Finite Group Databasebeta

← #11787682

Subgroups — #11787682

Order 1152.

753 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594990C211yesmaximal, characteristic
2C2×#63833C211yesmaximal, characteristic
3C2×#63652C211yesmaximal, characteristic
4C3×#5120C211yesmaximal, characteristic, complemented
5#10591221C211yesmaximal, characteristic, complemented
6C3×#4554C211yesmaximal, characteristic, complemented
7#10589160C211yesmaximal, characteristic, complemented
8#70348—31nomaximal
9#69004—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C4×C3⋊Dic3C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17#63952—22no—
18#63900—21no—
19#63894—21no—
20#63849—21no—
21#63844—21no—
22C2×#3719—21no—
23#63833—22no—
24#63833C414yescomplemented
25C2×#3717—21no—
26#63652—22no—
27C3×#816—21no—
28C3×#816—21no—
29#63579—22no—
30C2×#942—31no—
31C2×#909—31no—
32#5523—31no—
33#5504—31no—
34C3×#419—31no—
35#5229—31no—
36C2×#825—31no—
37C2×#825—31no—
38C2×#825—61no—
39#5120D311yescharacteristic, complemented
40#4973—31no—
41C3×#343—31no—
42#4555—31no—
43#4554D311yescharacteristic, complemented
44#4532—31no—
45C22×C32×C4C2311yescharacteristic
46C2×#638—21no—
47C22×C3⋊Dic3—21no—
48#3828—21no—
49#3828—21no—
50#3828C2×C412yes—
51#3828D412yes—
52C2×C4⋊(C3⋊D3)D411yescharacteristic
53C2×C4×C3⋊D3—21no—
54C2×#634D411yescharacteristic
55C22×C3×Dic3—21no—
56C22×C3×Dic3D412yes—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6C2×C412yescomplemented
63C2×#624—22no—
64C32×M16—21no—
65C32×M16—21no—
66C2×C32×C8—21no—
67C4×C3⋊Dic3—21no—
68C4×C3⋊Dic3—21no—
69C3×#217—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3D411yescharacteristic
73C3×C4⋊Dic3D411yescharacteristic
74C3×C4×Dic3—21no—
75C3×#208—21no—
76C3×#208C2×C412yescomplemented
77C2×C3×C3⋊C8—21no—
78#3724—22no—
79#3722—22no—
80Dic32—22no—
81#3719—24no—
82#3717—24no—
83#1882—91noSylow
84C22×C3×Q8—31no—
85C2×#235—31no—
86C2×#235—31no—
87C2×#235—61no—
88C22×Dic6C2×D311yescharacteristic
89C2×C3×M16—21no—
90C2×C3×M16—31no—
91C2×C3×M16C2×D311yescharacteristic
92C2×C3×M16C2×D311yescharacteristic
93C3×#117—31no—
94C2×C3×C4⋊C4—31no—
95C2×#216—61no—
96#942—34no—
97#942—62no—
98C2×#215—61no—
99C4×#67—62no—
100#926C2×D311yescharacteristic
101C2×C4⋊Dic3C2×D311yescharacteristic
102C2×C4×Dic3—31no—
103C2×C4×Dic3—31no—
104C2×C4×Dic3—61no—
105C2×#208C2×D311yescharacteristic
106#909—34no—
107#909—62no—
108C2×Dic12—61no—
109C2×#204—61no—
110#893—61no—
111#892—61no—
112#892—61no—
113D3×C4⋊C4—61no—
114D3×C4⋊C4—61no—
115#887—61no—
116#67⋊C4—62no—
117C4×D12—61no—
118C4×Dic6—61no—
119C3×#103—61no—
120C3×#103—61no—
121C3⋊#95—62no—
122C3⋊#95—62no—
123#825—62no—
124#825—62no—
125#825—64no—
126#816—21no—
127#816—21no—
128C23×C32C2×D411yescharacteristic
129C22×C3⋊D3—21no—
130C2×C32×C4C22×C411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4#4111yescharacteristic
135C2×C32×C4#4111yescharacteristic
136#638—22no—
137#638—22no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—41no—
141C2×C3⋊Dic3—41no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)C22⋊C412yes—
144C4×C3⋊D3—22no—
145C4×C3⋊D3—22no—
146#634—21no—
147#634C22⋊C412yes—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—41no—
153C2×C3×Dic3—41no—
154C2×C3×Dic3—42no—
155C3×Dic6—22no—
156C3×Dic6—22no—
157C3×Dic6—41no—
158C3×Dic6—41no—
159#624—82no—
160C32×C8—41no—
161C3×C3⋊C8—22no—
162C2×#137—91no—
163#520—91no—
164C2×#104—91no—
165#419—31no—
166#391—91no—
167#343—31no—
168#339—91no—
169C2×C3×Q8—32no—
170C2×C3×Q8—61no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×C3×Q8—61no—
174C2×C3×Q8—61no—
175C22×C3×C4—21no—
176C22×C3×C4—32no—
177C22×C3×C4—61no—
178C22×C3×C4C22×D311yescharacteristic
179C22×C3×C4C22×D311yescharacteristic
180C2×#67—61no—
181C2×#67—61no—
182C2×#67—121no—
183C22×Dic3—21no—
184C22×Dic3—61no—
185C22×Dic3—61no—
186C22×Dic3—121no—
187C22×Dic3D1212yes—
188#235—32no—
189#235—32no—
190#235—32no—
191#235—32no—
192#235—62no—
193#235—62no—
194#235—61no—
195#235—61no—
196#235—61no—
197#235—61no—
198#235—121no—
199#235—121no—
200C2×D12—31no—
201C2×D12—31no—
202C2×D12—61no—
203C2×C4×D3—61no—
204C2×C4×D3—61no—
205C2×C4×D3—62no—
206C2×C4×D3—62no—
207C2×C4×D3—121no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—31no—
214C2×Dic6—31no—
215C2×Dic6—61no—
216C2×Dic6C4×D312yescomplemented
217C3×M16—22no—
218C3×M16—21no—
219C3×M16—21no—
220C3×M16—21no—
221C3×M16—21no—
222C3×M16—32no—
223C3×M16—41no—
224C3×M16—61no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—41no—
228C2×C3×C8—61no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—61no—
233C3×C22⋊C4—61no—
234C3×C42—61no—
235#217—21no—
236#216—64no—
237#215—64no—
238(C2×D3)⋊C4—62no—
239(C2×D3)⋊C4—62no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3#6711yescharacteristic
243C4⋊Dic3#6711yescharacteristic
244Dic3⋊C4—62no—
245Dic3⋊C4—62no—
246C4×Dic3—21no—
247C4×Dic3—62no—
248C4×Dic3—62no—
249C4×Dic3—62no—
250C4×Dic3—61no—
251C4×Dic3—61no—
252C4×Dic3—61no—
253C4×Dic3—61no—
254C4×Dic3—121no—
255#208—21no—
256#208C4×D312yescomplemented
257C2×C3⋊C8—21no—
258Dic12—64no—
259#204—64no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32C4×D411yescharacteristic
263C22×C32#12011yescharacteristic
264C22×C32#12111yescharacteristic
265C2×C3⋊D3—22no—
266C2×C3⋊D3—41no—
267C2×C3⋊D3—41no—
268C32×C4C2×C22⋊C411yescharacteristic
269C32×C4C4×D411yescharacteristic
270C32×C4#12111yescharacteristic
271C32×C4#12311yescharacteristic
272C3⋊Dic3—22no—
273C3⋊Dic3—42no—
274C3×Dic3—42no—
275C3×Dic3—41no—
276C3×Dic3—42no—
277C3×Dic3—41no—
278C2×#41—91no—
279C22×Q8—31no—
280#137—94no—
281#137—182no—
282C2×Q16—181no—
283C2×SD16—181no—
284C2×M16—31no—
285C2×M16D3211yescharacteristic
286C4×Q8—181no—
287C4×D4—181no—
288C4×D4—182no—
289#117—31no—
290#117—181no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—181no—
293C2×C42—91no—
294#104—182no—
295#103—61no—
296#103—61no—
297#95—182no—
298C23×C3—21no—
299C23×C3C2×D1211yescharacteristic
300C23×C3D3×D411yescharacteristic
301C22×D3—61no—
302C22×D3—61no—
303C22×D3—121no—
304C3×Q8—62no—
305C3×Q8—62no—
306C3×Q8—121no—
307C3×Q8—121no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—62no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—62no—
318C2×C3×C4—121no—
319C2×C3×C4—121no—
320C2×C3×C4—122no—
321C2×C3×C4C2×C4×D311yescharacteristic
322C2×C3×C4C2×C4×D311yescharacteristic
323C2×C3×C4#23511yescharacteristic
324C2×C3×C4#23511yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4D3×D411yescharacteristic
327C2×C3×C4D3×D411yescharacteristic
328C2×C3×C4D3×D411yescharacteristic
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4#23711yescharacteristic
331C2×C3×C4#23911yescharacteristic
332C2×C3×C4C2×#6711yescharacteristic
333#67—62no—
334#67—62no—
335#67—62no—
336#67—62no—
337#67—122no—
338#67—122no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—42no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—62no—
349C2×Dic3—62no—
350C2×Dic3—121no—
351C2×Dic3—122no—
352C2×Dic3—122no—
353C2×Dic3—121no—
354C2×Dic3—121no—
355C2×Dic3—122no—
356C2×Dic3—121no—
357C2×Dic3—121no—
358D12—32no—
359D12—32no—
360D12—62no—
361D12—61no—
362D12—61no—
363D12—121no—
364C4×D3—62no—
365C4×D3—62no—
366C4×D3—62no—
367C4×D3—62no—
368C4×D3—122no—
369C4×D3—122no—
370C4×D3—242no—
371C4×D3—242no—
372Dic6—22no—
373Dic6—22no—
374Dic6—32no—
375Dic6—32no—
376Dic6—41no—
377Dic6—41no—
378Dic6—62no—
379Dic6—61no—
380Dic6—61no—
381Dic6—121no—
382C3×C8—42no—
383C3×C8—41no—
384C3×C8—41no—
385C3×C8—62no—
386C3⋊C8—22no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32#38611yescharacteristic
390C2×C32#45511yescharacteristic
391C2×C32#45611yescharacteristic
392C3⋊D3—42no—
393#41—92no—
394#41—92no—
395#41—181no—
396#41—181no—
397C2×Q8—32no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—91no—
404C2×D4—91no—
405C2×D4—181no—
406C22×C4—32no—
407C22×C4—61no—
408C22×C4—182no—
409C22×C4—181no—
410C22×C4—181no—
411C22×C4C2×D3211yescharacteristic
412Q16—184no—
413SD16—184no—
414M16—21no—
415M16—21no—
416M16—32no—
417M16—61no—
418C2×C8—21no—
419C2×C8—61no—
420C4⋊C4—31no—
421C4⋊C4—31no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—182no—
425C22⋊C4—61no—
426C22⋊C4—182no—
427C42—61no—
428C42—182no—
429C42—181no—
430C42—181no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3C4×D1211yescharacteristic
441C22×C3#67⋊C411yescharacteristic
442C22×C3#88211yescharacteristic
443C22×C3#88411yescharacteristic
444C22×C3#89511yescharacteristic
445C22×C3#93711yescharacteristic
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—122no—
449C2×D3—121no—
450C2×D3—121no—
451C2×D3—121no—
452C2×D3—121no—
453C2×D3—241no—
454C2×D3—241no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—122no—
460C3×C4—122no—
461C3×C4—121no—
462C3×C4—121no—
463C3×C4D3×C22⋊C411yescharacteristic
464C3×C4D3×C22⋊C411yescharacteristic
465C3×C4#88311yescharacteristic
466C3×C4#88411yescharacteristic
467C3×C4#89311yescharacteristic
468C3×C4#89411yescharacteristic
469C3×C4C4×#6711yescharacteristic
470C3×C4#93811yescharacteristic
471Dic3—41no—
472Dic3—41no—
473Dic3—42no—
474Dic3—42no—
475Dic3—62no—
476Dic3—62no—
477Dic3—122no—
478Dic3—122no—
479Dic3—122no—
480Dic3—124no—
481C32#188211yescharacteristic, complemented, Sylow
482C23—181no—
483C23D3×D1211yescharacteristic
484Q8—62no—
485Q8—62no—
486Q8—92no—
487Q8—121no—
488Q8—121no—
489Q8—181no—
490D4—92no—
491D4—182no—
492D4—182no—
493D4—181no—
494C2×C4—62no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—122no—
501C2×C4—181no—
502C2×C4—182no—
503C2×C4—182no—
504C2×C4—182no—
505C2×C4—361no—
506C2×C4—361no—
507C2×C4—722no—
508C2×C4#379911yescharacteristic
509C2×C4C4×D3211yescharacteristic
510C2×C4#380211yescharacteristic
511C2×C4#380411yescharacteristic
512C2×C4D3×#6711yescharacteristic
513C2×C4#381111yescharacteristic
514C8—41no—
515C8—62no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3#466511yescharacteristic
526C2×C3#474611yescharacteristic
527C2×C3#475211yescharacteristic
528C2×C3#495611yescharacteristic
529C2×C3#520211yescharacteristic
530C2×C3#520511yescharacteristic
531D3—122no—
532D3—122no—
533D3—242no—
534C22—21no—
535C22—21no—
536C22—182no—
537C22—361no—
538C22—361no—
539C22#6388011yescharacteristic
540C22#6391011yescharacteristic
541C22#6392211yescharacteristic
542C4—122no—
543C4—121no—
544C4—121no—
545C4—122no—
546C4—182no—
547C4—362no—
548C4#6385411yescharacteristic
549C4#6390411yescharacteristic
550C4#6392111yescharacteristic
551C4#6395811yescharacteristic
552C3—21no—
553C3#6900411yescharacteristic, complemented
554C3#7034811yescharacteristic, complemented
555C2—21no—
556C2—21no—
557C2—362no—
558C2#1059101211yescharacteristic
559C2#1059101811yescharacteristic
560C2#1059127511yescharacteristic
subgroupG/H
centerC22#63922
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#120
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1882
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
4. C3×#5120C2semidirect
5. #10591221C2semidirect
6. C3×#4554C2semidirect
7. #10589160C2semidirect
24. #63833C4semidirect
39. #5120D3semidirect
43. #4554D3semidirect
62. C2×C3×Dic6C2×C4semidirect
76. C3×#208C2×C4semidirect
216. C2×Dic6C4×D3semidirect
256. #208C4×D3semidirect
481. C32#1882semidirect
553. C3#69004semidirect
554. C3#70348semidirect

Maximal subgroups

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

subgroupindex
#105949902
C2×#638332
C2×#636522
C3×#51202
#105912212
C3×#45542
#105891602
#703483
#690043