Finite Group Databasebeta

← #11898300

Subgroups — #11898300

Order 1152.

908 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596192C211yesmaximal, characteristic, complemented
2#10593906C211yesmaximal, characteristic
3C3×#5085C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic, complemented
5#10591622C212yesmaximal, complemented
6#10591620C212yesmaximal, complemented
7#10591571C212yesmaximal, complemented
8#10591569C212yesmaximal, complemented
9#10590377C211yesmaximal, characteristic, complemented
10C4×#3716C211yesmaximal, characteristic, complemented
11#10590254C211yesmaximal, characteristic, complemented
12#81755—31nomaximal
13#80420—31nomaximal
14C2×#3795—21no—
15C32×C4⋊Q8C2211yescharacteristic
16Q8×C3⋊Dic3C2212yes—
17#64153C2212yes—
18#64099—21no—
19#64098—22no—
20#64096—21no—
21C4×#634C2211yescharacteristic
22C4×#634C2211yescharacteristic
23C2×C3×#215C2212yes—
24C2×C3×#215C2212yes—
25C2×C3×#215C2212yes—
26C3×#874C2211yescharacteristic
27C3×C4×D12C2211yescharacteristic
28#63914C2212yes—
29#63908—21no—
30#63898C2212yes—
31#63885—22no—
32C2×#3716C2211yescharacteristic
33C2×#3716C2212yes—
34C3×#835C2212yes—
35C3×C4⋊(C3⋊C8)C2211yescharacteristic
36C3×C4×C3⋊C8C2211yescharacteristic
37#63573C2211yescharacteristic
38#63572C2212yes—
39#63567C2211yescharacteristic
40#63563C2212yes—
41#63563C2211yescharacteristic
42#63553C2212yes—
43#5721—31no—
44#5721—31no—
45#5721—61no—
46#5718—31no—
47#5553—31no—
48C3×#492—31no—
49C3×#461—31no—
50#5175—32no—
51#5175—32no—
52#5163—32no—
53#5163—32no—
54#5162—32no—
55Dic3×SD16—32no—
56#5085D311yescharacteristic, complemented
57#5083D311yescharacteristic, complemented
58#5075—31no—
59C4×#214—31no—
60#4848—31no—
61#4830—31no—
62#4818—32no—
63#4788—32no—
64C4×#204—31no—
65C8×Dic6—31no—
66C2×C32×Q8C2312yes—
67C2×#634D411yescharacteristic
68C2×#634C2311yescharacteristic
69C2×C3×D12—21no—
70C2×C3×D12—21no—
71C2×C3×D12C2312yes—
72C2×C3×D12C2311yescharacteristic
73C2×C3×C4×D3—21no—
74C2×#625—22no—
75C2×#625—22no—
76C2×D3×Dic3—22no—
77#3795—44no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C42C2311yescharacteristic
81C4⋊(C3⋊Dic3)—22no—
82C4⋊(C3⋊Dic3)D411yescharacteristic
83C4⋊(C3⋊Dic3)C2311yescharacteristic
84#3750—21no—
85#3750—22no—
86#3750—22no—
87#3750D412yes—
88C4×C3⋊Dic3—21no—
89C4×C3⋊Dic3—22no—
90C4×C3⋊Dic3C2312yes—
91C3×#215—24no—
92C3×#215—42no—
93C3×#215—42no—
94C3×(C2×D3)⋊C4—21no—
95C3×C4⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2312yes—
98C2×C3×C3⋊C8C2311yescharacteristic
99(C2×D3)⋊Dic3—22no—
100#3716—22no—
101#3716—42no—
102#3392—91noSylow
103C2×#237—61no—
104C2×#235—61no—
105C2×C3×SD16—32no—
106C2×C3×SD16—32no—
107C2×C3×SD16—32no—
108C3×C4⋊Q8—21no—
109C3×C4⋊Q8C2×D311yescharacteristic
110C3×C4⋊Q8C2×D311yescharacteristic
111C3×#127—31no—
112C3×C4×D4—31no—
113Q8×Dic3—32no—
114Q8×Dic3—32no—
115Q8×Dic3—62no—
116#944—32no—
117#944—32no—
118#944—62no—
119C2×#215C2×D312yes—
120C2×#215C2×D312yes—
121C2×#215C2×D312yes—
122#940—32no—
123#940—32no—
124#939—62no—
125#938—61no—
126D4×Dic3—32no—
127C2×#214—31no—
128C2×#214—32no—
129#930—61no—
130C2×#204—31no—
131C2×#204—32no—
132#893—32no—
133#890—61no—
134#890—61no—
135#890—62no—
136#889—62no—
137#889—62no—
138#889—64no—
139#887—61no—
140#887—61no—
141#887—62no—
142#884—62no—
143#874C2×D311yescharacteristic
144C4×D12C2×D311yescharacteristic
145C4×Dic6—31no—
146C4×Dic6—31no—
147C4×Dic6—31no—
148C4×Dic6—31no—
149C4×Dic6—61no—
150C4×Dic6—61no—
151C3×C4⋊C8—31no—
152C3×#103—32no—
153C3×C4×C8—31no—
154#835C2×D312yes—
155#832—32no—
156#832—31no—
157#821—31no—
158#821—32no—
159#817—31no—
160#816—31no—
161Dic3⋊C8—32no—
162Dic3⋊C8—32no—
163C8×Dic3—32no—
164C8×Dic3—32no—
165#810—31no—
166#808—31no—
167C4⋊(C3⋊C8)C2×D311yescharacteristic
168C4×C3⋊C8C2×D311yescharacteristic
169C22×C3×D3—22no—
170C22×C3×D3—21no—
171C32×Q8—22no—
172C32×Q8—22no—
173C2×C32×C4—22no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2411yescharacteristic
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3C2×D412yes—
180C2×C3⋊Dic3C2×D412yes—
181#634—21no—
182#634—21no—
183#634SD1614yescomplemented
184C2×C3×Dic3—21no—
185C3×D12—21no—
186C3×D12—22no—
187C3×D12—22no—
188C3×D12—21no—
189C3×D12—42no—
190C3×D12—81no—
191C3×C4×D3—42no—
192#625—44no—
193#625—82no—
194D3×Dic3—44no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8#4112yescomplemented
198#557—91no—
199#550—91no—
200#492—31no—
201#492—92no—
202#461—31no—
203#461—92no—
204#460—91no—
205#460—92no—
206C8×Q8—91no—
207C4×SD16—91no—
208C4×SD16—92no—
209C2×C3×Q8—22no—
210C2×C3×Q8C22×D312yes—
211C2×C3×Q8C22×D312yes—
212C2×C3×D4—32no—
213C2×C3×D4—31no—
214C2×C3×D4—61no—
215C2×C3×D4—61no—
216C22×C3×C4—61no—
217C2×#67—62no—
218C2×#67—62no—
219C2×#67—62no—
220C2×#67—62no—
221C22×Dic3—62no—
222#237—124no—
223#235—124no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12C22×D311yescharacteristic
227C2×D12C22×D312yes—
228C2×C4×D3—21no—
229C2×C4×D3—62no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—31no—
233C2×Dic6—31no—
234C2×Dic6—61no—
235C2×Dic6—61no—
236C3×SD16—64no—
237C3×SD16—122no—
238C3×SD16—122no—
239C2×C3×C8—31no—
240C2×C3×C8—32no—
241C2×C3×C8—31no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—31no—
249C3×C22⋊C4—61no—
250C3×C42—21no—
251C3×C42C22×D311yescharacteristic
252C3×C42C22×D311yescharacteristic
253#217—62no—
254#215—24no—
255#215—42no—
256#215—42no—
257#214—62no—
258#214—122no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—62no—
261C4⋊Dic3—31no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3—61no—
267C4⋊Dic3—62no—
268C4⋊Dic3—62no—
269C4⋊Dic3—64no—
270C4⋊Dic3C22×D311yescharacteristic
271Dic3⋊C4—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—62no—
274Dic3⋊C4—62no—
275Dic3⋊C4—62no—
276Dic3⋊C4—62no—
277Dic3⋊C4—62no—
278Dic3⋊C4—62no—
279Dic3⋊C4—64no—
280Dic3⋊C4—64no—
281Dic3⋊C4—61no—
282Dic3⋊C4—61no—
283C4×Dic3—32no—
284C4×Dic3—32no—
285C4×Dic3—62no—
286C4×Dic3—62no—
287C4×Dic3—62no—
288C4×Dic3—62no—
289C4×Dic3—64no—
290C4×Dic3—61no—
291C4×Dic3—61no—
292C2×C3⋊C8C22×D311yescharacteristic
293C2×C3⋊C8C22×D312yes—
294C2×C3⋊C8C22×D311yescharacteristic
295#204—62no—
296#204—122no—
297C22×C32C22×D411yescharacteristic
298C2×C3×D3—42no—
299C2×C3×D3—41no—
300C2×C3×D3—42no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—42no—
304C32×C4—21no—
305C32×C4—42no—
306C32×C4C2×SD1612yes—
307C32×C4C2×#4111yescharacteristic
308C32×C4H(5,𝔽2)11yescharacteristic
309C3⋊Dic3—22no—
310C3⋊Dic3—41no—
311C3⋊Dic3—41no—
312C3⋊Dic3C2×SD1614yes—
313C3×Dic3—41no—
314C2×#41—181no—
315C2×SD16—32no—
316C2×SD16—32no—
317C2×SD16—32no—
318C2×SD16—91no—
319C2×SD16—92no—
320C4⋊Q8—92no—
321C4⋊Q8D3211yescharacteristic
322#127—31no—
323#127—92no—
324#125—182no—
325#125—181no—
326#121—182no—
327#121—181no—
328C4×Q8—91no—
329C4×Q8—92no—
330C4×Q8—91no—
331C4×Q8—181no—
332C4×D4—31no—
333C4×D4—92no—
334#106—91no—
335C4⋊C8—31no—
336C4⋊C8—92no—
337#103—32no—
338#103—91no—
339#102—91no—
340#102—92no—
341C4×C8—31no—
342C4×C8—92no—
343C23×C3—62no—
344C23×C3—61no—
345C22×D3—21no—
346C22×D3—22no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—22no—
350C3×Q8—22no—
351C3×Q8—42no—
352C3×Q8—42no—
353C3×D4—61no—
354C3×D4—62no—
355C3×D4—61no—
356C3×D4—62no—
357C3×D4—122no—
358C3×D4—241no—
359C2×C3×C4—24no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—61no—
366C2×C3×C4—122no—
367C2×C3×C4D3×D411yescharacteristic
368C2×C3×C4D3×D411yescharacteristic
369C2×C3×C4D3×D411yescharacteristic
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4C23×D311yescharacteristic
372C2×C3×C4C23×D311yescharacteristic
373#67—124no—
374#67—124no—
375#67—242no—
376#67—242no—
377C2×Dic3—21no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—62no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—124no—
391D12—22no—
392D12—22no—
393D12—21no—
394D12—21no—
395D12—42no—
396D12—81no—
397C4×D3—42no—
398C4×D3—124no—
399Dic6—34no—
400Dic6—34no—
401Dic6—64no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—61no—
406Dic6—121no—
407Dic6—121no—
408C3×C8—32no—
409C3×C8—62no—
410C3×C8—61no—
411C3⋊C8—22no—
412C3⋊C8—21no—
413C3⋊C8#23712yescomplemented
414C2×C32#55011yescharacteristic
415C2×C32C22×SD1611yescharacteristic
416C2×C32#57711yescharacteristic
417C3×D3—81no—
418C3×D3—82no—
419#41—364no—
420C2×Q8—91no—
421C2×Q8—91no—
422C2×Q8C2×D3212yes—
423C2×D4—31no—
424C2×D4—32no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—182no—
428C2×D4—182no—
429C22×C4—61no—
430C22×C4—182no—
431SD16—64no—
432SD16—122no—
433SD16—122no—
434SD16—182no—
435SD16—362no—
436C2×C8—31no—
437C2×C8—31no—
438C2×C8—32no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—31no—
442C4⋊C4—92no—
443C4⋊C4—91no—
444C4⋊C4—91no—
445C4⋊C4—182no—
446C4⋊C4—182no—
447C4⋊C4—182no—
448C4⋊C4—181no—
449C22⋊C4—61no—
450C22⋊C4—182no—
451C42—92no—
452C42—182no—
453C42—181no—
454C42C2×D3211yescharacteristic
455C22×C3—21no—
456C22×C3—121no—
457C22×C3—122no—
458C22×C3—121no—
459C22×C3—121no—
460C22×C3—122no—
461C22×C3—122no—
462C22×C3C2×D3×D411yescharacteristic
463C22×C3C2×D3×D411yescharacteristic
464C2×D3—42no—
465C2×D3—41no—
466C2×D3—42no—
467C2×D3—42no—
468C2×D3—41no—
469C2×D3—41no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—22no—
475C3×C4—41no—
476C3×C4—44no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—121no—
480C3×C4D3×SD1612yes—
481C3×C4D3×SD1612yes—
482C3×C4C2×#23711yescharacteristic
483C3×C4#100411yescharacteristic
484C3×C4#100411yescharacteristic
485C3×C4D3×#4111yescharacteristic
486Dic3—34no—
487Dic3—34no—
488Dic3—41no—
489Dic3—64no—
490Dic3—62no—
491Dic3—62no—
492Dic3—122no—
493Dic3—122no—
494Dic3—122no—
495Dic3—121no—
496Dic3—121no—
497Dic3—121no—
498Dic3—121no—
499C32#339211yescharacteristic, complemented, Sylow
500C23—61no—
501C23—62no—
502Q8—22no—
503Q8—22no—
504Q8—94no—
505Q8—181no—
506Q8—181no—
507D4—62no—
508D4—61no—
509D4—61no—
510D4—62no—
511D4—122no—
512D4—241no—
513D4—364no—
514D4—722no—
515C2×C4—22no—
516C2×C4—61no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4—122no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—364no—
523C2×C4#381111yescharacteristic
524C2×C4#381111yescharacteristic
525C2×C4C22×D3211yescharacteristic
526C8—32no—
527C8—61no—
528C8—62no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—242no—
533C2×C3—241no—
534C2×C3#559911yescharacteristic
535C2×C3#571811yescharacteristic
536C2×C3C2×D3×SD1611yescharacteristic
537C2×C3C2×D3×SD1611yescharacteristic
538C2×C3#576211yescharacteristic
539C2×C3#577811yescharacteristic
540D3—81no—
541D3—82no—
542C22—122no—
543C22—121no—
544C22—121no—
545C22—122no—
546C22—122no—
547C22—121no—
548C22C2×#381111yescharacteristic
549C4—21no—
550C4—42no—
551C4—94no—
552C4—121no—
553C4—182no—
554C4—361no—
555C4—361no—
556C4#6394012yes—
557C4D3×#23711yescharacteristic
558C4#6431311yescharacteristic
559C3—21no—
560C3#8042011yescharacteristic, complemented
561C3#8175511yescharacteristic, complemented
562C2—241no—
563C2—242no—
564C2#1059516411yescharacteristic
565C2C2×#6394011yescharacteristic
566C2#1059532411yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radical#10593906C2
hypercenterC4⋊Q8D32
hypocenterC32#3392
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #10596192C2semidirect
3. C3×#5085C2semidirect
4. C3×#5083C2semidirect
5. #10591622C2semidirect
6. #10591620C2semidirect
7. #10591571C2semidirect
8. #10591569C2semidirect
9. #10590377C2semidirect
10. C4×#3716C2semidirect
11. #10590254C2semidirect
56. #5085D3semidirect
57. #5083D3semidirect
183. #634SD16semidirect
197. C3×C3⋊C8#41semidirect
413. C3⋊C8#237semidirect
499. C32#3392semidirect
560. C3#80420semidirect
561. C3#81755semidirect

Maximal subgroups

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

subgroupindex
#105961922
#105939062
C3×#50852
C3×#50832
#105916222
#105916202
#105915712
#105915692
#105903772
C4×#37162
#105902542
#817553
#804203