Finite Group Databasebeta

← #26316238

Subgroups — #26316238

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253911C211yesmaximal, characteristic, complemented
2C2×#15689243C211yesmaximal, characteristic
3#20244562C212yesmaximal, complemented
4#20244559C212yesmaximal, complemented
5#20244283C212yesmaximal
6#20243510C212yesmaximal, complemented
7#20243442C211yesmaximal, characteristic, complemented
8#20243439C211yesmaximal, characteristic, complemented
9#20243431C211yesmaximal, characteristic, complemented
10#20243274C211yesmaximal, characteristic, complemented
11C47×#1812C211yesmaximal, characteristic, complemented
12C2×#13094044C2211yescharacteristic
13C2×#13094030C2211yescharacteristic
14#15689676C2212yes—
15#15689531—22no—
16#15689530—22no—
17#15689243—22no—
18#15689243—22no—
19C2×#13093960C2211yescharacteristic
20C2×#13093959C2211yescharacteristic
21#15689161C2211yescharacteristic
22#15689159C2212yes—
23#15689141C2212yes—
24#15689140C2212yes—
25C2×#13093951C2211yescharacteristic
26C2×#13093950C2211yescharacteristic
27#15689096C2212yes—
28#15689079C2212yes—
29C2×#13093934C2211yescharacteristic
30#15688895C2212yes—
31#15688893C2212yes—
32#15688882C2212yes—
33#15688879C2212yes—
34C2×#131×C47C2211yescharacteristic
35C47×#427C2211yescharacteristic
36C47×#427C2211yescharacteristic
37C47×#410C2212yes—
38C47×H(3,ℤ/4)C2212yes—
39C2×#10625350C2311yescharacteristic
40#13094044—22no—
41#13094044—22no—
42C2×#10625337Q811yescharacteristic, complemented
43C2×#10625335Q811yescharacteristic, complemented
44C2×#10625336Q811yescharacteristic, complemented
45C2×#10625334Q811yescharacteristic, complemented
46#13094030—21no—
47#13094030—21no—
48#13094030—21no—
49#13094030—21no—
50C2×#10625329C2312yes—
51C2×#10625329C2311yescharacteristic
52C2×#10625324—21no—
53C22×C47⋊C8C2311yescharacteristic
54C2×C8×D47—21no—
55#13094023—22no—
56#13094015—22no—
57#13094001—42no—
58C4×Dic94—42no—
59#13093983C2312yes—
60C2×C4×Dic47C2311yescharacteristic
61#13093960—21no—
62#13093960—21no—
63#13093959—22no—
64#13093958—21no—
65#13093957—22no—
66#13093956—21no—
67#13093955—22no—
68#13093952—42no—
69#13093951—22no—
70#13093951—42no—
71#13093950—22no—
72#13093938—42no—
73#13093935—22no—
74#13093934D412yes—
75#13093934D412yes—
76C4×C47⋊C8—22no—
77C2×#41×C47C2311yescharacteristic
78#131×C47—21no—
79#131×C47—21no—
80#131×C47—21no—
81#131×C47—21no—
82C2×M16×C47—21no—
83C2×M16×C47C2311yescharacteristic
84C22×C8×C47—21no—
85C22×C8×C47C2311yescharacteristic
86#117×C47C2312yes—
87#117×C47C2312yes—
88#107×C47—21no—
89#107×C47—21no—
90#106×C47—21no—
91#106×C47—21no—
92#99×C47—42no—
93#41×C47—21no—
94#41×C47—21no—
95#41×C47—21no—
96#41×C47—21no—
97C2×Q8×C47C2×Q811yescharacteristic
98C2×D4×C47—21no—
99C2×D4×C47C2×Q811yescharacteristic
100C22×C4×C47—21no—
101C22×C4×C47C2411yescharacteristic
102C2×C22⋊D47—21no—
103C22×Dic47—21no—
104#10625350—21no—
105#10625350—21no—
106#10625350—21no—
107#10625350—21no—
108C2×D188C2×Q811yescharacteristic
109C2×C4×D47—21no—
110C2×Dic94—42no—
111C2×Dic94C2×Q811yescharacteristic
112M16×C47—21no—
113M16×C47—21no—
114M16×C47—41no—
115M16×C47—41no—
116C2×C8×C47—21no—
117C2×C8×C47—41no—
118C2×C8×C47—41no—
119C2×C8×C47C2×D411yescharacteristic
120C2×C8×C47C2×D411yescharacteristic
121C2×C8×C47#4111yescharacteristic
122C2×C8×C47#4111yescharacteristic
123C2×C8×C47#4111yescharacteristic
124C2×C8×C47#4111yescharacteristic
125C4⋊C4×C47—22no—
126C4⋊C4×C47—22no—
127C22⋊C4×C47—22no—
128C22⋊C4×C47—22no—
129C42×C47—22no—
130C42×C47—22no—
131C22⋊Dic47—22no—
132#10625337—22no—
133#10625336—22no—
134#10625335—22no—
135#10625334—22no—
136C4⋊Dic47—22no—
137C4⋊Dic47—42no—
138Dic47⋊C4—42no—
139Dic47⋊C4—42no—
140C4×Dic47—21no—
141C4×Dic47—22no—
142C4×Dic47C2×D412yes—
143#10625329—21no—
144#10625329—22no—
145#10625329—21no—
146#10625329—22no—
147C2×C47⋊C8—22no—
148C2×C47⋊C8—21no—
149C2×C47⋊C8C2×D412yes—
150C2×C47⋊C8C2×Q811yescharacteristic
151C2×C47⋊C8C2×Q811yescharacteristic
152C2×C47⋊C8#4112yes—
153#10625324—41no—
154#10625324—41no—
155C8×D47—41no—
156C8×D47—41no—
157C23×C47—21no—
158C23×C47H(5,𝔽2)11yescharacteristic
159C22×D47—21no—
160Q8×C47—21no—
161Q8×C47—21no—
162D4×C47—21no—
163D4×C47—21no—
164D4×C47—41no—
165D4×C47—41no—
166C2×C4×C47—21no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—42no—
170C2×C4×C47—42no—
171C2×C4×C47C22×D411yescharacteristic
172C2×C4×C47C22×Q811yescharacteristic
173C2×C4×C47C2×#4111yescharacteristic
174C2×C4×C47C2×#4111yescharacteristic
175C2×C4×C47C2×#4111yescharacteristic
176C2×C4×C47H(5,𝔽2)11yescharacteristic
177C22⋊D47—41no—
178C22⋊D47—41no—
179C2×Dic47—22no—
180C2×Dic47—21no—
181C2×Dic47—42no—
182C2×Dic47—81no—
183D188—21no—
184D188—21no—
185C4×D47—41no—
186C4×D47—41no—
187Dic94—21no—
188Dic94—21no—
189Dic94—84no—
190C8×C47—41no—
191C8×C47#12411yescharacteristic
192C8×C47#12411yescharacteristic
193C8×C47#12411yescharacteristic
194C8×C47#12411yescharacteristic
195C47⋊C8—41no—
196C47⋊C8—42no—
197C47⋊C8C22⋊Q812yes—
198C47⋊C8C22⋊Q812yes—
199#29155—471nomaximal, Sylow
200C22×C47—41no—
201C22×C47—41no—
202C22×C47—81no—
203C22×C47#53511yescharacteristic
204C22×C47#55411yescharacteristic
205C22×C47#55511yescharacteristic
206C2×D47—41no—
207C2×D47—81no—
208C4×C47—41no—
209C4×C47—82no—
210C4×C47—82no—
211C4×C47C2×C22⋊Q811yescharacteristic
212C4×C47C2×#12411yescharacteristic
213C4×C47#55411yescharacteristic
214C4×C47#55511yescharacteristic
215Dic47—42no—
216Dic47—41no—
217Dic47—82no—
218#3162—471no—
219C2×#444—471no—
220#2031—472no—
221#2031—472no—
222#1976—472no—
223#1833—472no—
224#1814—471no—
225#1814—471no—
226#1812—471no—
227#1812D4711yescharacteristic, complemented
228#1773—471no—
229C2×C47—21no—
230C2×C47—21no—
231C2×C47—21no—
232C2×C47—81no—
233C2×C47#256311yescharacteristic
234D47—81no—
235C2×#135—471no—
236C2×#131—471no—
237C2×#131C2×D4711yescharacteristic
238#533—472no—
239#501—942no—
240#501—942no—
241#444—942no—
242#444—942no—
243C2×#108—471no—
244C2×#108—471no—
245#427—471no—
246#427—472no—
247#427C2×D4711yescharacteristic
248#427C2×D4711yescharacteristic
249#421—472no—
250#421—472no—
251C2×#104—471no—
252C2×#104—471no—
253#410—472no—
254#410C2×D4712yes—
255#405—472no—
256C2×#97—471no—
257#344—472no—
258#344—472no—
259#339—472no—
260H(3,ℤ/4)—472no—
261H(3,ℤ/4)C2×D4712yes—
262C47#2915511yescharacteristic, complemented, Sylow
263C2×#41—471no—
264C2×#41C22×D4711yescharacteristic
265#135—942no—
266#135—942no—
267C2×Q16—471no—
268C2×SD16—471no—
269C2×SD16—471no—
270C2×D8—471no—
271#131—21no—
272#131—21no—
273#131—21no—
274#131—21no—
275#131—941no—
276#131—941no—
277#131—941no—
278#131—941no—
279C2×M16—21no—
280C2×M16—471no—
281C2×M16—472no—
282C2×M16—941no—
283C2×M16C22×D4711yescharacteristic
284C22×C8—21no—
285C22×C8—471no—
286C22×C8—941no—
287C22×C8C22×D4711yescharacteristic
288C4⋊Q8—942no—
289#125—942no—
290C22⋊Q8—1882no—
291C4×Q8—1882no—
292#117—472no—
293#117C22×D4712yes—
294#117C22×D4712yes—
295C2×C42—471no—
296#108—942no—
297#108—941no—
298#108—941no—
299#107—21no—
300#107—21no—
301#107—942no—
302#107—941no—
303#106—21no—
304#106—21no—
305#106—942no—
306#106—941no—
307#104—942no—
308#104—942no—
309#104—1882no—
310#104—1882no—
311#99—42no—
312#99—1882no—
313#97—472no—
314#97—472no—
315#97—942no—
316C4×C8—942no—
317#41—21no—
318#41—21no—
319#41—21no—
320#41—21no—
321#41—941no—
322#41—941no—
323#41—941no—
324#41—941no—
325C2×Q8—471no—
326C2×Q8—1882no—
327C2×Q8Q8×D4711yescharacteristic
328C2×D4—21no—
329C2×D4—471no—
330C2×D4—941no—
331C2×D4Q8×D4711yescharacteristic
332C22×C4—21no—
333C22×C4—941no—
334C22×C4—941no—
335C22×C4C23×D4711yescharacteristic
336Q16—942no—
337SD16—942no—
338SD16—942no—
339D8—942no—
340M16—21no—
341M16—21no—
342M16—41no—
343M16—41no—
344M16—941no—
345M16—941no—
346M16—942no—
347M16—942no—
348M16—1881no—
349M16—1881no—
350C2×C8—21no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—472no—
354C2×C8—471no—
355C2×C8—471no—
356C2×C8—472no—
357C2×C8—942no—
358C2×C8—941no—
359C2×C8—1881no—
360C2×C8—1881no—
361C2×C8#1062535011yescharacteristic
362C2×C8#1062535011yescharacteristic
363C2×C8D4×D4711yescharacteristic
364C2×C8D4×D4711yescharacteristic
365C2×C8#1062535211yescharacteristic
366C2×C8#1062535211yescharacteristic
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—942no—
370C4⋊C4—1882no—
371C4⋊C4—1882no—
372C4⋊C4—1882no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—942no—
376C42—22no—
377C42—22no—
378C42—472no—
379C42—941no—
380C42—942no—
381C23—21no—
382C23—941no—
383C23#1309406911yescharacteristic
384Q8—21no—
385Q8—21no—
386Q8—941no—
387Q8—941no—
388Q8—3764no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—941no—
394D4—941no—
395D4—1881no—
396D4—1881no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—941no—
403C2×C4—942no—
404C2×C4—3761no—
405C2×C4—1882no—
406C2×C4—1881no—
407C2×C4—1881no—
408C2×C4C2×D4×D4711yescharacteristic
409C2×C4C2×Q8×D4711yescharacteristic
410C2×C4#41×D4711yescharacteristic
411C2×C4C2×#1062535211yescharacteristic
412C2×C4C2×#1062535011yescharacteristic
413C2×C4#1309407011yescharacteristic
414C8—41no—
415C8—472no—
416C8—472no—
417C8—1882no—
418C8—1881no—
419C8#1309401011yescharacteristic
420C8#1309401011yescharacteristic
421C8#1309401011yescharacteristic
422C8#1309401011yescharacteristic
423C22—41no—
424C22—41no—
425C22—81no—
426C22—3761no—
427C22—1881no—
428C22#1568968911yescharacteristic
429C22#1568982811yescharacteristic
430C22#1568983711yescharacteristic
431C4—41no—
432C4—82no—
433C4—82no—
434C4—3762no—
435C4—1882no—
436C4—1881no—
437C4C22⋊Q8×D4711yescharacteristic
438C4C2×#1309401011yescharacteristic
439C4#1568983111yescharacteristic
440C4#1568983611yescharacteristic
441C2—21no—
442C2—21no—
443C2—21no—
444C2—81no—
445C2—3761no—
446C2#2024855811yescharacteristic
subgroupG/H
centerC4#15689836
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1812C2
socleC2×C47#2563
prime radicalC2×#13094044C22
hypercenter#1812D47
hypocenterC47#29155
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1812

Complements

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

NH ≅ G/Ntype
1. #20253911C2semidirect
3. #20244562C2semidirect
4. #20244559C2semidirect
6. #20243510C2semidirect
7. #20243442C2semidirect
8. #20243439C2semidirect
9. #20243431C2semidirect
10. #20243274C2semidirect
11. C47×#1812C2semidirect
42. C2×#10625337Q8semidirect
43. C2×#10625335Q8semidirect
44. C2×#10625336Q8semidirect
45. C2×#10625334Q8semidirect
227. #1812D47semidirect
262. C47#29155semidirect

Maximal subgroups

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

subgroupindex
#202539112
C2×#156892432
#202445622
#202445592
#202442832
#202435102
#202434422
#202434392
#202434312
#202432742
C47×#18122
#2915547