Finite Group Databasebeta

← #11876951

Subgroups — #11876951

Order 1152.

783 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594444C212yesmaximal, complemented
2#10592557C211yesmaximal, characteristic, complemented
3C3×#4869C211yesmaximal, characteristic, complemented
4C3×#4867C211yesmaximal, characteristic, complemented
5#10590869C211yesmaximal, characteristic, complemented
6#10590836C211yesmaximal, characteristic
7#10590812C211yesmaximal, characteristic, complemented
8#10590667C211yesmaximal, characteristic, complemented
9#10590539C212yesmaximal, complemented
10#10590503C212yesmaximal, complemented
11#10590478C212yesmaximal, complemented
12#80892—31nomaximal
13#79862—31nomaximal
14#64099C2212yes—
15C2×C3×Dic12C2211yescharacteristic
16C2×C3×D24C2211yescharacteristic
17C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
18C3×#897C2212yes—
19C3×#893C2212yes—
20#63908C2212yes—
21#63891—22no—
22#63879—22no—
23C4×D3×Dic3C2211yescharacteristic
24#63863C2212yes—
25#63858—22no—
26Dic3×Dic6C2212yes—
27#63841—22no—
28C2×#3715C2212yes—
29C2×#3711C2211yescharacteristic
30C32×#106C2211yescharacteristic
31#63644C2211yescharacteristic
32C8×C3⋊Dic3C2211yescharacteristic
33#63634C2212yes—
34C3×#815C2211yescharacteristic
35C3×#810C2212yes—
36C3×#809C2212yes—
37#63566C2212yes—
38#63563C2212yes—
39#63554C2211yescharacteristic
40#63553C2211yescharacteristic
41#5624—32no—
42#5557—32no—
43C3×#470—31no—
44C3×#442—31no—
45#5191—31no—
46#5184—31no—
47#5159—31no—
48#5155—31no—
49#5135—31no—
50Dic3×M16—31no—
51#5049—32no—
52#5018—32no—
53#4869D311yescharacteristic, complemented
54#4867D311yescharacteristic, complemented
55#4858—31no—
56#4858—31no—
57#4858—61no—
58#4817—32no—
59#4800—32no—
60#4780—32no—
61#4768—32no—
62#4716—31no—
63#4707—31no—
64C2×C3×D12C2312yes—
65C2×C3×C4×D3—22no—
66C2×C3×C4×D3C2311yescharacteristic
67C2×C3×Dic6C2312yes—
68C2×#625—22no—
69C2×D3×Dic3D412yes—
70C2×C32×C8C2311yescharacteristic
71C32×C4⋊C4C2312yes—
72C4⋊(C3⋊Dic3)—22no—
73C4⋊(C3⋊Dic3)C2312yes—
74#3750—22no—
75C4×C3⋊Dic3C2311yescharacteristic
76C2×C3⋊(C3⋊C8)C2311yescharacteristic
77C3×(C2×D3)⋊C4—22no—
78C3×(C2×D3)⋊C4—22no—
79C3×C4⋊Dic3—22no—
80C3×C4×Dic3C2311yescharacteristic
81C2×C3×C3⋊C8C2311yescharacteristic
82C2×C3×C3⋊C8C2311yescharacteristic
83C3×Dic12—22no—
84C3×D24—22no—
85C3×ΓL(1,𝔽25)—22no—
86Dic3⋊Dic3—22no—
87(C2×D3)⋊Dic3—22no—
88(C2×D3)⋊Dic3—22no—
89(C2×D3)⋊Dic3—22no—
90Dic32—22no—
91Dic32D412yes—
92#3715—24no—
93#3711—22no—
94#3183—91noSylow
95C2×C3×Q16—31no—
96C2×C3×D8—31no—
97C2×C3×M16—31no—
98C3×C22⋊Q8—32no—
99C3×C4×D4—32no—
100#947—32no—
101Q8×Dic3—32no—
102C2×#215—32no—
103#938—32no—
104#936—32no—
105C2×#214—32no—
106#930—32no—
107C4×#67—62no—
108#926—62no—
109C2×C4×Dic3—31no—
110C2×#208—31no—
111C2×Dic12C2×D311yescharacteristic
112C2×D24C2×D311yescharacteristic
113C2×ΓL(1,𝔽25)—31no—
114C2×ΓL(1,𝔽25)C2×D311yescharacteristic
115#898—62no—
116#897C2×D312yes—
117#893C2×D312yes—
118#892—62no—
119#890—32no—
120#890—32no—
121#890—62no—
122#886—62no—
123#883—62no—
124#67⋊C4—62no—
125#879—62no—
126C42×D3—31no—
127C4×Dic6—32no—
128C3×#106—21no—
129C3×#106C2×D311yescharacteristic
130C3×#106C2×D311yescharacteristic
131C3×#103—32no—
132C3×#102—32no—
133C3×#97—31no—
134#835—32no—
135#832—32no—
136#821—32no—
137#817—31no—
138#817—31no—
139#817—61no—
140#816—32no—
141#815—31no—
142#815C2×D311yescharacteristic
143C8×Dic3—31no—
144C8×Dic3—31no—
145C8×Dic3—31no—
146C8×Dic3—31no—
147C8×Dic3—61no—
148#810C2×D312yes—
149#809C2×D312yes—
150#807—32no—
151#807—32no—
152#807—62no—
153#803—31no—
154C22×C3×D3—22no—
155C22×C3×D3C2×D411yescharacteristic
156C2×C32×C4—22no—
157C2×C32×C4C2411yescharacteristic
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3C2×D412yes—
160C2×C3×Dic3—22no—
161C2×C3×Dic3C2×D412yes—
162C2×C3×Dic3C2×D411yescharacteristic
163C3×D12—22no—
164C3×D12—22no—
165C3×C4×D3—21no—
166C3×C4×D3—21no—
167C3×C4×D3—44no—
168C3×Dic6—22no—
169C3×Dic6—22no—
170#625—44no—
171D3×Dic3—42no—
172C32×C8#4112yes—
173C3⋊(C3⋊C8)#4112yes—
174C3×C3⋊C8—21no—
175C3×C3⋊C8—21no—
176#529—92no—
177#499—91no—
178#495—91no—
179#488—92no—
180#470—31no—
181#469—91no—
182#468—92no—
183#442—31no—
184#441—91no—
185#440—92no—
186C4×M16—91no—
187C2×C3×Q8—32no—
188C2×C3×D4—32no—
189C22×C3×C4—31no—
190C22×C3×C4—62no—
191C2×#67—62no—
192C2×#67—62no—
193C22×Dic3—32no—
194C2×D12C22×D312yes—
195C2×C4×D3—22no—
196C2×C4×D3—32no—
197C2×C4×D3C22×D311yescharacteristic
198C2×Dic6C22×D312yes—
199C3×Q16—62no—
200C3×D8—62no—
201C3×M16—62no—
202C2×C3×C8—21no—
203C2×C3×C8—31no—
204C2×C3×C8—31no—
205C2×C3×C8C22×D311yescharacteristic
206C2×C3×C8C22×D311yescharacteristic
207C3×C4⋊C4—22no—
208C3×C4⋊C4—62no—
209C3×C4⋊C4C22×D312yes—
210C3×C4⋊C4C22×D312yes—
211C3×C22⋊C4—62no—
212C3×C22⋊C4—62no—
213C3×C42—31no—
214#217—62no—
215#217—62no—
216#217—62no—
217#215—64no—
218#214—64no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—62no—
222(C2×D3)⋊C4—62no—
223(C2×D3)⋊C4—62no—
224C4⋊Dic3—22no—
225C4⋊Dic3—32no—
226C4⋊Dic3—32no—
227C4⋊Dic3—62no—
228C4⋊Dic3—62no—
229C4⋊Dic3—62no—
230C4⋊Dic3—62no—
231C4⋊Dic3—122no—
232Dic3⋊C4—62no—
233Dic3⋊C4—62no—
234Dic3⋊C4—62no—
235Dic3⋊C4—122no—
236C4×Dic3—31no—
237C4×Dic3—31no—
238C4×Dic3—32no—
239C4×Dic3—32no—
240C4×Dic3—61no—
241C4×Dic3—62no—
242C4×Dic3—62no—
243C4×Dic3C22×D311yescharacteristic
244#208—62no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8C22×D311yescharacteristic
249C2×C3⋊C8C22×D311yescharacteristic
250Dic12—22no—
251D24—22no—
252ΓL(1,𝔽25)—22no—
253ΓL(1,𝔽25)—62no—
254C22×C32C22×D411yescharacteristic
255C2×C3×D3—21no—
256C2×C3×D3—22no—
257C2×C3×D3—42no—
258C2×C3×D3—42no—
259C2×C3×D3—42no—
260C32×C4—42no—
261C32×C4C2×#4111yescharacteristic
262C32×C4C2×#4111yescharacteristic
263C3⋊Dic3—22no—
264C3⋊Dic3—42no—
265C3×Dic3—21no—
266C3×Dic3—22no—
267C3×Dic3—42no—
268C2×Q16—31no—
269C2×SD16—92no—
270C2×D8—31no—
271C2×M16—31no—
272C2×M16—91no—
273#126—182no—
274#125—92no—
275#124—92no—
276#123—182no—
277C22⋊Q8—32no—
278#121—92no—
279C4×Q8—92no—
280C4×D4—32no—
281C4×D4—182no—
282#117—182no—
283C2×C42—91no—
284#107—92no—
285#106—91no—
286#106D3211yescharacteristic
287#103—32no—
288#103—92no—
289#102—32no—
290#102—92no—
291#97—31no—
292#97—91no—
293C4×C8—91no—
294C4×C8—91no—
295C23×C3—31no—
296C23×C3—62no—
297C22×D3—22no—
298C22×D3D3×D411yescharacteristic
299C3×Q8—62no—
300C3×Q8—62no—
301C3×D4—62no—
302C3×D4—62no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—31no—
307C2×C3×C4—32no—
308C2×C3×C4—42no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—62no—
312C2×C3×C4—124no—
313C2×C3×C4C23×D311yescharacteristic
314C2×C3×C4C23×D311yescharacteristic
315#67—124no—
316#67—124no—
317C2×Dic3—22no—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—62no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—122no—
324C2×Dic3—122no—
325C2×Dic3D3×D411yescharacteristic
326C2×Dic3D3×D412yes—
327D12—22no—
328D12—22no—
329C4×D3—21no—
330C4×D3—21no—
331C4×D3—44no—
332C4×D3—122no—
333Dic6—22no—
334Dic6—22no—
335C3×C8—22no—
336C3×C8—61no—
337C3×C8—61no—
338C3×C8#23512yes—
339C3×C8#23712yes—
340C3⋊C8—21no—
341C3⋊C8—21no—
342C3⋊C8—32no—
343C3⋊C8—32no—
344C3⋊C8—62no—
345C2×C32#53211yescharacteristic
346C2×C32#57512yes—
347C3×D3—41no—
348C3×D3—82no—
349C2×Q8—32no—
350C2×D4—32no—
351C2×D4—182no—
352C22×C4—31no—
353C22×C4—62no—
354C22×C4—92no—
355Q16—62no—
356SD16—184no—
357D8—62no—
358M16—62no—
359M16—182no—
360C2×C8—31no—
361C2×C8—31no—
362C2×C8—91no—
363C2×C8C2×D3211yescharacteristic
364C4⋊C4—62no—
365C4⋊C4—92no—
366C4⋊C4—182no—
367C4⋊C4—182no—
368C4⋊C4—182no—
369C4⋊C4C2×D3212yes—
370C22⋊C4—62no—
371C22⋊C4—62no—
372C22⋊C4—182no—
373C22⋊C4—182no—
374C22⋊C4—182no—
375C42—31no—
376C42—92no—
377C42—91no—
378C42—182no—
379C22×C3—21no—
380C22×C3—62no—
381C22×C3—61no—
382C22×C3—122no—
383C22×C3—122no—
384C22×C3—122no—
385C22×C3C2×D3×D411yescharacteristic
386C22×C3C2×D3×D411yescharacteristic
387C2×D3—22no—
388C2×D3—21no—
389C2×D3—42no—
390C2×D3—42no—
391C2×D3—42no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—62no—
397C3×C4—61no—
398C3×C4—82no—
399C3×C4—122no—
400C3×C4C2×#23511yescharacteristic
401C3×C4C2×#23711yescharacteristic
402C3×C4D3×#4111yescharacteristic
403C3×C4D3×#4111yescharacteristic
404Dic3—22no—
405Dic3—21no—
406Dic3—42no—
407Dic3—62no—
408Dic3—62no—
409Dic3—122no—
410Dic3—122no—
411Dic3—122no—
412Dic3—242no—
413C32#318311yescharacteristic, complemented, Sylow
414C23—31no—
415C23—62no—
416Q8—62no—
417Q8—62no—
418D4—62no—
419D4—62no—
420D4—364no—
421C2×C4—22no—
422C2×C4—32no—
423C2×C4—31no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—92no—
428C2×C4—124no—
429C2×C4—182no—
430C2×C4—362no—
431C2×C4C22×D3211yescharacteristic
432C8—61no—
433C8—61no—
434C8—92no—
435C8#379512yes—
436C2×C3—21no—
437C2×C3—22no—
438C2×C3—121no—
439C2×C3—242no—
440C2×C3#554811yescharacteristic
441C2×C3#559611yescharacteristic
442C2×C3#577312yes—
443C2×C3#577712yes—
444D3—41no—
445D3—82no—
446C22—61no—
447C22—62no—
448C22—122no—
449C22—122no—
450C22—122no—
451C22D32×D411yescharacteristic
452C4—42no—
453C4—61no—
454C4—62no—
455C4—122no—
456C4—182no—
457C4—362no—
458C4C2×#379511yescharacteristic
459C4#6431911yescharacteristic
460C3—21no—
461C3#7986211yescharacteristic, complemented
462C3#8089211yescharacteristic, complemented
463C2—121no—
464C2—242no—
465C2#1059436711yescharacteristic
466C2#1059474412yes—
subgroupG/H
centerC22D32×D4
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#106C22
socleC22×C32C22×D4
prime radical#10590836C2
hypercenter#106D32
hypocenterC32#3183
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. #10594444C2semidirect
2. #10592557C2semidirect
3. C3×#4869C2semidirect
4. C3×#4867C2semidirect
5. #10590869C2semidirect
7. #10590812C2semidirect
8. #10590667C2semidirect
9. #10590539C2semidirect
10. #10590503C2semidirect
11. #10590478C2semidirect
53. #4869D3semidirect
54. #4867D3semidirect
413. C32#3183semidirect
461. C3#79862semidirect
462. C3#80892semidirect

Maximal subgroups

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

subgroupindex
#105944442
#105925572
C3×#48692
C3×#48672
#105908692
#105908362
#105908122
#105906672
#105905392
#105905032
#105904782
#808923
#798623