Finite Group Databasebeta

← #26133440

Subgroups — #26133440

Order 12032.

484 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256522C211yesmaximal, characteristic
2#20245123C211yesmaximal, characteristic, complemented
3#20245116C211yesmaximal, characteristic, complemented
4#20244276C211yesmaximal, characteristic, complemented
5#20241504C211yesmaximal, characteristic, complemented
6#20241470C211yesmaximal, characteristic, complemented
7C4×#135×C47C211yesmaximal, characteristic, complemented
8C2×#13094046C2211yescharacteristic
9#15689564—21no—
10#15689563C412yescomplemented
11#15689559—21no—
12#15689558C412yescomplemented
13#15689553—22no—
14C2×#13093964C2211yescharacteristic
15C2×#13093964C2211yescharacteristic
16C2×#13093960C2211yescharacteristic
17C2×#13093952—21no—
18#15689089—21no—
19#15688932—22no—
20#15688930—22no—
21C2×#135×C47C2211yescharacteristic
22C4×#41×C47—21no—
23C4×Q16×C47—21no—
24C4×SD16×C47—22no—
25C4×D8×C47—21no—
26C47×#427C2211yescharacteristic
27C47×#416—21no—
28C2×C4×C8×C47C2211yescharacteristic
29C2×#10625350—21no—
30#13094046—21no—
31#13094046—21no—
32#13094046—21no—
33#13094046—21no—
34C2×Dic188C2×C411yescharacteristic, complemented
35C2×#10625325—21no—
36C2×D376C2×C411yescharacteristic, complemented
37C2×#10625329—21no—
38#13093970—22no—
39#13093968—22no—
40#13093967—22no—
41#13093965—22no—
42#13093964—21no—
43#13093964—21no—
44#13093964—21no—
45#13093964C2×C412yescomplemented
46C2×C47⋊C16—21no—
47C2×C47⋊C16—21no—
48#13093960—21no—
49#13093960—21no—
50#13093952—41no—
51#13093952—41no—
52(C2×D47)⋊C8—42no—
53C2×#41×C47—21no—
54#135×C47—21no—
55#135×C47—21no—
56#135×C47C2×C412yes—
57#135×C47D412yes—
58C2×Q16×C47D411yescharacteristic
59C2×SD16×C47—21no—
60C2×D8×C47D411yescharacteristic
61C22×C8×C47D411yescharacteristic
62C22×C8×C47C2311yescharacteristic
63C4×Q8×C47—22no—
64C4×D4×C47—22no—
65C4×D4×C47—42no—
66#117×C47—21no—
67#117×C47—41no—
68C2×C42×C47—41no—
69C2×C42×C47D411yescharacteristic
70#107×C47—21no—
71#106×C47D412yes—
72#103×C47—22no—
73#102×C47—22no—
74C4×C8×C47—21no—
75C4×C8×C47—21no—
76#41×C47—22no—
77#41×C47—22no—
78#41×C47—41no—
79#41×C47—41no—
80C2×Q8×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—41no—
83C22×C4×C47—21no—
84C22×C4×C47—42no—
85C22×C4×C47—41no—
86C22×C4×C47C2×D411yescharacteristic
87C2×C22⋊D47—41no—
88#10625350—41no—
89#10625350—41no—
90#10625350—41no—
91#10625350—41no—
92C2×D188—21no—
93C2×C4×D47—41no—
94C2×Dic94—21no—
95Q16×C47—21no—
96Q16×C47C22⋊C412yes—
97SD16×C47—22no—
98SD16×C47—22no—
99D8×C47—21no—
100D8×C47C22⋊C412yes—
101C2×C8×C47—21no—
102C2×C8×C47—21no—
103C2×C8×C47—21no—
104C2×C8×C47C22×C411yescharacteristic
105C2×C8×C47C2×D411yescharacteristic
106C2×C8×C47C2×D411yescharacteristic
107C2×C8×C47C2×D411yescharacteristic
108C2×C8×C47#4111yescharacteristic
109C2×C8×C47#4111yescharacteristic
110C4⋊C4×C47—22no—
111C4⋊C4×C47—42no—
112C22⋊C4×C47—42no—
113C22⋊C4×C47—41no—
114C42×C47—21no—
115C42×C47—42no—
116C42×C47—41no—
117C42×C47D811yescharacteristic
118C42×C47D811yescharacteristic
119#10625329—41no—
120#10625329—41no—
121C2×C47⋊C8—41no—
122Dic188—21no—
123Dic188—21no—
124D376—21no—
125D376—21no—
126#10625325—41no—
127#10625325—41no—
128C47⋊C16—22no—
129C47⋊C16—41no—
130C23×C47—41no—
131C23×C47AGL(1,ℤ/8)11yescharacteristic
132C22×D47—41no—
133Q8×C47—22no—
134Q8×C47—41no—
135D4×C47—22no—
136D4×C47—42no—
137D4×C47—42no—
138D4×C47—41no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—42no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—84no—
147C2×C4×C47C4×D411yescharacteristic
148C2×C4×C47#12011yescharacteristic
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47C2×D811yescharacteristic
151C2×C4×C47#13511yescharacteristic
152C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
153C22⋊D47—81no—
154C22⋊D47—81no—
155C2×Dic47—41no—
156D188—41no—
157D188—41no—
158C4×D47—81no—
159C4×D47—81no—
160Dic94—41no—
161Dic94—41no—
162C8×C47—41no—
163C8×C47C2×C22⋊C411yescharacteristic
164C8×C47C4×D411yescharacteristic
165C8×C47#12111yescharacteristic
166C8×C47#12311yescharacteristic
167C47⋊C8—81no—
168#13027—471nomaximal, Sylow
169C22×C47—41no—
170C22×C47—42no—
171C22×C47—82no—
172C22×C47#44211yescharacteristic
173C22×C47#44711yescharacteristic
174C22×C47#46311yescharacteristic
175C2×D47—82no—
176C2×D47—81no—
177C4×C47—22no—
178C4×C47—41no—
179C4×C47—42no—
180C4×C47—82no—
181C4×C47#38611yescharacteristic
182C4×C47C4×D811yescharacteristic
183C4×C47#44911yescharacteristic
184C4×C47#45911yescharacteristic
185Dic47—81no—
186#3414—471no—
187C4×#135D4711yescharacteristic, complemented
188#2145—471no—
189#2143—471no—
190#1974—471no—
191#1389—471no—
192#1376—471no—
193C2×C47—21no—
194C2×C47—21no—
195C2×C47—21no—
196C2×C47—82no—
197C2×C47#187911yescharacteristic
198D47—161no—
199C2×#135—471no—
200C2×#135C2×D4711yescharacteristic
201C4×#41—21no—
202#510—472no—
203#510—941no—
204#509—472no—
205#509—941no—
206#508—942no—
207C2×M32—471no—
208C2×M32—471no—
209C4×Q16—21no—
210C4×SD16—22no—
211C4×D8—21no—
212C2×#108—471no—
213#427C2×D4711yescharacteristic
214C2×#104—941no—
215#416—21no—
216#408—941no—
217C2×C4×C8C2×D4711yescharacteristic
218#360—942no—
219#359—942no—
220C47#1302711yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—941no—
223#135—21no—
224#135—21no—
225#135—941no—
226#135—941no—
227#135—941no—
228#135—941no—
229#135C4×D4712yes—
230#135D18812yes—
231C2×Q16—471no—
232C2×Q16C22⋊D4711yescharacteristic
233C2×SD16—21no—
234C2×SD16—941no—
235C2×D8—471no—
236C2×D8C22⋊D4711yescharacteristic
237C2×M16—941no—
238C22×C8C22⋊D4711yescharacteristic
239C22×C8C22×D4711yescharacteristic
240C4×Q8—22no—
241C4×D4—22no—
242C4×D4—42no—
243#117—21no—
244#117—41no—
245C2×C42—41no—
246C2×C42C22⋊D4711yescharacteristic
247Q32—942no—
248SD32—942no—
249SD32—942no—
250D16—942no—
251M32—472no—
252M32—941no—
253M32—941no—
254M32—941no—
255C2×C16—941no—
256C2×C16—941no—
257#108—941no—
258#108—941no—
259#107—21no—
260#106C22⋊D4712yes—
261#104—1881no—
262#104—1881no—
263#103—22no—
264#102—22no—
265C22⋊C8—1882no—
266C4×C8—21no—
267C4×C8—21no—
268#41—22no—
269#41—22no—
270#41—41no—
271#41—41no—
272#41—1881no—
273#41—1881no—
274#41—1881no—
275#41—1881no—
276C2×Q8—21no—
277C2×Q8—941no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—941no—
281C2×D4—1881no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—1881no—
286C22×C4C2×C22⋊D4711yescharacteristic
287Q16—21no—
288Q16—941no—
289Q16—941no—
290Q16(C2×D47)⋊C412yes—
291SD16—22no—
292SD16—22no—
293SD16—1881no—
294SD16—1881no—
295D8—21no—
296D8—941no—
297D8—941no—
298D8(C2×D47)⋊C412yes—
299M16—1881no—
300M16—1881no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—1881no—
305C2×C8C2×C4×D4711yescharacteristic
306C2×C8C2×D18811yescharacteristic
307C2×C8#1062535011yescharacteristic
308C2×C8#1062535011yescharacteristic
309C2×C8C2×C22⋊D4711yescharacteristic
310C2×C8C2×C22⋊D4711yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C42—21no—
316C42—42no—
317C42—41no—
318C42#1062533411yescharacteristic
319C42#1062533411yescharacteristic
320C16—942no—
321C16—1881no—
322C23—41no—
323C23—1881no—
324C23#1309405011yescharacteristic
325Q8—22no—
326Q8—41no—
327Q8—1881no—
328Q8—1881no—
329D4—22no—
330D4—42no—
331D4—42no—
332D4—41no—
333D4—3761no—
334D4—3761no—
335D4—1881no—
336D4—1881no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—84no—
345C2×C4—3761no—
346C2×C4—3761no—
347C2×C4—1881no—
348C2×C4C4×C22⋊D4711yescharacteristic
349C2×C4#1309399311yescharacteristic
350C2×C4#1309399811yescharacteristic
351C2×C4C2×#1062533411yescharacteristic
352C2×C4#1309404411yescharacteristic
353C2×C4#1309404911yescharacteristic
354C8—41no—
355C8—3761no—
356C8C2×(C2×D47)⋊C411yescharacteristic
357C8C4×C22⋊D4711yescharacteristic
358C8#1309399811yescharacteristic
359C8#1309400711yescharacteristic
360C22—41no—
361C22—42no—
362C22—82no—
363C22—3762no—
364C22—3761no—
365C22#1568923311yescharacteristic
366C22#1568925711yescharacteristic
367C22#1568933811yescharacteristic
368C4—22no—
369C4—41no—
370C4—42no—
371C4—82no—
372C4—3761no—
373C4#1568900211yescharacteristic
374C4C4×#1062533411yescharacteristic
375C4#1568926811yescharacteristic
376C4#1568931811yescharacteristic
377C2—21no—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—7521no—
382C2#2024373611yescharacteristic
subgroupG/H
centerC4#15689318
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC4×#135×C47C2
socleC2×C47#1879
prime radical#20256522C2
cosocleC22×C8×C47C23
hypercenterC4×#135D47
hypocenterC47#13027
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
2. #20245123C2semidirect
3. #20245116C2semidirect
4. #20244276C2semidirect
5. #20241504C2semidirect
6. #20241470C2semidirect
7. C4×#135×C47C2semidirect
10. #15689563C4semidirect
12. #15689558C4semidirect
34. C2×Dic188C2×C4semidirect
36. C2×D376C2×C4semidirect
45. #13093964C2×C4semidirect
187. C4×#135D47semidirect
220. C47#13027semidirect

Maximal subgroups

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

subgroupindex
#202565222
#202451232
#202451162
#202442762
#202415042
#202414702
C4×#135×C472
#1302747