Finite Group Databasebeta

← #11895423

Subgroups — #11895423

Order 1152.

940 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596024C211yesmaximal, characteristic
2#10593856C211yesmaximal, characteristic, complemented
3C3×C4×#214C211yesmaximal, characteristic, complemented
4C3×#5008C211yesmaximal, characteristic, complemented
5#10591444C212yesmaximal, complemented
6#10591402C212yesmaximal, complemented
7#10590483C212yesmaximal, complemented
8#10590433C212yesmaximal, complemented
9#10590367C211yesmaximal, characteristic, complemented
10#10590281C211yesmaximal, characteristic, complemented
11#10590254C211yesmaximal, characteristic, complemented
12#81090—31nomaximal
13#79622—31nomaximal
14C2×#3828—21no—
15C32×C4×D4C2211yescharacteristic
16#64149C2212yes—
17#64139—21no—
18#64102C2212yes—
19#64093—22no—
20#64083C2211yescharacteristic
21C4×#634C2211yescharacteristic
22C2×C3×#214C2211yescharacteristic
23C3×#869C2211yescharacteristic
24C3×C4×Dic6C2211yescharacteristic
25#63919C2212yes—
26#63918—21no—
27#63844—21no—
28#63843C2212yes—
29#63839—22no—
30C2×#3717C2212yes—
31C3×#832C2212yes—
32C3×#832C2211yescharacteristic
33C3×#810C2211yescharacteristic
34C3×#808C2212yes—
35C3×#808C2211yescharacteristic
36C3×C4⋊(C3⋊C8)C2211yescharacteristic
37C3×C4×C3⋊C8C2211yescharacteristic
38#63572C2212yes—
39#63564C2212yes—
40#63564C2212yes—
41#63564C2212yes—
42#63553C2212yes—
43#5722—31no—
44Q8×Dic6—31no—
45#5553—31no—
46#5553—31no—
47#5553—61no—
48C3×#476—31no—
49C3×C4×SD16—31no—
50#5163—32no—
51#5088—31no—
52#5027—31no—
53C4×#214D311yescharacteristic, complemented
54#5008D311yescharacteristic, complemented
55#4870—32no—
56#4870—32no—
57#4852—32no—
58#4848—31no—
59#4830—31no—
60#4788—32no—
61#4762—32no—
62#4760—32no—
63#4760—32no—
64#4694—31no—
65C8×Dic6—31no—
66C2×C32×D4C2311yescharacteristic
67C22×C32×C4—21no—
68C2×#638—22no—
69C2×#638—22no—
70#3828—44no—
71C2×C4⋊(C3⋊D3)—21no—
72C2×C4⋊(C3⋊D3)—21no—
73C2×C4⋊(C3⋊D3)C2312yes—
74C2×C4×C3⋊D3—22no—
75C2×#634D411yescharacteristic
76C2×C3×Dic6—21no—
77C2×C3×Dic6C2311yescharacteristic
78C2×#627—22no—
79C32×C4⋊C4C2311yescharacteristic
80C32×C22⋊C4—21no—
81C32×C42C2311yescharacteristic
82#3752—22no—
83C4⋊(C3⋊Dic3)D411yescharacteristic
84#3750D412yes—
85C4×C3⋊Dic3C2312yes—
86C3×#214—22no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3C2312yes—
89C3×C4⋊Dic3C2311yescharacteristic
90C3×Dic3⋊C4—21no—
91C3×C4×Dic3—21no—
92C2×C3×C3⋊C8C2311yescharacteristic
93C2×C3×C3⋊C8C2311yescharacteristic
94C2×C3×C3⋊C8C2312yes—
95#3724—22no—
96Dic3⋊Dic3—22no—
97Dic32—22no—
98#3717—24no—
99#3362—91noSylow
100C2×#235—61no—
101C2×#235—61no—
102C2×#235—62no—
103C2×C3×SD16—31no—
104C3×C4⋊Q8—31no—
105C3×C4×Q8—31no—
106C3×C4×D4—21no—
107C3×C4×D4C2×D311yescharacteristic
108C3×C4×D4C2×D311yescharacteristic
109Q8×Dic3—32no—
110#944—62no—
111C2×#215—32no—
112#940—32no—
113#940—32no—
114#940—62no—
115C2×#214C2×D311yescharacteristic
116#930—61no—
117#930—61no—
118#930—62no—
119C2×#204—32no—
120#893—32no—
121#893—32no—
122#893—62no—
123#888—32no—
124#888—32no—
125#888—61no—
126#888—62no—
127#887—32no—
128#887—61no—
129#884—62no—
130#884—62no—
131#884—64no—
132#874—31no—
133#874—31no—
134#874—61no—
135#869—61no—
136#869C2×D311yescharacteristic
137C4×Dic6—31no—
138C4×Dic6—31no—
139C4×Dic6—61no—
140C4×Dic6—61no—
141C4×Dic6C2×D311yescharacteristic
142C3×#106—31no—
143C3×#106—32no—
144C3×C4⋊C8—31no—
145C3×#103—31no—
146C3×#102—31no—
147C3×#102—32no—
148C3×C4×C8—31no—
149#832C2×D312yes—
150#832C2×D311yescharacteristic
151#821—32no—
152#821—32no—
153#821—32no—
154Dic3⋊C8—32no—
155Dic3⋊C8—32no—
156C8×Dic3—32no—
157C8×Dic3—32no—
158#810C2×D311yescharacteristic
159#809—32no—
160#809—32no—
161#809—32no—
162#808C2×D311yescharacteristic
163#808C2×D312yes—
164C4⋊(C3⋊C8)C2×D311yescharacteristic
165C4×C3⋊C8C2×D311yescharacteristic
166C23×C32—21no—
167C22×C3⋊D3—22no—
168C32×D4—21no—
169C32×D4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—42no—
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2411yescharacteristic
175#638—44no—
176#638—82no—
177C2×C3⋊Dic3C2×D412yes—
178C2×C3⋊Dic3C2×D412yes—
179C4⋊(C3⋊D3)—22no—
180C4⋊(C3⋊D3)—22no—
181C4⋊(C3⋊D3)—42no—
182C4⋊(C3⋊D3)—81no—
183C4×C3⋊D3—44no—
184#634SD1614yescomplemented
185C2×C3×Dic3—21no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C3×Dic6—21no—
189C3×Dic6—81no—
190C3×Dic6#4112yescomplemented
191#627—82no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195Q82—91no—
196#550—91no—
197#486—91no—
198#486—92no—
199#476—31no—
200#476—92no—
201#460—92no—
202#460—91no—
203C8×Q8—91no—
204C4×SD16—31no—
205C4×SD16—92no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C2×C3×D4—21no—
209C2×C3×D4C22×D311yescharacteristic
210C2×C3×D4C22×D311yescharacteristic
211C22×C3×C4—22no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C2×#67—62no—
215C2×#67—62no—
216C2×#67—62no—
217C2×#67—62no—
218C2×#67—64no—
219C2×#67—64no—
220#235—124no—
221#235—124no—
222#235—128no—
223C2×D12—32no—
224C2×D12—32no—
225C2×D12—62no—
226C2×D12—62no—
227C2×D12—61no—
228C2×D12—61no—
229C2×D12—62no—
230C2×D12—61no—
231C2×D12—61no—
232C2×C4×D3—62no—
233C2×C4×D3—62no—
234C2×C4×D3—64no—
235C2×Dic6—21no—
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—61no—
239C2×Dic6—62no—
240C2×Dic6—62no—
241C2×Dic6C22×D311yescharacteristic
242C3×SD16—62no—
243C2×C3×C8—31no—
244C2×C3×C8—32no—
245C2×C3×C8—31no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4C22×D311yescharacteristic
252C3×C4⋊C4C22×D311yescharacteristic
253C3×C22⋊C4—22no—
254C3×C22⋊C4—21no—
255C3×C22⋊C4—21no—
256C3×C42—21no—
257C3×C42—61no—
258C3×C42C22×D311yescharacteristic
259C3×C42C22×D311yescharacteristic
260#215—64no—
261#214—22no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4—62no—
264(C2×D3)⋊C4—64no—
265C4⋊Dic3—21no—
266C4⋊Dic3—31no—
267C4⋊Dic3—31no—
268C4⋊Dic3—61no—
269C4⋊Dic3—62no—
270C4⋊Dic3C22×D311yescharacteristic
271C4⋊Dic3C22×D312yes—
272Dic3⋊C4—21no—
273Dic3⋊C4—32no—
274Dic3⋊C4—32no—
275Dic3⋊C4—62no—
276Dic3⋊C4—62no—
277Dic3⋊C4—62no—
278Dic3⋊C4—62no—
279C4×Dic3—21no—
280C4×Dic3—32no—
281C4×Dic3—32no—
282C4×Dic3—62no—
283C4×Dic3—62no—
284C4×Dic3—62no—
285C2×C3⋊C8C22×D311yescharacteristic
286C2×C3⋊C8C22×D311yescharacteristic
287C2×C3⋊C8C22×D312yes—
288#204—64no—
289C22×C32—41no—
290C22×C32—41no—
291C22×C32—41no—
292C22×C32C22×D411yescharacteristic
293C2×C3⋊D3—42no—
294C2×C3⋊D3—42no—
295C2×C3⋊D3—42no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4C2×SD1612yes—
299C32×C4C2×#4111yescharacteristic
300C32×C42−1+411yescharacteristic
301C3⋊Dic3—22no—
302C3⋊Dic3C2×SD1614yes—
303C3×Dic3—22no—
304C3×Dic3—41no—
305C3×Dic3—42no—
306C2×#41—181no—
307C2×SD16—31no—
308C2×SD16—92no—
309C4⋊Q8—31no—
310C4⋊Q8—92no—
311C4⋊Q8—181no—
312C4⋊Q8—182no—
313#127—92no—
314#127—91no—
315#121—182no—
316#121—181no—
317C4×Q8—31no—
318C4×Q8—91no—
319C4×Q8—92no—
320C4×Q8—181no—
321C4×D4—92no—
322C4×D4D3211yescharacteristic
323#106—31no—
324#106—32no—
325C4⋊C8—31no—
326C4⋊C8—92no—
327#103—31no—
328#102—31no—
329#102—32no—
330#102—92no—
331#102—92no—
332#102—92no—
333C4×C8—31no—
334C4×C8—92no—
335C23×C3—22no—
336C23×C3—21no—
337C23×C3—21no—
338C22×D3—64no—
339C22×D3—62no—
340C22×D3—62no—
341C3×Q8—32no—
342C3×Q8—61no—
343C3×Q8—241no—
344C3×D4—21no—
345C3×D4—21no—
346C3×D4—21no—
347C3×D4—21no—
348C3×D4—41no—
349C3×D4—41no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—44no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—61no—
360C2×C3×C4—62no—
361C2×C3×C4—61no—
362C2×C3×C4D3×D411yescharacteristic
363C2×C3×C4D3×D411yescharacteristic
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4D3×D411yescharacteristic
366C2×C3×C4C23×D311yescharacteristic
367C2×C3×C4C23×D311yescharacteristic
368#67—124no—
369#67—124no—
370#67—128no—
371#67—242no—
372#67—242no—
373#67—244no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—62no—
382C2×Dic3—62no—
383D12—62no—
384D12—62no—
385D12—62no—
386D12—62no—
387D12—122no—
388D12—122no—
389D12—124no—
390D12—122no—
391D12—122no—
392D12—241no—
393D12—241no—
394D12—242no—
395C4×D3—124no—
396C4×D3—124no—
397C4×D3—128no—
398Dic6—21no—
399Dic6—34no—
400Dic6—34no—
401Dic6—64no—
402Dic6—81no—
403Dic6—242no—
404Dic6—242no—
405Dic6#23512yescomplemented
406C3×C8—62no—
407C3×C8—61no—
408C3×C8—61no—
409C3⋊C8—21no—
410C3⋊C8—22no—
411C3⋊C8—21no—
412C2×C32—81no—
413C2×C32#54811yescharacteristic
414C2×C32C22×SD1611yescharacteristic
415C2×C32#57611yescharacteristic
416C3⋊D3—82no—
417#41—364no—
418C2×Q8—31no—
419C2×Q8—61no—
420C2×Q8—91no—
421C2×Q8—182no—
422C2×D4—92no—
423C2×D4—182no—
424C2×D4—182no—
425C2×D4—181no—
426C2×D4—181no—
427C2×D4C2×D3211yescharacteristic
428C22×C4—21no—
429C22×C4—182no—
430SD16—62no—
431SD16—184no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—32no—
435C4⋊C4—31no—
436C4⋊C4—32no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—92no—
440C4⋊C4—91no—
441C4⋊C4—182no—
442C4⋊C4—182no—
443C4⋊C4C2×D3211yescharacteristic
444C22⋊C4—21no—
445C22⋊C4—182no—
446C42—61no—
447C42—92no—
448C42—182no—
449C42C2×D3211yescharacteristic
450C22×C3—21no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3C2×D3×D411yescharacteristic
461C22×C3C2×D3×D411yescharacteristic
462C2×D3—124no—
463C2×D3—124no—
464C2×D3—124no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—122no—
470C2×D3—122no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—62no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4D3×SD1612yes—
484C3×C4D3×SD1612yes—
485C3×C4C2×#23511yescharacteristic
486C3×C4#100711yescharacteristic
487C3×C4D3×#4111yescharacteristic
488C3×C4#101011yescharacteristic
489Dic3—22no—
490Dic3—34no—
491Dic3—34no—
492Dic3—41no—
493Dic3—42no—
494Dic3—64no—
495Dic3—62no—
496Dic3—62no—
497Dic3—122no—
498C32#336211yescharacteristic, complemented, Sylow
499C23—21no—
500C23—182no—
501Q8—32no—
502Q8—61no—
503Q8—94no—
504Q8—241no—
505Q8—722no—
506D4—21no—
507D4—21no—
508D4—182no—
509D4—182no—
510D4—364no—
511D4—362no—
512D4—721no—
513D4—722no—
514C2×C4—21no—
515C2×C4—42no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—62no—
519C2×C4—92no—
520C2×C4—92no—
521C2×C4—364no—
522C2×C4#381111yescharacteristic
523C2×C4#381111yescharacteristic
524C2×C4C22×D3211yescharacteristic
525C8—61no—
526C8—61no—
527C8—62no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—82no—
532C2×C3—81no—
533C2×C3—81no—
534C2×C3#555211yescharacteristic
535C2×C3#563311yescharacteristic
536C2×C3C2×D3×SD1611yescharacteristic
537C2×C3C2×D3×SD1611yescharacteristic
538C2×C3#576911yescharacteristic
539C2×C3#577411yescharacteristic
540D3—244no—
541D3—242no—
542D3—242no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—362no—
547C22—362no—
548C22—362no—
549C22C2×#381111yescharacteristic
550C4—21no—
551C4—41no—
552C4—62no—
553C4—94no—
554C4—122no—
555C4—121no—
556C4—182no—
557C4#6393012yes—
558C4#6429711yescharacteristic
559C4D3×#23511yescharacteristic
560C3—21no—
561C3#7962211yescharacteristic, complemented
562C3#8109011yescharacteristic, complemented
563C2—81no—
564C2—722no—
565C2#1059491611yescharacteristic
566C2C2×#6393011yescharacteristic
567C2#1059546111yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×D4C22
socleC22×C32C22×D4
prime radical#10596024C2
hypercenterC4×D4D32
hypocenterC32#3362
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #10593856C2semidirect
3. C3×C4×#214C2semidirect
4. C3×#5008C2semidirect
5. #10591444C2semidirect
6. #10591402C2semidirect
7. #10590483C2semidirect
8. #10590433C2semidirect
9. #10590367C2semidirect
10. #10590281C2semidirect
11. #10590254C2semidirect
53. C4×#214D3semidirect
54. #5008D3semidirect
184. #634SD16semidirect
190. C3×Dic6#41semidirect
405. Dic6#235semidirect
498. C32#3362semidirect
561. C3#79622semidirect
562. C3#81090semidirect

Maximal subgroups

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

subgroupindex
#105960242
#105938562
C3×C4×#2142
C3×#50082
#105914442
#105914022
#105904832
#105904332
#105903672
#105902812
#105902542
#810903
#796223