Finite Group Databasebeta

← #26268393

Subgroups — #26268393

Order 12032.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253181C211yesmaximal, characteristic
2C2×#15689400C211yesmaximal, characteristic
3#20252337C211yesmaximal, characteristic, complemented
4#20244896C212yesmaximal, complemented
5#20244745C212yesmaximal
6#20244378C211yesmaximal, characteristic, complemented
7#20244170C212yesmaximal, complemented
8#20244163C211yesmaximal, characteristic, complemented
9#20244049C212yesmaximal
10C4×#13093940C211yesmaximal, characteristic
11C47×#3165C211yesmaximal, characteristic, complemented
12C2×C4×D188C2211yescharacteristic
13#15689420—22no—
14#15689410—21no—
15#15689400—21no—
16#15689400—21no—
17#15689400—22no—
18#15689400—21no—
19#15689254—21no—
20#15689254—22no—
21Q8×C47⋊C8—21no—
22C2×#13093959C2212yes—
23#15689166C2212yes—
24C2×#13093957C2212yes—
25#15689150C2212yes—
26#15689147C2211yescharacteristic
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28#15689127C2212yes—
29C2×#13093944C2212yes—
30C2×#13093940C2211yescharacteristic
31C2×#13093940C2211yescharacteristic
32#15689079C2212yes—
33C4×#10625329C2211yescharacteristic
34#15688892C2212yes—
35#15688892C2212yes—
36#15688891C2211yescharacteristic
37#15688891C2211yescharacteristic
38#15688887C2212yes—
39C47×#533C2211yescharacteristic
40C2×C4⋊Q8×C47C2211yescharacteristic
41C47×#501—22no—
42C47×#500—21no—
43C47×#498—21no—
44C47×#428C2212yes—
45C47×#428C2212yes—
46C4×M16×C47C2211yescharacteristic
47C22×D188C2311yescharacteristic
48C22×C4×D47—21no—
49C2×#10625329C2311yescharacteristic
50C2×#10625329C2312yes—
51C2×#10625329C2311yescharacteristic
52C22×C47⋊C8C2312yes—
53C4×D188—21no—
54C4×D188—21no—
55C4×D188—21no—
56C4×D188—21no—
57C2×C4⋊Dic47C2311yescharacteristic
58C2×(C2×D47)⋊C4—21no—
59#13093959—24no—
60#13093957—22no—
61#13093957—22no—
62#13093957—22no—
63#13093955—22no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)—21no—
66C4⋊(C47⋊C8)—22no—
67C4⋊(C47⋊C8)—22no—
68C4⋊(C47⋊C8)D411yescharacteristic
69C4⋊(C47⋊C8)D412yes—
70C4⋊(C47⋊C8)D411yescharacteristic
71#13093944—22no—
72#13093944—22no—
73#13093944—22no—
74#13093944—22no—
75#13093940—22no—
76#13093940Q814yescomplemented
77#13093935—21no—
78#13093935—22no—
79C4×C47⋊C8—21no—
80C4×C47⋊C8—22no—
81C22×Q8×C47—21no—
82C2×M16×C47C2311yescharacteristic
83C2×M16×C47C2311yescharacteristic
84C4⋊Q8×C47—21no—
85C4⋊Q8×C47—21no—
86C4⋊Q8×C47—21no—
87C4⋊Q8×C47—21no—
88C4⋊Q8×C47D411yescharacteristic
89C4⋊Q8×C47D411yescharacteristic
90#125×C47D412yes—
91C22⋊Q8×C47—21no—
92C22⋊Q8×C47—21no—
93C4×Q8×C47—21no—
94C4×Q8×C47—21no—
95#117×C47C2312yes—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47C2312yes—
98C2×C42×C47C2311yescharacteristic
99#107×C47—22no—
100#107×C47—22no—
101#106×C47—22no—
102#106×C47—22no—
103#97×C47—21no—
104C4×C8×C47—21no—
105C23×D47—21no—
106C2×Q8×C47—21no—
107C2×Q8×C47—21no—
108C2×Q8×C47—41no—
109C2×Q8×C47—41no—
110C2×Q8×C47—82no—
111C22×C4×C47—22no—
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47C2411yescharacteristic
115C22×Dic47—21no—
116C2×D188—21no—
117C2×D188—21no—
118C2×D188—21no—
119C2×D188—21no—
120C2×D188—21no—
121C2×D188C2×Q812yes—
122C2×C4×D47—41no—
123C2×C4×D47—41no—
124C2×C4×D47—82no—
125M16×C47—21no—
126M16×C47—21no—
127M16×C47C2×Q812yes—
128M16×C47#4112yes—
129C2×C8×C47—21no—
130C2×C8×C47—21no—
131C4⋊C4×C47—22no—
132C4⋊C4×C47—22no—
133C4⋊C4×C47—21no—
134C4⋊C4×C47—21no—
135C4⋊C4×C47—22no—
136C4⋊C4×C47—22no—
137C4⋊C4×C47—41no—
138C4⋊C4×C47—41no—
139C4⋊C4×C47C2×D412yes—
140C4⋊C4×C47C2×D412yes—
141C22⋊C4×C47—22no—
142C42×C47—22no—
143C42×C47C2×D411yescharacteristic
144C42×C47C2×D411yescharacteristic
145C42×C47C2×D411yescharacteristic
146C42×C47C2×D411yescharacteristic
147(C2×D47)⋊C4—41no—
148(C2×D47)⋊C4—41no—
149C4⋊Dic47—21no—
150C4⋊Dic47—21no—
151#10625329—21no—
152#10625329—21no—
153#10625329—22no—
154#10625329—21no—
155#10625329—22no—
156#10625329C2×Q812yes—
157C2×C47⋊C8—21no—
158C2×C47⋊C8—22no—
159C2×C47⋊C8—22no—
160C2×C47⋊C8—21no—
161C2×C47⋊C8—22no—
162C2×C47⋊C8C2×D412yes—
163C2×C47⋊C8C2×D412yes—
164C23×C47C2×#4111yescharacteristic
165C22×D47—22no—
166C22×D47—41no—
167C22×D47—41no—
168C22×D47—81no—
169C22×D47—81no—
170Q8×C47—42no—
171Q8×C47—81no—
172Q8×C47—82no—
173Q8×C47—82no—
174C2×C4×C47—22no—
175C2×C4×C47—22no—
176C2×C4×C47—22no—
177C2×C4×C47—41no—
178C2×C4×C47—42no—
179C2×C4×C47—82no—
180C2×C4×C47#12111yescharacteristic
181C2×C4×C47#12111yescharacteristic
182C2×C4×C47#12111yescharacteristic
183C2×C4×C47#12111yescharacteristic
184C2×C4×C47C22×D411yescharacteristic
185C2×C4×C47C22×D411yescharacteristic
186C2×C4×C47C22×Q811yescharacteristic
187C2×C4×C47C2×#4111yescharacteristic
188C2×C4×C472−1+411yescharacteristic
189C2×C4×C472−1+411yescharacteristic
190C2×Dic47—41no—
191C2×Dic47—81no—
192D188—22no—
193D188—22no—
194D188—41no—
195D188—41no—
196C4×D47—82no—
197C4×D47—82no—
198C8×C47—22no—
199C8×C47—22no—
200C47⋊C8—22no—
201C47⋊C8—22no—
202C47⋊C8—22no—
203C47⋊C8—41no—
204C47⋊C8—42no—
205#25489—471nomaximal, Sylow
206C22×C47—21no—
207C22×C47—21no—
208C22×C47C2×#12111yescharacteristic
209C22×C47#53311yescharacteristic
210C22×C47#54111yescharacteristic
211C2×D47—42no—
212C2×D47—82no—
213C2×D47—81no—
214C2×D47—81no—
215C2×D47—82no—
216C4×C47—21no—
217C4×C47—21no—
218C4×C47—21no—
219C4×C47—42no—
220C4×C47—42no—
221C4×C47—82no—
222C4×C47#47212yes—
223C4×C47#54711yescharacteristic
224C4×C47#54811yescharacteristic
225C4×C47D4×Q811yescharacteristic
226C4×C47D4×Q811yescharacteristic
227Dic47—81no—
228#3165D4711yescharacteristic, complemented
229#3077—471no—
230C2×#474—471no—
231#2962—471no—
232#2089—472no—
233#2063—472no—
234#1997—471no—
235#1952—472no—
236#1950—471no—
237#1931—472no—
238C4×#100—471no—
239C2×C47—21no—
240C2×C47—21no—
241C2×C47#242911yescharacteristic
242C2×C47C2×#47211yescharacteristic
243C2×C47#306911yescharacteristic
244D47—82no—
245#533C2×D4711yescharacteristic
246C2×C4⋊Q8C2×D4711yescharacteristic
247C2×C4×D4—471no—
248#501—22no—
249#500—21no—
250#498—21no—
251#478—942no—
252#476—941no—
253#474—941no—
254#474—942no—
255#474—941no—
256#474—941no—
257#446—942no—
258#446—941no—
259C8×Q8—941no—
260C2×#108—472no—
261#428—472no—
262#428C2×D4712yes—
263#428C2×D4712yes—
264C2×#107—472no—
265#424—472no—
266#423—471no—
267C2×C4⋊C8—471no—
268#418—472no—
269C2×#102—472no—
270C2×#100—471no—
271C2×#100—471no—
272#405—472no—
273C4×M16—471no—
274C4×M16C2×D4711yescharacteristic
275#343—471no—
276#343—471no—
277#343—472no—
278#343—472no—
279#341—472no—
280C47#2548911yescharacteristic, complemented, Sylow
281C22×Q8—21no—
282C22×D4—471no—
283C23×C4—941no—
284C2×M16—471no—
285C2×M16—472no—
286C2×M16—471no—
287C2×M16C22×D4711yescharacteristic
288C2×M16C22×D4711yescharacteristic
289C22×C8—472no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C22⋊D4711yescharacteristic
295C4⋊Q8C22⋊D4711yescharacteristic
296#125C22⋊D4712yes—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×D4—941no—
302C4×D4—941no—
303C4×D4—941no—
304C4×D4—941no—
305#117C22×D4712yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—471no—
308C2×C4⋊C4C22×D4712yes—
309C2×C22⋊C4—941no—
310C2×C42C22×D4711yescharacteristic
311#108—944no—
312#107—22no—
313#107—22no—
314#107—942no—
315#107—942no—
316#107—942no—
317#106—22no—
318#106—22no—
319#106—942no—
320C4⋊C8—471no—
321C4⋊C8—471no—
322C4⋊C8—472no—
323C4⋊C8—942no—
324C4⋊C8—942no—
325C4⋊C8—941no—
326C4⋊C8—941no—
327#102—942no—
328#102—942no—
329#102—942no—
330#102—942no—
331#100—474no—
332#100—942no—
333#97—21no—
334#97—942no—
335#97—941no—
336C4×C8—21no—
337C4×C8—942no—
338C4×C8—941no—
339C24—941no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×D4—472no—
346C2×D4—941no—
347C2×D4—941no—
348C2×D4—941no—
349C2×D4—941no—
350C2×D4—941no—
351C22×C4—22no—
352C22×C4—941no—
353C22×C4—3762no—
354C22×C4—1881no—
355C22×C4—1881no—
356C22×C4#1062535211yescharacteristic
357C22×C4#1062535211yescharacteristic
358C22×C4C23×D4711yescharacteristic
359M16—21no—
360M16—21no—
361M16—472no—
362M16—941no—
363M16—941no—
364M16—942no—
365M16—941no—
366M16—942no—
367M16Q8×D4712yes—
368M16#1062535412yes—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—472no—
372C2×C8—472no—
373C2×C8—941no—
374C2×C8—942no—
375C2×C8—942no—
376C2×C8—941no—
377C2×C8—942no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—941no—
387C4⋊C4—941no—
388C4⋊C4C2×C22⋊D4712yes—
389C4⋊C4C2×C22⋊D4712yes—
390C22⋊C4—22no—
391C22⋊C4—1881no—
392C22⋊C4—1881no—
393C42—22no—
394C42D4×D4711yescharacteristic
395C42D4×D4711yescharacteristic
396C42C2×C22⋊D4711yescharacteristic
397C42C2×C22⋊D4711yescharacteristic
398C23—942no—
399C23—3761no—
400C23—3761no—
401C23—1881no—
402C23—1881no—
403C23C2×#1062535211yescharacteristic
404Q8—42no—
405Q8—81no—
406Q8—82no—
407Q8—82no—
408D4—942no—
409D4—942no—
410D4—1881no—
411D4—1881no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—3762no—
419C2×C4—3762no—
420C2×C4—3761no—
421C2×C4—1881no—
422C2×C4#1309399611yescharacteristic
423C2×C4#1309399611yescharacteristic
424C2×C4#1309399611yescharacteristic
425C2×C4#1309399611yescharacteristic
426C2×C4C2×D4×D4711yescharacteristic
427C2×C4C22×C22⋊D4711yescharacteristic
428C2×C4C2×Q8×D4711yescharacteristic
429C2×C4C2×#1062535411yescharacteristic
430C2×C4#1309407211yescharacteristic
431C2×C4#1309407211yescharacteristic
432C8—22no—
433C8—22no—
434C8—942no—
435C8—942no—
436C8—942no—
437C8—1881no—
438C8—1882no—
439C22—21no—
440C22—21no—
441C22—3762no—
442C22—3762no—
443C22—3761no—
444C22—3761no—
445C22—1882no—
446C22C2×#1309399611yescharacteristic
447C22#1568967811yescharacteristic
448C22#1568973411yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—3761no—
456C4#1568939112yes—
457C4#1568977611yescharacteristic
458C4#1568978411yescharacteristic
459C4Q8×C22⋊D4711yescharacteristic
460C4#1568978811yescharacteristic
461C2—21no—
462C2—21no—
463C2—3762no—
464C2#2024699011yescharacteristic
465C2C2×#1568939111yescharacteristic
466C2#2025312211yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#3165C2
socleC22×C47C2×#121
prime radicalC22×D188C23
hypercenter#3165D47
hypocenterC47#25489
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. #20252337C2semidirect
4. #20244896C2semidirect
6. #20244378C2semidirect
7. #20244170C2semidirect
8. #20244163C2semidirect
11. C47×#3165C2semidirect
76. #13093940Q8semidirect
228. #3165D47semidirect
280. C47#25489semidirect

Maximal subgroups

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

subgroupindex
#202531812
C2×#156894002
#202523372
#202448962
#202447452
#202443782
#202441702
#202441632
#202440492
C4×#130939402
C47×#31652
#2548947