Finite Group Databasebeta

← #11857313

Subgroups — #11857313

Order 1152.

1245 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64116C211yesmaximal, characteristic
2#10594954C211yesmaximal, characteristic
3C2×#63817C211yesmaximal, characteristic
4C3×#5122C211yesmaximal, characteristic, complemented
5C3×Dic3×M16C211yesmaximal, characteristic, complemented
6C2×#63554C211yesmaximal, characteristic, complemented
7#10591057C211yesmaximal, characteristic, complemented
8#10590910C212yesmaximal, complemented
9#10590700C212yesmaximal, complemented
10#10590668C212yesmaximal, complemented
11#10590664C212yesmaximal, complemented
12#78550—31nomaximal
13#78516—31nomaximal
14C2×#3828C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C22×C3⋊(C3⋊C8)C2211yescharacteristic
17#64116C418yescomplemented
18C2×#3743C2212yes—
19C2×C8×C3⋊D3C2212yes—
20C3×#926C2211yescharacteristic
21C2×C3×C4×Dic3C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24#63952C414yes—
25#63894C412yes—
26C4×#624C2212yes—
27#63874C2212yes—
28#63844C412yes—
29C2×#3712C2212yes—
30#63817C418yescomplemented
31C2×#3710C2212yes—
32C3×#815C2212yes—
33C3×#815C2212yes—
34C3×C8×Dic3C2212yes—
35C3×C8×Dic3C2212yes—
36#63554C2212yes—
37#63554C2211yescharacteristic
38#63554C2211yescharacteristic
39#63554C2211yescharacteristic
40#63554C2211yescharacteristic
41Dic3×C3⋊C8C2212yes—
42C2×#907—31no—
43C2×#907—31no—
44C2×#907—31no—
45C2×#907—61no—
46C2×#901—31no—
47#5522—31no—
48C4×#235—31no—
49C3×#405—31no—
50C3×C4×M16—31no—
51#5122—31no—
52#5122D311yescharacteristic, complemented
53Dic3×M16D311yescharacteristic, complemented
54C2×#815—31no—
55#4997—31no—
56C2×#803—31no—
57#4708—32no—
58#4707—32no—
59D3×#97—32no—
60D3×#97—32no—
61#4689—32no—
62#4689—32no—
63#4689—32no—
64C4×ΓL(1,𝔽25)—32no—
65C22×C32×C4C2311yescharacteristic
66C2×#638C2×C412yes—
67#3828C2×C414yes—
68#3828C2×C414yes—
69C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
70C2×C4×C3⋊D3C2312yes—
71C2×#634C2×C411yescharacteristic
72C22×C3×Dic3C2×C412yes—
73C2×#624C2×C414yes—
74C32×M16C2×C414yescomplemented
75C2×C32×C8C2312yes—
76C2×C3⋊(C3⋊C8)C2×C414yescomplemented
77C2×C3⋊(C3⋊C8)C2311yescharacteristic
78C2×C3⋊(C3⋊C8)C2311yescharacteristic
79#3743C2×C418yescomplemented
80C8×C3⋊D3C2×C418yescomplemented
81C3×#217C2×C412yes—
82C3×C4⋊Dic3C2×C412yes—
83C3×C4×Dic3C2×C412yes—
84C3×C4×Dic3C2311yescharacteristic
85C3×C4×Dic3C2311yescharacteristic
86C3×C4×Dic3C2312yes—
87C3×#208C2×C414yescomplemented
88C2×C3×C3⋊C8C2×C414yescomplemented
89C2×C3×C3⋊C8C2311yescharacteristic
90C2×C3×C3⋊C8C2311yescharacteristic
91C2×C3×C3⋊C8C2312yes—
92#3724C2×C414yes—
93#3722C2×C414yes—
94Dic32C2×C414yes—
95#3712C2×C418yescomplemented
96#3710C2×C418yescomplemented
97#2875—91noSylow
98C2×#235—31no—
99C2×#235—31no—
100C2×#235—61no—
101C2×C3×M16—21no—
102C2×C3×M16—31no—
103C2×C3×M16C2×D311yescharacteristic
104C2×C3×M16C2×D311yescharacteristic
105C22×C3×C8—31no—
106C3×#117—31no—
107C2×C3×C42—31no—
108C4×#67—34no—
109#926C2×D311yescharacteristic
110C2×C4×Dic3C2×D311yescharacteristic
111C2×#208C2×D311yescharacteristic
112C22×C3⋊C8—31no—
113C22×C3⋊C8—31no—
114C22×C3⋊C8—61no—
115C22×C3⋊C8C2×D311yescharacteristic
116#907—38no—
117#907—38no—
118#907—38no—
119#907—68no—
120#901—38no—
121C2×ΓL(1,𝔽25)—32no—
122C2×ΓL(1,𝔽25)—32no—
123C2×ΓL(1,𝔽25)—32no—
124C2×ΓL(1,𝔽25)—32no—
125C2×ΓL(1,𝔽25)—62no—
126C2×D3×C8—32no—
127C2×D3×C8—32no—
128C2×D3×C8—32no—
129C2×D3×C8—32no—
130C2×D3×C8—62no—
131#893—32no—
132#887—32no—
133#67⋊C4—34no—
134C4×D12—32no—
135#872—32no—
136#872—32no—
137C42×D3—32no—
138C42×D3—32no—
139C4×Dic6—32no—
140C3×#97—32no—
141C3×#97—32no—
142C3×C4×C8—32no—
143C3×C4×C8—32no—
144#815—32no—
145#815—31no—
146#815—31no—
147#815—31no—
148#815—31no—
149#815C2×D312yes—
150#815C2×D312yes—
151C8×Dic3—32no—
152C8×Dic3C2×D312yes—
153C8×Dic3C2×D312yes—
154#803—31no—
155#803—31no—
156#803—31no—
157#803—31no—
158#803—32no—
159C4×C3⋊C8—32no—
160C23×C32C22×C411yescharacteristic
161C22×C3⋊D3C22×C412yes—
162C2×C32×C4C22×C411yescharacteristic
163C2×C32×C4C22×C412yes—
164C2×C32×C4C22×C412yes—
165C2×C32×C4C2411yescharacteristic
166#638C4218yes—
167C2×C3⋊Dic3C22×C412yes—
168C4⋊(C3⋊D3)C4214yes—
169C4×C3⋊D3C22×C414yes—
170C4×C3⋊D3C22×C414yes—
171#634C4214yes—
172C2×C3×Dic3—42no—
173C2×C3×Dic3C22×C412yes—
174C2×C3×Dic3C22×C414yes—
175C2×C3×Dic3C22×C412yes—
176#624—44no—
177C32×C8C22×C414yes—
178C3⋊(C3⋊C8)C22×C412yes—
179C3⋊(C3⋊C8)C22×C412yes—
180C3×C3⋊C8C22×C412yes—
181C3×C3⋊C8C22×C414yes—
182C3×C3⋊C8C22×C412yes—
183C2×#131—91no—
184C2×#131—91no—
185C4×#41—91no—
186#405—31no—
187#405—92no—
188#405—92no—
189#405—91no—
190C4×M16—31no—
191C4×M16—92no—
192C2×#97—92no—
193C2×#97—91no—
194C22×C3×C4—21no—
195C22×C3×C4—32no—
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C22×D311yescharacteristic
198C2×#67—32no—
199C2×#67—32no—
200C2×#67—62no—
201C22×Dic3C4×D312yes—
202#235—34no—
203#235—34no—
204#235—34no—
205#235—34no—
206#235—64no—
207#235—64no—
208C2×D12—31no—
209C2×D12—31no—
210C2×D12—61no—
211C2×C4×D3—32no—
212C2×C4×D3—32no—
213C2×C4×D3—34no—
214C2×C4×D3—34no—
215C2×C4×D3—62no—
216C2×Dic6—31no—
217C2×Dic6—31no—
218C2×Dic6—61no—
219C3×M16—24no—
220C3×M16—34no—
221C3×M16C4×D314yescomplemented
222C3×M16C4×D314yescomplemented
223C2×C3×C8—22no—
224C2×C3×C8—34no—
225C2×C3×C8—32no—
226C2×C3×C8—31no—
227C2×C3×C8—31no—
228C2×C3×C8C22×D312yes—
229C2×C3×C8C22×D312yes—
230C3×C4⋊C4—32no—
231C3×C22⋊C4—32no—
232C3×C42—32no—
233C3×C42—32no—
234C3×C42—31no—
235C3×C42—31no—
236#217C4×D312yes—
237(C2×D3)⋊C4—34no—
238(C2×D3)⋊C4—34no—
239C4⋊Dic3C4×D312yes—
240Dic3⋊C4—34no—
241Dic3⋊C4—34no—
242C4×Dic3—34no—
243C4×Dic3—34no—
244C4×Dic3C4×D312yes—
245C4×Dic3C22×D312yes—
246C4×Dic3C22×D311yescharacteristic
247C4×Dic3C22×D311yescharacteristic
248#208C4×D314yescomplemented
249C2×C3⋊C8—34no—
250C2×C3⋊C8—34no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8—61no—
257C2×C3⋊C8—64no—
258C2×C3⋊C8C4×D314yescomplemented
259C2×C3⋊C8C22×D312yes—
260C2×C3⋊C8C22×D311yescharacteristic
261C2×C3⋊C8C22×D311yescharacteristic
262ΓL(1,𝔽25)—38no—
263ΓL(1,𝔽25)—38no—
264ΓL(1,𝔽25)—38no—
265ΓL(1,𝔽25)—38no—
266ΓL(1,𝔽25)—68no—
267D3×C8—38no—
268D3×C8—38no—
269D3×C8—38no—
270D3×C8—38no—
271D3×C8—68no—
272C22×C32—22no—
273C22×C32C2×C4212yes—
274C22×C32C23×C411yescharacteristic
275C2×C3⋊D3—24no—
276C2×C3⋊D3C2×C4214yes—
277C32×C4C2×C4212yes—
278C32×C4C23×C411yescharacteristic
279C32×C4C23×C411yescharacteristic
280C3⋊Dic3C2×C4214yes—
281C3×Dic3—22no—
282C3×Dic3—24no—
283C3×Dic3—22no—
284C2×#41—91no—
285#131—98no—
286#131—98no—
287C2×M16—31no—
288C2×M16—92no—
289C2×M16—92no—
290C2×M16D3211yescharacteristic
291C22×C8—31no—
292C22×C8—92no—
293C22×C8—91no—
294C22×C8—92no—
295C4×Q8—92no—
296C4×D4—92no—
297C4×D4—94no—
298#117—31no—
299#117—92no—
300C2×C42—31no—
301C2×C42—92no—
302#97—32no—
303#97—32no—
304#97—91no—
305#97—91no—
306#97—91no—
307#97—91no—
308#97—92no—
309C4×C8—32no—
310C4×C8—32no—
311C4×C8—92no—
312C23×C3—21no—
313C23×C3C2×C4×D311yescharacteristic
314C23×C3C2×C4×D311yescharacteristic
315C22×D3—32no—
316C22×D3—32no—
317C22×D3—62no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—32no—
323C2×C3×C4—34no—
324C2×C3×C4—32no—
325C2×C3×C4—122no—
326C2×C3×C4C2×C4×D311yescharacteristic
327C2×C3×C4C2×C4×D311yescharacteristic
328C2×C3×C4C2×C4×D312yes—
329C2×C3×C4C2×C4×D312yes—
330C2×C3×C4C2×C4×D312yes—
331C2×C3×C4C2×C4×D312yes—
332C2×C3×C4C23×D311yescharacteristic
333C2×C3×C4C23×D311yescharacteristic
334#67—38no—
335#67—38no—
336#67—68no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—42no—
340C2×Dic3—62no—
341C2×Dic3C2×C4×D312yes—
342C2×Dic3C2×C4×D314yes—
343C2×Dic3C2×C4×D312yes—
344D12—34no—
345D12—34no—
346D12—64no—
347C4×D3—34no—
348C4×D3—34no—
349C4×D3—34no—
350C4×D3—34no—
351C4×D3—64no—
352C4×D3—64no—
353C4×D3—124no—
354C4×D3—124no—
355Dic6—34no—
356Dic6—34no—
357Dic6—64no—
358C3×C8—24no—
359C3×C8—32no—
360C3×C8—34no—
361C3×C8—32no—
362C3×C8C2×C4×D314yes—
363C3×C8C2×C4×D314yes—
364C3⋊C8—32no—
365C3⋊C8—32no—
366C3⋊C8—32no—
367C3⋊C8—32no—
368C3⋊C8—62no—
369C3⋊C8—62no—
370C3⋊C8C2×C4×D312yes—
371C3⋊C8C2×C4×D314yes—
372C3⋊C8C2×C4×D312yes—
373C2×C32—22no—
374C2×C32C22×C4211yescharacteristic
375C2×C32#56812yes—
376C3⋊D3—24no—
377#41—94no—
378#41—94no—
379C2×Q8—91no—
380C2×D4—92no—
381C2×D4—91no—
382C22×C4—32no—
383C22×C4—94no—
384C22×C4—92no—
385C22×C4C2×D3211yescharacteristic
386M16—34no—
387M16—98no—
388M16—98no—
389M16#62414yescomplemented
390C2×C8—31no—
391C2×C8—32no—
392C2×C8—31no—
393C2×C8—34no—
394C2×C8—98no—
395C2×C8—91no—
396C2×C8—91no—
397C2×C8—94no—
398C2×C8—98no—
399C2×C8C2×D3212yes—
400C4⋊C4—32no—
401C4⋊C4—94no—
402C22⋊C4—32no—
403C22⋊C4—94no—
404C42—31no—
405C42—32no—
406C42—31no—
407C42—32no—
408C42—94no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—42no—
414C22×C3C42×D312yes—
415C22×C3C42×D312yes—
416C22×C3C22×C4×D311yescharacteristic
417C22×C3C22×C4×D311yescharacteristic
418C2×D3—34no—
419C2×D3—34no—
420C2×D3—64no—
421C2×D3—64no—
422C2×D3—64no—
423C2×D3—124no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—22no—
427C3×C4—64no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4C42×D312yes—
431C3×C4C42×D312yes—
432C3×C4C22×C4×D311yescharacteristic
433C3×C4C22×C4×D311yescharacteristic
434C3×C4C22×C4×D311yescharacteristic
435C3×C4C22×C4×D311yescharacteristic
436Dic3—22no—
437Dic3—22no—
438Dic3—24no—
439Dic3—34no—
440Dic3—34no—
441Dic3—64no—
442C32#287511yescharacteristic, complemented, Sylow
443C23—92no—
444C23C4×D3211yescharacteristic
445Q8—94no—
446D4—98no—
447D4—94no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—34no—
451C2×C4—92no—
452C2×C4—94no—
453C2×C4—94no—
454C2×C4—122no—
455C2×C4—364no—
456C2×C4C4×D3212yes—
457C2×C4C4×D3211yescharacteristic
458C2×C4C2×#62412yes—
459C2×C4C22×D3211yescharacteristic
460C8—32no—
461C8—32no—
462C8—34no—
463C8—92no—
464C8—92no—
465C8C2×#62414yes—
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—21no—
469C2×C3—22no—
470C2×C3—42no—
471C2×C3C2×C42×D311yescharacteristic
472C2×C3C2×C42×D311yescharacteristic
473C2×C3#574512yes—
474C2×C3#575012yes—
475D3—64no—
476D3—64no—
477D3—124no—
478C22—22no—
479C22—94no—
480C22—184no—
481C22C4×#62412yes—
482C22C2×C4×D3211yescharacteristic
483C4—62no—
484C4—62no—
485C4—64no—
486C4—94no—
487C4C4×#62412yes—
488C4C2×C4×D3211yescharacteristic
489C4C22×#62411yescharacteristic
490C3—21no—
491C3#7851611yescharacteristic, complemented
492C3#7855011yescharacteristic, complemented
493C2—22no—
494C2—184no—
495C2#1059470612yes—
496C2C2×C4×#62411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutatorC2×C32C22×C42
FittingC2×C32×M16C22
socleC22×C32C23×C4
prime radicalC2×#3828C22
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. C3×#5122C2semidirect
5. C3×Dic3×M16C2semidirect
6. C2×#63554C2semidirect
7. #10591057C2semidirect
8. #10590910C2semidirect
9. #10590700C2semidirect
10. #10590668C2semidirect
11. #10590664C2semidirect
17. #64116C4semidirect
30. #63817C4semidirect
52. #5122D3semidirect
53. Dic3×M16D3semidirect
74. C32×M16C2×C4semidirect
76. C2×C3⋊(C3⋊C8)C2×C4semidirect
79. #3743C2×C4semidirect
80. C8×C3⋊D3C2×C4semidirect
87. C3×#208C2×C4semidirect
88. C2×C3×C3⋊C8C2×C4semidirect
95. #3712C2×C4semidirect
96. #3710C2×C4semidirect
221. C3×M16C4×D3semidirect
222. C3×M16C4×D3semidirect
248. #208C4×D3semidirect
258. C2×C3⋊C8C4×D3semidirect
389. M16#624semidirect
442. C32#2875semidirect
491. C3#78516semidirect
492. C3#78550semidirect

Maximal subgroups

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

subgroupindex
C2×#641162
#105949542
C2×#638172
C3×#51222
C3×Dic3×M162
C2×#635542
#105910572
#105909102
#105907002
#105906682
#105906642
#785503
#785163