Finite Group Databasebeta

← #11895866

Subgroups — #11895866

Order 1152.

884 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#5584C211yesmaximal, characteristic, complemented
3C3×#5024C211yesmaximal, characteristic, complemented
4#10590379C211yesmaximal, characteristic, complemented
5#10590362C212yesmaximal, complemented
6#10590356C212yesmaximal
7C4×#3714C211yesmaximal, characteristic
8#10590287C211yesmaximal, characteristic, complemented
9#10590283C212yesmaximal, complemented
10#10590276C212yesmaximal, complemented
11#10590272C211yesmaximal, characteristic, complemented
12#83668—31nomaximal
13#79970—31nomaximal
14C2×Q8×C3⋊D3—21no—
15C32×C4×Q8C2211yescharacteristic
16#64106—21no—
17#64106—22no—
18C4×C4⋊(C3⋊D3)C2211yescharacteristic
19C4×C4⋊(C3⋊D3)C2212yes—
20#64078C2211yescharacteristic
21#64078C2212yes—
22C3×#946—21no—
23C3×#898—22no—
24C3×#892—21no—
25C3×#875C2212yes—
26C3×C4×D12C2211yescharacteristic
27C3×#869C2211yescharacteristic
28C3×C4×Dic6C2212yes—
29C2×#3717C2212yes—
30C2×#3714C2211yescharacteristic
31C3×C4⋊(C3⋊C8)C2211yescharacteristic
32C3×C4⋊(C3⋊C8)C2212yes—
33C3×#803C2212yes—
34C3×C4×C3⋊C8C2211yescharacteristic
35#63575C2212yes—
36#63575C2211yescharacteristic
37#63566C2212yes—
38#63566C2212yes—
39#63566C2212yes—
40#63564C2211yescharacteristic
41#63563C2212yes—
42#63563C2211yescharacteristic
43C3×#556—31no—
44#5584D311yescharacteristic, complemented
45Q8×D12—31no—
46Q8×D12—31no—
47Q8×D12—61no—
48C3×#446—31no—
49#5083—31no—
50#5079—31no—
51#5057—32no—
52#5055—32no—
53#5026—32no—
54#5024D311yescharacteristic, complemented
55#5021—32no—
56C4×#213—31no—
57#5007—31no—
58#4850—32no—
59#4846—31no—
60#4844—32no—
61#4841—31no—
62#4714—32no—
63#4710—32no—
64C4×D24—31no—
65#4681—31no—
66C2×C32×Q8D411yescharacteristic
67Q8×C3⋊D3—44no—
68C2×C4⋊(C3⋊D3)C2311yescharacteristic
69C2×C4×C3⋊D3—22no—
70C2×C4×C3⋊D3—21no—
71C2×#634—22no—
72C2×#634—21no—
73C2×C3×D12C2311yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×Dic6C2312yes—
76C32×C4⋊C4D411yescharacteristic
77C32×C4⋊C4D412yes—
78C32×C42C2311yescharacteristic
79C32×C42C2312yes—
80#3752—22no—
81#3752—21no—
82C4⋊(C3⋊Dic3)—22no—
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)C2312yes—
85C4⋊(C3⋊Dic3)C2311yescharacteristic
86C3×(C2×D3)⋊C4—22no—
87C3×(C2×D3)⋊C4—21no—
88C3×(C2×D3)⋊C4—22no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2311yescharacteristic
92C3×Dic3⋊C4—22no—
93C3×C4×Dic3—22no—
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2312yes—
96C2×C3×C3⋊C8C2311yescharacteristic
97#3717—24no—
98#3714—22no—
99#3367—91noSylow
100C2×D3×Q8—62no—
101C2×D3×Q8—61no—
102C2×D3×Q8—61no—
103C3×C4⋊Q8—31no—
104C3×#126—62no—
105C3×#124—32no—
106C3×C22⋊Q8—61no—
107C3×C4×Q8—21no—
108C3×C4×Q8—32no—
109C3×C4×Q8C2×D311yescharacteristic
110C3×C4×Q8C2×D311yescharacteristic
111C3×C4×D4—31no—
112C3×#117—61no—
113#946—21no—
114C2×#215—32no—
115C2×#213—31no—
116C2×D24—31no—
117C2×#204—32no—
118#898—22no—
119#897—62no—
120#897—64no—
121#897—61no—
122#897—62no—
123#897—61no—
124#897—62no—
125#892—21no—
126#875C2×D312yes—
127C4×D12—31no—
128C4×D12—32no—
129C4×D12—31no—
130C4×D12—32no—
131C4×D12—61no—
132C4×D12—62no—
133C4×D12C2×D311yescharacteristic
134#869—31no—
135#869—32no—
136#869—31no—
137#869—32no—
138#869—61no—
139#869—62no—
140#869C2×D311yescharacteristic
141C4×Dic6C2×D312yes—
142C3×C4⋊C8—31no—
143C3×C4⋊C8—32no—
144C3×#97—32no—
145C3×C4×C8—31no—
146#835—32no—
147#835—32no—
148#835—32no—
149#832—32no—
150#832—31no—
151#821—31no—
152#821—32no—
153#821—31no—
154#818—31no—
155#818—32no—
156#816—32no—
157#816—32no—
158#816—32no—
159#809—31no—
160#807—32no—
161#807—31no—
162C4⋊(C3⋊C8)C2×D311yescharacteristic
163C4⋊(C3⋊C8)C2×D312yes—
164#803C2×D312yes—
165C4×C3⋊C8C2×D311yescharacteristic
166C22×C3⋊D3—21no—
167C22×C3×D3—21no—
168C32×Q8—22no—
169C2×C32×C4C2×D412yes—
170C2×C32×C4C2×D412yes—
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2411yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—22no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)#4112yes—
178C4×C3⋊D3—44no—
179C4×C3⋊D3—42no—
180#634—44no—
181#634—42no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—22no—
184C3×D12—21no—
185C3×D12—21no—
186C3×C4×D3—42no—
187C3×Dic6—22no—
188C3×Dic6—22no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8—21no—
192#556—31no—
193D4×Q8—91no—
194#489—92no—
195#487—91no—
196#477—92no—
197#474—91no—
198#463—92no—
199#461—91no—
200#446—31no—
201#440—92no—
202C4×D8—91no—
203C2×C3×Q8—21no—
204C2×C3×Q8—32no—
205C2×C3×Q8D1211yescharacteristic
206C2×C3×Q8#6711yescharacteristic
207C2×C3×D4—31no—
208C22×C3×C4—61no—
209D3×Q8—128no—
210D3×Q8—124no—
211D3×Q8—124no—
212C2×D12—31no—
213C2×D12—31no—
214C2×D12—61no—
215C2×D12C22×D311yescharacteristic
216C2×C4×D3—21no—
217C2×C4×D3—64no—
218C2×C4×D3—62no—
219C2×C4×D3—62no—
220C2×C4×D3—61no—
221C2×C4×D3—62no—
222C2×C4×D3—61no—
223C2×Dic6—64no—
224C2×Dic6—62no—
225C2×Dic6—62no—
226C2×Dic6—61no—
227C2×Dic6—62no—
228C2×Dic6—61no—
229C2×Dic6C22×D312yes—
230C2×C3×C8—31no—
231C2×C3×C8—31no—
232C2×C3×C8—32no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—62no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4D1211yescharacteristic
240C3×C4⋊C4D1212yes—
241C3×C4⋊C4#6711yescharacteristic
242C3×C4⋊C4#6712yes—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—61no—
245C3×C22⋊C4—62no—
246C3×C42—21no—
247C3×C42—22no—
248C3×C42—62no—
249C3×C42C22×D311yescharacteristic
250C3×C42C22×D312yes—
251C3×C42C22×D311yescharacteristic
252C3×C42C22×D312yes—
253#215—64no—
254#213—62no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—64no—
259(C2×D3)⋊C4—62no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—61no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4—61no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—32no—
267C4⋊Dic3—31no—
268C4⋊Dic3—32no—
269C4⋊Dic3—31no—
270C4⋊Dic3—61no—
271C4⋊Dic3—62no—
272C4⋊Dic3—64no—
273C4⋊Dic3—62no—
274C4⋊Dic3—62no—
275C4⋊Dic3—61no—
276C4⋊Dic3—62no—
277C4⋊Dic3—61no—
278C4⋊Dic3C22×D311yescharacteristic
279Dic3⋊C4—22no—
280C4×Dic3—22no—
281C2×C3⋊C8C22×D311yescharacteristic
282C2×C3⋊C8C22×D311yescharacteristic
283C2×C3⋊C8C22×D312yes—
284D24—62no—
285#204—64no—
286C22×C32C22×D411yescharacteristic
287C2×C3⋊D3—22no—
288C2×C3⋊D3—41no—
289C2×C3⋊D3—41no—
290C2×C3×D3—41no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C32×C4—22no—
294C32×C4—22no—
295C32×C4—21no—
296C32×C4AGL(1,ℤ/8)12yes—
297C32×C4C2×#4111yescharacteristic
298C32×C42−1+411yescharacteristic
299C3⋊Dic3—41no—
300C3⋊Dic3—42no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C22×Q8—181no—
304C2×SD16—92no—
305C2×D8—91no—
306C4⋊Q8—31no—
307C4⋊Q8—91no—
308C4⋊Q8—92no—
309#126—62no—
310#124—32no—
311C22⋊Q8—61no—
312C22⋊Q8—181no—
313C22⋊Q8—182no—
314C4×Q8—32no—
315C4×Q8D3211yescharacteristic
316C4×D4—31no—
317C4×D4—91no—
318C4×D4—92no—
319#117—61no—
320#107—91no—
321#107—92no—
322C4⋊C8—31no—
323C4⋊C8—32no—
324#103—92no—
325#103—92no—
326#103—92no—
327#102—91no—
328#102—91no—
329#102—92no—
330#97—32no—
331C4×C8—31no—
332C23×C3—61no—
333C22×D3—21no—
334C22×D3—62no—
335C22×D3—61no—
336C22×D3—61no—
337C3×Q8—24no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—62no—
341C3×Q8—62no—
342C3×D4—61no—
343C3×D4—61no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—61no—
350C2×C3×C4—62no—
351C2×C3×C4—122no—
352C2×C3×C4C2×D1212yes—
353C2×C3×C4C2×D1211yescharacteristic
354C2×C3×C4C2×D1212yes—
355C2×C3×C4C2×D1211yescharacteristic
356C2×C3×C4C2×#6712yes—
357C2×C3×C4C2×#6712yes—
358C2×C3×C4C2×#6711yescharacteristic
359C2×C3×C4C2×#6711yescharacteristic
360C2×C3×C4C23×D311yescharacteristic
361C2×C3×C4C23×D311yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—22no—
364C2×Dic3—64no—
365C2×Dic3—62no—
366C2×Dic3—61no—
367C2×Dic3—62no—
368C2×Dic3—61no—
369C2×Dic3—62no—
370D12—21no—
371D12—21no—
372D12—32no—
373D12—32no—
374D12—62no—
375D12—62no—
376D12—61no—
377D12—61no—
378C4×D3—42no—
379C4×D3—128no—
380C4×D3—124no—
381C4×D3—124no—
382C4×D3—124no—
383C4×D3—122no—
384C4×D3—122no—
385Dic6—22no—
386Dic6—22no—
387Dic6—128no—
388Dic6—124no—
389Dic6—124no—
390Dic6—124no—
391Dic6—122no—
392Dic6—122no—
393C3×C8—62no—
394C3×C8—61no—
395C3×C8—61no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—22no—
399C2×C32#54811yescharacteristic
400C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C32#57811yescharacteristic
402C3⋊D3—42no—
403C3×D3—81no—
404C2×Q8—32no—
405C2×Q8—182no—
406C2×Q8—181no—
407C2×Q8—364no—
408C2×Q8#62611yescharacteristic
409C2×D4—31no—
410C2×D4—91no—
411C22×C4—61no—
412C22×C4—182no—
413C22×C4—181no—
414SD16—184no—
415D8—182no—
416C2×C8—31no—
417C2×C8—31no—
418C2×C8—32no—
419C4⋊C4—31no—
420C4⋊C4—61no—
421C4⋊C4—62no—
422C4⋊C4—61no—
423C4⋊C4—91no—
424C4⋊C4—92no—
425C4⋊C4—182no—
426C4⋊C4—181no—
427C4⋊C4#62611yescharacteristic
428C4⋊C4#62612yes—
429C22⋊C4—62no—
430C22⋊C4—61no—
431C22⋊C4—62no—
432C22⋊C4—182no—
433C22⋊C4—181no—
434C42—62no—
435C42C2×D3211yescharacteristic
436C42C2×D3212yes—
437C22×C3—21no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3C22×D1211yescharacteristic
442C22×C3C22×#6711yescharacteristic
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—41no—
446C2×D3—64no—
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—122no—
450C2×D3—122no—
451C2×D3—121no—
452C2×D3—121no—
453C2×D3—121no—
454C2×D3—121no—
455C3×C4—24no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—22no—
465C3×C4—42no—
466C3×C4—41no—
467C3×C4—121no—
468C3×C4—122no—
469C3×C4#90812yes—
470C3×C4#94912yes—
471C3×C4#100711yescharacteristic
472C3×C4D3×#4111yescharacteristic
473C3×C4D3×#4111yescharacteristic
474C3×C4#101011yescharacteristic
475Dic3—41no—
476Dic3—42no—
477Dic3—124no—
478Dic3—122no—
479Dic3—121no—
480Dic3—122no—
481Dic3—121no—
482Dic3—122no—
483C32#336711yescharacteristic, complemented, Sylow
484C23—61no—
485C23—181no—
486Q8—22no—
487Q8—62no—
488Q8—62no—
489Q8—364no—
490Q8—362no—
491D4—61no—
492D4—61no—
493D4—92no—
494D4—181no—
495C2×C4—61no—
496C2×C4—62no—
497C2×C4—122no—
498C2×C4—182no—
499C2×C4—181no—
500C2×C4—364no—
501C2×C4—362no—
502C2×C4C2×#62612yes—
503C2×C4C2×#62611yescharacteristic
504C2×C4C2×#62612yes—
505C2×C4C2×#62611yescharacteristic
506C2×C4C22×D3211yescharacteristic
507C8—61no—
508C8—61no—
509C8—62no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—241no—
514C2×C3#557611yescharacteristic
515C2×C3C2×#90811yescharacteristic
516C2×C3#575311yescharacteristic
517C2×C3C2×#94911yescharacteristic
518C2×C3#583111yescharacteristic
519C2×C3#583811yescharacteristic
520D3—81no—
521D3—124no—
522D3—122no—
523D3—122no—
524C22—121no—
525C22—121no—
526C22—121no—
527C22—182no—
528C22—361no—
529C22—361no—
530C22C22×#62611yescharacteristic
531C4—22no—
532C4—21no—
533C4—22no—
534C4—122no—
535C4—121no—
536C4—362no—
537C4—361no—
538C4#6382512yes—
539C4#6431511yescharacteristic
540C4#6431911yescharacteristic
541C3—21no—
542C3#7997011yescharacteristic, complemented
543C3#8366811yescharacteristic, complemented
544C2—241no—
545C2—362no—
546C2C2×#6382511yescharacteristic
547C2#1059541411yescharacteristic
548C2#1059545011yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radicalC2×#3714C22
hypercenterC4×Q8D32
hypocenterC32#3367
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊(C3⋊D3)C2semidirect
2. C3×#5584C2semidirect
3. C3×#5024C2semidirect
4. #10590379C2semidirect
5. #10590362C2semidirect
8. #10590287C2semidirect
9. #10590283C2semidirect
10. #10590276C2semidirect
11. #10590272C2semidirect
44. #5584D3semidirect
54. #5024D3semidirect
483. C32#3367semidirect
542. C3#79970semidirect
543. C3#83668semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊(C3⋊D3)2
C3×#55842
C3×#50242
#105903792
#105903622
#105903562
C4×#37142
#105902872
#105902832
#105902762
#105902722
#836683
#799703