Finite Group Databasebeta

← #11887572

Subgroups — #11887572

Order 1152.

889 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5727C211yesmaximal, characteristic, complemented
2#10593895C211yesmaximal, characteristic
3#10592778C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic, complemented
5#10591634C212yesmaximal, complemented
6#10591583C212yesmaximal, complemented
7#10591549C212yesmaximal, complemented
8D3×#835C212yesmaximal, complemented
9C4×#3715C211yesmaximal, characteristic, complemented
10#10590329C211yesmaximal, characteristic, complemented
11#10590301C211yesmaximal, characteristic, complemented
12#83550—31nomaximal
13#80450—31nomaximal
14C2×C3×D3×Q8C2212yes—
15C2×#3795—21no—
16C32×C4⋊Q8C2211yescharacteristic
17#64078C2211yescharacteristic
18C3×#946C2212yes—
19C2×C3×#215C2212yes—
20C3×#896—22no—
21C3×#893—21no—
22C3×#888—21no—
23C3×C4×D12C2211yescharacteristic
24C3×C4×D12C2211yescharacteristic
25C3×C4×Dic6C2211yescharacteristic
26#63908C2212yes—
27#63891—22no—
28D3×C4⋊Dic3C2212yes—
29#63857—21no—
30C2×#3715C2211yescharacteristic
31C2×D3×C3⋊C8C2212yes—
32#63662C2212yes—
33#63662C2212yes—
34C4×C3⋊(C3⋊C8)C2211yescharacteristic
35C3×#835C2212yes—
36C3×C4⋊(C3⋊C8)C2211yescharacteristic
37#63574C2211yescharacteristic
38#63573C2212yes—
39#63566C2212yes—
40#63565C2211yescharacteristic
41#63562C2211yescharacteristic
42(C2×D3)⋊(C3⋊C8)C2212yes—
43C3×D4×Q8—31no—
44#5727D311yescharacteristic, complemented
45#5555—31no—
46#5551—31no—
47C3×#461—31no—
48#5227—32no—
49C2×#835—32no—
50#5171—32no—
51#5170—32no—
52#5083D311yescharacteristic, complemented
53#5079—31no—
54#5079—31no—
55#5079—61no—
56#5048—32no—
57#5043—32no—
58C4×#215—31no—
59#5020—31no—
60C4×#214—31no—
61D4×C3⋊C8—31no—
62#4851—31no—
63#4834—31no—
64#4805—32no—
65D3×#103—32no—
66C2×C32×Q8C2312yes—
67C2×#634—21no—
68C3×D3×Q8—24no—
69C3×D3×Q8—42no—
70C2×C3×D12D411yescharacteristic
71C2×C3×D12C2311yescharacteristic
72C2×C3×C4×D3—22no—
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3C2312yes—
75C2×C3×Dic6—22no—
76C2×C3×Dic6C2311yescharacteristic
77C2×#625—22no—
78C2×D3×Dic3—22no—
79#3795—44no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C42C2311yescharacteristic
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)C2312yes—
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2311yescharacteristic
87C3×#215—42no—
88C3×(C2×D3)⋊C4—22no—
89C3×(C2×D3)⋊C4—21no—
90C3×(C2×D3)⋊C4D412yes—
91C3×C4⋊Dic3D411yescharacteristic
92C3×C4⋊Dic3C2311yescharacteristic
93C3×C4⋊Dic3C2311yescharacteristic
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—22no—
96C3×C4×Dic3—21no—
97C2×C3×C3⋊C8C2312yes—
98Dic3⋊Dic3—22no—
99(C2×D3)⋊Dic3—22no—
100#3715—22no—
101D3×C3⋊C8—42no—
102#3298—91noSylow
103C22×C3×Q8—32no—
104C2×D3×Q8C2×D312yes—
105C2×#237—61no—
106C2×#235—61no—
107C2×C3×SD16—32no—
108C3×C4⋊Q8—21no—
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—31no—
117C3×C4×D4—61no—
118#946C2×D312yes—
119C2×#215—31no—
120C2×#215C2×D312yes—
121#938—32no—
122C2×#214—31no—
123#930—32no—
124C2×C4⋊Dic3—32no—
125#924—61no—
126C22×C3⋊C8—32no—
127C2×D3×C8—32no—
128#897—61no—
129#896—22no—
130#893—21no—
131D3×C4⋊C4—32no—
132#888—21no—
133#886—62no—
134#883—62no—
135C4×D12C2×D311yescharacteristic
136C4×D12C2×D311yescharacteristic
137#869—31no—
138#869—31no—
139#869—61no—
140C4×Dic6C2×D311yescharacteristic
141C3×C4⋊C8—31no—
142C3×#103—32no—
143#835—32no—
144#835—32no—
145#835—32no—
146#835—32no—
147#835—31no—
148#835—32no—
149#835—62no—
150#835—62no—
151#835C2×D312yes—
152#832—31no—
153#830—32no—
154(C2×D3)⋊C8—32no—
155#817—32no—
156#816—32no—
157#810—31no—
158#809—31no—
159#808—31no—
160#808—31no—
161#808—32no—
162C4⋊(C3⋊C8)C2×D311yescharacteristic
163C4×C3⋊C8—31no—
164C4×C3⋊C8—31no—
165C4×C3⋊C8—61no—
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—22no—
175#634—42no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3C2×D412yes—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12SD1614yescomplemented
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×C4×D3—44no—
186C3×C4×D3—42no—
187C3×Dic6—21no—
188C3×Dic6—44no—
189C3×Dic6#4112yescomplemented
190#625—44no—
191D3×Dic3—44no—
192C3⋊(C3⋊C8)—21no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—22no—
195D4×Q8—31no—
196#547—91no—
197#487—91no—
198#483—92no—
199#475—91no—
200#465—92no—
201#461—31no—
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—64no—
210C2×C3×Q8—122no—
211C2×C3×Q8C22×D312yes—
212C2×C3×Q8C22×D312yes—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C22×C3×C4—32no—
216C22×C3×C4—62no—
217C22×C3×C4—61no—
218C2×#67—62no—
219C2×#67—62no—
220C22×Dic3—62no—
221D3×Q8—24no—
222D3×Q8—42no—
223#237—124no—
224#235—124no—
225C2×D12#6711yescharacteristic
226C2×D12C22×D311yescharacteristic
227C2×C4×D3—22no—
228C2×C4×D3—21no—
229C2×C4×D3—62no—
230C2×C4×D3C22×D312yes—
231C2×Dic6—22no—
232C2×Dic6—61no—
233C2×Dic6—61no—
234C2×Dic6—121no—
235C2×Dic6C22×D311yescharacteristic
236C3×SD16—122no—
237C2×C3×C8—32no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—62no—
248C3×C4⋊C4—61no—
249C3×C22⋊C4—32no—
250C3×C22⋊C4—62no—
251C3×C22⋊C4—61no—
252C3×C42—21no—
253C3×C42—61no—
254C3×C42C22×D311yescharacteristic
255C3×C42C22×D311yescharacteristic
256#217—62no—
257#215—42no—
258#215—62no—
259#214—62no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—21no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4#6712yes—
264C4⋊Dic3—32no—
265C4⋊Dic3—32no—
266C4⋊Dic3—62no—
267C4⋊Dic3—62no—
268C4⋊Dic3—61no—
269C4⋊Dic3—61no—
270C4⋊Dic3—121no—
271C4⋊Dic3#6711yescharacteristic
272C4⋊Dic3C22×D311yescharacteristic
273C4⋊Dic3C22×D311yescharacteristic
274Dic3⋊C4—21no—
275Dic3⋊C4—22no—
276Dic3⋊C4—62no—
277C4×Dic3—21no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8—122no—
285C2×C3⋊C8C22×D312yes—
286D3×C8—122no—
287C22×C32C22×D411yescharacteristic
288C2×C3×D3—22no—
289C2×C3×D3—22no—
290C2×C3×D3—41no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C2×C3×D3C2×SD1614yes—
294C32×C4—21no—
295C32×C4—42no—
296C32×C4C2×SD1612yes—
297C32×C4C2×#4111yescharacteristic
298C32×C4H(5,𝔽2)11yescharacteristic
299C3⋊Dic3—42no—
300C3×Dic3—22no—
301C3×Dic3—22no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C2×#41—181no—
305C22×Q8—32no—
306C2×SD16—32no—
307C2×SD16—91no—
308C22×C8—92no—
309C4⋊Q8—61no—
310C4⋊Q8—91no—
311C4⋊Q8D3211yescharacteristic
312#123—182no—
313C22⋊Q8—32no—
314C22⋊Q8—62no—
315C22⋊Q8—181no—
316#121—92no—
317C4×Q8—31no—
318C4×D4—31no—
319C4×D4—31no—
320C4×D4—61no—
321C2×C4⋊C4—92no—
322#106—91no—
323#106—92no—
324C4⋊C8—31no—
325#103—32no—
326#103—91no—
327#103—92no—
328#103—92no—
329#103—92no—
330#102—91no—
331C22⋊C8—92no—
332C4×C8—91no—
333C23×C3—32no—
334C23×C3—61no—
335C22×D3—21no—
336C22×D3C2×#6712yes—
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—32no—
342C3×Q8—42no—
343C3×Q8—42no—
344C3×Q8—61no—
345C3×Q8—124no—
346C3×D4—34no—
347C3×D4—61no—
348C3×D4—61no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—32no—
355C2×C3×C4—42no—
356C2×C3×C4—61no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—124no—
362C2×C3×C4—122no—
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—
370#67—124no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—62no—
375C2×Dic3—62no—
376C2×Dic3—122no—
377C2×Dic3—124no—
378C2×Dic3C2×#6712yes—
379D12—21no—
380D12—21no—
381D12#21514yescomplemented
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—44no—
385C4×D3—42no—
386C4×D3—124no—
387Dic6—21no—
388Dic6—44no—
389Dic6—122no—
390Dic6—122no—
391Dic6—242no—
392Dic6#23712yescomplemented
393C3×C8—62no—
394C3⋊C8—22no—
395C3⋊C8—61no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—121no—
400C3⋊C8—121no—
401C2×C32#54611yescharacteristic
402C2×C32C22×SD1611yescharacteristic
403C2×C32#57811yescharacteristic
404C3×D3—24no—
405C3×D3—81no—
406#41—364no—
407C2×Q8—31no—
408C2×Q8—62no—
409C2×Q8—64no—
410C2×Q8—122no—
411C2×Q8—181no—
412C2×Q8C2×D3212yes—
413C2×D4—31no—
414C2×D4—31no—
415C2×D4—182no—
416C22×C4—32no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—182no—
420SD16—122no—
421SD16—182no—
422C2×C8—32no—
423C2×C8—91no—
424C2×C8—91no—
425C2×C8—362no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—31no—
429C4⋊C4—31no—
430C4⋊C4—31no—
431C4⋊C4—62no—
432C4⋊C4—61no—
433C4⋊C4—92no—
434C4⋊C4—182no—
435C4⋊C4—181no—
436C22⋊C4—32no—
437C22⋊C4—62no—
438C22⋊C4—61no—
439C22⋊C4—182no—
440C42—61no—
441C42C2×D3211yescharacteristic
442C22×C3—21no—
443C22×C3—34no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3C2×D3×D411yescharacteristic
450C22×C3C22×#6711yescharacteristic
451C2×D3—22no—
452C2×D3—22no—
453C2×D3—41no—
454C2×D3—41no—
455C2×D3—41no—
456C2×D3C2×#21514yes—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—62no—
466C3×C4—62no—
467C3×C4—82no—
468C3×C4—121no—
469C3×C4—121no—
470C3×C4D3×SD1612yes—
471C3×C4C2×#21512yes—
472C3×C4C2×#23711yescharacteristic
473C3×C4#100411yescharacteristic
474C3×C4#100411yescharacteristic
475C3×C4D3×#4111yescharacteristic
476Dic3—22no—
477Dic3—22no—
478Dic3—41no—
479Dic3—41no—
480Dic3—122no—
481Dic3—122no—
482Dic3—242no—
483C32#329811yescharacteristic, complemented, Sylow
484C23—32no—
485C23—61no—
486Q8—22no—
487Q8—22no—
488Q8—32no—
489Q8—61no—
490Q8—124no—
491Q8—362no—
492D4—34no—
493D4—61no—
494D4—61no—
495D4—364no—
496C2×C4—22no—
497C2×C4—32no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—124no—
504C2×C4—122no—
505C2×C4—182no—
506C2×C4—364no—
507C2×C4D3×#6711yescharacteristic
508C2×C4D3×#6711yescharacteristic
509C2×C4C22×D3211yescharacteristic
510C8—62no—
511C8—181no—
512C8—181no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—64no—
517C2×C3—241no—
518C2×C3#561211yescharacteristic
519C2×C3C2×D3×SD1611yescharacteristic
520C2×C3#577911yescharacteristic
521C2×C3#580611yescharacteristic
522C2×C3C22×#21511yescharacteristic
523C2×C3#583811yescharacteristic
524D3—24no—
525D3—81no—
526C22—34no—
527C22—62no—
528C22—62no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22C2×D3×#6711yescharacteristic
533C4—21no—
534C4—42no—
535C4—62no—
536C4—62no—
537C4—121no—
538C4—121no—
539C4—362no—
540C4D3×#21512yes—
541C4D3×#23711yescharacteristic
542C4#6431311yescharacteristic
543C3—21no—
544C3#8045011yescharacteristic, complemented
545C3#8355011yescharacteristic, complemented
546C2—64no—
547C2—241no—
548C2#1059522811yescharacteristic
549C2C2×D3×#21511yescharacteristic
550C2#1059532811yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radical#10593895C2
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. #10592778C2semidirect
4. C3×#5083C2semidirect
5. #10591634C2semidirect
6. #10591583C2semidirect
7. #10591549C2semidirect
8. D3×#835C2semidirect
9. C4×#3715C2semidirect
10. #10590329C2semidirect
11. #10590301C2semidirect
44. #5727D3semidirect
52. #5083D3semidirect
182. C3×D12SD16semidirect
189. C3×Dic6#41semidirect
381. D12#215semidirect
392. Dic6#237semidirect
483. C32#3298semidirect
544. C3#80450semidirect
545. C3#83550semidirect

Maximal subgroups

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

subgroupindex
C3×#57272
#105938952
#105927782
C3×#50832
#105916342
#105915832
#105915492
D3×#8352
C4×#37152
#105903292
#105903012
#835503
#804503