Finite Group Databasebeta

← #11788562

Subgroups — #11788562

Order 1152.

996 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×#41C211yesmaximal, characteristic, complemented
2#10596293C211yesmaximal, characteristic
3#10592935C211yesmaximal, characteristic, complemented
4#10592705C211yesmaximal, characteristic, complemented
5C2×#63632C211yesmaximal, characteristic, complemented
6#10589399C211yesmaximal, characteristic, complemented
7#10589389C211yesmaximal, characteristic, complemented
8#71645—34nomaximal
9C2×C32×#41C2211yescharacteristic
10C2×#3828C2211yescharacteristic
11C32×C4×Q8—21no—
12C32×C4×D4—21no—
13C32×C4×D4—22no—
14C32×#117—21no—
15C32×#117C2211yescharacteristic
16C2×C32×C42—21no—
17C2×C32×C42C2211yescharacteristic
18#64160C412yescomplemented
19#64159—22no—
20#64158C412yescomplemented
21#64151—21no—
22#64141—21no—
23#64135C2211yescharacteristic
24C2×#3748C2211yescharacteristic
25C2×#3748C2211yescharacteristic
26#63664—22no—
27#63633—22no—
28#63632—21no—
29#63632—21no—
30C3×C4×#41D314yescomplemented
31#5822—34no—
32#5236—34no—
33#5006—34no—
34C2×#805—34no—
35#4533—34no—
36#4523—34no—
37C32×#41—21no—
38C32×#41—21no—
39C32×#41C2×C412yes—
40C32×#41D412yes—
41C2×C32×Q8D411yescharacteristic
42C2×C32×D4—21no—
43C2×C32×D4D411yescharacteristic
44C22×C32×C4—21no—
45C22×C32×C4—22no—
46C22×C32×C4—21no—
47C22×C32×C4C2311yescharacteristic
48C2×#638—21no—
49#3828—21no—
50#3828—21no—
51#3828—21no—
52#3828—21no—
53C2×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
54C2×C4×C3⋊D3—21no—
55C2×#634C2×C411yescharacteristic, complemented
56C32×C4⋊C4—22no—
57C32×C4⋊C4—21no—
58C32×C22⋊C4—22no—
59C32×C22⋊C4D412yes—
60C32×C42—22no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42D411yescharacteristic
64C32×C42D411yescharacteristic
65#3757—21no—
66#3756—22no—
67#3755—22no—
68#3754—22no—
69#3753—22no—
70C4⋊(C3⋊Dic3)—21no—
71C4×C3⋊Dic3—21no—
72#3748—21no—
73#3748—21no—
74#3748—21no—
75#3748C2×C412yescomplemented
76C2×C3⋊(C3⋊C8)—21no—
77C2×C3⋊(C3⋊C8)—21no—
78#1906—91noSylow
79C2×C3×#41C2×D314yes—
80C2×#235—34no—
81C3×C4×Q8—24no—
82C3×C4×D4—24no—
83C3×C4×D4—28no—
84C3×#117—24no—
85C3×#117C2×D314yes—
86C2×C3×C42—24no—
87C2×C3×C42C2×D314yes—
88#951—38no—
89#950—68no—
90#949—38no—
91#942—64no—
92#932—64no—
93#926—34no—
94C2×#208—34no—
95C2×#208—34no—
96#837—68no—
97#806—68no—
98#805—64no—
99#805—64no—
100C23×C32—21no—
101C23×C32C2×D411yescharacteristic
102C22×C3⋊D3—21no—
103C32×Q8—21no—
104C32×Q8C22⋊C412yes—
105C32×D4—22no—
106C32×D4—22no—
107C32×D4—21no—
108C32×D4C22⋊C412yes—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—42no—
116C2×C32×C4—44no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123#638—41no—
124#638—41no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—41no—
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)—21no—
129C4×C3⋊D3—41no—
130C4×C3⋊D3—41no—
131#634—21no—
132#634—21no—
133C3⋊(C3⋊C8)—22no—
134C3⋊(C3⋊C8)—41no—
135#575—91no—
136C4×#41C3⋊D311yescharacteristic, complemented
137#421—91no—
138C2×#104—91no—
139#410—91no—
140#344—91no—
141#339—91no—
142C3×#41—24no—
143C3×#41—24no—
144C3×#41C4×D318yes—
145C3×#41D1218yes—
146C2×C3×Q8#6714yes—
147C2×C3×D4—24no—
148C2×C3×D4#6714yes—
149C22×C3×C4—24no—
150C22×C3×C4—28no—
151C22×C3×C4—24no—
152C22×C3×C4C22×D314yes—
153C2×#67—64no—
154#235—64no—
155#235—64no—
156#235—64no—
157#235—64no—
158C2×D12—34no—
159C2×C4×D3—64no—
160C2×Dic6—34no—
161C3×C4⋊C4—28no—
162C3×C4⋊C4—24no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4#6718yes—
165C3×C42—28no—
166C3×C42—24no—
167C3×C42—24no—
168C3×C42#6714yes—
169C3×C42#6714yes—
170#217—64no—
171#216—68no—
172#215—68no—
173#214—68no—
174#213—68no—
175C4⋊Dic3—64no—
176C4×Dic3—64no—
177#208—38no—
178#208—64no—
179#208—64no—
180#208—64no—
181C2×C3⋊C8—64no—
182C2×C3⋊C8—64no—
183C22×C32—22no—
184C22×C32—41no—
185C22×C32—42no—
186C22×C32C4×D411yescharacteristic
187C22×C32#12011yescharacteristic
188C22×C32#12111yescharacteristic
189C2×C3⋊D3—41no—
190C2×C3⋊D3—81no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—42no—
194C32×C4—41no—
195C32×C4C2×C22⋊C411yescharacteristic
196C32×C4C4×D411yescharacteristic
197C32×C4#12111yescharacteristic
198C32×C4#12311yescharacteristic
199C3⋊Dic3—41no—
200C3⋊Dic3—81no—
201C2×#41—91no—
202C2×#41C2×C3⋊D311yescharacteristic
203#137—92no—
204#137—181no—
205AGL(1,ℤ/8)—92no—
206AGL(1,ℤ/8)—181no—
207#135—182no—
208C2×M16—91no—
209C2×M16—91no—
210C4×Q8—21no—
211C4×D4—21no—
212C4×D4—22no—
213#117—21no—
214#117—91no—
215#117C2×C3⋊D311yescharacteristic
216C2×C42—21no—
217C2×C42C2×C3⋊D311yescharacteristic
218#104—182no—
219#104—181no—
220#104—181no—
221#99—182no—
222C23×C3—24no—
223C23×C3C2×#6714yes—
224C22×D3—64no—
225C3×Q8—24no—
226C3×Q8(C2×D3)⋊C418yes—
227C3×D4—28no—
228C3×D4—28no—
229C3×D4—24no—
230C3×D4(C2×D3)⋊C418yes—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—28no—
234C2×C3×C4—28no—
235C2×C3×C4—28no—
236C2×C3×C4—24no—
237C2×C3×C4—48no—
238C2×C3×C4—416no—
239C2×C3×C4C2×C4×D314yes—
240C2×C3×C4C2×D1214yes—
241C2×C3×C4#23514yes—
242C2×C3×C4#23514yes—
243C2×C3×C4C2×#6714yes—
244C2×C3×C4C2×#6714yes—
245#67—124no—
246#67—124no—
247C2×Dic3—64no—
248C2×Dic3—124no—
249D12—64no—
250D12—64no—
251C4×D3—124no—
252C4×D3—124no—
253Dic6—64no—
254Dic6—64no—
255C3⋊C8—68no—
256C3⋊C8—124no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—42no—
261C2×C32#38611yescharacteristic
262C3⋊D3—81no—
263#41—21no—
264#41—21no—
265#41—181no—
266#41—181no—
267#41—181no—
268#41—181no—
269#41C4×C3⋊D312yes—
270#41C4⋊(C3⋊D3)12yes—
271C2×Q8—91no—
272C2×Q8#63811yescharacteristic
273C2×D4—21no—
274C2×D4—91no—
275C2×D4—181no—
276C2×D4#63811yescharacteristic
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—181no—
281C22×C4C22×C3⋊D311yescharacteristic
282Q16—182no—
283SD16—182no—
284SD16—182no—
285D8—182no—
286M16—92no—
287M16—181no—
288M16—181no—
289M16—181no—
290C2×C8—181no—
291C2×C8—181no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—181no—
295C22⋊C4—22no—
296C22⋊C4—181no—
297C22⋊C4#63812yes—
298C42—22no—
299C42—21no—
300C42—21no—
301C42—181no—
302C42#63811yescharacteristic
303C42#63811yescharacteristic
304C22×C3—28no—
305C22×C3—44no—
306C22×C3—48no—
307C22×C3C4×#6714yes—
308C22×C3#93014yes—
309C22×C3#95314yes—
310C2×D3—124no—
311C2×D3—244no—
312C3×C4—28no—
313C3×C4—28no—
314C3×C4—48no—
315C3×C4—44no—
316C3×C4C2×(C2×D3)⋊C414yes—
317C3×C4C4×#6714yes—
318C3×C4#92914yes—
319C3×C4#93014yes—
320Dic3—124no—
321Dic3—244no—
322C32#190611yescharacteristic, complemented, Sylow
323C23—21no—
324C23—181no—
325C23C2×#63811yescharacteristic
326Q8—21no—
327Q8—181no—
328Q8—181no—
329Q8#375212yes—
330D4—22no—
331D4—22no—
332D4—21no—
333D4—181no—
334D4—181no—
335D4—361no—
336D4—361no—
337D4#375212yes—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—181no—
347C2×C4—361no—
348C2×C4—361no—
349C2×C4—361no—
350C2×C4C2×C4×C3⋊D311yescharacteristic
351C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
352C2×C4#382811yescharacteristic
353C2×C4#382811yescharacteristic
354C2×C4C2×#63811yescharacteristic
355C2×C4C2×#63811yescharacteristic
356C8—182no—
357C8—361no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—24no—
361C2×C3—48no—
362C2×C3#521314yes—
363D3—244no—
364C22—22no—
365C22—41no—
366C22—42no—
367C22—361no—
368C22—721no—
369C22C4×#63811yescharacteristic
370C22#6413911yescharacteristic
371C22#6416211yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—41no—
376C4—361no—
377C4—721no—
378C4C2×#375211yescharacteristic
379C4C4×#63811yescharacteristic
380C4#6413811yescharacteristic
381C4#6413911yescharacteristic
382C3#7164514yescomplemented
383C2—21no—
384C2—21no—
385C2—21no—
386C2—42no—
387C2—721no—
388C2#1059291211yescharacteristic
subgroupG/H
centerC4#64139
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×C4×#41C2
socleC2×C32#386
prime radical#10596293C2
cosocleC22×C32×C4C23
hypercenterC4×#41C3⋊D3
hypocenterC32#1906
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C32×C4×#41C2semidirect
3. #10592935C2semidirect
4. #10592705C2semidirect
5. C2×#63632C2semidirect
6. #10589399C2semidirect
7. #10589389C2semidirect
18. #64160C4semidirect
20. #64158C4semidirect
30. C3×C4×#41D3semidirect
53. C2×C4⋊(C3⋊D3)C2×C4semidirect
55. C2×#634C2×C4semidirect
75. #3748C2×C4semidirect
136. C4×#41C3⋊D3semidirect
322. C32#1906semidirect
382. C3#71645semidirect

Maximal subgroups

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

subgroupindex
C32×C4×#412
#105962932
#105929352
#105927052
C2×#636322
#105893992
#105893892
#716453