Finite Group Databasebeta

← #26195071

Subgroups — #26195071

Order 12032.

932 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254516C211yesmaximal, characteristic
2#20253419C211yesmaximal, characteristic
3#20252860C211yesmaximal, characteristic
4#20242665C214yesmaximal, complemented
5#20242484C214yesmaximal, complemented
6#20242073C211yesmaximal, characteristic, complemented
7#20241817C211yesmaximal, characteristic, complemented
8#20241783C211yesmaximal, characteristic, complemented
9C2×C47×#557C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12#15689519C2212yes—
13#15689471C2212yes—
14#15689441—22no—
15#15689441—21no—
16#15689410—22no—
17#15689408C2212yes—
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#13093955—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—21no—
110C2×Q8×C47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47—22no—
113C2×Q8×C47—22no—
114C2×Q8×C47—42no—
115C2×Q8×C47—44no—
116C2×Q8×C47SD1614yescomplemented
117C2×Q8×C47C2×D412yes—
118C2×Q8×C47C2×D412yes—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2×D411yescharacteristic
127C2×C22⋊D47—41no—
128#10625350—42no—
129#10625350—42no—
130C2×D188—21no—
131C2×D188—21no—
132C2×D188—21no—
133C2×C4×D47—41no—
134C2×Dic94—21no—
135C4⋊C4×C47—28no—
136C4⋊C4×C47—28no—
137C4⋊C4×C47—28no—
138C4⋊C4×C47—22no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47—22no—
141C4⋊C4×C47—21no—
142C4⋊C4×C47—21no—
143C4⋊C4×C47—22no—
144C4⋊C4×C47—22no—
145C4⋊C4×C47—22no—
146C4⋊C4×C47—22no—
147C4⋊C4×C47C2×D412yes—
148C4⋊C4×C47C2×D412yes—
149C4⋊C4×C47C2×D414yes—
150C42×C47—28no—
151C42×C47—22no—
152C42×C47—22no—
153C42×C47—22no—
154C42×C47—22no—
155C42×C47C2×D411yescharacteristic
156C42×C47C2×D411yescharacteristic
157C42×C47C2×D411yescharacteristic
158C42×C47C2411yescharacteristic
159#10625337—44no—
160#10625336—44no—
161#10625336—44no—
162(C2×D47)⋊C4—41no—
163(C2×D47)⋊C4—41no—
164C4⋊Dic47—21no—
165C4⋊Dic47—21no—
166#10625329—42no—
167C2×C47⋊C8—22no—
168C2×C47⋊C8—22no—
169C2×C47⋊C8—22no—
170C23×C47#12011yescharacteristic
171C22×D47—41no—
172Q8×C47—22no—
173Q8×C47—24no—
174Q8×C47—22no—
175Q8×C47—22no—
176Q8×C47—42no—
177Q8×C47—44no—
178Q8×C47—44no—
179Q8×C47C2×SD1614yes—
180C2×C4×C47—22no—
181C2×C4×C47—24no—
182C2×C4×C47—22no—
183C2×C4×C47—22no—
184C2×C4×C47—22no—
185C2×C4×C47—22no—
186C2×C4×C47—24no—
187C2×C4×C47—41no—
188C2×C4×C47—42no—
189C2×C4×C47—42no—
190C2×C4×C47—42no—
191C2×C4×C47—44no—
192C2×C4×C47#12011yescharacteristic
193C2×C4×C47#12011yescharacteristic
194C2×C4×C47#12011yescharacteristic
195C2×C4×C47C2×SD1612yes—
196C2×C4×C47#13712yes—
197C2×C4×C47C22×D411yescharacteristic
198C2×C4×C47C22×D411yescharacteristic
199C2×C4×C47C22×D411yescharacteristic
200C22⋊D47—82no—
201C2×Dic47—41no—
202D188—42no—
203D188—41no—
204D188—41no—
205D188—42no—
206C4×D47—82no—
207Dic94—41no—
208Dic94—41no—
209C47⋊C8—42no—
210C47⋊C8—42no—
211C47⋊C8—42no—
212#18778—471nomaximal, Sylow
213C22×C47—21no—
214C22×C47—21no—
215C22×C47#44812yes—
216C22×C47C2×#12011yescharacteristic
217C2×D47—81no—
218C2×D47—81no—
219C2×D47—81no—
220C4×C47—24no—
221C4×C47—21no—
222C4×C47—21no—
223C4×C47—21no—
224C4×C47—42no—
225C4×C47—44no—
226C4×C47—42no—
227C4×C47—42no—
228C4×C47#44812yes—
229C4×C47C22×SD1611yescharacteristic
230C4×C47C2×#13711yescharacteristic
231C4×C47#57512yes—
232Dic47—81no—
233C2×#557D4711yescharacteristic, complemented
234#3227—471no—
235#3103—471no—
236#3037—471no—
237#1653—474no—
238#1626—474no—
239#1549—471no—
240#1487—471no—
241#1479—471no—
242C2×C47—22no—
243C2×C47C2×#44811yescharacteristic
244C2×C47#300511yescharacteristic
245C2×C47#302011yescharacteristic
246D47—161no—
247C2×#137—941no—
248#557—22no—
249#557—22no—
250#557—22no—
251#557C2×D4714yes—
252#532—471no—
253C2×C4⋊Q8—21no—
254C2×C4⋊Q8C2×D4711yescharacteristic
255C2×#125—22no—
256C2×#125—21no—
257C2×C4×Q8—21no—
258C2×C4×Q8—21no—
259C2×C4×Q8C2×D4711yescharacteristic
260C2×C4×Q8C2×D4711yescharacteristic
261#498—472no—
262#486—472no—
263#481—941no—
264#481—942no—
265#476—942no—
266#475—472no—
267#464—942no—
268#462—472no—
269#460—472no—
270#460—472no—
271#448—941no—
272C4×SD16—942no—
273#431—471no—
274#431—471no—
275#423—471no—
276#417—941no—
277#332—474no—
278Q8⋊C8—474no—
279Q8⋊C8—474no—
280C47#1877811yescharacteristic, complemented, Sylow
281C2×#41—941no—
282C22×Q8—21no—
283C22×Q8C22⋊D4711yescharacteristic
284C22×Q8C22⋊D4711yescharacteristic
285#137—1884no—
286C2×Q16—942no—
287C2×SD16—942no—
288C2×SD16—942no—
289C2×M16—941no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—471no—
295C4⋊Q8C22⋊D4712yes—
296C4⋊Q8C22×D4712yes—
297#127—471no—
298#125—28no—
299#125—24no—
300#125—28no—
301#125—24no—
302#124—941no—
303#121—941no—
304#121—941no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8C22⋊D4712yes—
314C4×Q8C22⋊D4712yes—
315C4×Q8C22×D4712yes—
316C4×Q8C22×D4712yes—
317C4×D4—941no—
318C4×D4—941no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D4711yescharacteristic
326C2×C4⋊C4C22⋊D4711yescharacteristic
327C2×C4⋊C4C22⋊D4712yes—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C22×D4711yescharacteristic
332#106—942no—
333#106—944no—
334C4⋊C8—472no—
335C4⋊C8—472no—
336C4⋊C8—941no—
337C4⋊C8—941no—
338#103—942no—
339#103—942no—
340#102—942no—
341#102—942no—
342#102—944no—
343C22⋊C8—941no—
344C22⋊C8—941no—
345C4×C8—472no—
346C4×C8—941no—
347#41—1882no—
348#41—1882no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—941no—
357C2×Q8#1062533614yescomplemented
358C2×Q8C2×C22⋊D4712yes—
359C2×Q8C2×C22⋊D4712yes—
360C2×D4—941no—
361C2×D4—941no—
362C2×D4—941no—
363C2×D4—1881no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—1881no—
370C22×C4C2×C22⋊D4711yescharacteristic
371C22×C4C2×C22⋊D4711yescharacteristic
372C22×C4C2×C22⋊D4711yescharacteristic
373Q16—1884no—
374SD16—1884no—
375SD16—1884no—
376M16—1882no—
377C2×C8—942no—
378C2×C8—942no—
379C2×C8—942no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—941no—
393C4⋊C4—941no—
394C4⋊C4C2×C22⋊D4712yes—
395C4⋊C4C2×C22⋊D4712yes—
396C4⋊C4C2×C22⋊D4714yes—
397C22⋊C4—1881no—
398C22⋊C4—1881no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42C2×C22⋊D4711yescharacteristic
405C42C2×C22⋊D4711yescharacteristic
406C42C2×C22⋊D4711yescharacteristic
407C42C23×D4711yescharacteristic
408C23—1881no—
409C23#1309399311yescharacteristic
410Q8—24no—
411Q8—22no—
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—44no—
417Q8—1881no—
418Q8—1881no—
419Q8C2×#1062533614yes—
420D4—3762no—
421D4—1882no—
422D4—1881no—
423D4—1881no—
424D4—1882no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—3762no—
438C2×C4—1881no—
439C2×C4#1309399311yescharacteristic
440C2×C4#1309399311yescharacteristic
441C2×C4#1309399311yescharacteristic
442C2×C4C2×#1062533612yes—
443C2×C4#1309405512yes—
444C2×C4C22×C22⋊D4711yescharacteristic
445C2×C4C22×C22⋊D4711yescharacteristic
446C2×C4C22×C22⋊D4711yescharacteristic
447C8—1882no—
448C8—1882no—
449C8—1882no—
450C22—21no—
451C22—21no—
452C22—3761no—
453C22—3761no—
454C22—3761no—
455C22#1568926212yes—
456C22C2×#1309399311yescharacteristic
457C4—21no—
458C4—21no—
459C4—24no—
460C4—21no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—44no—
465C4—3761no—
466C4#1568926212yes—
467C4C22×#1062533611yescharacteristic
468C4C2×#1309405511yescharacteristic
469C4#1568992112yes—
470C2—22no—
471C2—7521no—
472C2C2×#1568926211yescharacteristic
473C2#2025263011yescharacteristic
474C2#2025275211yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#557C2
socleC22×C47C2×#120
prime radical#15689673C22
hypercenterC2×#557D47
hypocenterC47#18778
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. #20242665C2semidirect
5. #20242484C2semidirect
6. #20242073C2semidirect
7. #20241817C2semidirect
8. #20241783C2semidirect
9. C2×C47×#557C2semidirect
116. C2×Q8×C47SD16semidirect
233. C2×#557D47semidirect
280. C47#18778semidirect
357. C2×Q8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202545162
#202534192
#202528602
#202426652
#202424842
#202420732
#202418172
#202417832
C2×C47×#5572
#1877847