Finite Group Databasebeta

← #11857312

Subgroups — #11857312

Order 1152.

1183 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic
2#10594958C211yesmaximal, characteristic
3C2×#63814C211yesmaximal, characteristic
4M16×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#5122C211yesmaximal, characteristic, complemented
6C2×#63555C211yesmaximal, characteristic, complemented
7#10591055C211yesmaximal, characteristic, complemented
8#10590909C212yesmaximal, complemented
9#10590692C212yesmaximal, complemented
10Dic3×ΓL(1,𝔽25)C212yesmaximal, complemented
11D3×#815C212yesmaximal, complemented
12#83782—31nomaximal
13#78550—31nomaximal
14C2×C3×#235C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C2×C4×C3⋊Dic3C2211yescharacteristic
17C2×#3748C2211yescharacteristic
18C3×#926C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C3×#907C418yescomplemented
22C2×C3×ΓL(1,𝔽25)C2212yes—
23C2×C3×D3×C8C2212yes—
24#67⋊Dic3C414yes—
25#63898C412yes—
26C4×D3×Dic3C2212yes—
27#63855C2212yes—
28#63843C412yes—
29C2×#3711C2212yes—
30#63814C418yescomplemented
31C2×D3×C3⋊C8C2212yes—
32#63642C2212yes—
33C8×C3⋊Dic3C2212yes—
34C3×#815C2212yes—
35C3×C8×Dic3C2212yes—
36#63555C2211yescharacteristic
37#63555C2211yescharacteristic
38#63555C2211yescharacteristic
39#63555C2211yescharacteristic
40#63554C2212yes—
41Dic3×C3⋊C8C2212yes—
42C2×C3×#131—31no—
43Dic3×#41—31no—
44C2×#948—31no—
45C2×#907D311yescharacteristic
46C2×#901—31no—
47#5522—31no—
48C3×#405—31no—
49#5122—32no—
50#5122D311yescharacteristic, complemented
51Dic3×M16—31no—
52Dic3×M16—31no—
53Dic3×M16—32no—
54Dic3×M16—61no—
55#5101—31no—
56#5101—32no—
57C2×#815—31no—
58C2×#815—31no—
59C2×#815—32no—
60#4997—31no—
61#4708—32no—
62D3×#97—32no—
63#4689—32no—
64C4×ΓL(1,𝔽25)—32no—
65C22×C32×C4C2311yescharacteristic
66C22×C3⋊Dic3C2×C412yes—
67C2×C3×#67C2×C412yes—
68C3×#235C2×C414yes—
69C3×#235C2×C414yes—
70C2×C3×D12C2×C411yescharacteristic
71C2×C3×C4×D3C2312yes—
72C2×C3×Dic6C2×C411yescharacteristic
73C2×D3×Dic3C2×C414yes—
74C32×M16C2×C414yescomplemented
75C2×C32×C8C2312yes—
76C4×C3⋊Dic3C2×C412yes—
77C4×C3⋊Dic3C2311yescharacteristic
78C4×C3⋊Dic3C2311yescharacteristic
79#3748C2×C414yescomplemented
80C2×C3⋊(C3⋊C8)C2312yes—
81C3×#217C2×C412yes—
82C3×C4⋊Dic3C2×C412yes—
83C3×C4×Dic3C2312yes—
84C2×C3×C3⋊C8C2×C414yescomplemented
85C2×C3×C3⋊C8C2×C414yescomplemented
86C2×C3×C3⋊C8C2311yescharacteristic
87C2×C3×C3⋊C8C2311yescharacteristic
88C2×C3×C3⋊C8C2311yescharacteristic
89C2×C3×C3⋊C8C2311yescharacteristic
90C3×ΓL(1,𝔽25)C2×C418yescomplemented
91C3×D3×C8C2×C418yescomplemented
92Dic3⋊Dic3C2×C414yes—
93(C2×D3)⋊Dic3C2×C414yes—
94Dic32C2×C414yes—
95#3711C2×C418yescomplemented
96D3×C3⋊C8C2×C418yescomplemented
97#2875—91noSylow
98C2×C3×#41—31no—
99C2×#235C2×D311yescharacteristic
100C3×#131—38no—
101C2×C3×M16—21no—
102C2×C3×M16—32no—
103C2×C3×M16C2×D311yescharacteristic
104C2×C3×M16C2×D311yescharacteristic
105C22×C3×C8—31no—
106C22×C3×C8—31no—
107C22×C3×C8—32no—
108C3×#117—31no—
109#948—38no—
110Q8×Dic3—32no—
111D4×Dic3—34no—
112D4×Dic3—32no—
113#926—32no—
114#926C2×D311yescharacteristic
115C2×C4×Dic3—31no—
116C2×C4×Dic3—31no—
117C2×C4×Dic3—32no—
118C2×C4×Dic3—61no—
119C2×#208—31no—
120C2×#208—31no—
121C2×#208—32no—
122C2×#208—61no—
123C22×C3⋊C8—32no—
124C22×C3⋊C8C2×D311yescharacteristic
125C22×C3⋊C8C2×D311yescharacteristic
126#907Dic318yescomplemented
127#901—38no—
128C2×ΓL(1,𝔽25)—32no—
129C2×ΓL(1,𝔽25)C2×D312yes—
130C2×D3×C8—32no—
131C2×D3×C8C2×D312yes—
132#893—32no—
133#887—32no—
134#67⋊C4—34no—
135#872—32no—
136C42×D3—32no—
137C3×#97—32no—
138C3×C4×C8—32no—
139#815—32no—
140#815—32no—
141#815—32no—
142#815—31no—
143#815—31no—
144#815—31no—
145#815—31no—
146#815—31no—
147#815—31no—
148#815—31no—
149#815—31no—
150#815—62no—
151#815C2×D312yes—
152C8×Dic3—32no—
153C8×Dic3—32no—
154C8×Dic3—32no—
155C8×Dic3—62no—
156C8×Dic3C2×D312yes—
157#803—32no—
158C4×C3⋊C8—32no—
159C23×C32C22×C411yescharacteristic
160C22×C3×D3C22×C412yes—
161C2×C32×C4C22×C411yescharacteristic
162C2×C32×C4C22×C412yes—
163C2×C32×C4C22×C412yes—
164C2×C32×C4C2411yescharacteristic
165C2×C3⋊Dic3—42no—
166C2×C3⋊Dic3C22×C412yes—
167C2×C3⋊Dic3C22×C412yes—
168C3×#67C4218yes—
169C2×C3×Dic3C22×C414yes—
170C2×C3×Dic3C22×C412yes—
171C3×D12C4214yes—
172C3×C4×D3C22×C414yes—
173C3×C4×D3C22×C414yes—
174C3×Dic6C4214yes—
175D3×Dic3—44no—
176C32×C8C22×C414yes—
177C3⋊(C3⋊C8)C22×C414yes—
178C3×C3⋊C8C22×C412yes—
179C3×C3⋊C8C22×C412yes—
180C3×C3⋊C8C22×C412yes—
181C3×C3⋊C8C22×C412yes—
182C2×#131—31no—
183C2×#131—91no—
184C4×#41—91no—
185#405—31no—
186#405—92no—
187#405—92no—
188#405—91no—
189C4×M16—92no—
190C4×M16—91no—
191C2×#97—92no—
192C2×#97—91no—
193C3×#41—34no—
194C3×#41—34no—
195C2×C3×Q8—31no—
196C2×C3×D4—32no—
197C2×C3×D4—31no—
198C22×C3×C4—21no—
199C22×C3×C4—32no—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C22×D311yescharacteristic
202C2×#67C4×D312yes—
203C22×Dic3—32no—
204C22×Dic3—32no—
205C22×Dic3—34no—
206C22×Dic3—62no—
207#235C4×D314yes—
208#235C2×Dic314yes—
209C2×D12C4×D311yescharacteristic
210C2×C4×D3—34no—
211C2×C4×D3C22×D312yes—
212C2×Dic6C4×D311yescharacteristic
213C3×M16—24no—
214C3×M16—38no—
215C3×M16C4×D314yescomplemented
216C3×M16C2×Dic314yescomplemented
217C2×C3×C8—22no—
218C2×C3×C8—34no—
219C2×C3×C8—34no—
220C2×C3×C8—31no—
221C2×C3×C8—31no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—38no—
225C2×C3×C8C22×D312yes—
226C2×C3×C8C22×D312yes—
227C3×C4⋊C4—32no—
228C3×C22⋊C4—32no—
229C3×C42—32no—
230#217—34no—
231#217C4×D312yes—
232(C2×D3)⋊C4—34no—
233C4⋊Dic3—34no—
234C4⋊Dic3C4×D312yes—
235Dic3⋊C4—34no—
236C4×Dic3—32no—
237C4×Dic3—32no—
238C4×Dic3—31no—
239C4×Dic3—31no—
240C4×Dic3—31no—
241C4×Dic3—31no—
242C4×Dic3—34no—
243C4×Dic3—34no—
244C4×Dic3—61no—
245C4×Dic3—61no—
246C4×Dic3—62no—
247C4×Dic3C22×D312yes—
248#208—34no—
249#208—34no—
250#208—38no—
251#208—64no—
252C2×C3⋊C8—32no—
253C2×C3⋊C8—32no—
254C2×C3⋊C8—38no—
255C2×C3⋊C8—62no—
256C2×C3⋊C8C4×D314yescomplemented
257C2×C3⋊C8C2×Dic314yescomplemented
258C2×C3⋊C8C22×D311yescharacteristic
259C2×C3⋊C8C22×D311yescharacteristic
260C2×C3⋊C8C22×D311yescharacteristic
261C2×C3⋊C8C22×D311yescharacteristic
262ΓL(1,𝔽25)—38no—
263ΓL(1,𝔽25)C2×Dic318yescomplemented
264D3×C8—38no—
265D3×C8C2×Dic318yescomplemented
266C22×C32—22no—
267C22×C32C2×C4212yes—
268C22×C32C23×C411yescharacteristic
269C2×C3×D3—24no—
270C2×C3×D3C2×C4214yes—
271C32×C4C2×C4212yes—
272C32×C4C23×C411yescharacteristic
273C32×C4C23×C411yescharacteristic
274C3⋊Dic3—22no—
275C3⋊Dic3—22no—
276C3×Dic3—24no—
277C3×Dic3C2×C4214yes—
278C2×#41—31no—
279#131—38no—
280#131—98no—
281C2×M16—32no—
282C2×M16—92no—
283C2×M16—91no—
284C2×M16D3211yescharacteristic
285C22×C8—31no—
286C22×C8—31no—
287C22×C8—32no—
288C22×C8—92no—
289C4×Q8—92no—
290C4×D4—94no—
291C4×D4—92no—
292#117—31no—
293#117—92no—
294C2×C42—92no—
295C2×C42—91no—
296#97—32no—
297#97—92no—
298#97—91no—
299#97—91no—
300#97—91no—
301#97—91no—
302#97—92no—
303C4×C8—32no—
304C4×C8—92no—
305C4×C8—92no—
306C23×C3—21no—
307C23×C3—32no—
308C23×C3C2×C4×D311yescharacteristic
309C23×C3C2×C4×D311yescharacteristic
310C22×D3C2×C4×D312yes—
311C3×Q8—34no—
312C3×D4—38no—
313C3×D4—34no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—32no—
319C2×C3×C4—34no—
320C2×C3×C4—34no—
321C2×C3×C4—34no—
322C2×C3×C4C2×C4×D311yescharacteristic
323C2×C3×C4C2×C4×D312yes—
324C2×C3×C4C2×C4×D311yescharacteristic
325C2×C3×C4C2×C4×D312yes—
326C2×C3×C4C2×C4×D312yes—
327C2×C3×C4C22×Dic312yes—
328C2×C3×C4C23×D311yescharacteristic
329C2×C3×C4C23×D311yescharacteristic
330#67C4×Dic318yes—
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—124no—
338C2×Dic3—122no—
339C2×Dic3—122no—
340C2×Dic3—124no—
341C2×Dic3C2×C4×D314yes—
342C2×Dic3C2×C4×D312yes—
343D12C4×Dic314yes—
344C4×D3—124no—
345C4×D3C2×C4×D314yes—
346C4×D3C22×Dic314yes—
347Dic6C4×Dic314yes—
348C3×C8—24no—
349C3×C8—32no—
350C3×C8—32no—
351C3×C8—32no—
352C3×C8—32no—
353C3×C8C2×C4×D314yes—
354C3×C8C22×Dic314yes—
355C3⋊C8—34no—
356C3⋊C8—34no—
357C3⋊C8—64no—
358C3⋊C8C2×C4×D312yes—
359C3⋊C8C2×C4×D312yes—
360C3⋊C8C22×Dic312yes—
361C3⋊C8C22×Dic312yes—
362C2×C32—22no—
363C2×C32C22×C4211yescharacteristic
364C2×C32#56812yes—
365C3×D3—24no—
366#41—34no—
367#41—34no—
368C2×Q8—31no—
369C2×D4—32no—
370C2×D4—31no—
371C22×C4—32no—
372C22×C4—94no—
373C22×C4—92no—
374C22×C4C2×D3211yescharacteristic
375M16—38no—
376M16—98no—
377M16—94no—
378M16D3×Dic314yescomplemented
379C2×C8—31no—
380C2×C8—31no—
381C2×C8—31no—
382C2×C8—31no—
383C2×C8—34no—
384C2×C8—34no—
385C2×C8—38no—
386C2×C8—98no—
387C2×C8—92no—
388C2×C8C2×D3212yes—
389C4⋊C4—32no—
390C4⋊C4—94no—
391C22⋊C4—32no—
392C22⋊C4—94no—
393C42—32no—
394C42—94no—
395C42—91no—
396C42—91no—
397C42—92no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—34no—
403C22×C3—42no—
404C22×C3—64no—
405C22×C3C42×D312yes—
406C22×C3C2×C4×Dic312yes—
407C22×C3C22×C4×D311yescharacteristic
408C22×C3C22×C4×D311yescharacteristic
409C2×D3—24no—
410C2×D3C2×C4×Dic314yes—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—34no—
415C3×C4—64no—
416C3×C4C42×D312yes—
417C3×C4C2×C4×Dic312yes—
418C3×C4C22×C4×D311yescharacteristic
419C3×C4C22×C4×D311yescharacteristic
420C3×C4C22×C4×D311yescharacteristic
421C3×C4C23×Dic311yescharacteristic
422Dic3—24no—
423Dic3—64no—
424Dic3—62no—
425Dic3—62no—
426Dic3—62no—
427Dic3—62no—
428Dic3—122no—
429Dic3C2×C4×Dic314yes—
430C32#287511yescharacteristic, complemented, Sylow
431C23—32no—
432C23C4×D3211yescharacteristic
433Q8—34no—
434D4—38no—
435D4—34no—
436C2×C4—32no—
437C2×C4—34no—
438C2×C4—34no—
439C2×C4—34no—
440C2×C4—92no—
441C2×C4—92no—
442C2×C4—364no—
443C2×C4—362no—
444C2×C4C4×D3212yes—
445C2×C4C4×D3211yescharacteristic
446C2×C4C2×D3×Dic312yes—
447C2×C4C22×D3211yescharacteristic
448C8—32no—
449C8—32no—
450C8—32no—
451C8—32no—
452C8—94no—
453C8C2×D3×Dic314yes—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—42no—
459C2×C3—64no—
460C2×C3C2×C42×D311yescharacteristic
461C2×C3#575012yes—
462C2×C3#575012yes—
463C2×C3C22×C4×Dic311yescharacteristic
464D3—24no—
465C22—22no—
466C22—34no—
467C22—64no—
468C22C4×D3×Dic312yes—
469C22C2×C4×D3211yescharacteristic
470C4—34no—
471C4—64no—
472C4—182no—
473C4—182no—
474C4C4×D3×Dic312yes—
475C4C2×C4×D3211yescharacteristic
476C4C22×D3×Dic311yescharacteristic
477C3—21no—
478C3#7855011yescharacteristic, complemented
479C3#8378211yescharacteristic, complemented
480C2—22no—
481C2—64no—
482C2#1059470812yes—
483C2C2×C4×D3×Dic311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutatorC2×C32C22×C42
FittingC2×C32×M16C22
socleC22×C32C23×C4
prime radicalC2×C3×#235C22
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2875
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. M16×C3⋊Dic3C2semidirect
5. C3×#5122C2semidirect
6. C2×#63555C2semidirect
7. #10591055C2semidirect
8. #10590909C2semidirect
9. #10590692C2semidirect
10. Dic3×ΓL(1,𝔽25)C2semidirect
11. D3×#815C2semidirect
21. C3×#907C4semidirect
30. #63814C4semidirect
50. #5122D3semidirect
74. C32×M16C2×C4semidirect
79. #3748C2×C4semidirect
84. C2×C3×C3⋊C8C2×C4semidirect
85. C2×C3×C3⋊C8C2×C4semidirect
90. C3×ΓL(1,𝔽25)C2×C4semidirect
91. C3×D3×C8C2×C4semidirect
95. #3711C2×C4semidirect
96. D3×C3⋊C8C2×C4semidirect
126. #907Dic3semidirect
215. C3×M16C4×D3semidirect
216. C3×M16C2×Dic3semidirect
256. C2×C3⋊C8C4×D3semidirect
257. C2×C3⋊C8C2×Dic3semidirect
263. ΓL(1,𝔽25)C2×Dic3semidirect
265. D3×C8C2×Dic3semidirect
378. M16D3×Dic3semidirect
430. C32#2875semidirect
478. C3#78550semidirect
479. C3#83782semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9072
#105949582
C2×#638142
M16×C3⋊Dic32
C3×#51222
C2×#635552
#105910552
#105909092
#105906922
Dic3×ΓL(1,𝔽25)2
D3×#8152
#837823
#785503