Finite Group Databasebeta

← #11860476

Subgroups — #11860476

Order 1152.

1518 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#518C211yesmaximal, characteristic, complemented
2C2×#64141C211yesmaximal, characteristic
3#10595994C211yesmaximal, characteristic, complemented
4C2×#63659C211yesmaximal, characteristic, complemented
5#10592717C212yesmaximal, complemented
6C4×#3754C212yesmaximal, complemented
7#10592713C212yesmaximal, complemented
8C4×#3753C212yesmaximal, complemented
9#10592701C211yesmaximal, characteristic, complemented
10#10592696C211yesmaximal, characteristic, complemented
11#10592694C211yesmaximal, characteristic, complemented
12#82537—34nomaximal
13C22×C32×D4C2211yescharacteristic
14C2×#3828C2211yescharacteristic
15C32×C4×D4—21no—
16C32×C4×D4—21no—
17C32×C4×D4C2212yes—
18C32×C4×D4C2212yes—
19C32×#117C2211yescharacteristic
20C32×#117C2211yescharacteristic
21C2×C32×C22⋊C4—22no—
22C2×#3754C2212yes—
23#64141C418yescomplemented
24C2×#3753C2212yes—
25C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C2×#3748C2211yescharacteristic
27C22×C3⋊(C3⋊C8)C2211yescharacteristic
28#64101—21no—
29C4⋊C4×C3⋊D3—21no—
30#64090—22no—
31C4×C4⋊(C3⋊D3)C2212yescomplemented
32C4×#634C2212yescomplemented
33#63659C2212yes—
34#63659C2212yes—
35#63637C2211yescharacteristic, complemented
36#63637C2211yescharacteristic, complemented
37#63636C2211yescharacteristic, complemented
38#63636C2211yescharacteristic, complemented
39#63635C2212yescomplemented
40#63634C2212yescomplemented
41#63630C2212yescomplemented
42C4×C3⋊(C3⋊C8)C2212yescomplemented
43C3×#518D314yescomplemented
44C2×#932—34no—
45#5523—34no—
46C2×#832—34no—
47#5018—38no—
48C4×#214—38no—
49#5014—38no—
50C4×#213—38no—
51#5002—34no—
52#4997—34no—
53#4995—34no—
54C24×C32—21no—
55C2×C32×D4—21no—
56C2×C32×D4—22no—
57C2×C32×D4C2×C414yescomplemented
58C2×C32×D4C2312yes—
59C22×C32×C4—22no—
60C22×C32×C4—22no—
61C22×C32×C4C2311yescharacteristic
62C2×#638—21no—
63C22×C3⋊Dic3—21no—
64#3828—22no—
65#3828C2×C414yescomplemented
66C2×C4⋊(C3⋊D3)C2311yescharacteristic
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3—22no—
69C2×#634C2311yescharacteristic
70C32×C4⋊C4D411yescharacteristic
71C32×C4⋊C4D411yescharacteristic
72C32×C4⋊C4C2311yescharacteristic
73C32×C4⋊C4C2311yescharacteristic
74C32×C22⋊C4—21no—
75C32×C22⋊C4—22no—
76C32×C22⋊C4—22no—
77C32×C22⋊C4D412yes—
78C32×C42—21no—
79C32×C42C2312yes—
80#3754C2×C418yescomplemented
81#3753C2×C418yescomplemented
82#3752—22no—
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)C2312yes—
85#3750—22no—
86C4×C3⋊Dic3—22no—
87#3748C2×C414yescomplemented
88C2×C3⋊(C3⋊C8)—22no—
89C2×C3⋊(C3⋊C8)C2311yescharacteristic
90C2×C3⋊(C3⋊C8)C2311yescharacteristic
91C2×C3⋊(C3⋊C8)C2312yes—
92#2946—91noSylow
93C22×C3×D4C2×D314yes—
94C2×#235—34no—
95C3×C4×D4—24no—
96C3×C4×D4—24no—
97C3×C4×D4C2×D318yes—
98C3×C4×D4C2×D318yes—
99C3×#117C2×D314yes—
100C3×#117C2×D314yes—
101C2×C3×C22⋊C4—28no—
102C2×#214—38no—
103#932—332no—
104C2×#213—38no—
105C2×C4⋊Dic3—34no—
106C2×#208—34no—
107C22×C3⋊C8—34no—
108#892—64no—
109D3×C4⋊C4—64no—
110#67⋊C4—68no—
111C4×D12—38no—
112C4×Dic6—38no—
113#832—38no—
114#832—38no—
115#810—34no—
116#810—34no—
117#809—34no—
118#809—34no—
119#808—38no—
120#807—38no—
121#803—38no—
122C4×C3⋊C8—38no—
123C23×C32—22no—
124C23×C32—24no—
125C23×C32—81no—
126C23×C32—81no—
127C23×C32C2×D411yescharacteristic
128C22×C3⋊D3—21no—
129C32×D4—24no—
130C32×D4—22no—
131C32×D4C22×C414yes—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—82no—
135C2×C32×C4—82no—
136C2×C32×C4C22×C412yes—
137C2×C32×C4C2×D412yes—
138C2×C32×C4C2×D412yes—
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4#4112yes—
141C2×C32×C4C2411yescharacteristic
142#638—24no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—81no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)C22×C412yes—
148C4×C3⋊D3—24no—
149C4×C3⋊D3—82no—
150#634—21no—
151#634C22×C412yes—
152C3⋊(C3⋊C8)—21no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)C22×C414yes—
155C2×AGL(1,ℤ/8)—91no—
156#520—91no—
157#518C3⋊D311yescharacteristic, complemented
158#442—92no—
159#440—92no—
160C4×SD16—92no—
161C4×D8—92no—
162#428—91no—
163#416—91no—
164C2×#102—91no—
165#405—91no—
166C24×C3—24no—
167C2×C3×D4—28no—
168C2×C3×D4—24no—
169C2×C3×D4C4×D3116yescomplemented
170C2×C3×D4C22×D318yes—
171C22×C3×C4—28no—
172C22×C3×C4—28no—
173C22×C3×C4C22×D314yes—
174C2×#67—64no—
175C22×Dic3—64no—
176#235—316no—
177#235—68no—
178C2×D12—34no—
179C2×C4×D3—64no—
180C2×C4×D3—68no—
181C2×Dic6—34no—
182C3×C4⋊C4#6714yes—
183C3×C4⋊C4#6714yes—
184C3×C4⋊C4C22×D314yes—
185C3×C4⋊C4C22×D314yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—28no—
188C3×C22⋊C4—28no—
189C3×C22⋊C4#6718yes—
190C3×C42—24no—
191C3×C42C22×D318yes—
192#214—332no—
193#213—332no—
194(C2×D3)⋊C4—68no—
195C4⋊Dic3—38no—
196C4⋊Dic3—64no—
197Dic3⋊C4—68no—
198C4×Dic3—68no—
199#208—316no—
200C2×C3⋊C8—38no—
201C2×C3⋊C8—34no—
202C2×C3⋊C8—34no—
203C2×C3⋊C8—68no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—24no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32—84no—
210C22×C32C4×D412yes—
211C22×C32C22×D411yescharacteristic
212C2×C3⋊D3—22no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C32×C4—22no—
216C32×C4—22no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4C4×D412yes—
220C32×C4C23×C411yescharacteristic
221C32×C4C2×#4111yescharacteristic
222C3⋊Dic3—22no—
223C3⋊Dic3—42no—
224C2×#41—91no—
225C22×D4C2×C3⋊D311yescharacteristic
226AGL(1,ℤ/8)—98no—
227C2×SD16—92no—
228C2×D8—92no—
229C2×M16—91no—
230C22×C8—91no—
231C4×Q8—92no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—92no—
235C4×D4—182no—
236C4×D4C2×C3⋊D312yes—
237C4×D4C2×C3⋊D312yes—
238#117—181no—
239#117C2×C3⋊D311yescharacteristic
240#117C2×C3⋊D311yescharacteristic
241C2×C4⋊C4—91no—
242C2×C4⋊C4—181no—
243C2×C22⋊C4—22no—
244#107—92no—
245#106—92no—
246#103—91no—
247#103—91no—
248#102—92no—
249#102—92no—
250#102—91no—
251#102—91no—
252#97—92no—
253C4×C8—92no—
254C23×C3—28no—
255C23×C3—216no—
256C23×C3—84no—
257C23×C3—84no—
258C23×C3C2×#6714yes—
259C22×D3—64no—
260C3×D4—216no—
261C3×D4—28no—
262C3×D4C2×C4×D3116yes—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—88no—
266C2×C3×C4—88no—
267C2×C3×C4C2×C4×D318yes—
268C2×C3×C4#23518yes—
269C2×C3×C4C2×#6718yes—
270C2×C3×C4C2×#6718yes—
271C2×C3×C4C2×#6714yes—
272C2×C3×C4C23×D314yes—
273#67—616no—
274C2×Dic3—64no—
275C2×Dic3—68no—
276C2×Dic3—244no—
277D12—38no—
278D12—64no—
279C4×D3—616no—
280C4×D3—248no—
281Dic6—38no—
282Dic6—64no—
283C3⋊C8—316no—
284C3⋊C8—64no—
285C3⋊C8—64no—
286C2×C32—22no—
287C2×C32—44no—
288C2×C32C2×C4×D411yescharacteristic
289C2×C32#57611yescharacteristic
290C2×C32#57711yescharacteristic
291C3⋊D3—42no—
292C24—21no—
293#41—94no—
294#41—182no—
295C2×Q8—91no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—91no—
299C2×D4—181no—
300C2×D4C4×C3⋊D314yescomplemented
301C2×D4C22×C3⋊D312yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—181no—
305C22×C4—181no—
306C22×C4—182no—
307C22×C4C22×C3⋊D311yescharacteristic
308SD16—98no—
309D8—98no—
310M16—94no—
311C2×C8—92no—
312C2×C8—91no—
313C2×C8—91no—
314C2×C8—182no—
315C4⋊C4—92no—
316C4⋊C4—181no—
317C4⋊C4—182no—
318C4⋊C4#63811yescharacteristic
319C4⋊C4#63811yescharacteristic
320C4⋊C4C22×C3⋊D311yescharacteristic
321C4⋊C4C22×C3⋊D311yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—182no—
326C22⋊C4#63812yes—
327C42—21no—
328C42—182no—
329C42C22×C3⋊D312yes—
330C22×C3—24no—
331C22×C3—24no—
332C22×C3—216no—
333C22×C3—48no—
334C22×C3—48no—
335C22×C3—816no—
336C22×C3C4×#6718yes—
337C22×C3C22×#6714yes—
338C2×D3—68no—
339C2×D3—124no—
340C2×D3—124no—
341C3×C4—28no—
342C3×C4—28no—
343C3×C4—44no—
344C3×C4—44no—
345C3×C4C4×#6718yes—
346C3×C4C22×C4×D314yes—
347C3×C4C2×#23514yes—
348Dic3—68no—
349Dic3—128no—
350C32#294611yescharacteristic, complemented, Sylow
351C23—22no—
352C23—24no—
353C23—81no—
354C23—81no—
355C23—181no—
356C23C2×#63811yescharacteristic
357Q8—92no—
358Q8—181no—
359D4—22no—
360D4—24no—
361D4—92no—
362D4—184no—
363D4—181no—
364D4C2×C4×C3⋊D314yes—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—181no—
370C2×C4—184no—
371C2×C4—182no—
372C2×C4—722no—
373C2×C4—721no—
374C2×C4C2×C4×C3⋊D312yes—
375C2×C4#382812yes—
376C2×C4C2×#63812yes—
377C2×C4C2×#63812yes—
378C2×C4C2×#63811yescharacteristic
379C2×C4C23×C3⋊D311yescharacteristic
380C8—94no—
381C8—181no—
382C8—181no—
383C2×C3—28no—
384C2×C3—416no—
385C2×C3C2×C4×#6714yes—
386C2×C3#583514yes—
387C2×C3#583614yes—
388D3—128no—
389C22—21no—
390C22—21no—
391C22—24no—
392C22—42no—
393C22—42no—
394C22—84no—
395C22—182no—
396C22—361no—
397C22—361no—
398C22C4×#63812yes—
399C22C22×#63811yescharacteristic
400C4—22no—
401C4—22no—
402C4—41no—
403C4—41no—
404C4—182no—
405C4—362no—
406C4C4×#63812yes—
407C4C22×C4×C3⋊D311yescharacteristic
408C4C2×#382811yescharacteristic
409C3#8253714yescomplemented
410C2—22no—
411C2—44no—
412C2—362no—
413C2C2×C4×#63811yescharacteristic
414C2#1059630611yescharacteristic
415C2#1059630711yescharacteristic
subgroupG/H
centerC22C22×#638
commutatorC32×C4C23×C4
FrattiniC2×C4C23×C3⋊D3
FittingC32×#518C2
socleC22×C32C22×D4
prime radicalC2×#64141C2
cosocleC2×C32×C4C24
hypercenter#518C3⋊D3
hypocenterC32#2946
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. C32×#518C2semidirect
3. #10595994C2semidirect
4. C2×#63659C2semidirect
5. #10592717C2semidirect
6. C4×#3754C2semidirect
7. #10592713C2semidirect
8. C4×#3753C2semidirect
9. #10592701C2semidirect
10. #10592696C2semidirect
11. #10592694C2semidirect
23. #64141C4semidirect
31. C4×C4⋊(C3⋊D3)C22semidirect
32. C4×#634C22semidirect
35. #63637C22semidirect
36. #63637C22semidirect
37. #63636C22semidirect
38. #63636C22semidirect
39. #63635C22semidirect
40. #63634C22semidirect
41. #63630C22semidirect
42. C4×C3⋊(C3⋊C8)C22semidirect
43. C3×#518D3semidirect
57. C2×C32×D4C2×C4semidirect
65. #3828C2×C4semidirect
80. #3754C2×C4semidirect
81. #3753C2×C4semidirect
87. #3748C2×C4semidirect
157. #518C3⋊D3semidirect
169. C2×C3×D4C4×D3semidirect
300. C2×D4C4×C3⋊D3semidirect
350. C32#2946semidirect
409. C3#82537semidirect

Maximal subgroups

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

subgroupindex
C32×#5182
C2×#641412
#105959942
C2×#636592
#105927172
C4×#37542
#105927132
C4×#37532
#105927012
#105926962
#105926942
#825373