Finite Group Databasebeta

← #26194885

Subgroups — #26194885

Order 12032.

932 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254509C211yesmaximal, characteristic
2#20253419C211yesmaximal, characteristic
3#20252860C211yesmaximal, characteristic
4#20242630C214yesmaximal, complemented
5#20242484C214yesmaximal, complemented
6#20242060C211yesmaximal, characteristic, complemented
7#20241817C211yesmaximal, characteristic, complemented
8#20241783C211yesmaximal, characteristic, complemented
9C2×C47×#557C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12#15689526C2212yes—
13#15689471C2212yes—
14#15689441—21no—
15#15689436—22no—
16#15689408C2212yes—
17#15689400—22no—
18#15689345—22no—
19#15689333C2212yes—
20#15689323C2212yes—
21#15689323C2212yes—
22#15689262—21no—
23C4×#10625336—22no—
24#15689176C2211yescharacteristic
25#15689176C2211yescharacteristic
26#15689147C2211yescharacteristic
27#15689122—21no—
28#15688869C2214yes—
29Q8⋊(C47⋊C8)C2214yes—
30Q8⋊(C47⋊C8)C2214yes—
31C47×#557—22no—
32C47×#557—22no—
33C47×#557—22no—
34C47×#557C2214yes—
35C2×C4⋊Q8×C47—21no—
36C2×C4⋊Q8×C47C2211yescharacteristic
37C2×#125×C47—22no—
38C2×#125×C47—21no—
39C2×C4×Q8×C47—21no—
40C2×C4×Q8×C47—21no—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4×Q8×C47C2211yescharacteristic
43C2×#10625350—21no—
44#13094055—44no—
45C2×#10625337—22no—
46C2×#10625336—22no—
47C2×#10625336—22no—
48C2×#10625329—21no—
49#13094024C2311yescharacteristic
50#13094021C2311yescharacteristic
51#13094012—21no—
52#13093998—21no—
53#13093998—21no—
54C4×D188—21no—
55C4×D188—21no—
56#13093957—24no—
57#13093955—22no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)C2312yes—
61C4⋊(C47⋊C8)C2312yes—
62#13093948—22no—
63#13093947—22no—
64#13093944—22no—
65#13093944—22no—
66#13093944—24no—
67C22⋊(C47⋊C8)—21no—
68C22⋊(C47⋊C8)—21no—
69C4×C47⋊C8—21no—
70C4×C47⋊C8C2312yes—
71C22×Q8×C47—21no—
72C22×Q8×C47D411yescharacteristic
73C22×Q8×C47D411yescharacteristic
74C4⋊Q8×C47—24no—
75C4⋊Q8×C47—24no—
76C4⋊Q8×C47—21no—
77C4⋊Q8×C47—21no—
78C4⋊Q8×C47D412yes—
79C4⋊Q8×C47C2312yes—
80#125×C47—28no—
81#125×C47—24no—
82#125×C47—28no—
83#125×C47—24no—
84C4×Q8×C47—24no—
85C4×Q8×C47—24no—
86C4×Q8×C47—24no—
87C4×Q8×C47—24no—
88C4×Q8×C47—21no—
89C4×Q8×C47—21no—
90C4×Q8×C47—21no—
91C4×Q8×C47—21no—
92C4×Q8×C47D412yes—
93C4×Q8×C47D412yes—
94C4×Q8×C47C2312yes—
95C4×Q8×C47C2312yes—
96C2×C4⋊C4×C47—22no—
97C2×C4⋊C4×C47—22no—
98C2×C4⋊C4×C47—22no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47D411yescharacteristic
103C2×C4⋊C4×C47D411yescharacteristic
104C2×C4⋊C4×C47D412yes—
105C2×C42×C47—22no—
106C2×C42×C47—21no—
107C2×C42×C47—21no—
108C2×C42×C47C2311yescharacteristic
109C2×Q8×C47—24no—
110C2×Q8×C47—21no—
111C2×Q8×C47—21no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—22no—
116C2×Q8×C47—42no—
117C2×Q8×C47—42no—
118C2×Q8×C47C2×D412yes—
119C2×Q8×C47C2×D412yes—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47—22no—
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C2×C22⋊D47—41no—
129#10625350—42no—
130#10625350—42no—
131C2×D188—21no—
132C2×D188—21no—
133C2×D188—21no—
134C2×C4×D47—41no—
135C2×Dic94—21no—
136C4⋊C4×C47—28no—
137C4⋊C4×C47—28no—
138C4⋊C4×C47—28no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47—22no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47C2×D412yes—
149C4⋊C4×C47C2×D412yes—
150C4⋊C4×C47C2×D414yes—
151C42×C47—28no—
152C42×C47—22no—
153C42×C47—22no—
154C42×C47—22no—
155C42×C47—22no—
156C42×C47C2×D411yescharacteristic
157C42×C47C2×D411yescharacteristic
158C42×C47C2×D411yescharacteristic
159C42×C47C2411yescharacteristic
160#10625337—44no—
161#10625336—44no—
162#10625336—44no—
163(C2×D47)⋊C4—41no—
164(C2×D47)⋊C4—41no—
165C4⋊Dic47—21no—
166C4⋊Dic47—21no—
167#10625329—42no—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C23×C47#12011yescharacteristic
172C22×D47—41no—
173Q8×C47—24no—
174Q8×C47—22no—
175Q8×C47—22no—
176Q8×C47—22no—
177Q8×C47—42no—
178Q8×C47—42no—
179Q8×C47—42no—
180Q8×C47—44no—
181Q8×C47#13714yescomplemented
182C2×C4×C47—24no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—22no—
187C2×C4×C47—22no—
188C2×C4×C47—24no—
189C2×C4×C47—41no—
190C2×C4×C47—42no—
191C2×C4×C47—42no—
192C2×C4×C47—42no—
193C2×C4×C47—44no—
194C2×C4×C47#12011yescharacteristic
195C2×C4×C47#12011yescharacteristic
196C2×C4×C47#12011yescharacteristic
197C2×C4×C47#13512yes—
198C2×C4×C47AGL(1,ℤ/8)12yes—
199C2×C4×C47C22×D411yescharacteristic
200C2×C4×C47C22×D411yescharacteristic
201C2×C4×C47C22×D411yescharacteristic
202C22⋊D47—82no—
203C2×Dic47—41no—
204D188—42no—
205D188—41no—
206D188—41no—
207D188—42no—
208C4×D47—82no—
209Dic94—41no—
210Dic94—41no—
211C47⋊C8—42no—
212C47⋊C8—42no—
213C47⋊C8—42no—
214#18760—471nomaximal, Sylow
215C22×C47—21no—
216C22×C47—21no—
217C22×C47#44912yes—
218C22×C47C2×#12011yescharacteristic
219C2×D47—81no—
220C2×D47—81no—
221C2×D47—81no—
222C4×C47—24no—
223C4×C47—21no—
224C4×C47—21no—
225C4×C47—21no—
226C4×C47—42no—
227C4×C47—44no—
228C4×C47—42no—
229C4×C47—42no—
230C4×C47#44912yes—
231C4×C47C2×#13511yescharacteristic
232C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C47C2×#13712yes—
234Dic47—81no—
235C2×#557D4711yescharacteristic, complemented
236#3226—471no—
237#3103—471no—
238#3037—471no—
239#1648—474no—
240#1626—474no—
241#1546—471no—
242#1487—471no—
243#1479—471no—
244C2×C47—22no—
245C2×C47C2×#44911yescharacteristic
246C2×C47#300411yescharacteristic
247C2×C47#302011yescharacteristic
248D47—161no—
249C2×#137—941no—
250#557—22no—
251#557—22no—
252#557—22no—
253#557C2×D4714yes—
254#532—471no—
255C2×C4⋊Q8—21no—
256C2×C4⋊Q8C2×D4711yescharacteristic
257C2×#125—22no—
258C2×#125—21no—
259C2×C4×Q8—21no—
260C2×C4×Q8—21no—
261C2×C4×Q8C2×D4711yescharacteristic
262C2×C4×Q8C2×D4711yescharacteristic
263#500—472no—
264#486—472no—
265#481—941no—
266#480—942no—
267#475—472no—
268#474—942no—
269#464—942no—
270#462—472no—
271#460—472no—
272#460—472no—
273#448—941no—
274C4×SD16—942no—
275#431—471no—
276#431—471no—
277#423—471no—
278#417—941no—
279#332—474no—
280Q8⋊C8—474no—
281Q8⋊C8—474no—
282C47#1876011yescharacteristic, complemented, Sylow
283C2×#41—941no—
284C22×Q8—21no—
285C22×Q8C22⋊D4711yescharacteristic
286C22×Q8C22⋊D4711yescharacteristic
287#137—1884no—
288C2×Q16—942no—
289C2×SD16—942no—
290C2×SD16—942no—
291C2×M16—941no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—471no—
297C4⋊Q8C22⋊D4712yes—
298C4⋊Q8C22×D4712yes—
299#127—471no—
300#125—28no—
301#125—24no—
302#125—28no—
303#125—24no—
304#124—941no—
305#121—941no—
306#121—941no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8C22⋊D4712yes—
316C4×Q8C22⋊D4712yes—
317C4×Q8C22×D4712yes—
318C4×Q8C22×D4712yes—
319C4×D4—941no—
320C4×D4—941no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C22⋊D4711yescharacteristic
328C2×C4⋊C4C22⋊D4711yescharacteristic
329C2×C4⋊C4C22⋊D4712yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42C22×D4711yescharacteristic
334#107—944no—
335#106—942no—
336C4⋊C8—472no—
337C4⋊C8—472no—
338C4⋊C8—941no—
339C4⋊C8—941no—
340#103—942no—
341#103—942no—
342#102—942no—
343#102—942no—
344#102—944no—
345C22⋊C8—941no—
346C22⋊C8—941no—
347C4×C8—472no—
348C4×C8—941no—
349#41—1882no—
350#41—1882no—
351C2×Q8—24no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—941no—
361C2×Q8C2×C22⋊D4712yes—
362C2×Q8C2×C22⋊D4712yes—
363C2×D4—941no—
364C2×D4—941no—
365C2×D4—941no—
366C2×D4—1881no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—1881no—
373C22×C4C2×C22⋊D4711yescharacteristic
374C22×C4C2×C22⋊D4711yescharacteristic
375C22×C4C2×C22⋊D4711yescharacteristic
376Q16—1884no—
377SD16—1884no—
378SD16—1884no—
379M16—1882no—
380C2×C8—942no—
381C2×C8—942no—
382C2×C8—942no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—941no—
396C4⋊C4—941no—
397C4⋊C4C2×C22⋊D4712yes—
398C4⋊C4C2×C22⋊D4712yes—
399C4⋊C4C2×C22⋊D4714yes—
400C22⋊C4—1881no—
401C22⋊C4—1881no—
402C42—28no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42C2×C22⋊D4711yescharacteristic
408C42C2×C22⋊D4711yescharacteristic
409C42C2×C22⋊D4711yescharacteristic
410C42C23×D4711yescharacteristic
411C23—1881no—
412C23#1309399311yescharacteristic
413Q8—24no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—42no—
418Q8—44no—
419Q8—42no—
420Q8—42no—
421Q8—1881no—
422Q8—1881no—
423Q8#1309405514yescomplemented
424D4—3762no—
425D4—1882no—
426D4—1881no—
427D4—1881no—
428D4—1882no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—3762no—
442C2×C4—1881no—
443C2×C4#1309399311yescharacteristic
444C2×C4#1309399311yescharacteristic
445C2×C4#1309399311yescharacteristic
446C2×C4#1309404412yes—
447C2×C4#1309404912yes—
448C2×C4C22×C22⋊D4711yescharacteristic
449C2×C4C22×C22⋊D4711yescharacteristic
450C2×C4C22×C22⋊D4711yescharacteristic
451C8—1882no—
452C8—1882no—
453C8—1882no—
454C22—21no—
455C22—21no—
456C22—3761no—
457C22—3761no—
458C22—3761no—
459C22#1568926812yes—
460C22C2×#1309399311yescharacteristic
461C4—21no—
462C4—21no—
463C4—24no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—44no—
469C4—3761no—
470C4#1568926812yes—
471C4C2×#1309404411yescharacteristic
472C4C2×#1309404911yescharacteristic
473C4C2×#1309405512yes—
474C2—22no—
475C2—7521no—
476C2C2×#1568926811yescharacteristic
477C2#2025262211yescharacteristic
478C2#2025275211yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#557C2
socleC22×C47C2×#120
prime radical#15689673C22
hypercenterC2×#557D47
hypocenterC47#18760
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #20242630C2semidirect
5. #20242484C2semidirect
6. #20242060C2semidirect
7. #20241817C2semidirect
8. #20241783C2semidirect
9. C2×C47×#557C2semidirect
181. Q8×C47#137semidirect
235. C2×#557D47semidirect
282. C47#18760semidirect
423. Q8#13094055semidirect

Maximal subgroups

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

subgroupindex
#202545092
#202534192
#202528602
#202426302
#202424842
#202420602
#202418172
#202417832
C2×C47×#5572
#1876047