Finite Group Databasebeta

← #13147906

Subgroups — #13147906

Order 1728.

1592 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88655C212yesmaximal, complemented
2C2×#88654C212yesmaximal, complemented
3C4×C4⋊#272C211yesmaximal, characteristic, complemented
4#11721385C211yesmaximal, characteristic, complemented
5C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
6#11721162C214yesmaximal, complemented
7#11721159C212yesmaximal, complemented
8#11721141C212yesmaximal, complemented
9#10594004—31nomaximal
10D122—31nomaximal
11C22×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
12C2×C4⋊#272C2211yescharacteristic
13C2×C4×#272C2212yescomplemented
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
16C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
17C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
18C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
19C2×#6200C2214yescomplemented
20C2×#6198C2214yescomplemented
21#88655—28no—
22#88654—28no—
23C42×H(3,𝔽3)C2211yescharacteristic, complemented
24#88542C2212yescomplemented
25C4⋊#1084C2211yescharacteristic, complemented
26#3752⋊C3C2212yescomplemented
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
28#88450C2214yescomplemented
29C2×#3802—32no—
30C2×D3×D12—32no—
31C2×D3×D12—32no—
32C2×D3×D12—32no—
33#64083D311yescharacteristic, complemented
34C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
35C3×#874—31no—
36C3×C4×D12—31no—
37C3×C4×D12—31no—
38C3×C4×D12—31no—
39C3×C4×D12—61no—
40#63923—34no—
41#63922—34no—
42#63916—32no—
43#63909—32no—
44#63909—32no—
45#63908—32no—
46C22×#272C2312yes—
47C22×(C3⋊D3)⋊C3C2314yes—
48C22×(C3⋊D3)⋊C3C2312yes—
49C2×((C3⋊D3)⋊C3)⋊C2—28no—
50C2×((C3⋊D3)⋊C3)⋊C2—28no—
51C2×((C3⋊D3)⋊C3)⋊C2—44no—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×#1084C2312yes—
56C4⋊#272D414yescomplemented
57C4×#272—24no—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C4⋊((C3⋊D3)⋊C3)—24no—
60C4⋊((C3⋊D3)⋊C3)—24no—
61C4⋊((C3⋊D3)⋊C3)—42no—
62C4⋊((C3⋊D3)⋊C3)D414yescomplemented
63C4×(C3⋊D3)⋊C3—24no—
64#6200—28no—
65#6198—28no—
66#5717—91no—
67D4×D12—91no—
68D4×D12—91no—
69C22×D32—34no—
70C22×D32—34no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×C4⋊(C3⋊D3)C2×D312yescomplemented
73C2×C4⋊(C3⋊D3)C2×D312yescomplemented
74C2×C4⋊(C3⋊D3)C2×D312yescomplemented
75C2×C4×C3⋊D3C2×D312yescomplemented
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—32no—
80C2×C3×D12—32no—
81C2×C3×D12—32no—
82C2×C3×D12—61no—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—62no—
87C2×#626—34no—
88C2×#626—34no—
89C2×#626—34no—
90C2×#625—34no—
91#3802—68no—
92D3×D12—68no—
93D3×D12—68no—
94D3×D12—68no—
95C32×C42—21no—
96C32×C42C2×D311yescharacteristic, complemented
97C32×C42C2×D311yescharacteristic, complemented
98#3752C2×D312yescomplemented
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—62no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—61no—
108#3722—34no—
109(C2×D3)⋊Dic3—34no—
110C22×H(3,𝔽3)C2411yescharacteristic
111C2×#272—22no—
112C2×#272—22no—
113C2×#272C2×D414yescomplemented
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3—22no—
116C2×(C3⋊D3)⋊C3—24no—
117C2×(C3⋊D3)⋊C3—24no—
118C2×(C3⋊D3)⋊C3—24no—
119C2×(C3⋊D3)⋊C3C2×D414yescomplemented
120((C3⋊D3)⋊C3)⋊C2—416no—
121C4×H(3,𝔽3)—21no—
122C4×H(3,𝔽3)C2×D412yescomplemented
123C4×H(3,𝔽3)C2×D412yescomplemented
124#1084—22no—
125(C3⋊Dic3)⋊C3—22no—
126C2×D3×D4—92no—
127C2×D3×D4—92no—
128C2×D3×D4—92no—
129C2×D3×D4—92no—
130C22×D12—92no—
131C22×D12—92no—
132C3×#127—91no—
133C3×C4×D4—91no—
134C3×C4×D4—91no—
135#938—92no—
136#938—92no—
137#937—94no—
138#930—92no—
139#930—92no—
140#895—92no—
141#895—92no—
142#882—94no—
143#882—94no—
144#874—31no—
145#874—91no—
146C4×D12—31no—
147C4×D12—31no—
148C4×D12—31no—
149C4×D12—61no—
150C4×D12—91no—
151C22×C3⋊D3C22×D314yes—
152C22×C3⋊D3C22×D312yes—
153C22×C3×D3—32no—
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—34no—
157C22×C3×D3—62no—
158C2×D32—68no—
159C2×D32—68no—
160C2×D32—68no—
161C2×D32—68no—
162C2×D32—124no—
163C2×D32—124no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3C22×D312yes—
174C4⋊(C3⋊D3)—24no—
175C4⋊(C3⋊D3)—24no—
176C4⋊(C3⋊D3)—42no—
177C4⋊(C3⋊D3)D1214yescomplemented
178C4×C3⋊D3—24no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—62no—
183C3×D12—34no—
184C3×D12—34no—
185C3×D12—34no—
186C3×D12—64no—
187C3×D12—64no—
188C3×D12—64no—
189C3×D12—122no—
190C3×C4×D3—64no—
191C3×C4×D3—64no—
192C3×C4×D3—64no—
193C3×C4×D3—124no—
194#626—68no—
195#626—68no—
196#626—68no—
197#625—68no—
198D42—271noSylow
199C2×H(3,𝔽3)C22×D411yescharacteristic
200C2×H(3,𝔽3)C22×D411yescharacteristic
201C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
202#272—24no—
203(C3⋊D3)⋊C3—24no—
204(C3⋊D3)⋊C3—44no—
205C23×D3—94no—
206C23×D3—94no—
207C23×D3—94no—
208C2×C3×D4—91no—
209C2×C3×D4—92no—
210C2×C3×D4—92no—
211C2×C3×D4—92no—
212C2×C3×D4—91no—
213C22×C3×C4—92no—
214C22×C3×C4—92no—
215C2×#67—94no—
216C2×#67—94no—
217C2×#67—94no—
218C2×#67—94no—
219D3×D4—188no—
220D3×D4—188no—
221D3×D4—188no—
222D3×D4—188no—
223C2×D12—31no—
224C2×D12—31no—
225C2×D12—31no—
226C2×D12—32no—
227C2×D12—32no—
228C2×D12—32no—
229C2×D12—61no—
230C2×D12—91no—
231C2×D12—92no—
232C2×D12—92no—
233C2×D12—92no—
234C2×D12—94no—
235C2×D12—94no—
236C2×D12—188no—
237C2×D12—188no—
238C2×C4×D3—32no—
239C2×C4×D3—32no—
240C2×C4×D3—32no—
241C2×C4×D3—62no—
242C2×C4×D3—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C42—31no—
248C3×C42—31no—
249C3×C42—61no—
250C3×C42D3211yescharacteristic
251#217—94no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—62no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—94no—
258(C2×D3)⋊C4—94no—
259C4⋊Dic3—31no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—61no—
263C4⋊Dic3—91no—
264C22×C32—21no—
265C22×C32C23×D311yescharacteristic
266C22×C32C23×D311yescharacteristic
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—24no—
270C2×C3⋊D3—24no—
271C2×C3⋊D3—24no—
272C2×C3⋊D3C2×D1214yescomplemented
273C2×C3×D3—34no—
274C2×C3×D3—34no—
275C2×C3×D3—34no—
276C2×C3×D3—64no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C2×C3×D3—64no—
286C2×C3×D3—122no—
287C2×C3×D3—122no—
288D32—1216no—
289D32—1216no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4—41no—
295C32×C4C2×D1212yescomplemented
296C32×C4D3×D412yescomplemented
297C32×C4D3×D412yescomplemented
298C32×C4D3×D412yescomplemented
299C3⋊Dic3—22no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—122no—
304C22×D4—272no—
305C22×D4—272no—
306#127—91no—
307#121—272no—
308#121—272no—
309#120—274no—
310C4×D4—91no—
311C4×D4—91no—
312H(3,𝔽3)D4211yescharacteristic, complemented, Sylow
313C23×C3—92no—
314C23×C3—94no—
315C23×C3—92no—
316C22×D3—32no—
317C22×D3—32no—
318C22×D3—32no—
319C22×D3—34no—
320C22×D3—62no—
321C22×D3—92no—
322C22×D3—94no—
323C22×D3—188no—
324C22×D3—188no—
325C22×D3—188no—
326C22×D3—188no—
327C22×D3—188no—
328C22×D3—188no—
329C22×D3—364no—
330C22×D3—364no—
331C22×D3—364no—
332C3×D4—94no—
333C3×D4—94no—
334C3×D4—184no—
335C3×D4—184no—
336C3×D4—362no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—92no—
347C2×C3×C4—92no—
348C2×C3×C4—184no—
349C2×C3×C4—184no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353#67—188no—
354#67—188no—
355#67—188no—
356#67—188no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—62no—
361C2×Dic3—92no—
362D12—34no—
363D12—34no—
364D12—34no—
365D12—64no—
366D12—64no—
367D12—64no—
368D12—94no—
369D12—122no—
370D12—184no—
371D12—184no—
372D12—188no—
373D12—188no—
374D12—362no—
375C4×D3—64no—
376C4×D3—64no—
377C4×D3—64no—
378C4×D3—124no—
379C4×D3—184no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32C22×D1211yescharacteristic
384C2×C32C2×D3×D411yescharacteristic
385C2×C32C2×D3×D411yescharacteristic
386C2×C32C2×D3×D411yescharacteristic
387C2×C32#100411yescharacteristic
388C2×C32#100911yescharacteristic
389C3⋊D3—24no—
390C3⋊D3—44no—
391C3×D3—64no—
392C3×D3—64no—
393C3×D3—64no—
394C3×D3—124no—
395C3×D3—124no—
396C24—274no—
397C2×D4—91no—
398C2×D4—91no—
399C2×D4—92no—
400C2×D4—92no—
401C2×D4—92no—
402C2×D4—274no—
403C2×D4—274no—
404C2×D4—548no—
405C2×D4—548no—
406C22×C4—92no—
407C22×C4—92no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C22⋊C4—274no—
413C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
414C22×C3—31no—
415C22×C3—31no—
416C22×C3—61no—
417C22×C3—94no—
418C22×C3—94no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3—184no—
422C22×C3—184no—
423C22×C3—184no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3C22×D3211yescharacteristic
427C2×D3—34no—
428C2×D3—34no—
429C2×D3—34no—
430C2×D3—64no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—64no—
438C2×D3—64no—
439C2×D3—64no—
440C2×D3—94no—
441C2×D3—122no—
442C2×D3—122no—
443C2×D3—182no—
444C2×D3—182no—
445C2×D3—184no—
446C2×D3—184no—
447C2×D3—184no—
448C2×D3—3616no—
449C2×D3—3616no—
450C2×D3—3616no—
451C3×C4—21no—
452C3×C4—32no—
453C3×C4—32no—
454C3×C4—32no—
455C3×C4—32no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—121no—
461C3×C4—182no—
462C3×C4—182no—
463C3×C4D3×D1212yes—
464C3×C4#380212yes—
465Dic3—62no—
466Dic3—62no—
467Dic3—62no—
468Dic3—122no—
469Dic3—182no—
470C32—21no—
471C32D4×D1211yescharacteristic, complemented
472C32#571711yescharacteristic, complemented
473C23—92no—
474C23—92no—
475C23—94no—
476C23—548no—
477C23—548no—
478C23—1084no—
479D4—94no—
480D4—94no—
481D4—184no—
482D4—184no—
483D4—362no—
484D4—548no—
485D4—548no—
486C2×C4—92no—
487C2×C4—92no—
488C2×C4—184no—
489C2×C4—184no—
490C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
491C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
492C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
493C2×C3—31no—
494C2×C3—31no—
495C2×C3—31no—
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—61no—
500C2×C3—61no—
501C2×C3—61no—
502C2×C3—184no—
503C2×C3—184no—
504C2×C3—364no—
505C2×C3C2×D3×D1211yescharacteristic
506C2×C3C2×#380211yescharacteristic
507C2×C3#6430711yescharacteristic
508D3—64no—
509D3—64no—
510D3—64no—
511D3—124no—
512D3—124no—
513D3—184no—
514D3—364no—
515C22—94no—
516C22—94no—
517C22—182no—
518C22—182no—
519C22—182no—
520C22—182no—
521C22—184no—
522C22—184no—
523C22—184no—
524C22—10816no—
525C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526C4—21no—
527C4—182no—
528C4—182no—
529C4#8865412yescomplemented
530C4#8865512yescomplemented
531C3—31no—
532C3—31no—
533C3—61no—
534C3#1059400411yescharacteristic
535C2—184no—
536C2—184no—
537C2—364no—
538C2C2×#8865411yescharacteristic
539C2C2×#8865511yescharacteristic
540C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)D42
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88655C2semidirect
2. C2×#88654C2semidirect
3. C4×C4⋊#272C2semidirect
4. #11721385C2semidirect
5. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
6. #11721162C2semidirect
7. #11721159C2semidirect
8. #11721141C2semidirect
11. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
13. C2×C4×#272C22semidirect
15. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
18. C2×C4×(C3⋊D3)⋊C3C22semidirect
19. C2×#6200C22semidirect
20. C2×#6198C22semidirect
23. C42×H(3,𝔽3)C22semidirect
24. #88542C22semidirect
25. C4⋊#1084C22semidirect
26. #3752⋊C3C22semidirect
27. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
28. #88450C22semidirect
33. #64083D3semidirect
34. C4×C4⋊(C3⋊D3)D3semidirect
56. C4⋊#272D4semidirect
62. C4⋊((C3⋊D3)⋊C3)D4semidirect
72. C2×C4⋊(C3⋊D3)C2×D3semidirect
73. C2×C4⋊(C3⋊D3)C2×D3semidirect
74. C2×C4⋊(C3⋊D3)C2×D3semidirect
75. C2×C4×C3⋊D3C2×D3semidirect
96. C32×C42C2×D3semidirect
97. C32×C42C2×D3semidirect
98. #3752C2×D3semidirect
99. C4⋊(C3⋊Dic3)C2×D3semidirect
113. C2×#272C2×D4semidirect
119. C2×(C3⋊D3)⋊C3C2×D4semidirect
122. C4×H(3,𝔽3)C2×D4semidirect
123. C4×H(3,𝔽3)C2×D4semidirect
177. C4⋊(C3⋊D3)D12semidirect
272. C2×C3⋊D3C2×D12semidirect
295. C32×C4C2×D12semidirect
296. C32×C4D3×D4semidirect
297. C32×C4D3×D4semidirect
298. C32×C4D3×D4semidirect
312. H(3,𝔽3)D42semidirect
413. C42((C3⋊D3)⋊C3)⋊C2semidirect
471. C32D4×D12semidirect
472. C32#5717semidirect
529. C4#88654semidirect
530. C4#88655semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886542
C4×C4⋊#2722
#117213852
C4×C4⋊((C3⋊D3)⋊C3)2
#117211622
#117211592
#117211412
#105940043
D1223