Finite Group Databasebeta

← #11878654

Subgroups — #11878654

Order 1152.

1123 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5725C211yesmaximal, characteristic, complemented
2D3×#874C211yesmaximal, characteristic, complemented
3#10592787C211yesmaximal, characteristic, complemented
4C3×#5088C211yesmaximal, characteristic, complemented
5#10590595C214yesmaximal, complemented
6#10590547C214yesmaximal, complemented
7#10590352C211yesmaximal, characteristic, complemented
8#10590350C211yesmaximal, characteristic, complemented
9#10590261C211yesmaximal, characteristic, complemented
10#83302—31nomaximal
11#81566—31nomaximal
12C2×C3×#239—21no—
13C2×D3×D12—21no—
14C2×D3×D12—21no—
15C2×D3×D12C2212yes—
16C32×C4⋊Q8C2211yescharacteristic, complemented
17#64083C2211yescharacteristic
18C3×#947—21no—
19C3×#895—21no—
20C3×#895C2212yes—
21C3×#893—21no—
22C3×#893C2212yescomplemented
23C3×#874C2211yescharacteristic
24C3×#874C2211yescharacteristic, complemented
25C3×C42×D3C2211yescharacteristic
26#63911—21no—
27#63911—21no—
28#63911C2212yes—
29C2×#3714C2214yescomplemented
30C2×#3713C2214yescomplemented
31#63636C2214yescomplemented
32C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
33C3×#809C2214yescomplemented
34C3×C4×C3⋊C8C2211yescharacteristic, complemented
35Dic3⋊(C3⋊C8)C2212yescomplemented
36(C2×D3)⋊(C3⋊C8)C2212yes—
37C3×#532—31no—
38#5725D311yescharacteristic, complemented
39D3×#127—31no—
40C2×#874—31no—
41C3×#486—31no—
42#5088—31no—
43#5088—31no—
44#5088—61no—
45#5088D311yescharacteristic, complemented
46#5073—31no—
47#5073—31no—
48#5073—31no—
49#5036—34no—
50#5015—34no—
51#4925—31no—
52#4786—34no—
53#4773—34no—
54#4695—31no—
55#4685—31no—
56C22×D32—22no—
57C2×C32×Q8—21no—
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)—21no—
60C2×C4⋊(C3⋊D3)C2312yes—
61C3×#239—44no—
62C2×C3×D12—22no—
63C2×C3×D12—21no—
64C2×C3×D12—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12C2312yes—
68C2×C3×D12C2312yes—
69C2×C3×C4×D3—22no—
70C2×C3×C4×D3D411yescharacteristic
71C2×C3×C4×D3D411yescharacteristic
72C2×C3×C4×D3C2311yescharacteristic
73C2×#626—22no—
74C2×#626—22no—
75C2×#626—22no—
76D3×D12—44no—
77D3×D12—44no—
78D3×D12—44no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4C2312yes—
81C32×C42C2311yescharacteristic
82C2×C3⋊(C3⋊C8)C2312yes—
83C3×(C2×D3)⋊C4—22no—
84C3×(C2×D3)⋊C4—22no—
85C3×C4×Dic3D411yescharacteristic
86C3×C4×Dic3D411yescharacteristic
87C3×C4×Dic3C2311yescharacteristic
88C2×C3×C3⋊C8C2312yes—
89#3714—44no—
90#3713—44no—
91#3205—91noSylow
92C2×C3×#41—61no—
93C2×#239—21no—
94C2×D3×D4—32no—
95C2×D3×D4—61no—
96C2×D3×D4—61no—
97C22×D12—32no—
98C22×D12—61no—
99C22×D12—61no—
100C3×C4⋊Q8—21no—
101C3×C4⋊Q8C2×D311yescharacteristic, complemented
102C3×C4⋊Q8C2×D311yescharacteristic, complemented
103C3×#127—31no—
104C3×#127—31no—
105C3×#124—61no—
106C3×#121—32no—
107C3×#121—61no—
108C3×C4×D4—32no—
109C3×C4×D4—61no—
110C2×C3×C42—31no—
111#947—21no—
112#940—32no—
113#940—61no—
114#940—61no—
115C2×#213—34no—
116C2×#213—34no—
117C2×#213—34no—
118C2×D24—34no—
119#895—21no—
120#895C2×D312yes—
121#893—21no—
122#893C2×D312yescomplemented
123#874—32no—
124#874—31no—
125#874—31no—
126#874—61no—
127#874—61no—
128#874—61no—
129#874C2×D311yescharacteristic
130#874C2×D311yescharacteristic, complemented
131C42×D3C2×D311yescharacteristic
132C3×#102—34no—
133C3×C4×C8—31no—
134#830—32no—
135(C2×D3)⋊C8—32no—
136Dic3⋊C8—32no—
137#809—34no—
138#809—34no—
139#809—64no—
140#809C2×D314yescomplemented
141C4⋊(C3⋊C8)—32no—
142C4×C3⋊C8—31no—
143C4×C3⋊C8—31no—
144C4×C3⋊C8—61no—
145C4×C3⋊C8C2×D311yescharacteristic, complemented
146C22×C3⋊D3—22no—
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3C2×D411yescharacteristic
150C2×D32—44no—
151C2×D32—44no—
152C2×D32—44no—
153C32×Q8—42no—
154C2×C32×C4—22no—
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2411yescharacteristic
158C4⋊(C3⋊D3)—24no—
159C4⋊(C3⋊D3)—42no—
160C4⋊(C3⋊D3)—42no—
161C2×C3×Dic3C2×D411yescharacteristic
162C2×C3×Dic3C2×D411yescharacteristic
163C2×C3×Dic3C2×D411yescharacteristic
164C3×D12—24no—
165C3×D12—24no—
166C3×D12—44no—
167C3×D12—42no—
168C3×D12—42no—
169C3×D12—42no—
170C3×D12—42no—
171C3×C4×D3—22no—
172C3×C4×D3—22no—
173C3×C4×D3—44no—
174C3×C4×D3D814yescomplemented
175#626—44no—
176#626—44no—
177#626—44no—
178C3⋊(C3⋊C8)—22no—
179C3×C3⋊C8—22no—
180#532—31no—
181C2×#127—91no—
182#493—91no—
183#493—91no—
184#486—31no—
185#486—91no—
186#459—94no—
187#447—94no—
188#431—91no—
189C23×D3—62no—
190C23×D3—62no—
191C3×#41—124no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—41no—
195C2×C3×D4—32no—
196C2×C3×D4—32no—
197C2×C3×D4—62no—
198C2×C3×D4—61no—
199C2×C3×D4—61no—
200C2×C3×D4—61no—
201C2×C3×D4—61no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—62no—
206C2×#67—62no—
207C2×#67—62no—
208C2×#67—62no—
209#239—44no—
210D3×D4—124no—
211D3×D4—124no—
212D3×D4—124no—
213C2×D12—22no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—32no—
219C2×D12—32no—
220C2×D12—62no—
221C2×D12—62no—
222C2×D12—62no—
223C2×D12—62no—
224C2×D12—61no—
225C2×D12—61no—
226C2×D12—61no—
227C2×D12—61no—
228C2×D12—121no—
229C2×D12—121no—
230C2×D12—124no—
231C2×D12—124no—
232C2×D12—124no—
233C2×D12C22×D312yes—
234C2×D12C22×D312yes—
235C2×C4×D3—22no—
236C2×C4×D3#6711yescharacteristic
237C2×C4×D3#6711yescharacteristic
238C2×C4×D3C22×D311yescharacteristic
239C2×C3×C8—32no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4C22×D312yes—
245C3×C4⋊C4C22×D312yes—
246C3×C22⋊C4—62no—
247C3×C22⋊C4—62no—
248C3×C42—21no—
249C3×C42—31no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42C22×D311yescharacteristic
253C3×C42C22×D311yescharacteristic
254#213—124no—
255#213—124no—
256#213—124no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259C4×Dic3#6711yescharacteristic
260C4×Dic3#6711yescharacteristic
261C4×Dic3C22×D311yescharacteristic
262C2×C3⋊C8—32no—
263C2×C3⋊C8—32no—
264C2×C3⋊C8—62no—
265C2×C3⋊C8C22×D312yes—
266D24—124no—
267C22×C32C22×D411yescharacteristic
268C2×C3⋊D3—42no—
269C2×C3⋊D3—42no—
270C2×C3⋊D3—42no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—42no—
274C2×C3×D3—42no—
275C2×C3×D3—42no—
276C2×C3×D3—42no—
277C2×C3×D3—42no—
278C2×C3×D3—42no—
279C2×C3×D3C2×D812yes—
280D32—88no—
281C32×C4—42no—
282C32×C4C2×D812yes—
283C32×C4AGL(1,ℤ/8)12yes—
284C32×C4H(5,𝔽2)12yes—
285C3×Dic3—21no—
286C3×Dic3—21no—
287C3×Dic3C2×D812yes—
288C2×#41—61no—
289C22×D4—92no—
290C22×D4—181no—
291C22×D4—181no—
292C2×D8—94no—
293C2×D8—94no—
294C4⋊Q8D3211yescharacteristic, complemented
295#127—31no—
296#127—31no—
297#127—92no—
298#127—91no—
299#127—181no—
300#127—181no—
301#124—61no—
302#121—32no—
303#121—61no—
304C4×D4—32no—
305C4×D4—61no—
306C2×C42—31no—
307C4⋊C8—92no—
308#102—34no—
309#102—94no—
310C22⋊C8—92no—
311C4×C8—31no—
312C4×C8—91no—
313C23×C3—31no—
314C23×C3—62no—
315C23×C3—62no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—62no—
319C22×D3—62no—
320C22×D3—122no—
321C22×D3—124no—
322C22×D3—124no—
323C22×D3—124no—
324C22×D3—124no—
325C22×D3—124no—
326C22×D3—124no—
327C22×D3C2×#6711yescharacteristic
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—82no—
331C3×D4—64no—
332C3×D4—64no—
333C3×D4—124no—
334C3×D4—122no—
335C3×D4—122no—
336C3×D4—122no—
337C3×D4—122no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—31no—
344C2×C3×C4—34no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—42no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4—124no—
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4C2×#6711yescharacteristic
354C2×C3×C4C2×#6711yescharacteristic
355C2×C3×C4C23×D311yescharacteristic
356C2×C3×C4C23×D311yescharacteristic
357#67—124no—
358#67—124no—
359#67—124no—
360C2×Dic3C2×#6711yescharacteristic
361C2×Dic3C2×#6711yescharacteristic
362C2×Dic3C2×#6711yescharacteristic
363D12—24no—
364D12—24no—
365D12—44no—
366D12—42no—
367D12—42no—
368D12—42no—
369D12—42no—
370D12—64no—
371D12—64no—
372D12—124no—
373D12—124no—
374D12—124no—
375D12—124no—
376D12—122no—
377D12—122no—
378D12—122no—
379D12—122no—
380D12—242no—
381D12—242no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—44no—
385C4×D3#21314yescomplemented
386C3×C8—62no—
387C3⋊C8—22no—
388C3⋊C8—62no—
389C3⋊C8—62no—
390C3⋊C8—122no—
391C2×C32#53511yescharacteristic
392C2×C32C22×D811yescharacteristic
393C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
394C3⋊D3—82no—
395C3×D3—22no—
396C3×D3—82no—
397C3×D3—82no—
398C24—182no—
399#41—124no—
400C2×Q8—21no—
401C2×D4—32no—
402C2×D4—32no—
403C2×D4—62no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—92no—
409C2×D4—182no—
410C2×D4—182no—
411C2×D4—182no—
412C2×D4—181no—
413C2×D4—181no—
414C2×D4—364no—
415C2×D4—364no—
416C2×D4—364no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—62no—
421D8—364no—
422D8—364no—
423C2×C8—32no—
424C2×C8—92no—
425C4⋊C4—21no—
426C4⋊C4C2×D3212yes—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C42—31no—
430C42—31no—
431C42—31no—
432C42C2×D3211yescharacteristic
433C22×C3—21no—
434C22×C3—32no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—122no—
440C22×C3—122no—
441C22×C3—122no—
442C22×C3—122no—
443C22×C3C2×D3×D411yescharacteristic
444C22×C3C22×#6711yescharacteristic
445C2×D3—21no—
446C2×D3—21no—
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—122no—
454C2×D3—122no—
455C2×D3—122no—
456C2×D3—122no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—242no—
460C2×D3—242no—
461C2×D3—242no—
462C2×D3—248no—
463C2×D3—248no—
464C2×D3C2×#21312yes—
465C3×C4—22no—
466C3×C4—22no—
467C3×C4—22no—
468C3×C4—32no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—82no—
474C3×C4D3×D812yes—
475C3×C4#91412yes—
476C3×C4C2×#21312yes—
477C3×C4#94912yes—
478C3×C4#100912yes—
479C3×C4#100912yes—
480Dic3—21no—
481Dic3—21no—
482Dic3C2×#21312yes—
483C32#320511yescharacteristic, complemented, Sylow
484C23—31no—
485C23—62no—
486C23—62no—
487C23—182no—
488C23—364no—
489C23—364no—
490C23—364no—
491Q8—42no—
492D4—64no—
493D4—64no—
494D4—124no—
495D4—122no—
496D4—122no—
497D4—122no—
498D4—122no—
499D4—184no—
500D4—364no—
501D4—364no—
502D4—364no—
503D4—362no—
504D4—362no—
505C2×C4—22no—
506C2×C4—31no—
507C2×C4—34no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—124no—
513C2×C4D3×#6711yescharacteristic
514C2×C4D3×#6711yescharacteristic
515C2×C4C22×D3211yescharacteristic
516C8—62no—
517C8—182no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—62no—
522C2×C3—242no—
523C2×C3—242no—
524C2×C3#567411yescharacteristic
525C2×C3C2×D3×D811yescharacteristic
526C2×C3C2×#91411yescharacteristic
527C2×C3C22×#21311yescharacteristic
528C2×C3C2×#94911yescharacteristic
529C2×C3#583011yescharacteristic
530D3—22no—
531D3—82no—
532D3—82no—
533D3—242no—
534D3—242no—
535D3—482no—
536C22—32no—
537C22—61no—
538C22—61no—
539C22—122no—
540C22—122no—
541C22—122no—
542C22—122no—
543C22—122no—
544C22—122no—
545C22—362no—
546C22—362no—
547C22—362no—
548C22—728no—
549C22C2×D3×#6711yescharacteristic
550C4—32no—
551C4—42no—
552C4—61no—
553C4—61no—
554C4D3×#21312yes—
555C4#6393912yes—
556C4#6432012yes—
557C3—21no—
558C3#8156611yescharacteristic, complemented
559C3#8330211yescharacteristic, complemented
560C2—62no—
561C2—242no—
562C2—242no—
563C2—722no—
564C2C2×D3×#21311yescharacteristic
565C2C2×#6393911yescharacteristic
566C2#1059542611yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
hypercenterC4⋊Q8D32
hypocenterC32#3205
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×#5725C2semidirect
2. D3×#874C2semidirect
3. #10592787C2semidirect
4. C3×#5088C2semidirect
5. #10590595C2semidirect
6. #10590547C2semidirect
7. #10590352C2semidirect
8. #10590350C2semidirect
9. #10590261C2semidirect
16. C32×C4⋊Q8C22semidirect
22. C3×#893C22semidirect
24. C3×#874C22semidirect
29. C2×#3714C22semidirect
30. C2×#3713C22semidirect
31. #63636C22semidirect
32. C4×C3⋊(C3⋊C8)C22semidirect
33. C3×#809C22semidirect
34. C3×C4×C3⋊C8C22semidirect
35. Dic3⋊(C3⋊C8)C22semidirect
38. #5725D3semidirect
45. #5088D3semidirect
101. C3×C4⋊Q8C2×D3semidirect
102. C3×C4⋊Q8C2×D3semidirect
122. #893C2×D3semidirect
130. #874C2×D3semidirect
140. #809C2×D3semidirect
145. C4×C3⋊C8C2×D3semidirect
174. C3×C4×D3D8semidirect
294. C4⋊Q8D32semidirect
385. C4×D3#213semidirect
483. C32#3205semidirect
558. C3#81566semidirect
559. C3#83302semidirect

Maximal subgroups

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

subgroupindex
C3×#57252
D3×#8742
#105927872
C3×#50882
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