Finite Group Databasebeta

← #11868994

Subgroups — #11868994

Order 1152.

891 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10595051C211yesmaximal, characteristic, complemented
2#10594996C211yesmaximal, characteristic
3#10592830C211yesmaximal, characteristic
4C3×#5121C211yesmaximal, characteristic, complemented
5C3×#5119C211yesmaximal, characteristic, complemented
6#10591122C211yesmaximal, characteristic, complemented
7C2×#63564C211yesmaximal, characteristic
8#10590774C212yesmaximal, complemented
9#10590769C212yesmaximal, complemented
10#10590764C212yesmaximal, complemented
11#10590759C212yesmaximal, complemented
12#79242—31nomaximal
13#79090—31nomaximal
14C22×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16#64135C2211yescharacteristic
17C3×#926C2211yescharacteristic
18C3×#926C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21#63974—22no—
22#63964—22no—
23#63919—21no—
24#63918C2212yescomplemented
25#63894—21no—
26#63894—21no—
27#63894C2212yes—
28#63894C2212yes—
29#63872—21no—
30#63869C2212yescomplemented
31#63648C2212yes—
32#63648C2212yes—
33C3×#818C2212yescomplemented
34C3×#817C2212yescomplemented
35C3×Dic3⋊C8C2212yescomplemented
36C3×Dic3⋊C8C2212yescomplemented
37#63575C2211yescharacteristic, complemented
38#63575C2211yescharacteristic, complemented
39#63573C2211yescharacteristic, complemented
40#63573C2211yescharacteristic, complemented
41#63564C2212yes—
42#63564C2212yes—
43#5521—31no—
44#5521—31no—
45#5519—31no—
46#5500—31no—
47C3×#428—31no—
48C3×#424—31no—
49#5131—31no—
50#5131—31no—
51#5131—61no—
52#5121D311yescharacteristic, complemented
53#5119D311yescharacteristic, complemented
54C2×#821—31no—
55#5106—31no—
56#4998—31no—
57C2×#809—31no—
58#4891—32no—
59#4890—32no—
60#4871—32no—
61#4870—32no—
62#4847—32no—
63#4847—32no—
64#4846—32no—
65#4842—32no—
66C23×C3⋊D3—21no—
67C22×C32×C4C2311yescharacteristic
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)—22no—
70C2×C4⋊(C3⋊D3)Q814yescomplemented
71C2×C4⋊(C3⋊D3)C2312yes—
72C22×C3×Dic3—21no—
73C2×#627—22no—
74C2×#624—22no—
75C2×#624—22no—
76C32×M16—22no—
77C2×C32×C8C2312yes—
78#3757—21no—
79C4⋊(C3⋊Dic3)C2311yescharacteristic
80C4⋊(C3⋊Dic3)C2311yescharacteristic
81C4×C3⋊Dic3—21no—
82C3×#217—21no—
83C3×#217D412yes—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3D411yescharacteristic
86C3×C4⋊Dic3D411yescharacteristic
87C3×C4⋊Dic3C2312yes—
88C3×C4⋊Dic3C2311yescharacteristic
89C3×C4⋊Dic3C2311yescharacteristic
90C3×C4×Dic3—21no—
91C3×C4×Dic3C2312yes—
92C2×C3×C3⋊C8—22no—
93C2×C3×C3⋊C8C2311yescharacteristic
94C2×C3×C3⋊C8C2311yescharacteristic
95#3724—22no—
96Dic3⋊Dic3—22no—
97Dic3⋊Dic3—22no—
98#3722—22no—
99#3722—22no—
100#3081—91noSylow
101C22×D12—31no—
102C22×D12—31no—
103C22×D12—61no—
104C2×C3×M16—21no—
105C2×C3×M16C2×D311yescharacteristic
106C2×C3×M16C2×D311yescharacteristic
107C22×C3×C8—31no—
108C3×#117—31no—
109C3×#117—31no—
110C2×C3×C4⋊C4—31no—
111#926—31no—
112#926—31no—
113#926—61no—
114#926C2×D311yescharacteristic
115#926C2×D311yescharacteristic
116C2×C4⋊Dic3C2×D311yescharacteristic
117#924—62no—
118C22×C3⋊C8C2×D311yescharacteristic
119#893—32no—
120#893—32no—
121#893—61no—
122#893—61no—
123#890—32no—
124#890—61no—
125#890—61no—
126#888—32no—
127#888—61no—
128#888—61no—
129D3×C22⋊C4—62no—
130D3×C22⋊C4—62no—
131#878—62no—
132C4×D12—32no—
133C4×D12—32no—
134C4×D12—61no—
135C4×D12—61no—
136#870—32no—
137#869—32no—
138C3×#107—32no—
139C3×#106—32no—
140C3×C4⋊C8—32no—
141C3×C4⋊C8—32no—
142#821—32no—
143#821—32no—
144#821—32no—
145#821—32no—
146#821—32no—
147#821—32no—
148#821—62no—
149#821—62no—
150#818—31no—
151#818—31no—
152#818C2×D312yescomplemented
153#817—31no—
154#817—31no—
155#817C2×D312yescomplemented
156Dic3⋊C8C2×D312yescomplemented
157Dic3⋊C8C2×D312yescomplemented
158#809—32no—
159#809—32no—
160#808—31no—
161#808—31no—
162#807—31no—
163#807—31no—
164C23×C32C2×D411yescharacteristic
165C22×C3⋊D3—22no—
166C22×C3⋊D3—24no—
167C22×C3⋊D3—81no—
168C22×C3⋊D3—81no—
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×Q812yes—
171C2×C32×C4#4112yes—
172C2×C32×C4C2411yescharacteristic
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C4⋊(C3⋊D3)—24no—
176C4⋊(C3⋊D3)—22no—
177C4⋊(C3⋊D3)C2×Q814yes—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—81no—
182C2×C3×Dic3C2×D412yes—
183C2×C3×Dic3C2×D412yes—
184#627—82no—
185#624—82no—
186#624—82no—
187C32×C8—22no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—21no—
190#544—91no—
191#518—91no—
192#478—92no—
193#478—92no—
194#476—92no—
195#474—92no—
196#428—31no—
197#427—91no—
198#424—31no—
199#418—91no—
200C2×#102—91no—
201C23×D3—61no—
202C23×D3—61no—
203C23×D3—121no—
204C22×C3×C4—21no—
205C22×C3×C4—61no—
206C22×C3×C4C22×D311yescharacteristic
207C22×C3×C4C22×D311yescharacteristic
208C22×Dic3—21no—
209C2×D12—34no—
210C2×D12—34no—
211C2×D12—32no—
212C2×D12—32no—
213C2×D12—62no—
214C2×D12—64no—
215C2×D12—61no—
216C2×D12—62no—
217C2×D12—61no—
218C2×D12—62no—
219C2×D12—121no—
220C2×D12—122no—
221C2×C4×D3—62no—
222C2×C4×D3—62no—
223C2×C4×D3—62no—
224C2×C4×D3—62no—
225C2×Dic6—62no—
226C2×Dic6—62no—
227C3×M16—22no—
228C3×M16—22no—
229C3×M16—42no—
230C2×C3×C8—22no—
231C2×C3×C8—31no—
232C2×C3×C8—31no—
233C2×C3×C8—62no—
234C2×C3×C8C22×D312yes—
235C2×C3×C8C22×D312yes—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—61no—
242C3×C22⋊C4—32no—
243C3×C22⋊C4—61no—
244C3×C42—32no—
245C3×C42—61no—
246#217—21no—
247#217—61no—
248#217—61no—
249#217—121no—
250#217D1212yes—
251(C2×D3)⋊C4—62no—
252(C2×D3)⋊C4—62no—
253(C2×D3)⋊C4—62no—
254(C2×D3)⋊C4—62no—
255C4⋊Dic3—21no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—31no—
260C4⋊Dic3—61no—
261C4⋊Dic3—61no—
262C4⋊Dic3—62no—
263C4⋊Dic3—62no—
264C4⋊Dic3D1211yescharacteristic
265C4⋊Dic3D1211yescharacteristic
266C4⋊Dic3C22×D311yescharacteristic
267C4⋊Dic3C22×D311yescharacteristic
268C4⋊Dic3C22×D312yes—
269Dic3⋊C4—62no—
270Dic3⋊C4—62no—
271Dic3⋊C4—62no—
272Dic3⋊C4—62no—
273C4×Dic3—21no—
274C4×Dic3—61no—
275C4×Dic3—61no—
276C4×Dic3—121no—
277C4×Dic3C22×D312yes—
278C2×C3⋊C8—22no—
279C2×C3⋊C8C22×D311yescharacteristic
280C2×C3⋊C8C22×D311yescharacteristic
281C22×C32—21no—
282C22×C32—21no—
283C22×C32C22⋊Q812yes—
284C22×C32C22×D411yescharacteristic
285C2×C3⋊D3—24no—
286C2×C3⋊D3—42no—
287C2×C3⋊D3—42no—
288C2×C3⋊D3—84no—
289C32×C4C22⋊Q812yes—
290C32×C4C22×Q811yescharacteristic
291C32×C4C2×#4111yescharacteristic
292C3⋊Dic3—41no—
293C3⋊Dic3—41no—
294C3×Dic3—22no—
295C3×Dic3—22no—
296C3×Dic3—41no—
297C3×Dic3—42no—
298C3×Dic3—41no—
299C22×D4—91no—
300C2×M16D3211yescharacteristic
301C22×C8—31no—
302C4⋊Q8—92no—
303C4⋊Q8—181no—
304#125—92no—
305#125—181no—
306C22⋊Q8—182no—
307C4×D4—92no—
308C4×D4—92no—
309C4×D4—181no—
310C4×D4—181no—
311#117—31no—
312#117—31no—
313#117—91no—
314C2×C4⋊C4—31no—
315C2×C22⋊C4—182no—
316#107—32no—
317#107—91no—
318#107—91no—
319#106—32no—
320#106—91no—
321#106—91no—
322C4⋊C8—32no—
323C4⋊C8—32no—
324#102—92no—
325#102—92no—
326#102—92no—
327#102—92no—
328C23×C3—21no—
329C23×C3C2×D1211yescharacteristic
330C23×C3D3×D411yescharacteristic
331C22×D3—62no—
332C22×D3—62no—
333C22×D3—64no—
334C22×D3—64no—
335C22×D3—122no—
336C22×D3—124no—
337C22×D3—241no—
338C22×D3—241no—
339C22×D3—241no—
340C22×D3—241no—
341C22×D3—242no—
342C22×D3—242no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—62no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—241no—
353C2×C3×C4C2×D1211yescharacteristic
354C2×C3×C4#23512yes—
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4#23712yes—
357C2×C3×C4D3×Q812yes—
358C2×C3×C4D3×Q812yes—
359C2×C3×C4C23×D311yescharacteristic
360C2×C3×C4C23×D311yescharacteristic
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—81no—
369C2×Dic3—121no—
370C2×Dic3—121no—
371C2×Dic3C2×D1212yes—
372C2×Dic3C2×D1212yes—
373D12—34no—
374D12—34no—
375D12—64no—
376D12—64no—
377D12—64no—
378D12—62no—
379D12—62no—
380D12—122no—
381D12—124no—
382C4×D3—242no—
383C4×D3—242no—
384C4×D3—242no—
385C4×D3—242no—
386Dic6—242no—
387Dic6—242no—
388C3×C8—22no—
389C3×C8—22no—
390C3×C8—42no—
391C3×C8—61no—
392C3×C8—61no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C2×C32—22no—
396C2×C32C2×C22⋊Q811yescharacteristic
397C2×C32#57611yescharacteristic
398C2×C32#57711yescharacteristic
399C3⋊D3—44no—
400C24—181no—
401C2×Q8—182no—
402C2×D4—92no—
403C2×D4—94no—
404C2×D4—181no—
405C2×D4—182no—
406C22×C4—61no—
407C22×C4—182no—
408C22×C4—182no—
409C22×C4C2×D3211yescharacteristic
410M16—22no—
411C2×C8—31no—
412C2×C8—31no—
413C2×C8—62no—
414C2×C8C2×D3212yes—
415C4⋊C4—31no—
416C4⋊C4—31no—
417C4⋊C4—32no—
418C4⋊C4—31no—
419C4⋊C4—31no—
420C4⋊C4—61no—
421C4⋊C4—91no—
422C4⋊C4—91no—
423C4⋊C4—182no—
424C4⋊C4—182no—
425C4⋊C4—182no—
426C22⋊C4—32no—
427C22⋊C4—61no—
428C22⋊C4—182no—
429C22⋊C4—182no—
430C22⋊C4—181no—
431C42—32no—
432C42—61no—
433C42—181no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—22no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3#89612yes—
443C22×C3#89712yes—
444C22×C3C22×D1211yescharacteristic
445C22×C3C2×D3×D411yescharacteristic
446C2×D3—64no—
447C2×D3—64no—
448C2×D3—124no—
449C2×D3—122no—
450C2×D3—122no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—242no—
454C2×D3—242no—
455C2×D3—244no—
456C2×D3—244no—
457C2×D3—248no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—22no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—121no—
464C3×C4—122no—
465C3×C4—121no—
466C3×C4#89612yes—
467C3×C4#89712yes—
468C3×C4C2×#23511yescharacteristic
469C3×C4C2×#23711yescharacteristic
470C3×C4C2×D3×Q811yescharacteristic
471C3×C4C2×D3×Q811yescharacteristic
472Dic3—22no—
473Dic3—22no—
474Dic3—42no—
475Dic3—41no—
476Dic3—41no—
477Dic3—121no—
478Dic3—121no—
479Dic3—121no—
480Dic3—121no—
481Dic3—241no—
482Dic3—241no—
483C32#308111yescharacteristic, complemented, Sylow
484C23—182no—
485C23—184no—
486C23—721no—
487C23—721no—
488C23D3×D1211yescharacteristic
489Q8—722no—
490D4—94no—
491D4—182no—
492D4—184no—
493C2×C4—32no—
494C2×C4—32no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—181no—
499C2×C4—181no—
500C2×C4—241no—
501C2×C4—722no—
502C2×C4—722no—
503C2×C4#379712yes—
504C2×C4D3×D1211yescharacteristic
505C2×C4#380412yes—
506C2×C4C22×D3211yescharacteristic
507C8—22no—
508C8—61no—
509C8—61no—
510C2×C3—22no—
511C2×C3—22no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3C2×#89611yescharacteristic
517C2×C3C2×#89711yescharacteristic
518C2×C3#575111yescharacteristic
519C2×C3#575211yescharacteristic
520C2×C3#577411yescharacteristic
521C2×C3#577811yescharacteristic
522D3—124no—
523D3—124no—
524D3—244no—
525C22—21no—
526C22—21no—
527C22—184no—
528C22—362no—
529C22—362no—
530C22—724no—
531C22#6389512yes—
532C22C2×D3×D1211yescharacteristic
533C4—62no—
534C4—62no—
535C4—122no—
536C4—121no—
537C4—121no—
538C4—361no—
539C4—361no—
540C4#6389512yes—
541C4C2×#379711yescharacteristic
542C4C2×#380411yescharacteristic
543C3—21no—
544C3#7909011yescharacteristic, complemented
545C3#7924211yescharacteristic, complemented
546C2—22no—
547C2—364no—
548C2#1059471411yescharacteristic
549C2#1059471511yescharacteristic
550C2C2×#6389511yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×M16D32
hypocenterC32#3081
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. #10595051C2semidirect
4. C3×#5121C2semidirect
5. C3×#5119C2semidirect
6. #10591122C2semidirect
8. #10590774C2semidirect
9. #10590769C2semidirect
10. #10590764C2semidirect
11. #10590759C2semidirect
24. #63918C22semidirect
30. #63869C22semidirect
33. C3×#818C22semidirect
34. C3×#817C22semidirect
35. C3×Dic3⋊C8C22semidirect
36. C3×Dic3⋊C8C22semidirect
37. #63575C22semidirect
38. #63575C22semidirect
39. #63573C22semidirect
40. #63573C22semidirect
52. #5121D3semidirect
53. #5119D3semidirect
70. C2×C4⋊(C3⋊D3)Q8semidirect
152. #818C2×D3semidirect
155. #817C2×D3semidirect
156. Dic3⋊C8C2×D3semidirect
157. Dic3⋊C8C2×D3semidirect
483. C32#3081semidirect
544. C3#79090semidirect
545. C3#79242semidirect

Maximal subgroups

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

subgroupindex
#105950512
#105949962
#105928302
C3×#51212
C3×#51192
#105911222
C2×#635642
#105907742
#105907692
#105907642
#105907592
#792423
#790903