Finite Group Databasebeta

← #11861177

Subgroups — #11861177

Order 1152.

1047 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596219C211yesmaximal, characteristic
2#10594780C212yesmaximal, complemented
3C3×#5125C211yesmaximal, characteristic, complemented
4C3×Dic3×M16C211yesmaximal, characteristic, complemented
5C2×#63570C212yesmaximal, complemented
6#10590918C214yesmaximal, complemented
7#10590917C214yesmaximal, complemented
8#79258—31nomaximal
9#78541—31nomaximal
10C2×#3828C2212yes—
11C2×C32×M16C2211yescharacteristic
12#64118C414yescomplemented
13#64117C414yescomplemented
14#64113C2214yes—
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×#208C2212yes—
17C2×C3×#208C2211yescharacteristic
18#63818—24no—
19#63817—24no—
20#63817C2218yescomplemented
21C3×#819C2214yescomplemented
22C3×#815C2212yescomplemented
23C3×C8×Dic3C2212yescomplemented
24#63570C2214yescomplemented
25#63570C2214yescomplemented
26#5748—31no—
27#5748—31no—
28#5748—61no—
29#5745—32no—
30#5745—32no—
31C3×#430—31no—
32C3×C4×M16—31no—
33C2×#825—32no—
34#5125D311yescharacteristic, complemented
35Dic3×M16D311yescharacteristic, complemented
36C2×#805—32no—
37#4895—34no—
38#4894—34no—
39#4717—34no—
40#4700—34no—
41C22×C32×C4C2311yescharacteristic
42C2×#638—22no—
43#3828—22no—
44#3828—22no—
45#3828C2×C414yes—
46#3828C2314yes—
47C2×C4⋊(C3⋊D3)C2×C412yes—
48C2×C4×C3⋊D3—22no—
49C2×#634C2×C412yes—
50C22×C3×Dic3—21no—
51C32×M16C2×C414yes—
52C2×C32×C8C2312yes—
53#3746C2×C414yescomplemented
54#3745C2×C414yescomplemented
55#3744C2×C418yescomplemented
56C3×C4×Dic3—21no—
57C3×C4×Dic3C2311yescharacteristic
58C3×C4×Dic3C2311yescharacteristic
59C3×#208—22no—
60C3×#208D414yes—
61C3×#208C2314yes—
62C2×C3×C3⋊C8—22no—
63C2×C3×C3⋊C8C2312yes—
64#3712—28no—
65#3710—28no—
66#2956—91noSylow
67C2×#235—32no—
68C2×#235—32no—
69C2×#235—62no—
70C2×C3×M16—21no—
71C2×C3×M16—31no—
72C2×C3×M16—32no—
73C2×C3×M16C2×D311yescharacteristic
74C2×C3×M16C2×D311yescharacteristic
75C2×C3×C42—31no—
76C2×C4×Dic3C2×D311yescharacteristic
77C2×#208C2×D312yes—
78C2×#208C2×D311yescharacteristic
79#909—34no—
80#909—34no—
81#909—64no—
82#908—34no—
83#908—34no—
84#908—64no—
85#907—38no—
86#907—64no—
87D3×M16—64no—
88D3×M16—64no—
89#904—34no—
90#904—34no—
91#904—64no—
92#901—38no—
93#901—64no—
94C3×#108—34no—
95C3×#97—32no—
96C3×C4×C8—32no—
97#825—34no—
98#825—34no—
99#819C2×D314yescomplemented
100#815C2×D312yescomplemented
101C8×Dic3C2×D312yescomplemented
102#805—34no—
103#805—34no—
104C23×C32#4111yescharacteristic
105C22×C3⋊D3—22no—
106C2×C32×C4C22×C412yes—
107C2×C32×C4C2×D412yes—
108C2×C32×C4#4111yescharacteristic
109C2×C32×C4C2411yescharacteristic
110#638—24no—
111#638—24no—
112C2×C3⋊Dic3—22no—
113C4⋊(C3⋊D3)—22no—
114C4⋊(C3⋊D3)C22×C414yes—
115C4×C3⋊D3—24no—
116C4×C3⋊D3—24no—
117#634—22no—
118#634C22×C414yes—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—42no—
122C32×C8C22×C414yes—
123C3×C3⋊C8—24no—
124C3×C3⋊C8C2×D414yes—
125#575—91no—
126#568—92no—
127#444—94no—
128#443—94no—
129#430—31no—
130C2×#104—92no—
131C4×M16—31no—
132C22×C3×C4—21no—
133C22×C3×C4—61no—
134C22×C3×C4C22×D311yescharacteristic
135C22×C3×C4C22×D311yescharacteristic
136C2×#67—62no—
137C2×#67—62no—
138C2×#67—122no—
139C22×Dic3—21no—
140#235—34no—
141#235—34no—
142#235—34no—
143#235—34no—
144#235—64no—
145#235—64no—
146#235—62no—
147#235—62no—
148#235—62no—
149#235—62no—
150#235—122no—
151#235—122no—
152C2×D12—32no—
153C2×D12—32no—
154C2×D12—62no—
155C2×C4×D3—62no—
156C2×C4×D3—62no—
157C2×C4×D3—122no—
158C2×Dic6—32no—
159C2×Dic6—32no—
160C2×Dic6—62no—
161C3×M16—24no—
162C3×M16—34no—
163C3×M16—34no—
164C3×M16—62no—
165C3×M16C4×D314yes—
166C3×M16C4×D314yes—
167C2×C3×C8—22no—
168C2×C3×C8—32no—
169C2×C3×C8—62no—
170C2×C3×C8C22×D312yes—
171C2×C3×C8C22×D312yes—
172C3×C42—31no—
173C3×C42—31no—
174C3×C42—61no—
175C4×Dic3—21no—
176C4×Dic3C22×D311yescharacteristic
177C4×Dic3C22×D311yescharacteristic
178#208—22no—
179#208D1214yes—
180#208C22×D314yes—
181C2×C3⋊C8—22no—
182C2×C3⋊C8C22×D312yes—
183Dic12—34no—
184Dic12—34no—
185Dic12—64no—
186D24—34no—
187D24—34no—
188D24—64no—
189#204—38no—
190#204—38no—
191#204—68no—
192ΓL(1,𝔽25)—68no—
193ΓL(1,𝔽25)—68no—
194D3×C8—68no—
195D3×C8—68no—
196C22×C32—41no—
197C22×C32C4×D412yes—
198C22×C32C2×#4111yescharacteristic
199C2×C3⋊D3—24no—
200C2×C3⋊D3—82no—
201C32×C4C4×D412yes—
202C32×C4C23×C411yescharacteristic
203C32×C4C22×D411yescharacteristic
204C3⋊Dic3—24no—
205C3×Dic3—22no—
206C3×Dic3—41no—
207C2×#41—92no—
208#137—94no—
209AGL(1,ℤ/8)—94no—
210#135—94no—
211#131—98no—
212#131—184no—
213C2×M16—31no—
214C2×M16—32no—
215C2×M16—184no—
216C2×M16D3211yescharacteristic
217C2×C42—31no—
218#108—34no—
219#104—94no—
220#104—94no—
221#97—32no—
222C4×C8—32no—
223C23×C3—21no—
224C23×C3#23511yescharacteristic
225C23×C3#23911yescharacteristic
226C22×D3—62no—
227C22×D3—62no—
228C22×D3—122no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—122no—
236C2×C3×C4C2×C4×D312yes—
237C2×C3×C4C2×C4×D312yes—
238C2×C3×C4C2×D1212yes—
239C2×C3×C4#23511yescharacteristic
240C2×C3×C4D3×D412yes—
241C2×C3×C4#23911yescharacteristic
242C2×C3×C4C23×D311yescharacteristic
243C2×C3×C4C23×D311yescharacteristic
244#67—64no—
245#67—64no—
246#67—64no—
247#67—64no—
248#67—124no—
249#67—124no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—42no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—122no—
256D12—34no—
257D12—34no—
258D12—64no—
259D12—62no—
260D12—62no—
261D12—122no—
262C4×D3—64no—
263C4×D3—64no—
264C4×D3—64no—
265C4×D3—64no—
266C4×D3—124no—
267C4×D3—124no—
268Dic6—34no—
269Dic6—34no—
270Dic6—64no—
271Dic6—62no—
272Dic6—62no—
273Dic6—122no—
274C3×C8—24no—
275C3×C8—34no—
276C3×C8—64no—
277C3×C8C2×C4×D314yes—
278C3×C8C2×C4×D314yes—
279C3⋊C8—24no—
280C3⋊C8C2×D1214yes—
281C2×C32—21no—
282C2×C32—22no—
283C2×C32C2×C4×D411yescharacteristic
284C3⋊D3—44no—
285#41—94no—
286#41—94no—
287#41—182no—
288#41—182no—
289C2×Q8—92no—
290C2×D4—92no—
291C2×D4—182no—
292C22×C4—61no—
293C22×C4—182no—
294C22×C4C2×D3211yescharacteristic
295Q16—94no—
296SD16—98no—
297D8—94no—
298M16—34no—
299M16—34no—
300M16—62no—
301M16—188no—
302M16#62414yes—
303C2×C8—32no—
304C2×C8—62no—
305C2×C8—188no—
306C2×C8C2×D3212yes—
307C42—31no—
308C42—31no—
309C42—61no—
310C22×C3—21no—
311C22×C3—22no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—81no—
315C22×C3C4×D1212yes—
316C22×C3#89312yes—
317C22×C3C2×#23511yescharacteristic
318C22×C3C2×#23911yescharacteristic
319C2×D3—64no—
320C2×D3—64no—
321C2×D3—124no—
322C2×D3—242no—
323C2×D3—242no—
324C2×D3—244no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—22no—
328C3×C4—62no—
329C3×C4—121no—
330C3×C4C4×D1212yes—
331C3×C4#89312yes—
332C3×C4C22×C4×D311yescharacteristic
333C3×C4C22×C4×D311yescharacteristic
334C3×C4C22×D1211yescharacteristic
335C3×C4C2×D3×D411yescharacteristic
336Dic3—22no—
337Dic3—41no—
338Dic3—64no—
339Dic3—64no—
340Dic3—124no—
341C32#295611yescharacteristic, complemented, Sylow
342C23—182no—
343C23#379911yescharacteristic
344Q8—94no—
345Q8—182no—
346D4—94no—
347D4—184no—
348D4—184no—
349D4—182no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—122no—
353C2×C4—182no—
354C2×C4—184no—
355C2×C4—184no—
356C2×C4#379911yescharacteristic
357C2×C4D3×D1212yes—
358C2×C4C2×#62412yes—
359C2×C4C22×D3211yescharacteristic
360C8—34no—
361C8—64no—
362C8C2×#62414yes—
363C2×C3—21no—
364C2×C3—22no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—42no—
369C2×C3—41no—
370C2×C3C2×C4×D1211yescharacteristic
371C2×C3C2×#89311yescharacteristic
372D3—124no—
373D3—124no—
374D3—244no—
375C22—41no—
376C22—184no—
377C22—722no—
378C22#6389412yes—
379C22C2×#379911yescharacteristic
380C4—62no—
381C4—121no—
382C4—184no—
383C4#6389412yes—
384C4C2×D3×D1211yescharacteristic
385C4C22×#62411yescharacteristic
386C3—21no—
387C3#7854111yescharacteristic, complemented
388C3#7925811yescharacteristic, complemented
389C2—21no—
390C2—22no—
391C2—364no—
392C2C2×#6389411yescharacteristic
subgroupG/H
centerC4C2×D3×D12
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC2×C32C2×C4×D4
prime radical#10596219C2
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2956
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10594780C2semidirect
3. C3×#5125C2semidirect
4. C3×Dic3×M16C2semidirect
5. C2×#63570C2semidirect
6. #10590918C2semidirect
7. #10590917C2semidirect
12. #64118C4semidirect
13. #64117C4semidirect
20. #63817C22semidirect
21. C3×#819C22semidirect
22. C3×#815C22semidirect
23. C3×C8×Dic3C22semidirect
24. #63570C22semidirect
25. #63570C22semidirect
34. #5125D3semidirect
35. Dic3×M16D3semidirect
53. #3746C2×C4semidirect
54. #3745C2×C4semidirect
55. #3744C2×C4semidirect
99. #819C2×D3semidirect
100. #815C2×D3semidirect
101. C8×Dic3C2×D3semidirect
341. C32#2956semidirect
387. C3#78541semidirect
388. C3#79258semidirect

Maximal subgroups

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

subgroupindex
#105962192
#105947802
C3×#51252
C3×Dic3×M162
C2×#635702
#105909182
#105909172
#792583
#785413