Finite Group Databasebeta

← #26194969

Subgroups — #26194969

Order 12032.

976 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254313C211yesmaximal, characteristic
2#20253419C211yesmaximal, characteristic
3#20252860C211yesmaximal, characteristic
4#20242609C214yesmaximal, complemented
5#20242466C214yesmaximal, complemented
6#20242069C211yesmaximal, characteristic, complemented
7#20241817C211yesmaximal, characteristic, complemented
8#20241783C211yesmaximal, characteristic, complemented
9C2×Q82×C47C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12#15689497C2212yes—
13#15689473C2212yes—
14#15689441—21no—
15#15689408C2212yes—
16#15689345—22no—
17#15689333C2212yes—
18#15689325—22no—
19#15689323C2212yes—
20#15689323C2212yes—
21#15689262—22no—
22#15689262—21no—
23C4×#10625336—22no—
24#15689176C2211yescharacteristic
25#15689176C2211yescharacteristic
26#15689147C2211yescharacteristic
27#15689122—21no—
28#15688872C2214yes—
29Q8⋊(C47⋊C8)C2214yes—
30Q8⋊(C47⋊C8)C2214yes—
31Q82×C47—22no—
32Q82×C47—22no—
33Q82×C47—22no—
34Q82×C47C2214yes—
35C2×C4⋊Q8×C47—22no—
36C2×C4⋊Q8×C47—21no—
37C2×C4⋊Q8×C47—21no—
38C2×C4⋊Q8×C47C2211yescharacteristic
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×#10625336—24no—
49C2×#10625329—21no—
50#13094024C2311yescharacteristic
51#13094021C2311yescharacteristic
52#13094012—21no—
53#13093998—21no—
54#13093998—21no—
55C4×D188—21no—
56C4×D188—21no—
57#13093955—22no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)C2312yes—
61C4⋊(C47⋊C8)C2312yes—
62#13093948—24no—
63#13093948—22no—
64#13093947—22no—
65#13093944—22no—
66#13093944—22no—
67C22⋊(C47⋊C8)—21no—
68C22⋊(C47⋊C8)—21no—
69C4×C47⋊C8—21no—
70C4×C47⋊C8C2312yes—
71C22×Q8×C47—22no—
72C22×Q8×C47D411yescharacteristic
73C22×Q8×C47D411yescharacteristic
74C22×Q8×C47D412yes—
75C4⋊Q8×C47—28no—
76C4⋊Q8×C47—24no—
77C4⋊Q8×C47—24no—
78C4⋊Q8×C47—28no—
79C4⋊Q8×C47—24no—
80C4⋊Q8×C47—24no—
81C4⋊Q8×C47—21no—
82C4⋊Q8×C47—21no—
83C4⋊Q8×C47D412yes—
84C4⋊Q8×C47C2312yes—
85C4×Q8×C47—24no—
86C4×Q8×C47—24no—
87C4×Q8×C47—24no—
88C4×Q8×C47—24no—
89C4×Q8×C47—21no—
90C4×Q8×C47—21no—
91C4×Q8×C47—21no—
92C4×Q8×C47—21no—
93C4×Q8×C47D412yes—
94C4×Q8×C47D412yes—
95C4×Q8×C47C2312yes—
96C4×Q8×C47C2312yes—
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×C47—21no—
103C2×C4⋊C4×C47D411yescharacteristic
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C42×C47—22no—
106C2×C42×C47—21no—
107C2×C42×C47—21no—
108C2×C42×C47C2311yescharacteristic
109C2×Q8×C47—24no—
110C2×Q8×C47—28no—
111C2×Q8×C47—21no—
112C2×Q8×C47—21no—
113C2×Q8×C47—22no—
114C2×Q8×C47—22no—
115C2×Q8×C47—82no—
116C2×Q8×C47—84no—
117C2×Q8×C47SD1614yescomplemented
118C2×Q8×C47C2×D412yes—
119C2×Q8×C47C2×D412yes—
120C2×Q8×C47C2×D414yes—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47C2×D411yescharacteristic
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C2×C22⋊D47—41no—
130#10625350—42no—
131#10625350—42no—
132C2×D188—21no—
133C2×D188—21no—
134C2×D188—21no—
135C2×C4×D47—41no—
136C2×Dic94—21no—
137C4⋊C4×C47—28no—
138C4⋊C4×C47—28no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47—22no—
142C4⋊C4×C47—22no—
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—22no—
149C4⋊C4×C47C2×D412yes—
150C4⋊C4×C47C2×D412yes—
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#10625336—84no—
164(C2×D47)⋊C4—41no—
165(C2×D47)⋊C4—41no—
166C4⋊Dic47—21no—
167C4⋊Dic47—21no—
168#10625329—42no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C23×C47#12011yescharacteristic
173C22×D47—41no—
174Q8×C47—24no—
175Q8×C47—22no—
176Q8×C47—22no—
177Q8×C47—24no—
178Q8×C47—44no—
179Q8×C47—82no—
180Q8×C47—88no—
181Q8×C47C2×SD1614yes—
182Q8×C47#13714yescomplemented
183C2×C4×C47—24no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—24no—
187C2×C4×C47—22no—
188C2×C4×C47—24no—
189C2×C4×C47—22no—
190C2×C4×C47—22no—
191C2×C4×C47—41no—
192C2×C4×C47—44no—
193C2×C4×C47—42no—
194C2×C4×C47—42no—
195C2×C4×C47#12011yescharacteristic
196C2×C4×C47#12011yescharacteristic
197C2×C4×C47#12011yescharacteristic
198C2×C4×C47C2×SD1612yes—
199C2×C4×C47#13712yes—
200C2×C4×C47C22×D411yescharacteristic
201C2×C4×C47C22×D411yescharacteristic
202C2×C4×C47C22×D411yescharacteristic
203C22⋊D47—82no—
204C2×Dic47—41no—
205D188—42no—
206D188—41no—
207D188—42no—
208D188—41no—
209C4×D47—82no—
210Dic94—41no—
211Dic94—41no—
212C47⋊C8—42no—
213C47⋊C8—42no—
214C47⋊C8—42no—
215#18768—471nomaximal, Sylow
216C22×C47—21no—
217C22×C47—21no—
218C22×C47#44812yes—
219C22×C47C2×#12011yescharacteristic
220C2×D47—81no—
221C2×D47—81no—
222C2×D47—81no—
223C4×C47—24no—
224C4×C47—24no—
225C4×C47—21no—
226C4×C47—21no—
227C4×C47—21no—
228C4×C47—44no—
229C4×C47—42no—
230C4×C47—42no—
231C4×C47#44812yes—
232C4×C47C22×SD1611yescharacteristic
233C4×C47C2×#13711yescharacteristic
234C4×C47C2×#13712yes—
235Dic47—81no—
236C2×Q82D4711yescharacteristic, complemented
237#3206—471no—
238#3103—471no—
239#3037—471no—
240#1645—474no—
241#1623—474no—
242#1548—471no—
243#1487—471no—
244#1479—471no—
245C2×C47—22no—
246C2×C47C2×#44811yescharacteristic
247C2×C47#300411yescharacteristic
248C2×C47#301911yescharacteristic
249D47—161no—
250C2×#137—941no—
251Q82—22no—
252Q82—22no—
253Q82—22no—
254Q82C2×D4714yes—
255#532—471no—
256C2×C4⋊Q8—22no—
257C2×C4⋊Q8—21no—
258C2×C4⋊Q8—21no—
259C2×C4⋊Q8C2×D4711yescharacteristic
260C2×C4×Q8—21no—
261C2×C4×Q8—21no—
262C2×C4×Q8C2×D4711yescharacteristic
263C2×C4×Q8C2×D4711yescharacteristic
264#492—472no—
265#487—472no—
266#481—941no—
267#475—472no—
268#464—942no—
269#462—472no—
270#461—942no—
271#460—472no—
272#460—472no—
273#448—941no—
274#448—942no—
275C4×SD16—942no—
276#431—471no—
277#431—471no—
278#423—471no—
279#417—941no—
280#333—474no—
281Q8⋊C8—474no—
282Q8⋊C8—474no—
283C47#1876811yescharacteristic, complemented, Sylow
284C2×#41—941no—
285C22×Q8—22no—
286C22×Q8C22⋊D4711yescharacteristic
287C22×Q8C22⋊D4711yescharacteristic
288C22×Q8C22⋊D4712yes—
289#137—1884no—
290C2×Q16—942no—
291C2×SD16—944no—
292C2×SD16—942no—
293C2×SD16—942no—
294C2×M16—941no—
295C4⋊Q8—28no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—28no—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—471no—
304C4⋊Q8C22⋊D4712yes—
305C4⋊Q8C22×D4712yes—
306#127—471no—
307#124—941no—
308#121—941no—
309#121—941no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8C22⋊D4712yes—
319C4×Q8C22⋊D4712yes—
320C4×Q8C22×D4712yes—
321C4×Q8C22×D4712yes—
322C4×D4—941no—
323C4×D4—941no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C22⋊D4711yescharacteristic
331C2×C4⋊C4C22⋊D4711yescharacteristic
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×D4711yescharacteristic
336#106—942no—
337C4⋊C8—472no—
338C4⋊C8—472no—
339C4⋊C8—941no—
340C4⋊C8—941no—
341#103—942no—
342#103—944no—
343#103—942no—
344#102—942no—
345#102—942no—
346C22⋊C8—941no—
347C22⋊C8—941no—
348C4×C8—472no—
349C4×C8—941no—
350#41—1882no—
351#41—1882no—
352C2×Q8—24no—
353C2×Q8—28no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—82no—
359C2×Q8—84no—
360C2×Q8—941no—
361C2×Q8#1062533614yescomplemented
362C2×Q8C2×C22⋊D4712yes—
363C2×Q8C2×C22⋊D4712yes—
364C2×Q8C2×C22⋊D4714yes—
365C2×D4—941no—
366C2×D4—941no—
367C2×D4—941no—
368C2×D4—1881no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—1881no—
375C22×C4C2×C22⋊D4711yescharacteristic
376C22×C4C2×C22⋊D4711yescharacteristic
377C22×C4C2×C22⋊D4711yescharacteristic
378Q16—1884no—
379SD16—3764no—
380SD16—1884no—
381SD16—1884no—
382M16—1882no—
383C2×C8—942no—
384C2×C8—942no—
385C2×C8—942no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—941no—
399C4⋊C4—941no—
400C4⋊C4C2×C22⋊D4712yes—
401C4⋊C4C2×C22⋊D4712yes—
402C22⋊C4—1881no—
403C22⋊C4—1881no—
404C42—28no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42C2×C22⋊D4711yescharacteristic
410C42C2×C22⋊D4711yescharacteristic
411C42C2×C22⋊D4711yescharacteristic
412C42C23×D4711yescharacteristic
413C23—1881no—
414C23#1309399311yescharacteristic
415Q8—24no—
416Q8—24no—
417Q8—22no—
418Q8—22no—
419Q8—44no—
420Q8—82no—
421Q8—88no—
422Q8—1881no—
423Q8—1881no—
424Q8C2×#1062533614yes—
425Q8#1309405514yescomplemented
426D4—3762no—
427D4—1882no—
428D4—1881no—
429D4—1882no—
430D4—1881no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—3762no—
444C2×C4—1881no—
445C2×C4#1309399311yescharacteristic
446C2×C4#1309399311yescharacteristic
447C2×C4#1309399311yescharacteristic
448C2×C4C2×#1062533612yes—
449C2×C4#1309405512yes—
450C2×C4C22×C22⋊D4711yescharacteristic
451C2×C4C22×C22⋊D4711yescharacteristic
452C2×C4C22×C22⋊D4711yescharacteristic
453C8—1882no—
454C8—1882no—
455C8—1882no—
456C22—21no—
457C22—21no—
458C22—3761no—
459C22—3761no—
460C22—3761no—
461C22#1568926212yes—
462C22C2×#1309399311yescharacteristic
463C4—21no—
464C4—21no—
465C4—24no—
466C4—24no—
467C4—21no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—3761no—
472C4#1568926212yes—
473C4C22×#1062533611yescharacteristic
474C4C2×#1309405511yescharacteristic
475C4C2×#1309405512yes—
476C2—22no—
477C2—7521no—
478C2C2×#1568926211yescharacteristic
479C2#2025262211yescharacteristic
480C2#2025274511yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×Q82×C47C2
socleC22×C47C2×#120
prime radical#15689673C22
hypercenterC2×Q82D47
hypocenterC47#18768
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
4. #20242609C2semidirect
5. #20242466C2semidirect
6. #20242069C2semidirect
7. #20241817C2semidirect
8. #20241783C2semidirect
9. C2×Q82×C47C2semidirect
117. C2×Q8×C47SD16semidirect
182. Q8×C47#137semidirect
236. C2×Q82D47semidirect
283. C47#18768semidirect
361. C2×Q8#10625336semidirect
425. Q8#13094055semidirect

Maximal subgroups

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

subgroupindex
#202543132
#202534192
#202528602
#202426092
#202424662
#202420692
#202418172
#202417832
C2×Q82×C472
#1876847