Finite Group Databasebeta

← #11895878

Subgroups — #11895878

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
1#10596198C211yesmaximal, characteristic, complemented
2#10593863C211yesmaximal, characteristic, complemented
3C3×#5093C211yesmaximal, characteristic, complemented
4C3×#5087C211yesmaximal, characteristic, complemented
5#10590541C212yesmaximal, complemented
6#10590540C212yesmaximal
7#10590496C212yesmaximal, complemented
8#10590484C212yesmaximal, complemented
9#10590379C211yesmaximal, characteristic, complemented
10C4×#3714C211yesmaximal, characteristic
11#10590296C211yesmaximal, characteristic, complemented
12#81622—31nomaximal
13#80993—31nomaximal
14C2×Q8×C3⋊D3—21no—
15C32×C4⋊Q8C2211yescharacteristic
16#64155—21no—
17#64105—22no—
18#64102C2212yes—
19#64097C2212yes—
20C4×C4⋊(C3⋊D3)C2211yescharacteristic
21C4×#634C2211yescharacteristic
22C3×C4×D12C2211yescharacteristic
23C3×#869C2211yescharacteristic
24#63863C2212yes—
25#63858—22no—
26#63857—21no—
27#63843C2212yes—
28#63841—21no—
29C2×#3717C2212yes—
30C2×#3714C2211yescharacteristic
31C3×#810C2212yes—
32C3×#810C2212yes—
33C3×#809C2212yes—
34C3×#807C2212yes—
35C3×C4⋊(C3⋊C8)C2211yescharacteristic
36C3×C4×C3⋊C8C2211yescharacteristic
37#63575C2211yescharacteristic
38#63572C2212yes—
39#63566C2212yes—
40#63564C2211yescharacteristic
41#63563C2211yescharacteristic
42#63555C2212yes—
43#5732—31no—
44#5727—31no—
45#5727—31no—
46#5727—61no—
47#5542—31no—
48C3×#487—31no—
49C3×#474—31no—
50#5093D311yescharacteristic, complemented
51#5087D311yescharacteristic, complemented
52#5083—31no—
53C4×#213—31no—
54#4889—32no—
55#4875—32no—
56#4841—31no—
57#4831—31no—
58#4819—32no—
59#4817—32no—
60#4794—32no—
61#4793—32no—
62#4787—32no—
63#4765—32no—
64C4×D24—31no—
65#4679—31no—
66C2×C32×Q8—21no—
67Q8×C3⋊D3—44no—
68C2×C4⋊(C3⋊D3)C2311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3—22no—
71C2×#634—22no—
72C2×#634D411yescharacteristic
73C2×C3×D12C2311yescharacteristic
74C2×C3×C4×D3—21no—
75C2×C3×Dic6C2312yes—
76C32×C4⋊C4—21no—
77C32×C4⋊C4C2312yes—
78C32×C42C2311yescharacteristic
79#3752—21no—
80#3752—22no—
81C4⋊(C3⋊Dic3)D411yescharacteristic
82C4⋊(C3⋊Dic3)C2311yescharacteristic
83#3750—22no—
84#3750D412yes—
85C4×C3⋊Dic3C2312yes—
86C3×(C2×D3)⋊C4—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2311yescharacteristic
90C2×C3×C3⋊C8C2311yescharacteristic
91C2×C3×C3⋊C8C2312yes—
92C2×C3×C3⋊C8C2311yescharacteristic
93Dic3⋊Dic3—22no—
94(C2×D3)⋊Dic3—22no—
95(C2×D3)⋊Dic3—22no—
96Dic32—22no—
97#3717—24no—
98#3714—22no—
99#3367—91noSylow
100C2×D3×Q8—61no—
101C2×D3×Q8—61no—
102C2×D3×Q8—62no—
103C3×C4⋊Q8—21no—
104C3×C4⋊Q8—31no—
105C3×C4⋊Q8C2×D311yescharacteristic
106C3×C4⋊Q8C2×D311yescharacteristic
107C3×C4×D4—31no—
108#947—32no—
109#946—61no—
110#946—61no—
111#946—62no—
112Q8×Dic3—32no—
113C2×#215—32no—
114#936—32no—
115C2×#213—31no—
116#926—61no—
117#924—61no—
118C2×D24—31no—
119C2×#204—32no—
120#898—62no—
121#897—61no—
122#896—62no—
123#896—62no—
124#896—64no—
125#893—32no—
126#893—32no—
127#893—62no—
128#892—61no—
129#888—32no—
130#888—32no—
131#888—62no—
132#887—32no—
133#879—62no—
134C4×D12—31no—
135C4×D12—31no—
136C4×D12—61no—
137C4×D12C2×D311yescharacteristic
138#869C2×D311yescharacteristic
139C4×Dic6—31no—
140C4×Dic6—31no—
141C4×Dic6—61no—
142C3×#107—32no—
143C3×C4⋊C8—31no—
144C3×#103—32no—
145C3×#103—32no—
146C3×#102—32no—
147C3×C4×C8—31no—
148#835—32no—
149#832—31no—
150#821—31no—
151#821—31no—
152#818—31no—
153#816—32no—
154#815—32no—
155#815—32no—
156Dic3⋊C8—32no—
157Dic3⋊C8—32no—
158#810C2×D312yes—
159#810C2×D312yes—
160#809—31no—
161#809C2×D312yes—
162#807—31no—
163#807C2×D312yes—
164C4⋊(C3⋊C8)C2×D311yescharacteristic
165C4×C3⋊C8C2×D311yescharacteristic
166C22×C3⋊D3—21no—
167C22×C3×D3—21no—
168C32×Q8—42no—
169C2×C32×C4—22no—
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2411yescharacteristic
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3C2×D412yes—
175C2×C3⋊Dic3C2×D412yes—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)#4112yes—
178C4×C3⋊D3—44no—
179C4×C3⋊D3—42no—
180#634—22no—
181#634—44no—
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—91no—
193D4×Q8—91no—
194#489—92no—
195#487—31no—
196#477—92no—
197#474—31no—
198#463—92no—
199#461—91no—
200#446—91no—
201#440—92no—
202C4×D8—91no—
203C2×C3×Q8—22no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—32no—
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—62no—
218C2×C4×D3—62no—
219C2×C4×D3—62no—
220C2×C4×D3—64no—
221C2×C4×D3—61no—
222C2×C4×D3—61no—
223C2×Dic6—31no—
224C2×Dic6—31no—
225C2×Dic6—61no—
226C2×Dic6—62no—
227C2×Dic6—62no—
228C2×Dic6—64no—
229C2×Dic6C22×D312yes—
230C2×C3×C8—31no—
231C2×C3×C8—32no—
232C2×C3×C8—31no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4C22×D312yes—
241C3×C4⋊C4C22×D312yes—
242C3×C22⋊C4—61no—
243C3×C42—21no—
244C3×C42C22×D311yescharacteristic
245C3×C42C22×D311yescharacteristic
246#217—62no—
247#217—62no—
248#215—64no—
249#213—62no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—62no—
252(C2×D3)⋊C4—62no—
253(C2×D3)⋊C4—62no—
254(C2×D3)⋊C4—62no—
255(C2×D3)⋊C4—62no—
256(C2×D3)⋊C4—64no—
257(C2×D3)⋊C4—61no—
258(C2×D3)⋊C4—61no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—31no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3—61no—
267C4⋊Dic3—62no—
268C4⋊Dic3C22×D311yescharacteristic
269Dic3⋊C4—32no—
270Dic3⋊C4—32no—
271Dic3⋊C4—62no—
272Dic3⋊C4—62no—
273Dic3⋊C4—62no—
274Dic3⋊C4—62no—
275Dic3⋊C4—64no—
276C4×Dic3—32no—
277C4×Dic3—32no—
278C4×Dic3—62no—
279C4×Dic3—62no—
280C4×Dic3—62no—
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—21no—
294C32×C4—42no—
295C32×C4AGL(1,ℤ/8)12yes—
296C32×C4C2×#4111yescharacteristic
297C32×C42−1+411yescharacteristic
298C3⋊Dic3—22no—
299C3⋊Dic3—22no—
300C3⋊Dic3—41no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C22×Q8—181no—
304C2×SD16—92no—
305C2×D8—91no—
306C4⋊Q8—31no—
307C4⋊Q8—92no—
308C4⋊Q8D3211yescharacteristic
309#126—182no—
310#124—92no—
311C22⋊Q8—182no—
312C22⋊Q8—181no—
313C22⋊Q8—181no—
314C4×Q8—91no—
315C4×Q8—92no—
316C4×D4—31no—
317C4×D4—92no—
318C4×D4—91no—
319#117—181no—
320#107—32no—
321#107—91no—
322C4⋊C8—31no—
323C4⋊C8—92no—
324#103—32no—
325#103—32no—
326#103—92no—
327#102—32no—
328#102—91no—
329#102—91no—
330#97—92no—
331C4×C8—31no—
332C23×C3—61no—
333C22×D3—21no—
334C22×D3—62no—
335C22×D3—61no—
336C22×D3—61no—
337C3×Q8—44no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—62no—
341C3×Q8—62no—
342C3×D4—61no—
343C3×D4—61no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—122no—
353C2×C3×C4D3×D411yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4D3×D411yescharacteristic
357C2×C3×C4C23×D311yescharacteristic
358C2×C3×C4C23×D311yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—62no—
368C2×Dic3—61no—
369C2×Dic3—61no—
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—64no—
388Dic6—62no—
389Dic6—62no—
390Dic6—128no—
391Dic6—124no—
392Dic6—124no—
393C3×C8—61no—
394C3×C8—62no—
395C3×C8—61no—
396C3⋊C8—21no—
397C3⋊C8—22no—
398C3⋊C8—21no—
399C2×C32#54811yescharacteristic
400C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C32#57811yescharacteristic
402C3⋊D3—42no—
403C3×D3—81no—
404C2×Q8—21no—
405C2×Q8—32no—
406C2×Q8—91no—
407C2×Q8—182no—
408C2×Q8—364no—
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—21no—
420C4⋊C4—31no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—91no—
424C4⋊C4—92no—
425C4⋊C4—91no—
426C4⋊C4—182no—
427C4⋊C4—182no—
428C4⋊C4C2×D3212yes—
429C22⋊C4—61no—
430C22⋊C4—182no—
431C22⋊C4—182no—
432C22⋊C4—182no—
433C22⋊C4—181no—
434C42—92no—
435C42—182no—
436C42C2×D3211yescharacteristic
437C22×C3—21no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3C2×D3×D411yescharacteristic
442C22×C3C2×D3×D411yescharacteristic
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—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—22no—
460C3×C4—41no—
461C3×C4—44no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—122no—
465C3×C4—121no—
466C3×C4#91112yes—
467C3×C4#91412yes—
468C3×C4#100711yescharacteristic
469C3×C4D3×#4111yescharacteristic
470C3×C4D3×#4111yescharacteristic
471C3×C4#101011yescharacteristic
472Dic3—42no—
473Dic3—41no—
474Dic3—64no—
475Dic3—62no—
476Dic3—62no—
477Dic3—62no—
478Dic3—62no—
479Dic3—122no—
480Dic3—122no—
481Dic3—121no—
482Dic3—121no—
483C32#336711yescharacteristic, complemented, Sylow
484C23—61no—
485C23—181no—
486Q8—42no—
487Q8—62no—
488Q8—62no—
489Q8—182no—
490Q8—364no—
491D4—61no—
492D4—61no—
493D4—92no—
494D4—181no—
495C2×C4—22no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—122no—
501C2×C4—181no—
502C2×C4—364no—
503C2×C4—362no—
504C2×C4#381111yescharacteristic
505C2×C4#381111yescharacteristic
506C2×C4C22×D3211yescharacteristic
507C8—61no—
508C8—61no—
509C8—62no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—241no—
514C2×C3#563311yescharacteristic
515C2×C3#567711yescharacteristic
516C2×C3C2×#91111yescharacteristic
517C2×C3C2×#91411yescharacteristic
518C2×C3#576611yescharacteristic
519C2×C3#577911yescharacteristic
520D3—81no—
521D3—124no—
522D3—122no—
523D3—122no—
524C22—121no—
525C22—121no—
526C22—121no—
527C22—182no—
528C22—361no—
529C22—361no—
530C22C2×#381111yescharacteristic
531C4—21no—
532C4—42no—
533C4—122no—
534C4—121no—
535C4—182no—
536C4—182no—
537C4—361no—
538C4#6393712yes—
539C4#6431511yescharacteristic
540C4#6431911yescharacteristic
541C3—21no—
542C3#8099311yescharacteristic, complemented
543C3#8162211yescharacteristic, complemented
544C2—241no—
545C2—362no—
546C2C2×#6393711yescharacteristic
547C2#1059533711yescharacteristic
548C2#1059536211yescharacteristic
subgroupG/H
centerC22C2×#3811
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 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #10596198C2semidirect
2. #10593863C2semidirect
3. C3×#5093C2semidirect
4. C3×#5087C2semidirect
5. #10590541C2semidirect
7. #10590496C2semidirect
8. #10590484C2semidirect
9. #10590379C2semidirect
11. #10590296C2semidirect
50. #5093D3semidirect
51. #5087D3semidirect
483. C32#3367semidirect
542. C3#80993semidirect
543. C3#81622semidirect

Maximal subgroups

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

subgroupindex
#105961982
#105938632
C3×#50932
C3×#50872
#105905412
#105905402
#105904962
#105904842
#105903792
C4×#37142
#105902962
#816223
#809933