Finite Group Databasebeta

← #11887574

Subgroups — #11887574

Order 1152.

900 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5727C211yesmaximal, characteristic, complemented
2#10593896C211yesmaximal, characteristic
3#10592782C211yesmaximal, characteristic, complemented
4C3×#5079C211yesmaximal, characteristic, complemented
5#10591585C212yesmaximal, complemented
6#10591584C212yesmaximal, complemented
7#10591549C212yesmaximal, complemented
8D3×#835C212yesmaximal, complemented
9C4×#3717C211yesmaximal, characteristic, complemented
10C3⋊C8×D12C211yesmaximal, characteristic, complemented
11#10590285C211yesmaximal, characteristic, complemented
12#83550—31nomaximal
13#81612—31nomaximal
14C2×C3×D3×Q8C2212yes—
15C2×#3796—21no—
16C32×C4⋊Q8C2211yescharacteristic
17C2×#3755C2212yes—
18C4×C4⋊(C3⋊D3)C2211yescharacteristic
19C3×#946C2212yes—
20C3×#896—22no—
21C3×#893—21no—
22C3×#888—21no—
23C3×C4×D12C2211yescharacteristic
24C3×#869C2211yescharacteristic
25C3×C4×Dic6C2211yescharacteristic
26#63909C2212yes—
27#63890—22no—
28D3×C4⋊Dic3C2212yes—
29#63871—21no—
30C2×#3717C2211yescharacteristic
31C2×D3×C3⋊C8C2212yes—
32#63662C2212yes—
33C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
34C3×#835C2212yes—
35C3×#835C2212yes—
36C3×C4×C3⋊C8C2211yescharacteristic
37#63574C2212yes—
38#63573C2211yescharacteristic
39#63566C2211yescharacteristic
40#63565C2212yes—
41#63564C2211yescharacteristic
42(C2×D3)⋊(C3⋊C8)C2212yes—
43C3×D4×Q8—31no—
44#5727D311yescharacteristic, complemented
45#5726—31no—
46#5555—31no—
47C3×#487—31no—
48#5227—32no—
49C2×#835—32no—
50#5094—31no—
51#5083—31no—
52#5083—31no—
53#5083—61no—
54#5079D311yescharacteristic, complemented
55#5048—32no—
56#5043—32no—
57C4×#215—31no—
58#5020—31no—
59D4×C3⋊C8—31no—
60#4807—32no—
61#4806—32no—
62#4805—32no—
63D3×#103—32no—
64C4×#204—31no—
65C8×D12—31no—
66C2×C32×Q8C2312yes—
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C3×D3×Q8—24no—
70C3×D3×Q8—42no—
71C2×C3×D12D411yescharacteristic
72C2×C3×C4×D3—22no—
73C2×C3×C4×D3C2312yes—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—22no—
76C2×C3×Dic6C2311yescharacteristic
77C2×#626—22no—
78C2×D3×Dic3—22no—
79#3796—44no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C42C2311yescharacteristic
83#3755—42no—
84#3752—21no—
85C4⋊(C3⋊Dic3)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2312yes—
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4D412yes—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3D411yescharacteristic
91C3×C4⋊Dic3C2311yescharacteristic
92C3×C4⋊Dic3C2312yes—
93C3×Dic3⋊C4—21no—
94C3×Dic3⋊C4—22no—
95C3×C4×Dic3—21no—
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2311yescharacteristic
98Dic3⋊Dic3—22no—
99#3722—22no—
100#3717—22no—
101D3×C3⋊C8—42no—
102#3298—91noSylow
103C22×C3×Q8—32no—
104C2×#239—61no—
105C2×D3×Q8C2×D312yes—
106C2×#237—61no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8—31no—
109C3×C4⋊Q8—61no—
110C3×C4⋊Q8C2×D311yescharacteristic
111C3×C4⋊Q8C2×D311yescharacteristic
112C3×C22⋊Q8—32no—
113C3×C22⋊Q8—62no—
114C3×C4×Q8—31no—
115C3×C4×D4—31no—
116C3×C4×D4—61no—
117#946—61no—
118#946C2×D312yes—
119C2×#215—32no—
120C2×#215—32no—
121C2×#215—31no—
122C2×#215—62no—
123#930—32no—
124C2×C4⋊Dic3—32no—
125C22×C3⋊C8—32no—
126C2×#204—31no—
127C2×D3×C8—32no—
128#897—61no—
129#896—22no—
130#895—32no—
131#894—62no—
132#893—21no—
133D3×C4⋊C4—32no—
134#888—21no—
135#886—62no—
136C4×D12—31no—
137C4×D12—31no—
138C4×D12—61no—
139C4×D12C2×D311yescharacteristic
140#869C2×D311yescharacteristic
141C4×Dic6C2×D311yescharacteristic
142C3×#103—32no—
143C3×#103—32no—
144C3×C4×C8—31no—
145#835—32no—
146#835—32no—
147#835—32no—
148#835—31no—
149#835—62no—
150#835C2×D312yes—
151#835C2×D312yes—
152#830—32no—
153#821—31no—
154(C2×D3)⋊C8—32no—
155#817—31no—
156#816—31no—
157#810—32no—
158#809—31no—
159#808—32no—
160#808—32no—
161#808—31no—
162C4⋊(C3⋊C8)—31no—
163C4⋊(C3⋊C8)—31no—
164C4⋊(C3⋊C8)—61no—
165C4×C3⋊C8C2×D311yescharacteristic
166C22×C3⋊D3—21no—
167C22×C3×D3C2×D412yes—
168C32×Q8—22no—
169C32×Q8—22no—
170C2×C32×C4—22no—
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2411yescharacteristic
174C2×C3⋊Dic3—21no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177C4×C3⋊D3—42no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3C2×D412yes—
182C3×D12SD1614yescomplemented
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×C4×D3—44no—
186C3×Dic6—21no—
187C3×Dic6—44no—
188C3×Dic6—42no—
189C3×Dic6#4112yescomplemented
190#626—44no—
191D3×Dic3—44no—
192C3⋊(C3⋊C8)—22no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195D4×Q8—31no—
196#547—91no—
197#487—31no—
198#483—92no—
199#475—91no—
200#465—92no—
201#461—91no—
202#448—92no—
203C4×SD16—91no—
204C8×D4—91no—
205C2×#103—92no—
206C2×C3×Q8—22no—
207C2×C3×Q8—31no—
208C2×C3×Q8—62no—
209C2×C3×Q8—61no—
210C2×C3×Q8—64no—
211C2×C3×Q8—122no—
212C2×C3×Q8C22×D312yes—
213C2×C3×Q8C22×D312yes—
214C2×C3×D4—31no—
215C22×C3×C4—32no—
216C22×C3×C4—62no—
217C2×#67—62no—
218C22×Dic3—62no—
219#239—124no—
220D3×Q8—24no—
221D3×Q8—42no—
222#237—124no—
223C2×D12—31no—
224C2×D12—31no—
225C2×D12—61no—
226C2×D12—62no—
227C2×D12#6711yescharacteristic
228C2×C4×D3—22no—
229C2×C4×D3—62no—
230C2×C4×D3—61no—
231C2×C4×D3—61no—
232C2×C4×D3—121no—
233C2×C4×D3C22×D312yes—
234C2×Dic6—21no—
235C2×Dic6—22no—
236C2×Dic6C22×D311yescharacteristic
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—41no—
248C3×C4⋊C4—62no—
249C3×C4⋊C4—61no—
250C3×C4⋊C4—61no—
251C3×C22⋊C4—32no—
252C3×C22⋊C4—62no—
253C3×C42—21no—
254C3×C42—61no—
255C3×C42C22×D311yescharacteristic
256C3×C42C22×D311yescharacteristic
257#215—62no—
258#215—122no—
259#215—122no—
260#215—124no—
261(C2×D3)⋊C4—22no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4—61no—
264(C2×D3)⋊C4—61no—
265(C2×D3)⋊C4—62no—
266(C2×D3)⋊C4—121no—
267(C2×D3)⋊C4#6712yes—
268C4⋊Dic3—21no—
269C4⋊Dic3—31no—
270C4⋊Dic3—31no—
271C4⋊Dic3—61no—
272C4⋊Dic3—62no—
273C4⋊Dic3#6711yescharacteristic
274C4⋊Dic3C22×D311yescharacteristic
275C4⋊Dic3C22×D312yes—
276Dic3⋊C4—21no—
277Dic3⋊C4—22no—
278Dic3⋊C4—62no—
279C4×Dic3—21no—
280C2×C3⋊C8—32no—
281C2×C3⋊C8—32no—
282C2×C3⋊C8—62no—
283C2×C3⋊C8—122no—
284C2×C3⋊C8C22×D311yescharacteristic
285C2×C3⋊C8C22×D311yescharacteristic
286#204—62no—
287D3×C8—122no—
288C22×C32C22×D411yescharacteristic
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3C2×SD1614yes—
295C32×C4—21no—
296C32×C4—42no—
297C32×C4C2×SD1612yes—
298C32×C4C2×#4111yescharacteristic
299C32×C4H(5,𝔽2)11yescharacteristic
300C3⋊Dic3—41no—
301C3×Dic3—22no—
302C3×Dic3—22no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C2×#41—181no—
306C22×Q8—32no—
307C2×SD16—91no—
308C2×SD16—92no—
309C22×C8—92no—
310C4⋊Q8—31no—
311C4⋊Q8—61no—
312C4⋊Q8D3211yescharacteristic
313#123—182no—
314C22⋊Q8—32no—
315C22⋊Q8—62no—
316C22⋊Q8—181no—
317#121—92no—
318C4×Q8—31no—
319C4×D4—31no—
320C4×D4—61no—
321C4×D4—91no—
322C2×C4⋊C4—92no—
323#106—92no—
324#106—91no—
325C4⋊C8—91no—
326#103—32no—
327#103—32no—
328#103—92no—
329#103—91no—
330#103—92no—
331#102—91no—
332C22⋊C8—92no—
333C4×C8—31no—
334C23×C3—32no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—121no—
338C22×D3C2×#6712yes—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—32no—
344C3×Q8—42no—
345C3×Q8—42no—
346C3×Q8—61no—
347C3×Q8—124no—
348C3×Q8—122no—
349C3×D4—34no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—32no—
356C2×C3×C4—42no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—61no—
361C2×C3×C4—62no—
362C2×C3×C4—124no—
363C2×C3×C4D3×D411yescharacteristic
364C2×C3×C4D3×D411yescharacteristic
365C2×C3×C4C2×#6711yescharacteristic
366C2×C3×C4C2×#6711yescharacteristic
367C2×C3×C4C23×D311yescharacteristic
368C2×C3×C4C23×D311yescharacteristic
369#67—124no—
370C2×Dic3—21no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—121no—
376C2×Dic3—124no—
377C2×Dic3C2×#6712yes—
378D12—62no—
379D12—61no—
380D12—61no—
381D12—61no—
382D12—61no—
383D12—121no—
384D12—124no—
385D12#21514yescomplemented
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—44no—
389C4×D3—124no—
390C4×D3—122no—
391C4×D3—122no—
392C4×D3—242no—
393Dic6—21no—
394Dic6—44no—
395Dic6—42no—
396Dic6#23712yescomplemented
397C3×C8—61no—
398C3×C8—61no—
399C3⋊C8—21no—
400C3⋊C8—21no—
401C3⋊C8—62no—
402C3⋊C8—62no—
403C3⋊C8—122no—
404C2×C32#54611yescharacteristic
405C2×C32C22×SD1611yescharacteristic
406C2×C32#57811yescharacteristic
407C3⋊D3—81no—
408C3×D3—24no—
409#41—364no—
410C2×Q8—31no—
411C2×Q8—62no—
412C2×Q8—61no—
413C2×Q8—64no—
414C2×Q8—122no—
415C2×Q8C2×D3212yes—
416C2×D4—31no—
417C2×D4—91no—
418C2×D4—182no—
419C22×C4—32no—
420C22×C4—62no—
421C22×C4—182no—
422C22×C4—181no—
423SD16—182no—
424SD16—362no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—92no—
428C2×C8—362no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—31no—
432C4⋊C4—31no—
433C4⋊C4—32no—
434C4⋊C4—62no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—91no—
438C4⋊C4—182no—
439C22⋊C4—32no—
440C22⋊C4—62no—
441C22⋊C4—182no—
442C22⋊C4—181no—
443C42—61no—
444C42C2×D3211yescharacteristic
445C22×C3—21no—
446C22×C3—34no—
447C22×C3—62no—
448C22×C3—62no—
449C22×C3C2×D3×D411yescharacteristic
450C22×C3C22×#6711yescharacteristic
451C2×D3—22no—
452C2×D3—22no—
453C2×D3—122no—
454C2×D3—121no—
455C2×D3—121no—
456C2×D3—121no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—121no—
460C2×D3—241no—
461C2×D3—241no—
462C2×D3C2×#21514yes—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—22no—
468C3×C4—41no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—62no—
472C3×C4—62no—
473C3×C4—82no—
474C3×C4—121no—
475C3×C4—122no—
476C3×C4D3×SD1612yes—
477C3×C4C2×#21512yes—
478C3×C4C2×#23711yescharacteristic
479C3×C4#100411yescharacteristic
480C3×C4D3×#4111yescharacteristic
481C3×C4#100911yescharacteristic
482Dic3—22no—
483Dic3—22no—
484Dic3—42no—
485Dic3—41no—
486Dic3—121no—
487Dic3—121no—
488Dic3—241no—
489C32#329811yescharacteristic, complemented, Sylow
490C23—32no—
491C23—181no—
492Q8—22no—
493Q8—22no—
494Q8—32no—
495Q8—61no—
496Q8—124no—
497Q8—122no—
498D4—34no—
499D4—181no—
500D4—181no—
501D4—364no—
502C2×C4—22no—
503C2×C4—32no—
504C2×C4—61no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—61no—
508C2×C4—62no—
509C2×C4—124no—
510C2×C4—181no—
511C2×C4—364no—
512C2×C4—362no—
513C2×C4D3×#6711yescharacteristic
514C2×C4D3×#6711yescharacteristic
515C2×C4C22×D3211yescharacteristic
516C8—61no—
517C8—61no—
518C8—182no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—64no—
523C2×C3#567611yescharacteristic
524C2×C3C2×D3×SD1611yescharacteristic
525C2×C3#576611yescharacteristic
526C2×C3#580611yescharacteristic
527C2×C3C22×#21511yescharacteristic
528C2×C3#583811yescharacteristic
529D3—24no—
530D3—242no—
531D3—241no—
532D3—241no—
533C22—34no—
534C22—62no—
535C22—62no—
536C22—361no—
537C22—361no—
538C22—361no—
539C22C2×D3×#6711yescharacteristic
540C4—21no—
541C4—42no—
542C4—62no—
543C4—62no—
544C4—122no—
545C4—121no—
546C4—361no—
547C4D3×#21512yes—
548C4D3×#23711yescharacteristic
549C4#6431411yescharacteristic
550C3—21no—
551C3#8161211yescharacteristic, complemented
552C3#8355011yescharacteristic, complemented
553C2—64no—
554C2—721no—
555C2#1059522911yescharacteristic
556C2C2×D3×#21511yescharacteristic
557C2#1059532711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radical#10593896C2
hypercenterC4⋊Q8D32
hypocenterC32#3298
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. C3×#5727C2semidirect
3. #10592782C2semidirect
4. C3×#5079C2semidirect
5. #10591585C2semidirect
6. #10591584C2semidirect
7. #10591549C2semidirect
8. D3×#835C2semidirect
9. C4×#3717C2semidirect
10. C3⋊C8×D12C2semidirect
11. #10590285C2semidirect
44. #5727D3semidirect
54. #5079D3semidirect
182. C3×D12SD16semidirect
189. C3×Dic6#41semidirect
385. D12#215semidirect
396. Dic6#237semidirect
489. C32#3298semidirect
551. C3#81612semidirect
552. C3#83550semidirect

Maximal subgroups

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

subgroupindex
C3×#57272
#105938962
#105927822
C3×#50792
#105915852
#105915842
#105915492
D3×#8352
C4×#37172
C3⋊C8×D122
#105902852
#835503
#816123