Finite Group Databasebeta

← #11868995

Subgroups — #11868995

Order 1152.

855 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10595052C211yesmaximal, characteristic, complemented
2#10594982C211yesmaximal, characteristic
3#10592820C211yesmaximal, characteristic, complemented
4C3×#5131C211yesmaximal, characteristic
5C3×#5119C211yesmaximal, characteristic, complemented
6#10591122C211yesmaximal, characteristic, complemented
7C2×#63563C211yesmaximal, characteristic
8#10590772C212yesmaximal, complemented
9#10590767C212yesmaximal, complemented
10#10590762C212yesmaximal, complemented
11#10590757C212yesmaximal, complemented
12#79090—31nomaximal
13#78918—31nomaximal
14C22×C3×D12C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16#64135C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#926C2211yescharacteristic
19C3×#926C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21#63982—22no—
22D3×#217—22no—
23#63918—21no—
24#63918C2212yescomplemented
25#63898—21no—
26#63898—21no—
27Dic3×D12C2212yes—
28Dic3×D12C2212yes—
29#63869—21no—
30#63869C2212yescomplemented
31#63645C2212yescomplemented
32#63644C2212yescomplemented
33C3×#821C2212yes—
34C3×#821C2212yes—
35C3×Dic3⋊C8C2212yescomplemented
36C3×Dic3⋊C8C2212yescomplemented
37#63575C2211yescharacteristic, complemented
38#63575C2211yescharacteristic, complemented
39#63573C2211yescharacteristic, complemented
40#63573C2211yescharacteristic, complemented
41#63563C2212yes—
42#63563C2212yes—
43#5798—31no—
44#5521—31no—
45#5519—31no—
46#5519—31no—
47C3×#424—31no—
48C3×#418—31no—
49C2×#832—31no—
50#5131D311yescharacteristic
51#5121—31no—
52#5121—31no—
53#5121—61no—
54#5119D311yescharacteristic, complemented
55C2×#821—31no—
56#5106—31no—
57#4998—31no—
58#4847—32no—
59#4847—32no—
60#4846—32no—
61#4842—32no—
62#4766—32no—
63#4763—32no—
64#4761—32no—
65#4759—32no—
66C23×C3×D3—21no—
67C22×C32×C4C2311yescharacteristic
68C22×C3⋊Dic3—21no—
69C2×C3×D12—21no—
70C2×C3×D12—22no—
71C2×C3×D12Q814yescomplemented
72C2×C3×D12C2312yes—
73C2×#627—22no—
74C2×D3×Dic3—22no—
75C2×D3×Dic3—22no—
76C32×M16—22no—
77C2×C32×C8C2312yes—
78#3757—21no—
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)C2312yes—
81C4⋊(C3⋊Dic3)C2311yescharacteristic
82C4⋊(C3⋊Dic3)C2311yescharacteristic
83C4×C3⋊Dic3—21no—
84C3×#217—21no—
85C3×#217D412yes—
86C3×C4⋊Dic3D411yescharacteristic
87C3×C4⋊Dic3D411yescharacteristic
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(C2×D3)⋊Dic3—22no—
99(C2×D3)⋊Dic3—22no—
100#3081—91noSylow
101C22×C3×D4—31no—
102C22×D12C2×D311yescharacteristic
103C2×C3×M16—21no—
104C2×C3×M16C2×D311yescharacteristic
105C2×C3×M16C2×D311yescharacteristic
106C22×C3×C8—31no—
107C3×#117—31no—
108C3×#117—31no—
109C2×#217—62no—
110D4×Dic3—32no—
111D4×Dic3—32no—
112D4×Dic3—61no—
113D4×Dic3—61no—
114#926—31no—
115#926—31no—
116#926—61no—
117#926C2×D311yescharacteristic
118#926C2×D311yescharacteristic
119C2×C4⋊Dic3—31no—
120C2×C4⋊Dic3—31no—
121C2×C4⋊Dic3—61no—
122C22×C3⋊C8C2×D311yescharacteristic
123#893—61no—
124#893—61no—
125#890—32no—
126#890—61no—
127#888—32no—
128#888—61no—
129D3×C22⋊C4—62no—
130#878—62no—
131#878—62no—
132C4×D12—32no—
133C4×D12—32no—
134#870—32no—
135#870—61no—
136#869—32no—
137#869—61no—
138C3×C4⋊C8—32no—
139C3×C4⋊C8—32no—
140C3×#102—32no—
141C3×#102—32no—
142#832—32no—
143#832—32no—
144#821—32no—
145#821—32no—
146#821C2×D312yes—
147#821C2×D312yes—
148#818—32no—
149#818—32no—
150#818—31no—
151#818—31no—
152#818—62no—
153#817—32no—
154#817—32no—
155#817—31no—
156#817—31no—
157#817—62no—
158Dic3⋊C8C2×D312yescomplemented
159Dic3⋊C8C2×D312yescomplemented
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—
175C2×C3⋊Dic3—22no—
176C2×C3⋊Dic3—81no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3C2×D412yes—
180C2×C3×Dic3C2×D412yes—
181C3×D12—24no—
182C3×D12—22no—
183C3×D12C2×Q814yes—
184#627—82no—
185D3×Dic3—82no—
186D3×Dic3—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—91no—
197#427—91no—
198#424—31no—
199#418—31no—
200C2×#102—91no—
201C24×C3—61no—
202C23×D3—21no—
203C2×C3×D4—34no—
204C2×C3×D4—32no—
205C2×C3×D4—61no—
206C2×C3×D4—62no—
207C22×C3×C4—21no—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C22×D311yescharacteristic
210C22×Dic3—61no—
211C22×Dic3—61no—
212C22×Dic3—62no—
213C22×Dic3—62no—
214C22×Dic3—121no—
215C2×D12—21no—
216C2×D12—22no—
217C2×D12Dic614yescomplemented
218C2×D12C22×D312yes—
219C2×C4×D3—62no—
220C2×C4×D3—62no—
221C2×Dic6—62no—
222C2×Dic6—62no—
223C3×M16—24no—
224C3×M16—22no—
225C3×M16—22no—
226C2×C3×C8—22no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—62no—
230C2×C3×C8C22×D312yes—
231C2×C3×C8C22×D312yes—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C22⋊C4—32no—
237C3×C22⋊C4—61no—
238C3×C42—32no—
239C3×C42—61no—
240#217—21no—
241#217—61no—
242#217—61no—
243#217—62no—
244#217—62no—
245#217—121no—
246#217D1212yes—
247(C2×D3)⋊C4—62no—
248(C2×D3)⋊C4—62no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—31no—
253C4⋊Dic3—32no—
254C4⋊Dic3—32no—
255C4⋊Dic3—61no—
256C4⋊Dic3—62no—
257C4⋊Dic3—61no—
258C4⋊Dic3—62no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3—62no—
262C4⋊Dic3—121no—
263C4⋊Dic3D1211yescharacteristic
264C4⋊Dic3D1211yescharacteristic
265C4⋊Dic3C22×D311yescharacteristic
266C4⋊Dic3C22×D311yescharacteristic
267Dic3⋊C4—62no—
268Dic3⋊C4—62no—
269Dic3⋊C4—62no—
270Dic3⋊C4—62no—
271C4×Dic3—21no—
272C4×Dic3—61no—
273C4×Dic3—61no—
274C4×Dic3—121no—
275C4×Dic3C22×D312yes—
276C2×C3⋊C8—22no—
277C2×C3⋊C8C22×D311yescharacteristic
278C2×C3⋊C8C22×D311yescharacteristic
279C22×C32—21no—
280C22×C32—21no—
281C22×C32C22⋊Q812yes—
282C22×C32C22×D411yescharacteristic
283C2×C3×D3—24no—
284C2×C3×D3—42no—
285C2×C3×D3—42no—
286C2×C3×D3—84no—
287C32×C4C22⋊Q812yes—
288C32×C4C22×Q811yescharacteristic
289C32×C4C2×#4111yescharacteristic
290C3⋊Dic3—41no—
291C3⋊Dic3—42no—
292C3⋊Dic3—41no—
293C3×Dic3—22no—
294C3×Dic3—22no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C22×D4—31no—
298C2×M16D3211yescharacteristic
299C22×C8—31no—
300C4⋊Q8—92no—
301C4⋊Q8—181no—
302#125—92no—
303#125—181no—
304C22⋊Q8—182no—
305C4×D4—92no—
306C4×D4—92no—
307C4×D4—181no—
308C4×D4—181no—
309#117—31no—
310#117—31no—
311#117—91no—
312C2×C4⋊C4—91no—
313C2×C22⋊C4—182no—
314#107—91no—
315#107—91no—
316#107—92no—
317#106—91no—
318#106—91no—
319#106—92no—
320C4⋊C8—32no—
321C4⋊C8—32no—
322#102—32no—
323#102—32no—
324#102—92no—
325#102—92no—
326C23×C3—21no—
327C23×C3—62no—
328C23×C3—64no—
329C23×C3—241no—
330C23×C3—241no—
331C23×C3C2×D1211yescharacteristic
332C23×C3D3×D411yescharacteristic
333C22×D3—22no—
334C22×D3—24no—
335C22×D3—81no—
336C22×D3—81no—
337C3×D4—34no—
338C3×D4—64no—
339C3×D4—62no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4C2×Dic612yes—
349C2×C3×C4C2×D1211yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4#23712yes—
352C2×C3×C4#23712yes—
353C2×C3×C4D3×Q812yes—
354C2×C3×C4C23×D311yescharacteristic
355C2×C3×C4C23×D311yescharacteristic
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—62no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—62no—
364C2×Dic3—121no—
365C2×Dic3—121no—
366C2×Dic3—122no—
367C2×Dic3—242no—
368C2×Dic3—242no—
369C2×Dic3—241no—
370C2×Dic3—241no—
371C2×Dic3—242no—
372C2×Dic3C2×D1212yes—
373C2×Dic3C2×D1212yes—
374D12—24no—
375D12—22no—
376D12C2×Dic614yes—
377C4×D3—242no—
378C4×D3—242no—
379Dic6—242no—
380Dic6—242no—
381C3×C8—24no—
382C3×C8—22no—
383C3×C8—22no—
384C3×C8—61no—
385C3×C8—61no—
386C3⋊C8—21no—
387C3⋊C8—21no—
388C2×C32—22no—
389C2×C32C2×C22⋊Q811yescharacteristic
390C2×C32#57611yescharacteristic
391C2×C32#57711yescharacteristic
392C3×D3—44no—
393C24—61no—
394C2×Q8—182no—
395C2×D4—32no—
396C2×D4—34no—
397C2×D4—61no—
398C2×D4—62no—
399C22×C4—182no—
400C22×C4—182no—
401C22×C4—181no—
402C22×C4C2×D3211yescharacteristic
403M16—22no—
404C2×C8—31no—
405C2×C8—31no—
406C2×C8—62no—
407C2×C8C2×D3212yes—
408C4⋊C4—31no—
409C4⋊C4—31no—
410C4⋊C4—31no—
411C4⋊C4—31no—
412C4⋊C4—92no—
413C4⋊C4—91no—
414C4⋊C4—91no—
415C4⋊C4—182no—
416C4⋊C4—182no—
417C4⋊C4—182no—
418C4⋊C4—181no—
419C22⋊C4—32no—
420C22⋊C4—61no—
421C22⋊C4—182no—
422C22⋊C4—182no—
423C22⋊C4—181no—
424C42—32no—
425C42—61no—
426C42—181no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—64no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—244no—
439C22×C3#87812yes—
440C22×C3#89712yes—
441C22×C3C22×D1211yescharacteristic
442C22×C3C2×D3×D411yescharacteristic
443C2×D3—24no—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—84no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—22no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—121no—
453C3×C4—121no—
454C3×C4#87812yes—
455C3×C4#89712yes—
456C3×C4C22×Dic611yescharacteristic
457C3×C4C2×#23711yescharacteristic
458C3×C4C2×#23711yescharacteristic
459C3×C4C2×D3×Q811yescharacteristic
460Dic3—22no—
461Dic3—22no—
462Dic3—41no—
463Dic3—41no—
464Dic3—121no—
465Dic3—122no—
466Dic3—121no—
467Dic3—122no—
468Dic3—121no—
469Dic3—121no—
470Dic3—242no—
471Dic3—241no—
472Dic3—241no—
473C32#308111yescharacteristic, complemented, Sylow
474C23—62no—
475C23—64no—
476C23—241no—
477C23—241no—
478C23D3×D1211yescharacteristic
479Q8—722no—
480D4—34no—
481D4—62no—
482D4—64no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—181no—
488C2×C4—181no—
489C2×C4—182no—
490C2×C4—722no—
491C2×C4—722no—
492C2×C4—721no—
493C2×C4D3×Dic612yes—
494C2×C4D3×D1211yescharacteristic
495C2×C4#380512yes—
496C2×C4C22×D3211yescharacteristic
497C8—22no—
498C8—61no—
499C8—61no—
500C2×C3—22no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—42no—
506C2×C3—124no—
507C2×C3C2×#87811yescharacteristic
508C2×C3C2×#89711yescharacteristic
509C2×C3#575111yescharacteristic
510C2×C3#575211yescharacteristic
511C2×C3#577411yescharacteristic
512C2×C3#577511yescharacteristic
513D3—44no—
514C22—21no—
515C22—21no—
516C22—64no—
517C22—122no—
518C22—122no—
519C22—244no—
520C22#6389312yes—
521C22C2×D3×D1211yescharacteristic
522C4—62no—
523C4—62no—
524C4—121no—
525C4—121no—
526C4—361no—
527C4—361no—
528C4—362no—
529C4#6389312yes—
530C4C2×D3×Dic611yescharacteristic
531C4C2×#380511yescharacteristic
532C3—21no—
533C3#7891811yescharacteristic, complemented
534C3#7909011yescharacteristic, complemented
535C2—22no—
536C2—124no—
537C2#1059471311yescharacteristic
538C2#1059471411yescharacteristic
539C2C2×#6389311yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radicalC22×C3×D12C22
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. #10595052C2semidirect
3. #10592820C2semidirect
5. C3×#5119C2semidirect
6. #10591122C2semidirect
8. #10590772C2semidirect
9. #10590767C2semidirect
10. #10590762C2semidirect
11. #10590757C2semidirect
24. #63918C22semidirect
30. #63869C22semidirect
31. #63645C22semidirect
32. #63644C22semidirect
35. C3×Dic3⋊C8C22semidirect
36. C3×Dic3⋊C8C22semidirect
37. #63575C22semidirect
38. #63575C22semidirect
39. #63573C22semidirect
40. #63573C22semidirect
54. #5119D3semidirect
71. C2×C3×D12Q8semidirect
158. Dic3⋊C8C2×D3semidirect
159. Dic3⋊C8C2×D3semidirect
217. C2×D12Dic6semidirect
473. C32#3081semidirect
533. C3#78918semidirect
534. C3#79090semidirect

Maximal subgroups

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

subgroupindex
#105950522
#105949822
#105928202
C3×#51312
C3×#51192
#105911222
C2×#635632
#105907722
#105907672
#105907622
#105907572
#790903
#789183