Finite Group Databasebeta

← #26199896

Subgroups — #26199896

Order 12032.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254326C211yesmaximal, characteristic
2#20253764C211yesmaximal, characteristic
3#20252337C211yesmaximal, characteristic, complemented
4#20242958C212yesmaximal, complemented
5#20242930C212yesmaximal, complemented
6#20242834C212yesmaximal, complemented
7#20242806C212yesmaximal, complemented
8#20242138C211yesmaximal, characteristic, complemented
9#20242045C211yesmaximal, characteristic, complemented
10C2×#15688866C211yesmaximal, characteristic
11C47×#3249C211yesmaximal, characteristic, complemented
12C2×#13094021C2211yescharacteristic
13#15689497—21no—
14#15689489—22no—
15#15689471—21no—
16#15689471C2212yes—
17#15689471C2212yes—
18#15689323C2212yes—
19#15689323C2212yes—
20#15689273—22no—
21#15689254—22no—
22#15689254C2212yescomplemented
23Q8×C47⋊C8C2212yescomplemented
24#15689176C2211yescharacteristic
25#15689150—21no—
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27#15689127—21no—
28#15689127—21no—
29#15689079—21no—
30C4×#10625329C2211yescharacteristic
31#15688875C2212yescomplemented
32#15688873C2212yescomplemented
33#15688872C2211yescharacteristic, complemented
34#15688872C2211yescharacteristic, complemented
35#15688869C2211yescharacteristic, complemented
36#15688869C2211yescharacteristic, complemented
37#15688866C2212yes—
38#15688866C2212yes—
39C47×#533C2211yescharacteristic
40C47×#533C2211yescharacteristic
41C47×#477—21no—
42C47×#477C2212yescomplemented
43C47×#467—22no—
44C47×#462—21no—
45C47×#462C2212yescomplemented
46C2×C4⋊C8×C47C2211yescharacteristic
47C22×D188—21no—
48C22×D188—21no—
49C22×D188—21no—
50C2×#10625336—22no—
51C2×#10625336—22no—
52C2×#10625329D411yescharacteristic
53C2×#10625329D411yescharacteristic
54C22×C47⋊C8—21no—
55#13094021—21no—
56#13094021—21no—
57#13094021—21no—
58#13094021C2312yes—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)—22no—
61C4⋊(C47⋊C8)—22no—
62C4⋊(C47⋊C8)C2311yescharacteristic
63C4⋊(C47⋊C8)C2311yescharacteristic
64C4⋊(C47⋊C8)C2311yescharacteristic
65C4⋊(C47⋊C8)C2311yescharacteristic
66#13093944—22no—
67#13093944—22no—
68#13093944—22no—
69#13093944—22no—
70#13093944—22no—
71#13093944—22no—
72C22⋊(C47⋊C8)—21no—
73#13093935—22no—
74#13093935D412yes—
75C4×C47⋊C8—21no—
76C4×C47⋊C8—22no—
77C4×C47⋊C8C2312yes—
78C2×Q16×C47—22no—
79C22×C8×C47—21no—
80C4⋊Q8×C47D411yescharacteristic
81C4⋊Q8×C47D411yescharacteristic
82C4⋊Q8×C47C2311yescharacteristic
83C4⋊Q8×C47C2311yescharacteristic
84#125×C47—21no—
85#125×C47—21no—
86C22⋊Q8×C47—21no—
87C22⋊Q8×C47—21no—
88C22⋊Q8×C47—21no—
89C22⋊Q8×C47D412yes—
90C4×Q8×C47—21no—
91C4×Q8×C47—21no—
92C4×Q8×C47—21no—
93C4×Q8×C47C2312yes—
94#117×C47—21no—
95#117×C47—21no—
96C2×C42×C47C2311yescharacteristic
97#107×C47—22no—
98C4⋊C8×C47—21no—
99C4⋊C8×C47C2312yes—
100#103×C47—22no—
101#103×C47—22no—
102C23×D47—41no—
103C2×Q8×C47—21no—
104C2×Q8×C47—21no—
105C2×Q8×C47—21no—
106C2×Q8×C47C2×D412yes—
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C2×D188—22no—
111C2×D188—22no—
112C2×D188—22no—
113C2×D188—42no—
114C2×D188—42no—
115C2×D188—41no—
116C2×D188—41no—
117C2×D188—41no—
118C2×D188—81no—
119C2×D188—81no—
120C2×D188—82no—
121Q16×C47—82no—
122C2×C8×C47—22no—
123C2×C8×C47—81no—
124C4⋊C4×C47—21no—
125C4⋊C4×C47—21no—
126C4⋊C4×C47—21no—
127C4⋊C4×C47—22no—
128C4⋊C4×C47—22no—
129C4⋊C4×C47—21no—
130C4⋊C4×C47—21no—
131C4⋊C4×C47—21no—
132C4⋊C4×C47—21no—
133C4⋊C4×C47—41no—
134C4⋊C4×C47—41no—
135C4⋊C4×C47C2×D412yes—
136C22⋊C4×C47—22no—
137C22⋊C4×C47—41no—
138C42×C47—22no—
139C42×C47—41no—
140C42×C47C2×D411yescharacteristic
141C42×C47C2×D411yescharacteristic
142C42×C47#4111yescharacteristic
143C42×C47C2411yescharacteristic
144#10625336—82no—
145#10625336—82no—
146#10625329—22no—
147#10625329—22no—
148C2×C47⋊C8—21no—
149C2×C47⋊C8—21no—
150C2×C47⋊C8—21no—
151C2×C47⋊C8—21no—
152C2×C47⋊C8—42no—
153C2×C47⋊C8C2×D412yes—
154C2×C47⋊C8C2×D412yes—
155C23×C47#12111yescharacteristic
156C22×D47—42no—
157C22×D47—82no—
158C22×D47—82no—
159C22×D47—82no—
160Q8×C47—22no—
161Q8×C47—22no—
162Q8×C47—41no—
163Q8×C47—41no—
164Q8×C47—81no—
165Q8×C47—81no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—22no—
169C2×C4×C47—22no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47#12111yescharacteristic
174C2×C4×C47#12111yescharacteristic
175C2×C4×C47#12111yescharacteristic
176C2×C4×C47#13512yes—
177C2×C4×C47#13712yes—
178C2×C4×C47C22×D411yescharacteristic
179C2×C4×C47C22×D411yescharacteristic
180C2×C4×C47C2×#4111yescharacteristic
181D188—42no—
182D188—42no—
183D188—42no—
184D188—42no—
185D188—82no—
186D188—82no—
187D188—82no—
188D188—82no—
189C8×C47—42no—
190C47⋊C8—22no—
191C47⋊C8—22no—
192C47⋊C8—41no—
193C47⋊C8—41no—
194C47⋊C8—41no—
195C47⋊C8—41no—
196#19242—471nomaximal, Sylow
197C22×C47—21no—
198C22×C47—21no—
199C22×C47#46412yes—
200C22×C47C2×#12111yescharacteristic
201C2×D47—82no—
202C2×D47—82no—
203C2×D47—82no—
204C2×D47—164no—
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—42no—
209C4×C47—42no—
210C4×C47—81no—
211C4×C47—81no—
212C4×C47#46412yes—
213C4×C47C2×#13511yescharacteristic
214C4×C47C2×#13711yescharacteristic
215C4×C47#57611yescharacteristic
216C4×C47#57711yescharacteristic
217#3249D4711yescharacteristic, complemented
218#3208—471no—
219#3138—471no—
220#2962—471no—
221#1702—472no—
222#1698—472no—
223#1682—472no—
224#1678—472no—
225#1565—471no—
226#1542—471no—
227C2×#331—471no—
228C2×C47—22no—
229C2×C47C2×#46411yescharacteristic
230C2×C47#306211yescharacteristic
231C2×C47#306811yescharacteristic
232D47—162no—
233#533C2×D4711yescharacteristic
234#533C2×D4711yescharacteristic
235C2×#127—471no—
236#492—941no—
237#490—942no—
238#486—472no—
239#486—472no—
240#486—941no—
241#477—21no—
242#477C2×D4712yescomplemented
243#467—22no—
244#462—21no—
245#462C2×D4712yescomplemented
246#460—472no—
247#460—472no—
248#450—942no—
249#446—472no—
250#446—942no—
251C8×Q8—472no—
252#431—471no—
253#424—941no—
254C2×C4⋊C8—471no—
255C2×C4⋊C8C2×D4711yescharacteristic
256#418—941no—
257#418—941no—
258#405—941no—
259C4×M16—471no—
260#335—472no—
261#334—472no—
262#333—471no—
263#333—471no—
264#332—471no—
265#332—471no—
266#331—472no—
267#331—472no—
268C47#1924211yescharacteristic, complemented, Sylow
269C22×D4—941no—
270C22×D4—941no—
271C22×D4—941no—
272C2×Q16—22no—
273C2×SD16—942no—
274C2×SD16—942no—
275C2×M16—471no—
276C2×M16—471no—
277C22×C8—21no—
278C22×C8—941no—
279C4⋊Q8C22⋊D4711yescharacteristic
280C4⋊Q8C22⋊D4711yescharacteristic
281C4⋊Q8C22×D4711yescharacteristic
282C4⋊Q8C22×D4711yescharacteristic
283#127—472no—
284#127—941no—
285#127—941no—
286#127—941no—
287#125—21no—
288#125—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8C22⋊D4712yes—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8C22×D4712yes—
297#117—21no—
298#117—21no—
299C2×C42C22×D4711yescharacteristic
300#107—22no—
301C4⋊C8—21no—
302C4⋊C8—471no—
303C4⋊C8—471no—
304C4⋊C8—471no—
305C4⋊C8—471no—
306C4⋊C8—942no—
307C4⋊C8—942no—
308C4⋊C8—941no—
309C4⋊C8C22×D4712yes—
310#103—22no—
311#103—22no—
312#102—942no—
313#102—942no—
314#102—942no—
315#102—942no—
316#102—942no—
317#102—942no—
318C22⋊C8—941no—
319#97—472no—
320#97—942no—
321C4×C8—472no—
322C4×C8—942no—
323C4×C8—941no—
324C24—1881no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8C2×C22⋊D4712yes—
329C2×D4—942no—
330C2×D4—942no—
331C2×D4—942no—
332C2×D4—3761no—
333C2×D4—3761no—
334C2×D4—3762no—
335C2×D4—1881no—
336C2×D4—1881no—
337C2×D4—1881no—
338C2×D4—1882no—
339C2×D4—1882no—
340C22×C4D4×D4711yescharacteristic
341C22×C4#1062535211yescharacteristic
342C22×C4C2×C22⋊D4711yescharacteristic
343Q16—82no—
344SD16—3762no—
345SD16—3762no—
346M16—942no—
347M16—942no—
348C2×C8—22no—
349C2×C8—81no—
350C2×C8—472no—
351C2×C8—472no—
352C2×C8—941no—
353C2×C8—941no—
354C2×C8—941no—
355C2×C8—941no—
356C2×C8—1882no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4C2×C22⋊D4712yes—
369C22⋊C4—22no—
370C22⋊C4—41no—
371C42—22no—
372C42—41no—
373C42D4×D4711yescharacteristic
374C42#1062535211yescharacteristic
375C42C2×C22⋊D4711yescharacteristic
376C42C23×D4711yescharacteristic
377C23—3762no—
378C23—3762no—
379C23—3762no—
380C23—1882no—
381C23#1309399611yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—41no—
385Q8—41no—
386Q8—81no—
387Q8—81no—
388D4—3762no—
389D4—3762no—
390D4—3762no—
391D4—3762no—
392D4—1882no—
393D4—1882no—
394D4—1882no—
395D4—1882no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4#1309399611yescharacteristic
404C2×C4#1309399611yescharacteristic
405C2×C4#1309399611yescharacteristic
406C2×C4#1309404712yes—
407C2×C4#1309405612yes—
408C2×C4C2×D4×D4711yescharacteristic
409C2×C4C22×C22⋊D4711yescharacteristic
410C2×C4C2×#1062535211yescharacteristic
411C8—42no—
412C8—942no—
413C8—942no—
414C8—1881no—
415C8—1881no—
416C8—1881no—
417C8—1881no—
418C22—21no—
419C22—21no—
420C22—3762no—
421C22—3762no—
422C22—3762no—
423C22—7524no—
424C22#1568934712yes—
425C22C2×#1309399611yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4#1568934712yes—
434C4C2×#1309404711yescharacteristic
435C4C2×#1309405611yescharacteristic
436C4#1568992611yescharacteristic
437C4#1568993111yescharacteristic
438C2—22no—
439C2—7522no—
440C2C2×#1568934711yescharacteristic
441C2#2025307011yescharacteristic
442C2#2025311811yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3249C2
socleC22×C47C2×#121
prime radicalC2×#13094021C22
hypercenter#3249D47
hypocenterC47#19242
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
3. #20252337C2semidirect
4. #20242958C2semidirect
5. #20242930C2semidirect
6. #20242834C2semidirect
7. #20242806C2semidirect
8. #20242138C2semidirect
9. #20242045C2semidirect
11. C47×#3249C2semidirect
22. #15689254C22semidirect
23. Q8×C47⋊C8C22semidirect
31. #15688875C22semidirect
32. #15688873C22semidirect
33. #15688872C22semidirect
34. #15688872C22semidirect
35. #15688869C22semidirect
36. #15688869C22semidirect
42. C47×#477C22semidirect
45. C47×#462C22semidirect
217. #3249D47semidirect
242. #477C2×D47semidirect
245. #462C2×D47semidirect
268. C47#19242semidirect

Maximal subgroups

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

subgroupindex
#202543262
#202537642
#202523372
#202429582
#202429302
#202428342
#202428062
#202421382
#202420452
C2×#156888662
C47×#32492
#1924247