Finite Group Databasebeta

← #11887565

Subgroups — #11887565

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
1Dic6×D12C211yesmaximal, characteristic, complemented
2#10593896C211yesmaximal, characteristic
3#10592540C211yesmaximal, characteristic, complemented
4C3×#4851C211yesmaximal, characteristic, complemented
5C3×#4834C211yesmaximal, characteristic, complemented
6#10590951C212yesmaximal, complemented
7#10590830C212yesmaximal, complemented
8#10590674C212yesmaximal, complemented
9D3×#816C212yesmaximal, complemented
10C4×#3717C211yesmaximal, characteristic, complemented
11#10590288C211yesmaximal, characteristic, complemented
12#81022—31nomaximal
13#79447—31nomaximal
14C2×#3796—21no—
15C2×D3×Dic6C2212yes—
16C2×#3744C2212yes—
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18#64078C2211yescharacteristic
19C2×C3×D3×C8C2212yes—
20C3×C4×D12C2211yescharacteristic
21C3×#869C2211yescharacteristic
22C3×C4×Dic6C2211yescharacteristic
23#63918—21no—
24#63909C2212yes—
25#63893—22no—
26Dic3×D12—21no—
27#63890—22no—
28D3×C4⋊Dic3C2212yes—
29#63871—21no—
30#63857C2212yes—
31C2×#3717C2211yescharacteristic
32C32×C4⋊C8C2211yescharacteristic
33#63643C2212yes—
34C3×(C2×D3)⋊C8C2212yes—
35C3×#817C2212yes—
36C3×#816C2212yes—
37C3×C4×C3⋊C8C2211yescharacteristic
38#63573C2211yescharacteristic
39#63567C2212yes—
40#63567C2212yes—
41#63566C2211yescharacteristic
42#63564C2211yescharacteristic
43#5726—31no—
44Q8×D12—31no—
45#5555—31no—
46D4×Dic6—31no—
47C3×#475—31no—
48C3×C8×D4—31no—
49#5114—32no—
50C2×#816—32no—
51#5093—31no—
52C4×#215—31no—
53#4864—32no—
54#4862—32no—
55#4851D311yescharacteristic, complemented
56#4841—31no—
57#4841—31no—
58#4841—61no—
59#4834D311yescharacteristic, complemented
60#4806—32no—
61D3×#103—32no—
62#4738—32no—
63#4721—32no—
64C4×#204—31no—
65#4681—31no—
66C2×C4⋊(C3⋊D3)C2311yescharacteristic
67C2×C4×C3⋊D3—21no—
68C2×#634C2312yes—
69C2×C3×D12D411yescharacteristic
70C2×C3×C4×D3C2312yes—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2311yescharacteristic
73C2×#627—22no—
74C2×#626—22no—
75C2×D3×Dic3—22no—
76C2×D3×Dic3—22no—
77#3796—44no—
78D3×Dic6—24no—
79D3×Dic6—42no—
80C2×C32×C8C2312yes—
81C32×C42C2311yescharacteristic
82#3752—21no—
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)—21no—
85C4⋊(C3⋊Dic3)C2311yescharacteristic
86#3744—42no—
87C3×(C2×D3)⋊C4D412yes—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3D411yescharacteristic
90C3×C4⋊Dic3C2311yescharacteristic
91C3×C4⋊Dic3C2312yes—
92C3×Dic3⋊C4—21no—
93C3×C4×Dic3—21no—
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2311yescharacteristic
96C3×D3×C8—42no—
97Dic3⋊Dic3—22no—
98Dic3⋊Dic3—22no—
99#3722—22no—
100(C2×D3)⋊Dic3—22no—
101#3717—22no—
102#3298—91noSylow
103C2×#239—61no—
104C2×D3×Q8—32no—
105C2×#237—61no—
106C22×Dic6—32no—
107C22×C3×C8—32no—
108C3×C4⋊Q8—31no—
109C3×C4×Q8—31no—
110C3×C4×D4—31no—
111#946—61no—
112C2×#215—31no—
113D4×Dic3—61no—
114#930—32no—
115C2×C4⋊Dic3—32no—
116#924—32no—
117C2×#204—31no—
118C2×#204—32no—
119C2×#204—32no—
120C2×#204—62no—
121C2×D3×C8C2×D312yes—
122#897—32no—
123#897—61no—
124#897—62no—
125#895—32no—
126#894—62no—
127D3×C4⋊C4—32no—
128#888—61no—
129#886—62no—
130#878—62no—
131C4×D12—31no—
132C4×D12—31no—
133C4×D12—61no—
134C4×D12—61no—
135C4×D12C2×D311yescharacteristic
136#869—31no—
137#869—31no—
138#869—61no—
139#869—61no—
140#869C2×D311yescharacteristic
141C4×Dic6C2×D311yescharacteristic
142C3×#106—32no—
143C3×C4⋊C8—21no—
144C3×C4⋊C8C2×D311yescharacteristic
145C3×C4⋊C8C2×D311yescharacteristic
146C3×#103—32no—
147C3×C22⋊C8—32no—
148C3×C4×C8—31no—
149#835—31no—
150#821—31no—
151(C2×D3)⋊C8C2×D312yes—
152#817—31no—
153#817C2×D312yes—
154#816—31no—
155#816—32no—
156#816—32no—
157#816—32no—
158#816—32no—
159#816—62no—
160#816C2×D312yes—
161#810—32no—
162#810—32no—
163#809—31no—
164#808—31no—
165C4×C3⋊C8C2×D311yescharacteristic
166C22×C3⋊D3—21no—
167C22×C3×D3C2×D412yes—
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2411yescharacteristic
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—21no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)—21no—
175C4×C3⋊D3—42no—
176#634—22no—
177#634—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3C2×D412yes—
182C3×D12SD1614yescomplemented
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×Dic6—21no—
186C3×Dic6—42no—
187C3×Dic6#4112yescomplemented
188#627—44no—
189#626—44no—
190D3×Dic3—44no—
191D3×Dic3—44no—
192C32×C8—22no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195D4×Q8—91no—
196#547—91no—
197#487—91no—
198#483—92no—
199#475—31no—
200#465—92no—
201#461—91no—
202#448—92no—
203C4×SD16—91no—
204C8×D4—31no—
205C2×#103—92no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C2×C3×D4—31no—
209C22×C3×C4—32no—
210C2×#67—62no—
211C22×Dic3—62no—
212C22×Dic3—62no—
213#239—124no—
214D3×Q8—64no—
215D3×Q8—122no—
216#237—124no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×D12—62no—
221C2×D12D1211yescharacteristic
222C2×C4×D3—62no—
223C2×C4×D3—62no—
224C2×C4×D3—61no—
225C2×C4×D3—61no—
226C2×C4×D3—121no—
227C2×C4×D3C22×D312yes—
228C2×Dic6—21no—
229C2×Dic6—32no—
230C2×Dic6—32no—
231C2×Dic6—62no—
232C2×Dic6—64no—
233C2×Dic6—62no—
234C2×Dic6—62no—
235C2×Dic6—122no—
236C2×Dic6C22×D311yescharacteristic
237C2×C3×C8—22no—
238C2×C3×C8—31no—
239C2×C3×C8—31no—
240C2×C3×C8—122no—
241C2×C3×C8C22×D312yes—
242C2×C3×C8C22×D312yes—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—61no—
248C3×C22⋊C4—32no—
249C3×C42—21no—
250C3×C42—61no—
251C3×C42C22×D311yescharacteristic
252C3×C42C22×D311yescharacteristic
253#217—62no—
254#215—62no—
255(C2×D3)⋊C4—62no—
256(C2×D3)⋊C4—62no—
257(C2×D3)⋊C4—61no—
258(C2×D3)⋊C4—61no—
259(C2×D3)⋊C4—62no—
260(C2×D3)⋊C4—121no—
261(C2×D3)⋊C4D1212yes—
262C4⋊Dic3—21no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3—62no—
267C4⋊Dic3—61no—
268C4⋊Dic3—61no—
269C4⋊Dic3—61no—
270C4⋊Dic3—61no—
271C4⋊Dic3—62no—
272C4⋊Dic3—121no—
273C4⋊Dic3—121no—
274C4⋊Dic3D1211yescharacteristic
275C4⋊Dic3C22×D312yes—
276C4⋊Dic3C22×D311yescharacteristic
277Dic3⋊C4—21no—
278Dic3⋊C4—62no—
279Dic3⋊C4—62no—
280C4×Dic3—21no—
281C2×C3⋊C8C22×D311yescharacteristic
282C2×C3⋊C8C22×D311yescharacteristic
283#204—62no—
284#204—122no—
285#204—122no—
286#204—124no—
287D3×C8—42no—
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×C4C2×SD1612yes—
297C32×C4C2×#4111yescharacteristic
298C32×C4H(5,𝔽2)11yescharacteristic
299C3⋊Dic3—42no—
300C3⋊Dic3—41no—
301C3×Dic3—22no—
302C3×Dic3—22no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C2×#41—181no—
306C22×Q8—92no—
307C2×SD16—91no—
308C2×SD16—92no—
309C22×C8—32no—
310C4⋊Q8—31no—
311C4⋊Q8—91no—
312C4⋊Q8—181no—
313#123—182no—
314C22⋊Q8—92no—
315C22⋊Q8—182no—
316C22⋊Q8—181no—
317#121—92no—
318C4×Q8—31no—
319C4×D4—31no—
320C4×D4—91no—
321C4×D4—181no—
322C2×C4⋊C4—92no—
323#106—32no—
324#106—91no—
325C4⋊C8D3211yescharacteristic
326#103—32no—
327#103—92no—
328#103—92no—
329#103—91no—
330#103—92no—
331#102—91no—
332C22⋊C8—32no—
333C4×C8—31no—
334C23×C3—32no—
335C22×D3—61no—
336C22×D3—61no—
337C22×D3—121no—
338C22×D3C2×D1212yes—
339C3×Q8—32no—
340C3×Q8—61no—
341C3×Q8—122no—
342C3×D4—34no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—32no—
347C2×C3×C4—61no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—61no—
352C2×C3×C4C2×D1211yescharacteristic
353C2×C3×C4C2×D1211yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4C23×D311yescharacteristic
357C2×C3×C4C23×D311yescharacteristic
358#67—124no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—22no—
362C2×Dic3—61no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—61no—
366C2×Dic3—121no—
367C2×Dic3—122no—
368C2×Dic3—124no—
369C2×Dic3—124no—
370C2×Dic3C2×D1212yes—
371D12—62no—
372D12—61no—
373D12—61no—
374D12—61no—
375D12—61no—
376D12—121no—
377D12—124no—
378D12#20414yescomplemented
379C4×D3—22no—
380C4×D3—22no—
381C4×D3—124no—
382C4×D3—124no—
383C4×D3—122no—
384C4×D3—122no—
385C4×D3—242no—
386Dic6—21no—
387Dic6—42no—
388Dic6—62no—
389Dic6—62no—
390Dic6—62no—
391Dic6—62no—
392Dic6—122no—
393Dic6—122no—
394Dic6—124no—
395Dic6—124no—
396Dic6#23912yescomplemented
397C3×C8—22no—
398C3×C8—22no—
399C3×C8—42no—
400C3×C8—61no—
401C3×C8—61no—
402C3⋊C8—21no—
403C3⋊C8—21no—
404C2×C32#54611yescharacteristic
405C2×C32C22×SD1611yescharacteristic
406C2×C32#57811yescharacteristic
407C3⋊D3—81no—
408C3×D3—24no—
409#41—364no—
410C2×Q8—31no—
411C2×Q8—61no—
412C2×Q8—92no—
413C2×Q8—182no—
414C2×Q8—184no—
415C2×Q8—362no—
416C2×D4—31no—
417C2×D4—91no—
418C2×D4—182no—
419C22×C4—32no—
420C22×C4—182no—
421C22×C4—182no—
422C22×C4—181no—
423SD16—182no—
424SD16—362no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—122no—
428C2×C8C2×D3212yes—
429C4⋊C4—31no—
430C4⋊C4—31no—
431C4⋊C4—32no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—91no—
435C4⋊C4—182no—
436C4⋊C4—182no—
437C4⋊C4—181no—
438C4⋊C4—181no—
439C22⋊C4—32no—
440C22⋊C4—182no—
441C22⋊C4—182no—
442C22⋊C4—181no—
443C42—61no—
444C42C2×D3211yescharacteristic
445C22×C3—21no—
446C22×C3—34no—
447C22×C3—62no—
448C22×C3—62no—
449C22×C3C22×D1211yescharacteristic
450C22×C3C2×D3×D411yescharacteristic
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×#20414yes—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—22no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—41no—
469C3×C4—62no—
470C3×C4—62no—
471C3×C4—122no—
472C3×C4—121no—
473C3×C4C2×#20412yes—
474C3×C4D3×SD1612yes—
475C3×C4#100411yescharacteristic
476C3×C4C2×#23911yescharacteristic
477C3×C4D3×#4111yescharacteristic
478C3×C4#100911yescharacteristic
479Dic3—22no—
480Dic3—22no—
481Dic3—42no—
482Dic3—41no—
483Dic3—121no—
484Dic3—121no—
485Dic3—122no—
486Dic3—122no—
487Dic3—242no—
488Dic3—241no—
489C32#329811yescharacteristic, complemented, Sylow
490C23—32no—
491C23—181no—
492Q8—32no—
493Q8—61no—
494Q8—122no—
495Q8—182no—
496Q8—182no—
497Q8—364no—
498D4—34no—
499D4—181no—
500D4—181no—
501D4—364no—
502C2×C4—32no—
503C2×C4—61no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—61no—
507C2×C4—62no—
508C2×C4—181no—
509C2×C4—182no—
510C2×C4—364no—
511C2×C4—364no—
512C2×C4—362no—
513C2×C4D3×D1211yescharacteristic
514C2×C4D3×D1211yescharacteristic
515C2×C4C22×D3211yescharacteristic
516C8—22no—
517C8—61no—
518C8—61no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—64no—
523C2×C3#555411yescharacteristic
524C2×C3#563011yescharacteristic
525C2×C3C22×#20411yescharacteristic
526C2×C3#575311yescharacteristic
527C2×C3C2×D3×SD1611yescharacteristic
528C2×C3#577911yescharacteristic
529D3—24no—
530D3—242no—
531D3—241no—
532D3—241no—
533C22—34no—
534C22—62no—
535C22—62no—
536C22—361no—
537C22—361no—
538C22—361no—
539C22C2×D3×D1211yescharacteristic
540C4—21no—
541C4—62no—
542C4—62no—
543C4—122no—
544C4—121no—
545C4—362no—
546C4—361no—
547C4D3×#20412yes—
548C4#6431411yescharacteristic
549C4D3×#23911yescharacteristic
550C3—21no—
551C3#7944711yescharacteristic, complemented
552C3#8102211yescharacteristic, complemented
553C2—64no—
554C2—721no—
555C2#1059462911yescharacteristic
556C2C2×D3×#20411yescharacteristic
557C2#1059472111yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊C8C22
socleC22×C32C22×D4
prime radical#10593896C2
hypercenterC4⋊C8D32
hypocenterC32#3298
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. Dic6×D12C2semidirect
3. #10592540C2semidirect
4. C3×#4851C2semidirect
5. C3×#4834C2semidirect
6. #10590951C2semidirect
7. #10590830C2semidirect
8. #10590674C2semidirect
9. D3×#816C2semidirect
10. C4×#3717C2semidirect
11. #10590288C2semidirect
55. #4851D3semidirect
59. #4834D3semidirect
182. C3×D12SD16semidirect
187. C3×Dic6#41semidirect
378. D12#204semidirect
396. Dic6#239semidirect
489. C32#3298semidirect
551. C3#79447semidirect
552. C3#81022semidirect

Maximal subgroups

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

subgroupindex
Dic6×D122
#105938962
#105925402
C3×#48512
C3×#48342
#105909512
#105908302
#105906742
D3×#8162
C4×#37172
#105902882
#810223
#794473