Finite Group Databasebeta

← #26113136

Subgroups — #26113136

Order 12032.

580 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252868C211yesmaximal, characteristic
2#20252499C211yesmaximal, characteristic, complemented
3#20244315C211yesmaximal, characteristic, complemented
4#20242057C211yesmaximal, characteristic, complemented
5#20241291C211yesmaximal, characteristic, complemented
6#20241262C211yesmaximal, characteristic, complemented
7C47×#1806C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C2×#13094021C2211yescharacteristic
10#15689323—21no—
11#15689323C412yescomplemented
12#15689318—21no—
13#15689318C412yescomplemented
14Q8×C47⋊C8—21no—
15D4×C47⋊C8—22no—
16D4×C47⋊C8—21no—
17#15689176—21no—
18#15689176C2211yescharacteristic
19#15689147C2211yescharacteristic
20C2×C4⋊(C47⋊C8)—21no—
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22#15689127—21no—
23C2×#13093944—22no—
24#15688866—21no—
25#15688866—21no—
26C4×#41×C47C2211yescharacteristic
27C47×#423C2211yescharacteristic
28C2×#104×C47—21no—
29C4×C4⋊C4×C47C2211yescharacteristic
30C47×#339—21no—
31C22×D188—21no—
32C22×D188—21no—
33C22×D188—21no—
34#13094049—42no—
35#13094049—42no—
36C2×#10625336—22no—
37C2×#10625334—22no—
38C2×#10625329—21no—
39C22×C47⋊C8—22no—
40C22×C47⋊C8D412yes—
41#13094021—21no—
42#13094021—21no—
43#13094021—21no—
44#13094021C2×C412yes—
45C4⋊(C47⋊C8)—22no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)C2×C412yes—
50C4⋊(C47⋊C8)D411yescharacteristic
51C4⋊(C47⋊C8)D411yescharacteristic
52#13093944—22no—
53#13093944—22no—
54#13093944—44no—
55C22⋊(C47⋊C8)—22no—
56C22⋊(C47⋊C8)—21no—
57C4×C47⋊C8—22no—
58C4×C47⋊C8—21no—
59C2×#41×C47D411yescharacteristic
60C2×M16×C47—21no—
61C4×Q8×C47C2×C411yescharacteristic
62C4×Q8×C47D411yescharacteristic
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47C2×C411yescharacteristic
66C4×D4×C47D411yescharacteristic
67#117×C47—21no—
68#117×C47D411yescharacteristic
69C2×C4⋊C4×C47—21no—
70C2×C42×C47—21no—
71C2×C42×C47—21no—
72C2×C42×C47C2311yescharacteristic
73C4⋊C8×C47—21no—
74C4⋊C8×C47—21no—
75#104×C47—42no—
76#95×C47—21no—
77C23×D47—41no—
78#41×C47—21no—
79#41×C47—21no—
80#41×C47D812yes—
81#41×C47SD1612yes—
82C2×Q8×C47C22⋊C411yescharacteristic
83C2×D4×C47—21no—
84C2×D4×C47C22⋊C411yescharacteristic
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—41no—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47#4111yescharacteristic
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—22no—
95C2×D188—42no—
96C2×D188—42no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—41no—
100C2×D188—41no—
101C2×D188—41no—
102C2×D188—41no—
103C2×D188—41no—
104C2×D188—82no—
105M16×C47—41no—
106M16×C47—41no—
107C2×C8×C47—41no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47C22⋊C411yescharacteristic
113C4⋊C4×C47C22⋊C411yescharacteristic
114C22⋊C4×C47—21no—
115C22⋊C4×C47—21no—
116C22⋊C4×C47—21no—
117C42×C47—21no—
118C42×C47—21no—
119C42×C47—21no—
120C42×C47—41no—
121C42×C47—41no—
122C42×C47C22×C411yescharacteristic
123C42×C47C2×D411yescharacteristic
124C42×C47C2×D411yescharacteristic
125C42×C47#4111yescharacteristic
126#10625336—44no—
127#10625334—44no—
128#10625329—41no—
129#10625329—41no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—21no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—21no—
134C2×C47⋊C8—42no—
135C2×C47⋊C8—82no—
136C23×C47—21no—
137C23×C47#12111yescharacteristic
138C22×D47—42no—
139C22×D47—82no—
140C22×D47—82no—
141C22×D47—81no—
142C22×D47—81no—
143Q8×C47—21no—
144Q8×C47#10212yescomplemented
145D4×C47—22no—
146D4×C47—22no—
147D4×C47—21no—
148D4×C47#10212yescomplemented
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—22no—
153C2×C4×C47—22no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—42no—
160C2×C4×C47—81no—
161C2×C4×C47—81no—
162C2×C4×C47—81no—
163C2×C4×C47C2×C22⋊C411yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47C4×D411yescharacteristic
166C2×C4×C47#12011yescharacteristic
167C2×C4×C47#12111yescharacteristic
168C2×C4×C47#12311yescharacteristic
169C2×C4×C47C2×D811yescharacteristic
170C2×C4×C47C2×SD1611yescharacteristic
171C2×C4×C47#13511yescharacteristic
172C2×C4×C47#13511yescharacteristic
173D188—44no—
174D188—42no—
175D188—42no—
176D188—82no—
177D188—82no—
178D188—81no—
179D188—81no—
180D188—81no—
181D188—81no—
182C8×C47—81no—
183C47⋊C8—42no—
184C47⋊C8—42no—
185C47⋊C8—41no—
186C47⋊C8—41no—
187#9318—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—22no—
191C22×C47—41no—
192C22×C47—41no—
193C22×C47#38611yescharacteristic
194C22×C47#46511yescharacteristic
195C22×C47#46611yescharacteristic
196C2×D47—82no—
197C2×D47—82no—
198C2×D47—82no—
199C2×D47—162no—
200C2×D47—162no—
201C4×C47—21no—
202C4×C47—22no—
203C4×C47—21no—
204C4×C47—21no—
205C4×C47—41no—
206C4×C47—41no—
207C4×C47—82no—
208C4×C47C2×#10211yescharacteristic
209C4×C47#41611yescharacteristic
210C4×C47#45311yescharacteristic
211C4×C47#45411yescharacteristic
212C4×C47#46511yescharacteristic
213C4×C47#46611yescharacteristic
214#3038—471no—
215#2988—471no—
216#1982—471no—
217#1806D4711yescharacteristic, complemented
218#1545—471no—
219#1303—471no—
220#1292—471no—
221C2×C47—21no—
222C2×C47—21no—
223C2×C47—42no—
224C2×C47#188311yescharacteristic
225C2×C47#189311yescharacteristic
226C2×C47#190311yescharacteristic
227D47—162no—
228C2×AGL(1,ℤ/8)—941no—
229C2×#127—471no—
230C4×#41C2×D4711yescharacteristic
231#460—472no—
232#460—941no—
233#459—472no—
234#459—941no—
235C8×Q8—941no—
236C8×D4—942no—
237C8×D4—941no—
238#431—471no—
239#431—941no—
240#423—471no—
241#423C2×D4711yescharacteristic
242C2×C4⋊C8—471no—
243C2×C4⋊C8—941no—
244C2×#104—21no—
245#418—941no—
246C2×#102—942no—
247C4×C4⋊C4C2×D4711yescharacteristic
248#339—21no—
249#331—941no—
250#331—941no—
251C47#931811yescharacteristic, complemented, Sylow
252C2×#41C22⋊D4711yescharacteristic
253C22×D4—941no—
254C22×D4—941no—
255C22×D4—941no—
256AGL(1,ℤ/8)—1882no—
257AGL(1,ℤ/8)—1882no—
258C2×SD16—942no—
259C2×D8—942no—
260C2×M16—21no—
261C2×M16—941no—
262C22×C8—472no—
263C22×C8—942no—
264#127—472no—
265#127—941no—
266#127—941no—
267#127—941no—
268C4×Q8C4×D4711yescharacteristic
269C4×Q8D18811yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D4711yescharacteristic
273C4×D4D18811yescharacteristic
274#117—21no—
275#117C22⋊D4711yescharacteristic
276C2×C4⋊C4—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C22×D4711yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—472no—
283C4⋊C8—471no—
284C4⋊C8—471no—
285C4⋊C8—942no—
286C4⋊C8—941no—
287C4⋊C8—941no—
288C4⋊C8—941no—
289#104—42no—
290#102—942no—
291#102—942no—
292#102—1884no—
293C22⋊C8—942no—
294C22⋊C8—941no—
295C4×C8—942no—
296C4×C8—941no—
297#95—21no—
298C24—1881no—
299#41—21no—
300#41—21no—
301#41#1062533412yes—
302#41#1062533612yes—
303C2×Q8(C2×D47)⋊C411yescharacteristic
304C2×D4—21no—
305C2×D4—942no—
306C2×D4—942no—
307C2×D4—942no—
308C2×D4—3762no—
309C2×D4—1882no—
310C2×D4—1882no—
311C2×D4—1881no—
312C2×D4—1881no—
313C2×D4—1881no—
314C2×D4—1881no—
315C2×D4—1881no—
316C2×D4—1881no—
317C2×D4—1881no—
318C2×D4(C2×D47)⋊C411yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4D4×D4711yescharacteristic
324C22×C4#1062535211yescharacteristic
325C22×C4C2×C22⋊D4711yescharacteristic
326SD16—1884no—
327D8—1884no—
328M16—41no—
329M16—41no—
330M16—1881no—
331M16—1881no—
332C2×C8—41no—
333C2×C8—942no—
334C2×C8—941no—
335C2×C8—941no—
336C2×C8—942no—
337C2×C8—3762no—
338C2×C8—1882no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—41no—
343C4⋊C4(C2×D47)⋊C411yescharacteristic
344C4⋊C4(C2×D47)⋊C411yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—41no—
352C42—41no—
353C42C2×C4×D4711yescharacteristic
354C42C2×D18811yescharacteristic
355C42D4×D4711yescharacteristic
356C42#1062535211yescharacteristic
357C23—21no—
358C23—3762no—
359C23—3762no—
360C23—3761no—
361C23—3761no—
362C23—1882no—
363C23#1309399611yescharacteristic
364Q8—21no—
365Q8#1309394412yescomplemented
366D4—22no—
367D4—22no—
368D4—21no—
369D4—3762no—
370D4—3762no—
371D4—3761no—
372D4—3761no—
373D4—3761no—
374D4—3761no—
375D4—1884no—
376D4—1882no—
377D4—1882no—
378D4#1309394412yescomplemented
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C2×(C2×D47)⋊C411yescharacteristic
394C2×C4(C22⋊D47)⋊C411yescharacteristic
395C2×C4(C22⋊D47)⋊C411yescharacteristic
396C2×C4#1309399411yescharacteristic
397C2×C4#1309399611yescharacteristic
398C2×C4#1309400811yescharacteristic
399C2×C4C2×#1062533411yescharacteristic
400C2×C4C2×#1062533611yescharacteristic
401C2×C4#1309404411yescharacteristic
402C2×C4#1309404411yescharacteristic
403C8—81no—
404C8—1881no—
405C8—1882no—
406C8—1881no—
407C8—1882no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—3762no—
414C22—3762no—
415C22—3762no—
416C22—7522no—
417C22—7522no—
418C22#1568900011yescharacteristic
419C22#1568935411yescharacteristic
420C22#1568935711yescharacteristic
421C4—21no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4C2×#1309394411yescharacteristic
429C4#1568911711yescharacteristic
430C4#1568929211yescharacteristic
431C4#1568929711yescharacteristic
432C4#1568935411yescharacteristic
433C4#1568935711yescharacteristic
434C2—21no—
435C2—21no—
436C2—42no—
437C2—7522no—
438C2#2024376211yescharacteristic
439C2#2024381411yescharacteristic
440C2#2024386611yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1806C2
socleC22×C47#386
prime radical#20252868C2
cosocleC2×C42×C47C23
hypercenter#1806D47
hypocenterC47#9318
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20252499C2semidirect
3. #20244315C2semidirect
4. #20242057C2semidirect
5. #20241291C2semidirect
6. #20241262C2semidirect
7. C47×#1806C2semidirect
11. #15689323C4semidirect
13. #15689318C4semidirect
144. Q8×C47#102semidirect
148. D4×C47#102semidirect
217. #1806D47semidirect
251. C47#9318semidirect
365. Q8#13093944semidirect
378. D4#13093944semidirect

Maximal subgroups

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

subgroupindex
#202528682
#202524992
#202443152
#202420572
#202412912
#202412622
C47×#18062
#931847