Finite Group Databasebeta

← #11857239

Subgroups — #11857239

Order 1152.

1193 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×M16C211yesmaximal, characteristic
2D3×#926C211yesmaximal, characteristic
3C22×#3711C211yesmaximal, characteristic
4#10592821C211yesmaximal, characteristic, complemented
5C3×#5122C211yesmaximal, characteristic, complemented
6#10591059C211yesmaximal, characteristic, complemented
7#10591057C211yesmaximal, characteristic, complemented
8#10590693C212yesmaximal, complemented
9#10590692C212yesmaximal, complemented
10#10590667C212yesmaximal, complemented
11Dic3×ΓL(1,𝔽25)C212yesmaximal, complemented
12#83747—31nomaximal
13#78548—31nomaximal
14C22×C3×C4×D3C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16#64135C2211yescharacteristic
17C22×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#926C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C3×D3×M16C418yescomplemented
22C2×C3×ΓL(1,𝔽25)C2212yes—
23C2×C3×D3×C8C2212yes—
24D3×#217C412yes—
25#63955C412yes—
26D3×C4⋊Dic3C412yes—
27C4×D3×Dic3C2212yes—
28#63855C2212yes—
29#63841C412yes—
30C2×#3711C418yescomplemented
31C2×#3711C2211yescharacteristic
32C2×#3711C2211yescharacteristic
33C2×#3711C2211yescharacteristic
34C2×#3711C2211yescharacteristic
35#63642C2212yescomplemented
36C8×C3⋊Dic3C2212yescomplemented
37C3×#815C2212yescomplemented
38C3×C8×Dic3C2212yescomplemented
39#63555C2212yescomplemented
40#63554C2212yescomplemented
41#63553C2212yescomplemented
42Dic3×C3⋊C8C2212yescomplemented
43C22×C3×M16—31no—
44C2×#926—31no—
45C22×#208—31no—
46C2×D3×M16D311yescharacteristic
47C22×ΓL(1,𝔽25)—31no—
48D3×#117—31no—
49C3×#405—31no—
50#5122—31no—
51#5122—31no—
52#5122—31no—
53#5122—31no—
54#5122—61no—
55#5122D311yescharacteristic, complemented
56Dic3×M16—32no—
57Dic3×M16—32no—
58#5101—31no—
59#5101—32no—
60#5101—32no—
61#4997—31no—
62#4708—32no—
63#4707—32no—
64#4689—32no—
65C4×ΓL(1,𝔽25)—32no—
66C23×C3×D3C2×C411yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C3×Dic3C2×C411yescharacteristic
69C2×C3×C4×D3C2×C411yescharacteristic
70C2×C3×C4×D3C2×C411yescharacteristic
71C2×C3×C4×D3C2×C414yes—
72C2×C3×C4×D3C2×C414yes—
73C2×C3×C4×D3C2311yescharacteristic
74C2×C3×C4×D3C2311yescharacteristic
75C2×D3×Dic3C2×C414yes—
76C32×M16C2×C414yescomplemented
77C2×C32×C8C2312yes—
78#3757C2×C412yes—
79C4⋊(C3⋊Dic3)C2×C412yes—
80C4×C3⋊Dic3C2312yes—
81C2×C3⋊(C3⋊C8)C2×C414yescomplemented
82C2×C3⋊(C3⋊C8)C2311yescharacteristic
83C2×C3⋊(C3⋊C8)C2311yescharacteristic
84C3×#217C2×C412yes—
85C3×C4⋊Dic3C2×C412yes—
86C3×C4×Dic3C2312yes—
87C3×#208C2×C414yescomplemented
88C2×C3×C3⋊C8C2×C414yescomplemented
89C2×C3×C3⋊C8C2312yes—
90C2×C3×C3⋊C8C2311yescharacteristic
91C2×C3×C3⋊C8C2311yescharacteristic
92C3×ΓL(1,𝔽25)C2×C418yes—
93C3×D3×C8C2×C418yescomplemented
94Dic3⋊Dic3C2×C414yes—
95(C2×D3)⋊Dic3C2×C414yes—
96Dic32C2×C414yes—
97#3711C2×C414yes—
98#3711C2×C414yes—
99#3711C2×C414yescomplemented
100#3711C2×C414yescomplemented
101#2873—91noSylow
102C23×C3×C4—31no—
103C22×C4×D3C2×D311yescharacteristic
104C2×C3×M16—21no—
105C2×C3×M16—31no—
106C2×C3×M16—38no—
107C2×C3×M16—32no—
108C2×C3×M16C2×D311yescharacteristic
109C2×C3×M16C2×D311yescharacteristic
110C22×C3×C8—31no—
111C22×C3×C8—32no—
112C3×#117—31no—
113C2×#217—32no—
114#926—31no—
115#926—31no—
116#926—32no—
117#926—32no—
118#926—32no—
119#926—61no—
120#926C2×D311yescharacteristic
121C2×C4⋊Dic3—32no—
122C2×C4×Dic3—32no—
123C2×#208—38no—
124C2×#208—31no—
125C2×#208—31no—
126C2×#208—31no—
127C2×#208—31no—
128C2×#208C2×D311yescharacteristic
129C22×C3⋊C8—31no—
130C22×C3⋊C8—31no—
131C22×C3⋊C8—61no—
132C22×C3⋊C8C2×D311yescharacteristic
133D3×M16Dic318yescomplemented
134C2×ΓL(1,𝔽25)—38no—
135C2×ΓL(1,𝔽25)—31no—
136C2×ΓL(1,𝔽25)—31no—
137C2×ΓL(1,𝔽25)—31no—
138C2×ΓL(1,𝔽25)—31no—
139C2×ΓL(1,𝔽25)C2×D312yes—
140C2×D3×C8C2×D312yes—
141#892—32no—
142D3×C4⋊C4—32no—
143D3×C22⋊C4—32no—
144#877—32no—
145#872—32no—
146C42×D3—32no—
147C3×#97—32no—
148C3×C4×C8—32no—
149#815—32no—
150#815—32no—
151#815—32no—
152#815—32no—
153#815—32no—
154#815—62no—
155#815C2×D312yescomplemented
156C8×Dic3—32no—
157C8×Dic3—32no—
158C8×Dic3—32no—
159C8×Dic3—32no—
160C8×Dic3—32no—
161C8×Dic3—62no—
162C8×Dic3C2×D312yescomplemented
163#803—32no—
164C4×C3⋊C8—32no—
165C23×C32C22×C411yescharacteristic
166C22×C3×D3—42no—
167C22×C3×D3C4214yes—
168C22×C3×D3C22×C411yescharacteristic
169C22×C3×D3C22×C411yescharacteristic
170C2×C32×C4C22×C411yescharacteristic
171C2×C32×C4C22×C412yes—
172C2×C32×C4C22×C412yes—
173C2×C32×C4C2411yescharacteristic
174C2×C3⋊Dic3C22×C414yes—
175C2×C3×Dic3C4214yes—
176C2×C3×Dic3C22×C414yes—
177C2×C3×Dic3C22×C411yescharacteristic
178C2×C3×Dic3C22×C411yescharacteristic
179C3×C4×D3C4214yes—
180C3×C4×D3C4214yes—
181C3×C4×D3C22×C412yes—
182C3×C4×D3C22×C412yes—
183C3×C4×D3C22×C412yes—
184C3×C4×D3C22×C412yes—
185D3×Dic3—44no—
186C32×C8C22×C414yes—
187C3⋊(C3⋊C8)C22×C412yes—
188C3⋊(C3⋊C8)C22×C412yes—
189C3×C3⋊C8C22×C412yes—
190C3×C3⋊C8C22×C412yes—
191C3×C3⋊C8C22×C414yes—
192C22×M16—31no—
193C22×M16—91no—
194C2×#117—91no—
195#405—31no—
196#405—92no—
197#405—92no—
198#405—91no—
199#405—91no—
200#405—91no—
201C4×M16—92no—
202C4×M16—92no—
203C24×C3—31no—
204C23×D3C4×D311yescharacteristic
205C22×C3×C4—21no—
206C22×C3×C4—31no—
207C22×C3×C4—34no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—34no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D311yescharacteristic
215C22×Dic3—34no—
216C22×Dic3C4×D311yescharacteristic
217C2×C4×D3—34no—
218C2×C4×D3C4×D311yescharacteristic
219C2×C4×D3C4×D311yescharacteristic
220C2×C4×D3C4×D314yes—
221C2×C4×D3C2×Dic314yes—
222C2×C4×D3C22×D311yescharacteristic
223C2×C4×D3C22×D311yescharacteristic
224C3×M16—24no—
225C3×M16—34no—
226C3×M16—38no—
227C3×M16C4×D314yescomplemented
228C3×M16C2×Dic314yescomplemented
229C2×C3×C8—22no—
230C2×C3×C8—34no—
231C2×C3×C8—32no—
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—38no—
235C2×C3×C8C22×D312yes—
236C2×C3×C8C22×D312yes—
237C3×C4⋊C4—32no—
238C3×C22⋊C4—32no—
239C3×C42—32no—
240#217—32no—
241#217—32no—
242#217—34no—
243#217—62no—
244#217C4×D312yes—
245(C2×D3)⋊C4—34no—
246C4⋊Dic3—32no—
247C4⋊Dic3—32no—
248C4⋊Dic3—34no—
249C4⋊Dic3—62no—
250C4⋊Dic3C4×D312yes—
251Dic3⋊C4—34no—
252C4×Dic3—32no—
253C4×Dic3—32no—
254C4×Dic3—34no—
255C4×Dic3—34no—
256C4×Dic3—62no—
257C4×Dic3C22×D312yes—
258#208—34no—
259#208—34no—
260#208—34no—
261#208—34no—
262#208C2×Dic314yescomplemented
263C2×C3⋊C8—34no—
264C2×C3⋊C8—34no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—31no—
269C2×C3⋊C8—61no—
270C2×C3⋊C8—61no—
271C2×C3⋊C8—64no—
272C2×C3⋊C8C4×D314yescomplemented
273C2×C3⋊C8C22×D311yescharacteristic
274C2×C3⋊C8C22×D311yescharacteristic
275C2×C3⋊C8C22×D312yes—
276ΓL(1,𝔽25)—34no—
277ΓL(1,𝔽25)—34no—
278ΓL(1,𝔽25)—34no—
279ΓL(1,𝔽25)—34no—
280ΓL(1,𝔽25)C2×Dic318yes—
281D3×C8C2×Dic318yescomplemented
282C22×C32—22no—
283C22×C32C2×C4212yes—
284C22×C32C23×C411yescharacteristic
285C2×C3×D3—22no—
286C2×C3×D3—22no—
287C2×C3×D3—44no—
288C2×C3×D3C2×C4212yes—
289C2×C3×D3C2×C4212yes—
290C32×C4C2×C4212yes—
291C32×C4C23×C411yescharacteristic
292C32×C4C23×C411yescharacteristic
293C3⋊Dic3—24no—
294C3×Dic3—24no—
295C3×Dic3C2×C4212yes—
296C3×Dic3C2×C4212yes—
297C23×C4—31no—
298C2×M16—31no—
299C2×M16—38no—
300C2×M16—32no—
301C2×M16—91no—
302C2×M16—91no—
303C2×M16—91no—
304C2×M16—91no—
305C2×M16—98no—
306C2×M16D3211yescharacteristic
307C22×C8—31no—
308C22×C8—32no—
309C22×C8—91no—
310#117—31no—
311#117—92no—
312#117—92no—
313#117—92no—
314#117—91no—
315C2×C4⋊C4—92no—
316C2×C22⋊C4—92no—
317C2×C42—92no—
318#97—32no—
319#97—92no—
320#97—92no—
321#97—92no—
322C4×C8—32no—
323C4×C8—92no—
324C4×C8—92no—
325C4×C8—92no—
326C23×C3—21no—
327C23×C3—31no—
328C23×C3—31no—
329C23×C3—34no—
330C23×C3—122no—
331C23×C3C2×C4×D311yescharacteristic
332C23×C3C2×C4×D311yescharacteristic
333C22×D3—42no—
334C22×D3C4×Dic314yes—
335C22×D3C2×C4×D311yescharacteristic
336C22×D3C2×C4×D311yescharacteristic
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—34no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—34no—
345C2×C3×C4—34no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—34no—
351C2×C3×C4C2×C4×D311yescharacteristic
352C2×C3×C4C2×C4×D312yes—
353C2×C3×C4C2×C4×D311yescharacteristic
354C2×C3×C4C2×C4×D312yes—
355C2×C3×C4C2×C4×D312yes—
356C2×C3×C4C22×Dic312yes—
357C2×C3×C4C23×D311yescharacteristic
358C2×C3×C4C23×D311yescharacteristic
359C2×Dic3—34no—
360C2×Dic3—34no—
361C2×Dic3—64no—
362C2×Dic3—124no—
363C2×Dic3C4×Dic314yes—
364C2×Dic3C2×C4×D314yes—
365C2×Dic3C2×C4×D311yescharacteristic
366C2×Dic3C2×C4×D311yescharacteristic
367C4×D3—124no—
368C4×D3C4×Dic314yes—
369C4×D3C4×Dic314yes—
370C4×D3C2×C4×D312yes—
371C4×D3C2×C4×D312yes—
372C4×D3C22×Dic312yes—
373C4×D3C22×Dic312yes—
374C3×C8—24no—
375C3×C8—32no—
376C3×C8—32no—
377C3×C8—34no—
378C3×C8C2×C4×D314yes—
379C3×C8C22×Dic314yes—
380C3⋊C8—32no—
381C3⋊C8—32no—
382C3⋊C8—32no—
383C3⋊C8—32no—
384C3⋊C8—62no—
385C3⋊C8—62no—
386C3⋊C8C2×C4×D312yes—
387C3⋊C8C2×C4×D312yes—
388C3⋊C8C22×Dic314yes—
389C2×C32—22no—
390C2×C32C22×C4211yescharacteristic
391C2×C32#56812yes—
392C3×D3—22no—
393C3×D3—22no—
394C24—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—34no—
401C22×C4—34no—
402C22×C4—94no—
403C22×C4C2×D3211yescharacteristic
404M16—34no—
405M16—38no—
406M16—94no—
407M16—94no—
408M16—94no—
409M16—94no—
410M16D3×Dic314yescomplemented
411C2×C8—31no—
412C2×C8—31no—
413C2×C8—32no—
414C2×C8—34no—
415C2×C8—38no—
416C2×C8—91no—
417C2×C8—91no—
418C2×C8—94no—
419C2×C8C2×D3212yes—
420C4⋊C4—32no—
421C4⋊C4—94no—
422C4⋊C4—92no—
423C22⋊C4—32no—
424C22⋊C4—94no—
425C22⋊C4—92no—
426C42—32no—
427C42—94no—
428C42—92no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—32no—
434C22×C3—32no—
435C22×C3—42no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—124no—
439C22×C3C42×D312yes—
440C22×C3C2×C4×Dic312yes—
441C22×C3C22×C4×D311yescharacteristic
442C22×C3C22×C4×D311yescharacteristic
443C2×D3—22no—
444C2×D3—22no—
445C2×D3—44no—
446C2×D3C2×C4×Dic312yes—
447C2×D3C2×C4×Dic312yes—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—22no—
451C3×C4—32no—
452C3×C4—32no—
453C3×C4—64no—
454C3×C4C42×D312yes—
455C3×C4C2×C4×Dic312yes—
456C3×C4C22×C4×D311yescharacteristic
457C3×C4C22×C4×D311yescharacteristic
458C3×C4C22×C4×D311yescharacteristic
459C3×C4C23×Dic311yescharacteristic
460Dic3—24no—
461Dic3—64no—
462Dic3—64no—
463Dic3—124no—
464Dic3C2×C4×Dic312yes—
465Dic3C2×C4×Dic312yes—
466C32#287311yescharacteristic, complemented, Sylow
467C23—31no—
468C23—31no—
469C23—34no—
470C23—122no—
471C23C4×D3211yescharacteristic
472C2×C4—31no—
473C2×C4—31no—
474C2×C4—34no—
475C2×C4—32no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—34no—
480C2×C4—34no—
481C2×C4—34no—
482C2×C4—94no—
483C2×C4—364no—
484C2×C4C4×D3212yes—
485C2×C4C4×D3211yescharacteristic
486C2×C4C2×D3×Dic312yes—
487C2×C4C22×D3211yescharacteristic
488C8—32no—
489C8—32no—
490C8—34no—
491C8—92no—
492C8—92no—
493C8C2×D3×Dic314yes—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—42no—
499C2×C3—62no—
500C2×C3—62no—
501C2×C3C2×C42×D311yescharacteristic
502C2×C3#574512yes—
503C2×C3#575012yes—
504C2×C3C22×C4×Dic311yescharacteristic
505D3—22no—
506D3—22no—
507C22—22no—
508C22—32no—
509C22—32no—
510C22—62no—
511C22—62no—
512C22—124no—
513C22C4×D3×Dic312yes—
514C22C2×C4×D3211yescharacteristic
515C4—32no—
516C4—32no—
517C4—64no—
518C4—184no—
519C4C4×D3×Dic312yes—
520C4C2×C4×D3211yescharacteristic
521C4C22×D3×Dic311yescharacteristic
522C3—21no—
523C3#7854811yescharacteristic, complemented
524C3#8374711yescharacteristic, complemented
525C2—22no—
526C2—62no—
527C2—62no—
528C2#1059470612yes—
529C2C2×C4×D3×Dic311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutatorC2×C32C22×C42
FittingC2×C32×M16C22
socleC22×C32C23×C4
prime radicalC23×C3×D3C2×C4
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2873
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. #10592821C2semidirect
5. C3×#5122C2semidirect
6. #10591059C2semidirect
7. #10591057C2semidirect
8. #10590693C2semidirect
9. #10590692C2semidirect
10. #10590667C2semidirect
11. Dic3×ΓL(1,𝔽25)C2semidirect
21. C3×D3×M16C4semidirect
30. C2×#3711C4semidirect
35. #63642C22semidirect
36. C8×C3⋊Dic3C22semidirect
37. C3×#815C22semidirect
38. C3×C8×Dic3C22semidirect
39. #63555C22semidirect
40. #63554C22semidirect
41. #63553C22semidirect
42. Dic3×C3⋊C8C22semidirect
55. #5122D3semidirect
76. C32×M16C2×C4semidirect
81. C2×C3⋊(C3⋊C8)C2×C4semidirect
87. C3×#208C2×C4semidirect
88. C2×C3×C3⋊C8C2×C4semidirect
93. C3×D3×C8C2×C4semidirect
99. #3711C2×C4semidirect
100. #3711C2×C4semidirect
133. D3×M16Dic3semidirect
155. #815C2×D3semidirect
162. C8×Dic3C2×D3semidirect
227. C3×M16C4×D3semidirect
228. C3×M16C2×Dic3semidirect
262. #208C2×Dic3semidirect
272. C2×C3⋊C8C4×D3semidirect
281. D3×C8C2×Dic3semidirect
410. M16D3×Dic3semidirect
466. C32#2873semidirect
523. C3#78548semidirect
524. C3#83747semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×M162
D3×#9262
C22×#37112
#105928212
C3×#51222
#105910592
#105910572
#105906932
#105906922
#105906672
Dic3×ΓL(1,𝔽25)2
#837473
#785483