Finite Group Databasebeta

← #11876989

Subgroups — #11876989

Order 1152.

941 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594367C212yesmaximal, complemented
2#10592554C211yesmaximal, characteristic, complemented
3C3×#4869C211yesmaximal, characteristic, complemented
4C3×#4865C211yesmaximal, characteristic, complemented
5#10590886C211yesmaximal, characteristic, complemented
6#10590836C211yesmaximal, characteristic, complemented
7#10590771C211yesmaximal, characteristic, complemented
8#10590667C211yesmaximal, characteristic, complemented
9#10590552C212yesmaximal, complemented
10#10590504C212yesmaximal, complemented
11#10590455C212yesmaximal, complemented
12#80853—31nomaximal
13#79877—31nomaximal
14C2×#3795—22no—
15#64097C2212yes—
16C2×C3×D24C2211yescharacteristic
17C2×C3×D24C2211yescharacteristic
18C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
19C3×#895C2212yes—
20C3×#893C2212yes—
21#63914C2212yes—
22#63913—22no—
23#63908C2212yes—
24Dic3×D12C2212yes—
25#63885—22no—
26#63879—22no—
27C4×D3×Dic3C2211yescharacteristic
28C2×#3713C2212yes—
29C2×#3711C2211yescharacteristic
30C32×#106C2211yescharacteristic
31#63644C2211yescharacteristic
32C8×C3⋊Dic3C2211yescharacteristic
33#63635C2212yes—
34C3×#815C2211yescharacteristic
35C3×#809C2212yes—
36C3×#809C2212yes—
37#63563C2212yes—
38#63563C2212yes—
39#63554C2211yescharacteristic
40#63553C2211yescharacteristic
41#5596—32no—
42#5548—32no—
43C3×#469—31no—
44C3×#442—31no—
45#5159—31no—
46#5152—31no—
47#5151—31no—
48#5151—31no—
49#5135—31no—
50Dic3×M16—31no—
51#5039—32no—
52#5014—32no—
53#4869D311yescharacteristic, complemented
54#4865D311yescharacteristic, complemented
55#4855—31no—
56#4855—31no—
57#4855—61no—
58#4785—32no—
59#4782—32no—
60#4760—32no—
61#4760—32no—
62#4711—31no—
63#4707—31no—
64C2×#634—22no—
65C2×C3×D12—22no—
66C2×C3×D12C2312yes—
67C2×C3×D12C2312yes—
68C2×C3×C4×D3—22no—
69C2×C3×C4×D3C2311yescharacteristic
70C2×#625—22no—
71C2×#625—22no—
72C2×#625—22no—
73C2×D3×Dic3—22no—
74C2×D3×Dic3D412yes—
75#3795—48no—
76C2×C32×C8C2311yescharacteristic
77C32×C4⋊C4C2312yes—
78C4⋊(C3⋊Dic3)C2312yes—
79#3750—22no—
80C4×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊(C3⋊C8)C2311yescharacteristic
82C3×(C2×D3)⋊C4—22no—
83C3×(C2×D3)⋊C4—22no—
84C3×C4×Dic3C2311yescharacteristic
85C2×C3×C3⋊C8C2311yescharacteristic
86C2×C3×C3⋊C8C2311yescharacteristic
87C3×D24—22no—
88C3×D24—22no—
89C3×ΓL(1,𝔽25)—22no—
90(C2×D3)⋊Dic3—22no—
91(C2×D3)⋊Dic3—22no—
92Dic32D412yes—
93#3713—24no—
94#3711—22no—
95#3184—91noSylow
96C2×#237—62no—
97C2×#235—62no—
98C2×C3×D8—31no—
99C2×C3×D8—31no—
100C2×C3×M16—31no—
101C3×#121—32no—
102C3×C4×D4—32no—
103#940—32no—
104#940—32no—
105#939—62no—
106#938—32no—
107D4×Dic3—32no—
108C2×#213—32no—
109C2×#213—32no—
110#930—32no—
111C4×#67—62no—
112C2×C4×Dic3—31no—
113C2×#208—31no—
114C2×D24C2×D311yescharacteristic
115C2×D24C2×D311yescharacteristic
116C2×ΓL(1,𝔽25)—31no—
117C2×ΓL(1,𝔽25)C2×D311yescharacteristic
118#895C2×D312yes—
119#893C2×D312yes—
120#888—32no—
121#888—32no—
122#888—62no—
123#885—62no—
124#885—62no—
125#884—62no—
126#67⋊C4—62no—
127C4×D12—32no—
128C42×D3—31no—
129C3×#106—21no—
130C3×#106C2×D311yescharacteristic
131C3×#106C2×D311yescharacteristic
132C3×#102—32no—
133C3×#102—32no—
134C3×#97—31no—
135#832—32no—
136#832—32no—
137#821—32no—
138#821—32no—
139#817—31no—
140#817—31no—
141#817—61no—
142#815—31no—
143#815C2×D311yescharacteristic
144C8×Dic3—31no—
145C8×Dic3—31no—
146C8×Dic3—31no—
147C8×Dic3—31no—
148C8×Dic3—61no—
149#809C2×D312yes—
150#809C2×D312yes—
151#808—32no—
152#808—32no—
153#808—62no—
154#803—31no—
155C22×C3×D3—22no—
156C22×C3×D3—22no—
157C22×C3×D3C2×D411yescharacteristic
158C2×C32×C4—22no—
159C2×C32×C4C2411yescharacteristic
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3C2×D412yes—
162#634—44no—
163C2×C3×Dic3C2×D411yescharacteristic
164C2×C3×Dic3C2×D412yes—
165C3×D12—22no—
166C3×D12—22no—
167C3×D12—22no—
168C3×D12—22no—
169C3×D12—44no—
170C3×C4×D3—21no—
171C3×C4×D3—21no—
172C3×C4×D3—44no—
173#625—44no—
174#625—44no—
175#625—44no—
176D3×Dic3—24no—
177D3×Dic3—44no—
178C32×C8#4112yescomplemented
179C3⋊(C3⋊C8)#4112yescomplemented
180C3×C3⋊C8—21no—
181C3×C3⋊C8—21no—
182#532—92no—
183#498—91no—
184#493—91no—
185#486—92no—
186#469—31no—
187#469—92no—
188#469—91no—
189#442—31no—
190#442—91no—
191#442—92no—
192C4×M16—91no—
193C2×C3×D4—32no—
194C2×C3×D4—32no—
195C2×C3×D4—62no—
196C22×C3×C4—31no—
197C22×C3×C4—62no—
198C2×#67—62no—
199C2×#67—62no—
200C2×#67—62no—
201C2×#67—62no—
202C2×#67—62no—
203C2×#67—62no—
204C22×Dic3—32no—
205C22×Dic3—62no—
206#237—128no—
207#235—128no—
208C2×D12—22no—
209C2×D12C22×D312yes—
210C2×D12C22×D312yes—
211C2×C4×D3—22no—
212C2×C4×D3—32no—
213C2×C4×D3—62no—
214C2×C4×D3C22×D311yescharacteristic
215C2×Dic6—62no—
216C2×Dic6—62no—
217C2×Dic6—122no—
218C3×D8—62no—
219C3×D8—62no—
220C3×M16—62no—
221C2×C3×C8—21no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8C22×D311yescharacteristic
225C2×C3×C8C22×D311yescharacteristic
226C3×C4⋊C4—22no—
227C3×C4⋊C4C22×D312yes—
228C3×C4⋊C4C22×D312yes—
229C3×C22⋊C4—62no—
230C3×C22⋊C4—62no—
231C3×C42—31no—
232#217—62no—
233#217—62no—
234#213—64no—
235#213—64no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—62no—
239(C2×D3)⋊C4—62no—
240C4⋊Dic3—32no—
241C4⋊Dic3—32no—
242C4⋊Dic3—62no—
243Dic3⋊C4—62no—
244Dic3⋊C4—62no—
245Dic3⋊C4—122no—
246C4×Dic3—31no—
247C4×Dic3—31no—
248C4×Dic3—32no—
249C4×Dic3—32no—
250C4×Dic3—61no—
251C4×Dic3C22×D311yescharacteristic
252#208—62no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8C22×D311yescharacteristic
257C2×C3⋊C8C22×D311yescharacteristic
258D24—22no—
259D24—22no—
260ΓL(1,𝔽25)—22no—
261ΓL(1,𝔽25)—62no—
262C22×C32C22×D411yescharacteristic
263C2×C3×D3—21no—
264C2×C3×D3—22no—
265C2×C3×D3—42no—
266C2×C3×D3—42no—
267C2×C3×D3—42no—
268C2×C3×D3—42no—
269C2×C3×D3—42no—
270C2×C3×D3—42no—
271C32×C4—42no—
272C32×C4C2×#4111yescharacteristic
273C32×C4C2×#4111yescharacteristic
274C3⋊Dic3—42no—
275C3⋊Dic3AGL(1,ℤ/8)14yes—
276C3×Dic3—21no—
277C3×Dic3—22no—
278C2×#41—182no—
279C2×D8—31no—
280C2×D8—31no—
281C2×D8—92no—
282C2×M16—31no—
283C2×M16—91no—
284C4⋊Q8—92no—
285#127—92no—
286#124—182no—
287#121—32no—
288#121—92no—
289#121—182no—
290C4×D4—32no—
291C4×D4—92no—
292C4×D4—182no—
293C2×C42—91no—
294#106—91no—
295#106—92no—
296#106D3211yescharacteristic
297#102—32no—
298#102—32no—
299#102—92no—
300#102—92no—
301#97—31no—
302#97—91no—
303C4×C8—91no—
304C4×C8—91no—
305C23×C3—31no—
306C23×C3—62no—
307C23×C3—62no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3D3×D411yescharacteristic
311C3×D4—62no—
312C3×D4—62no—
313C3×D4—62no—
314C3×D4—62no—
315C3×D4—124no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—31no—
320C2×C3×C4—32no—
321C2×C3×C4—42no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—124no—
325C2×C3×C4C23×D311yescharacteristic
326C2×C3×C4C23×D311yescharacteristic
327#67—124no—
328#67—124no—
329#67—124no—
330#67—124no—
331#67—124no—
332#67—124no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—62no—
336C2×Dic3—64no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—122no—
340C2×Dic3—124no—
341C2×Dic3D3×D411yescharacteristic
342C2×Dic3D3×D412yes—
343D12—22no—
344D12—22no—
345D12—22no—
346D12—22no—
347D12—44no—
348C4×D3—21no—
349C4×D3—21no—
350C4×D3—44no—
351C4×D3—64no—
352C4×D3—124no—
353Dic6—124no—
354Dic6—124no—
355Dic6—244no—
356C3×C8—22no—
357C3×C8—61no—
358C3×C8—61no—
359C3×C8#23512yescomplemented
360C3×C8#23712yescomplemented
361C3⋊C8—21no—
362C3⋊C8—21no—
363C3⋊C8—32no—
364C3⋊C8—32no—
365C3⋊C8—62no—
366C2×C32#53211yescharacteristic
367C2×C32C2×AGL(1,ℤ/8)12yes—
368C3×D3—41no—
369C3×D3—82no—
370C3×D3—82no—
371#41—368no—
372C2×Q8—182no—
373C2×D4—32no—
374C2×D4—32no—
375C2×D4—62no—
376C2×D4—182no—
377C2×D4—182no—
378C2×D4—182no—
379C22×C4—31no—
380C22×C4—62no—
381C22×C4—92no—
382C22×C4—182no—
383D8—62no—
384D8—62no—
385D8—184no—
386M16—62no—
387M16—182no—
388C2×C8—31no—
389C2×C8—31no—
390C2×C8—91no—
391C2×C8C2×D3211yescharacteristic
392C4⋊C4—92no—
393C4⋊C4—182no—
394C4⋊C4C2×D3212yes—
395C22⋊C4—62no—
396C22⋊C4—62no—
397C22⋊C4—182no—
398C22⋊C4—182no—
399C42—31no—
400C42—92no—
401C42—91no—
402C22×C3—21no—
403C22×C3—62no—
404C22×C3—61no—
405C22×C3—122no—
406C22×C3—122no—
407C22×C3—122no—
408C22×C3—122no—
409C22×C3—122no—
410C22×C3—122no—
411C22×C3C2×D3×D411yescharacteristic
412C22×C3C2×D3×D411yescharacteristic
413C2×D3—22no—
414C2×D3—21no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—42no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—62no—
426C3×C4—61no—
427C3×C4—82no—
428C3×C4C2×#23511yescharacteristic
429C3×C4C2×#23711yescharacteristic
430C3×C4D3×#4111yescharacteristic
431C3×C4D3×#4111yescharacteristic
432Dic3—22no—
433Dic3—21no—
434Dic3—34no—
435Dic3—34no—
436Dic3—64no—
437Dic3—122no—
438Dic3—122no—
439Dic3—242no—
440C32#318411yescharacteristic, complemented, Sylow
441C23—31no—
442C23—62no—
443C23—62no—
444Q8—364no—
445D4—62no—
446D4—62no—
447D4—62no—
448D4—62no—
449D4—124no—
450D4—364no—
451D4—364no—
452D4—364no—
453C2×C4—22no—
454C2×C4—32no—
455C2×C4—31no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—92no—
459C2×C4—124no—
460C2×C4—184no—
461C2×C4—182no—
462C2×C4—364no—
463C2×C4C22×D3211yescharacteristic
464C8—61no—
465C8—61no—
466C8—92no—
467C8#379512yescomplemented
468C2×C3—21no—
469C2×C3—22no—
470C2×C3—121no—
471C2×C3—242no—
472C2×C3—242no—
473C2×C3#554811yescharacteristic
474C2×C3#559611yescharacteristic
475C2×C3D3×AGL(1,ℤ/8)12yes—
476C2×C3D3×AGL(1,ℤ/8)12yes—
477D3—41no—
478D3—82no—
479D3—82no—
480C22—61no—
481C22—62no—
482C22—122no—
483C22—122no—
484C22—122no—
485C22—122no—
486C22—122no—
487C22—122no—
488C22D32×D411yescharacteristic
489C4—42no—
490C4—61no—
491C4—62no—
492C4—94no—
493C4—362no—
494C4C2×#379511yescharacteristic
495C4#6431911yescharacteristic
496C3—21no—
497C3#7987711yescharacteristic, complemented
498C3#8085311yescharacteristic, complemented
499C2—121no—
500C2—242no—
501C2—242no—
502C2#1059436711yescharacteristic
503C2#1059474012yes—
subgroupG/H
centerC22D32×D4
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#106C22
socleC22×C32C22×D4
hypercenter#106D32
hypocenterC32#3184
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
1. #10594367C2semidirect
2. #10592554C2semidirect
3. C3×#4869C2semidirect
4. C3×#4865C2semidirect
5. #10590886C2semidirect
6. #10590836C2semidirect
7. #10590771C2semidirect
8. #10590667C2semidirect
9. #10590552C2semidirect
10. #10590504C2semidirect
11. #10590455C2semidirect
53. #4869D3semidirect
54. #4865D3semidirect
178. C32×C8#41semidirect
179. C3⋊(C3⋊C8)#41semidirect
359. C3×C8#235semidirect
360. C3×C8#237semidirect
440. C32#3184semidirect
467. C8#3795semidirect
497. C3#79877semidirect
498. C3#80853semidirect

Maximal subgroups

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

subgroupindex
#105943672
#105925542
C3×#48692
C3×#48652
#105908862
#105908362
#105907712
#105906672
#105905522
#105905042
#105904552
#808533
#798773