no—
49#5091—31no—
50#5088—31no—
51#5068—32no—
52#5066—32no—
53Q8×C3⋊C8D311yescharacteristic, complemented
54C4×#214—31no—
55#5016—32no—
56#5015—31no—
57C4×#213—32no—
58#4850—31no—
59#4847—32no—
60#4844—32no—
61#4843—31no—
62#4697—32no—
63#4694—31no—
64C4×D24—32no—
65C4×#204—31no—
66C2×C32×Q8D411yescharacteristic
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)C2312yes—
70C2×C4×C3⋊D3—22no—
71C2×#634C2311yescharacteristic
72C2×C3×D12C2311yescharacteristic
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3—22no—
75C2×C3×Dic6—21no—
76C32×C4⋊C4D411yescharacteristic
77C32×C4⋊C4D412yes—
78C32×C42C2311yescharacteristic
79C32×C42C2312yes—
80#3752—22no—
81#3752—22no—
82#3752—22no—
83C4⋊(C3⋊Dic3)C2311yescharacteristic
84#3750—21no—
85C4×C3⋊Dic3—21no—
86C3×(C2×D3)⋊C4—21no—
87C3×(C2×D3)⋊C4—22no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2311yescharacteristic
90C3×C4⋊Dic3C2312yes—
91C3×Dic3⋊C4—22no—
92C3×Dic3⋊C4—22no—
93C2×C3×C3⋊C8C2311yescharacteristic
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2312yes—
96#3716—22no—
97#3714—24no—
98#3361—91noSylow
99C3×C4⋊Q8—31no—
100C3×#125—32no—
101C3×C22⋊Q8—61no—
102C3×C22⋊Q8—62no—
103C3×C4×Q8—21no—
104C3×C4×Q8C2×D311yescharacteristic
105C3×C4×Q8C2×D311yescharacteristic
106C3×C4×D4—32no—
107C3×C4×D4—31no—
108C2×C3×C4⋊C4—61no—
109#947—62no—
110#947—61no—
111#947—61no—
112#946—21no—
113C2×#214—31no—
114C2×#213—32no—
115C2×D24—32no—
116C2×#204—31no—
117#896—22no—
118#894—64no—
119#894—62no—
120#894—62no—
121#893—62no—
122#893—61no—
123#893—61no—
124D3×C4⋊C4—21no—
125#875—32no—
126#875—32no—
127#875—62no—
128#874—31no—
129#874—31no—
130#874—61no—
131C4×D12—32no—
132C4×D12—32no—
133C4×D12—62no—
134C4×D12C2×D311yescharacteristic
135C4×D12C2×D312yes—
136#870C2×D312yes—
137#869C2×D311yescharacteristic
138C4×Dic6—31no—
139C4×Dic6—31no—
140C4×Dic6—61no—
141C3×C4⋊C8—31no—
142C3×C4⋊C8—32no—
143C3×C4×C8—31no—
144C3×C4×C8—32no—
145#832—31no—
146#821—31no—
147#821—32no—
148#821—32no—
149#821—32no—
150#818—32no—
151#817—31no—
152#816—31no—
153#816—32no—
154#810—32no—
155#810—31no—
156#809—32no—
157#809—32no—
158#809—32no—
159#808—31no—
160#807—32no—
161C4⋊(C3⋊C8)C2×D311yescharacteristic
162C4⋊(C3⋊C8)C2×D312yes—
163C4×C3⋊C8C2×D312yes—
164C4×C3⋊C8C2×D311yescharacteristic
165C22×C3⋊D3—22no—
166C22×C3×D3—21no—
167C32×Q8—22no—
168C2×C32×C4C2×D412yes—
169C2×C32×C4C2×D412yes—
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2411yescharacteristic
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C4⋊(C3⋊D3)—22no—
176C4⋊(C3⋊D3)—22no—
177C4⋊(C3⋊D3)—42no—
178C4⋊(C3⋊D3)—81no—
179C4×C3⋊D3—82no—
180#634—21no—
181#634—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—22no—
184C3×D12—21no—
185C3×D12#4112yes—
186C3×C4×D3—42no—
187C3×C4×D3—82no—
188C3×Dic6—81no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192#559—91no—
193#554—31no—
194#488—92no—
195#486—91no—
196#478—92no—
197#477—91no—
198#463—92no—
199#459—91no—
200C8×Q8—31no—
201C4×SD16—91no—
202C4×D8—92no—
203C2×C3×Q8—21no—
204C2×C3×Q8—61no—
205C2×C3×Q8D1211yescharacteristic
206C2×C3×Q8#6711yescharacteristic
207C2×C3×D4—31no—
208C22×C3×C4—61no—
209C22×C3×C4—62no—
210C2×D12—32no—
211C2×D12—32no—
212C2×D12—62no—
213C2×D12—62no—
214C2×D12—61no—
215C2×D12—61no—
216C2×D12—62no—
217C2×D12—61no—
218C2×D12—61no—
219C2×D12C22×D311yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—22no—
222C2×C4×D3—64no—
223C2×C4×D3—62no—
224C2×C4×D3—62no—
225C2×Dic6—21no—
226C2×Dic6—31no—
227C2×Dic6—31no—
228C2×Dic6—61no—
229C2×C3×C8—32no—
230C2×C3×C8—31no—
231C2×C3×C8—31no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—32no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—62no—
237C3×C4⋊C4—62no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4D1211yescharacteristic
240C3×C4⋊C4D1212yes—
241C3×C4⋊C4#6711yescharacteristic
242C3×C4⋊C4#6712yes—
243C3×C22⋊C4—61no—
244C3×C22⋊C4—62no—
245C3×C42—21no—
246C3×C42—22no—
247C3×C42C22×D311yescharacteristic
248C3×C42C22×D312yes—
249C3×C42C22×D311yescharacteristic
250C3×C42C22×D312yes—
251#214—62no—
252#213—64no—
253(C2×D3)⋊C4—21no—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—64no—
256(C2×D3)⋊C4—64no—
257(C2×D3)⋊C4—64no—
258(C2×D3)⋊C4—62no—
259(C2×D3)⋊C4—62no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—62no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4—62no—
264C4⋊Dic3—21no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—61no—
268C4⋊Dic3C22×D311yescharacteristic
269C4⋊Dic3C22×D312yes—
270Dic3⋊C4—22no—
271Dic3⋊C4—22no—
272Dic3⋊C4—62no—
273Dic3⋊C4—61no—
274Dic3⋊C4—61no—
275C4×Dic3—62no—
276C4×Dic3—61no—
277C4×Dic3—61no—
278C2×C3⋊C8C22×D311yescharacteristic
279C2×C3⋊C8C22×D312yes—
280C2×C3⋊C8C22×D311yescharacteristic
281D24—64no—
282#204—62no—
283C22×C32C22×D411yescharacteristic
284C2×C3⋊D3—42no—
285C2×C3⋊D3—42no—
286C2×C3⋊D3—42no—
287C2×C3×D3—22no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C32×C4—22no—
291C32×C4—21no—
292C32×C4—22no—
293C32×C4#13512yes—
294C32×C4C2×#4111yescharacteristic
295C32×C42−1+411yescharacteristic
296C3⋊Dic3—41no—
297C3⋊Dic3—41no—
298C3×Dic3—41no—
299C3×Dic3—42no—
300C2×SD16—91no—
301C2×D8—92no—
302C4⋊Q8—31no—
303#127—91no—
304#125—32no—
305#124—92no—
306#124—181no—
307#123—182no—
308C22⋊Q8—61no—
309C22⋊Q8—62no—
310C4×Q8—91no—
311C4×Q8D3211yescharacteristic
312C4×D4—32no—
313C4×D4—31no—
314C4×D4—92no—
315C4×D4—181no—
316C2×C4⋊C4—61no—
317#107—92no—
318#106—91no—
319C4⋊C8—31no—
320C4⋊C8—32no—
321#103—91no—
322#103—92no—
323#102—92no—
324#102—92no—
325#102—92no—
326#102—91no—
327C4×C8—32no—
328C4×C8—31no—
329C23×C3—61no—
330C22×D3—21no—
331C22×D3—64no—
332C22×D3—62no—
333C22×D3—62no—
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—42no—
337C3×Q8—241no—
338C3×D4—32no—
339C3×D4—61no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—61no—
346C2×C3×C4—62no—
347C2×C3×C4—122no—
348C2×C3×C4—242no—
349C2×C3×C4C2×D1211yescharacteristic
350C2×C3×C4C2×D1212yes—
351C2×C3×C4C2×D1212yes—
352C2×C3×C4C2×D1211yescharacteristic
353C2×C3×C4C2×#6712yes—
354C2×C3×C4C2×#6712yes—
355C2×C3×C4C2×#6711yescharacteristic
356C2×C3×C4C2×#6711yescharacteristic
357C2×C3×C4C23×D311yescharacteristic
358C2×C3×C4C23×D311yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—62no—
362C2×Dic3—62no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367D12—21no—
368D12—62no—
369D12—62no—
370D12—62no—
371D12—62no—
372D12—122no—
373D12—122no—
374D12—124no—
375D12—122no—
376D12—122no—
377D12—241no—
378D12—241no—
379D12—242no—
380D12#23912yes—
381C4×D3—42no—
382C4×D3—82no—
383C4×D3—242no—
384C4×D3—242no—
385C4×D3—244no—
386Dic6—61no—
387Dic6—61no—
388Dic6—61no—
389Dic6—61no—
390Dic6—81no—
391Dic6—121no—
392Dic6—121no—
393C3×C8—61no—
394C3×C8—62no—
395C3×C8—61no—
396C3⋊C8—21no—
397C3⋊C8—22no—
398C3⋊C8—21no—
399C2×C32#54811yescharacteristic
400C2×C32C2×#13511yescharacteristic
401C2×C32#57611yescharacteristic
402C3⋊D3—82no—
403C3×D3—42no—
404C2×Q8—61no—
405C2×Q8—91no—
406C2×Q8#62611yescharacteristic
407C2×D4—31no—
408C2×D4—92no—
409C2×D4—181no—
410C2×D4—181no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—182no—
414SD16—182no—
415D8—184no—
416C2×C8—31no—
417C2×C8—31no—
418C2×C8—32no—
419C4⋊C4—31no—
420C4⋊C4—32no—
421C4⋊C4—62no—
422C4⋊C4—61no—
423C4⋊C4—62no—
424C4⋊C4—91no—
425C4⋊C4—181no—
426C4⋊C4#62611yescharacteristic
427C4⋊C4#62612yes—
428C22⋊C4—61no—
429C22⋊C4—62no—
430C22⋊C4—182no—
431C22⋊C4—182no—
432C22⋊C4—182no—
433C42—181no—
434C42C2×D3212yes—
435C42C2×D3211yescharacteristic
436C22×C3—21no—
437C22×C3—62no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3C22×D1211yescharacteristic
441C22×C3C22×#6711yescharacteristic
442C2×D3—22no—
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—124no—
446C2×D3—124no—
447C2×D3—124no—
448C2×D3—122no—
449C2×D3—122no—
450C2×D3—122no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—122no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—21no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—121no—
467C3×C4—122no—
468C3×C4#90412yes—
469C3×C4#95012yes—
470C3×C4C2×#23911yescharacteristic
471C3×C4#100711yescharacteristic
472C3×C4#100711yescharacteristic
473C3×C4D3×#4111yescharacteristic
474Dic3—41no—
475Dic3—42no—
476Dic3—122no—
477Dic3—122no—
478Dic3—121no—
479Dic3—121no—
480Dic3—121no—
481Dic3—121no—
482C32#336111yescharacteristic, complemented, Sylow
483C23—61no—
484C23—182no—
485Q8—22no—
486Q8—181no—
487Q8—181no—
488Q8—241no—
489D4—32no—
490D4—61no—
491D4—182no—
492D4—182no—
493D4—362no—
494D4—721no—
495C2×C4—61no—
496C2×C4—62no—
497C2×C4—122no—
498C2×C4—181no—
499C2×C4—181no—
500C2×C4—242no—
501C2×C4—722no—
502C2×C4C2×#62612yes—
503C2×C4C2×#62611yescharacteristic
504C2×C4C2×#62612yes—
505C2×C4C2×#62611yescharacteristic
506C2×C4C22×D3211yescharacteristic
507C8—61no—
508C8—62no—
509C8—61no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—122no—
514C2×C3#557611yescharacteristic
515C2×C3C2×#90411yescharacteristic
516C2×C3#575211yescharacteristic
517C2×C3#581711yescharacteristic
518C2×C3C2×#95011yescharacteristic
519C2×C3#583511yescharacteristic
520D3—42no—
521D3—244no—
522D3—242no—
523D3—242no—
524C22—62no—
525C22—121no—
526C22—121no—
527C22—362no—
528C22—362no—
529C22—362no—
530C22C22×#62611yescharacteristic
531C4—22no—
532C4—21no—
533C4—22no—
534C4—122no—
535C4—121no—
536C4—361no—
537C4—361no—
538C4#6382912yes—
539C4#6431611yescharacteristic
540C4D3×#23911yescharacteristic
541C3—21no—
542C3#7992711yescharacteristic, complemented
543C3#8364311yescharacteristic, complemented
544C2—122no—
545C2—722no—
546C2C2×#6382911yescharacteristic
547C2#1059534211yescharacteristic
548C2#1059544911yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radical#10590349C2
hypercenterC4×Q8D32
hypocenterC32#3361
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #10596056C2semidirect
2. C3×#5579C2semidirect
3. C3×Q8×C3⋊C8C2semidirect
4. #10590366C2semidirect
5. C4×#3716C2semidirect
7. C4×#3714C2semidirect
8. #10590308C2semidirect
9. #10590293C2semidirect
10. #10590284C2semidirect
11. #10590281C2semidirect
47. #5579D3semidirect
53. Q8×C3⋊C8D3semidirect
482. C32#3361semidirect
542. C3#79927semidirect
543. C3#83643semidirect

Maximal subgroups

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

subgroupindex
#105960562
C3×#55792
C3×Q8×C3⋊C82
#105903662
C4×#37162
#105903492
C4×#37142
#105903082
#105902932
#105902842
#105902812
#836433
#799273