Finite Group Databasebeta

← #26370866

Subgroups — #26370866

Order 12032.

656 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689558C211yesmaximal, characteristic, complemented
2#20256491C211yesmaximal, characteristic, complemented
3C2×#15689393C211yesmaximal, characteristic, complemented
4#20245520C214yesmaximal, complemented
5#20245506C214yesmaximal, complemented
6#20245182C211yesmaximal, characteristic, complemented
7#20245137C211yesmaximal, characteristic, complemented
8C2×#15688935C211yesmaximal, characteristic, complemented
9C47×#3067C211yesmaximal, characteristic, complemented
10C2×#13094049—21no—
11C22×D376C2211yescharacteristic
12#15689558—22no—
13#15689558—22no—
14C2×#13093967C2212yescomplemented
15C2×#13093965C2212yescomplemented
16#15689540—22no—
17#15689540C2214yescomplemented
18C2×#13093964—22no—
19C2×#13093964C2211yescharacteristic
20C2×#13093964C2211yescharacteristic
21#15689393—22no—
22#15689393C2214yes—
23C2×#13093960C2211yescharacteristic
24C2×#13093940—21no—
25#15688952C2214yescomplemented
26#15688939C2212yescomplemented
27#15688936C2212yescomplemented
28#15688935C2214yescomplemented
29C47×#575—21no—
30C2×#135×C47C2211yescharacteristic
31C2×#131×C47C2211yescharacteristic
32C47×#473C2212yescomplemented
33C47×#472C2212yescomplemented
34C47×#471—22no—
35C2×#108×C47C2211yescharacteristic
36C47×#413—21no—
37C22×D188—21no—
38#13094049—24no—
39#13094049—42no—
40C2×#10625336—41no—
41C2×D376—21no—
42C2×D376—21no—
43C2×D376—21no—
44C2×D376—21no—
45C2×D376—21no—
46C2×D376C2312yes—
47C2×#10625334—41no—
48C2×#10625329—21no—
49#13093967—24no—
50#13093965—24no—
51#13093964—21no—
52#13093964—24no—
53#13093964D412yes—
54#13093964D412yes—
55#13093964C2312yes—
56C2×C47⋊C16—21no—
57C2×C47⋊C16—24no—
58C2×C47⋊C16C2312yes—
59#13093960—21no—
60#13093960C2312yes—
61#13093940—24no—
62C2×#41×C47—21no—
63C2×#41×C47D411yescharacteristic
64#137×C47—22no—
65#137×C47—41no—
66AGL(1,ℤ/8)×C47—22no—
67AGL(1,ℤ/8)×C47—41no—
68#135×C47—21no—
69#135×C47—21no—
70#135×C47—21no—
71#135×C47—21no—
72#135×C47—42no—
73C2×Q16×C47C2311yescharacteristic
74C2×SD16×C47—21no—
75C2×D8×C47C2311yescharacteristic
76#131×C47—21no—
77#131×C47—21no—
78#131×C47D412yes—
79#131×C47C2312yes—
80C2×M16×C47—21no—
81C2×M16×C47—21no—
82C2×M16×C47D411yescharacteristic
83C22×C8×C47—21no—
84C22×C8×C47C2311yescharacteristic
85#108×C47—21no—
86#108×C47C2312yes—
87#101×C47—22no—
88#100×C47—22no—
89C23×D47—41no—
90#41×C47—21no—
91#41×C47—21no—
92#41×C47—41no—
93#41×C47—41no—
94#41×C47—41no—
95#41×C47—41no—
96#41×C47D812yes—
97#41×C47C2×D412yes—
98C2×Q8×C47—21no—
99C2×Q8×C47D811yescharacteristic
100C2×D4×C47—21no—
101C2×D4×C47—21no—
102C2×D4×C47—41no—
103C2×D4×C47D811yescharacteristic
104C22×C4×C47—21no—
105C22×C4×C47—41no—
106C22×C4×C47C2×D411yescharacteristic
107C2×D188—22no—
108C2×D188—41no—
109C2×D188—41no—
110C2×D188—41no—
111C2×D188—41no—
112C2×D188—41no—
113Q16×C47—21no—
114Q16×C47—21no—
115Q16×C47—42no—
116SD16×C47—42no—
117SD16×C47—42no—
118SD16×C47—41no—
119SD16×C47—41no—
120D8×C47—21no—
121D8×C47—21no—
122D8×C47—42no—
123M16×C47—21no—
124M16×C47—22no—
125M16×C47—22no—
126M16×C47—22no—
127M16×C47—41no—
128M16×C47C2×D412yes—
129C2×C8×C47—21no—
130C2×C8×C47—22no—
131C2×C8×C47—22no—
132C2×C8×C47—41no—
133C2×C8×C47C2×D411yescharacteristic
134C2×C8×C47C2×D411yescharacteristic
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47#4111yescharacteristic
137C2×C8×C47#4111yescharacteristic
138C2×C8×C47C2411yescharacteristic
139#10625336—44no—
140#10625334—44no—
141#10625329—22no—
142#10625329—41no—
143C2×C47⋊C8—41no—
144D376—22no—
145D376—22no—
146D376—41no—
147D376—41no—
148C47⋊C16—22no—
149C47⋊C16C2×D414yes—
150C23×C47—21no—
151C23×C47—41no—
152C23×C47#13511yescharacteristic
153C22×D47—42no—
154C22×D47—81no—
155C22×D47—81no—
156C22×D47—162no—
157Q8×C47—21no—
158Q8×C47—41no—
159Q8×C47—41no—
160Q8×C47C2×D812yes—
161D4×C47—22no—
162D4×C47—22no—
163D4×C47—21no—
164D4×C47—41no—
165D4×C47—41no—
166D4×C47—81no—
167D4×C47—81no—
168D4×C47C2×D812yes—
169C2×C4×C47—21no—
170C2×C4×C47—22no—
171C2×C4×C47—22no—
172C2×C4×C47—41no—
173C2×C4×C47—81no—
174C2×C4×C47—81no—
175C2×C4×C47#12111yescharacteristic
176C2×C4×C47#12111yescharacteristic
177C2×C4×C47C2×D811yescharacteristic
178C2×C4×C47C2×D811yescharacteristic
179C2×C4×C47#13511yescharacteristic
180C2×C4×C47C22×D411yescharacteristic
181D188—42no—
182D188—42no—
183D188—81no—
184D188—81no—
185C8×C47—22no—
186C8×C47—42no—
187C8×C47#12111yescharacteristic
188C8×C47#12111yescharacteristic
189C8×C47C22×D411yescharacteristic
190C8×C47C2×#4111yescharacteristic
191C47⋊C8—42no—
192#33618—471nomaximal, Sylow
193C22×C47—22no—
194C22×C47—41no—
195C22×C47—81no—
196C22×C47—81no—
197C22×C47—161no—
198C22×C47#46611yescharacteristic
199C22×C47#46611yescharacteristic
200C22×C47C2×#13511yescharacteristic
201C2×D47—82no—
202C2×D47—162no—
203C2×D47—162no—
204C2×D47—162no—
205C4×C47—22no—
206C4×C47—81no—
207C4×C47#46611yescharacteristic
208C4×C47#46611yescharacteristic
209C4×C47C2×#12111yescharacteristic
210C4×C47C22×D811yescharacteristic
211C2×#509—471no—
212#3407—471no—
213#3067D4711yescharacteristic, complemented
214C2×#472—471no—
215#2223—474no—
216#2221—474no—
217#2161—471no—
218#2148—471no—
219C2×#361—471no—
220C2×C47—21no—
221C2×C47—21no—
222C2×C47—21no—
223C2×C47—42no—
224C2×C47—161no—
225C2×C47C2×#46611yescharacteristic
226D47—162no—
227#575—21no—
228C2×AGL(1,ℤ/8)—941no—
229C2×#135C2×D4711yescharacteristic
230C22×D8—471no—
231C2×#131C2×D4711yescharacteristic
232#509—942no—
233#509—942no—
234C2×SD32—472no—
235C2×D16—472no—
236#504—474no—
237#504—942no—
238C2×M32—471no—
239C2×M32—471no—
240C2×M32—942no—
241#473C2×D4712yescomplemented
242#472—474no—
243#472—942no—
244#472C2×D4712yescomplemented
245#471—22no—
246C2×#108—471no—
247C2×#108C2×D4711yescharacteristic
248#413—21no—
249C2×#100—941no—
250#368—474no—
251#362—472no—
252#361—472no—
253#361—474no—
254C47#3361811yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41C22⋊D4711yescharacteristic
257C22×D4—941no—
258#137—22no—
259#137—41no—
260AGL(1,ℤ/8)—22no—
261AGL(1,ℤ/8)—41no—
262AGL(1,ℤ/8)—944no—
263AGL(1,ℤ/8)—1882no—
264#135—21no—
265#135—21no—
266#135—21no—
267#135—21no—
268#135—42no—
269C2×Q16C22×D4711yescharacteristic
270C2×SD16—21no—
271C2×SD16—1881no—
272C2×D8—472no—
273C2×D8—941no—
274C2×D8—941no—
275C2×D8—941no—
276C2×D8—941no—
277C2×D8—941no—
278C2×D8—1881no—
279C2×D8C22×D4711yescharacteristic
280#131—21no—
281#131—21no—
282#131C22⋊D4712yes—
283#131C22×D4712yes—
284C2×M16—21no—
285C2×M16—21no—
286C2×M16—941no—
287C2×M16C22⋊D4711yescharacteristic
288C22×C8—21no—
289C22×C8C22×D4711yescharacteristic
290SD32—944no—
291D16—944no—
292M32—472no—
293M32—472no—
294M32—472no—
295M32—944no—
296M32—941no—
297C2×C16—472no—
298C2×C16—944no—
299C2×C16—941no—
300#108—21no—
301#108—472no—
302#108—941no—
303#108C22×D4712yes—
304#101—22no—
305#100—22no—
306#100—944no—
307C24—1881no—
308#41—21no—
309#41—21no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41#1062533412yes—
315#41C2×C22⋊D4712yes—
316C2×Q8—21no—
317C2×Q8#1062533411yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—942no—
322C2×D4—1881no—
323C2×D4—1881no—
324C2×D4—1881no—
325C2×D4—1881no—
326C2×D4—1881no—
327C2×D4#1062533411yescharacteristic
328C22×C4—21no—
329C22×C4—41no—
330C22×C4C2×C22⋊D4711yescharacteristic
331Q16—21no—
332Q16—21no—
333Q16—42no—
334SD16—42no—
335SD16—42no—
336SD16—41no—
337SD16—41no—
338SD16—1884no—
339D8—21no—
340D8—21no—
341D8—42no—
342D8—942no—
343D8—942no—
344D8—1884no—
345D8—1881no—
346D8—1881no—
347M16—21no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—41no—
352M16—942no—
353M16—1881no—
354M16C2×C22⋊D4712yes—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—1881no—
360C2×C8D4×D4711yescharacteristic
361C2×C8D4×D4711yescharacteristic
362C2×C8#1062535211yescharacteristic
363C2×C8#1062535211yescharacteristic
364C2×C8C2×C22⋊D4711yescharacteristic
365C2×C8C23×D4711yescharacteristic
366C16—474no—
367C16—942no—
368C23—21no—
369C23—41no—
370C23—3761no—
371C23—3761no—
372C23—1882no—
373C23—7522no—
374C23#1309404411yescharacteristic
375Q8—21no—
376Q8—41no—
377Q8—41no—
378Q8C2×#1062533412yes—
379D4—22no—
380D4—22no—
381D4—21no—
382D4—41no—
383D4—41no—
384D4—81no—
385D4—81no—
386D4—3761no—
387D4—3761no—
388D4—1882no—
389D4—1882no—
390D4C2×#1062533412yes—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4#1309399611yescharacteristic
398C2×C4#1309399611yescharacteristic
399C2×C4C2×#1062533411yescharacteristic
400C2×C4C2×#1062533411yescharacteristic
401C2×C4#1309404411yescharacteristic
402C2×C4C22×C22⋊D4711yescharacteristic
403C8—22no—
404C8—42no—
405C8—1882no—
406C8#1309399611yescharacteristic
407C8#1309399611yescharacteristic
408C8C2×D4×D4711yescharacteristic
409C8C2×#1062535211yescharacteristic
410C22—22no—
411C22—41no—
412C22—81no—
413C22—81no—
414C22—161no—
415C22—3762no—
416C22—7522no—
417C22—7522no—
418C22—7522no—
419C22#1568935711yescharacteristic
420C22#1568935711yescharacteristic
421C22C2×#1309404411yescharacteristic
422C4—22no—
423C4—81no—
424C4#1568935711yescharacteristic
425C4#1568935711yescharacteristic
426C4C2×#1309399611yescharacteristic
427C4C22×#1062533411yescharacteristic
428C2—21no—
429C2—21no—
430C2—21no—
431C2—42no—
432C2—161no—
433C2—7522no—
434C2C2×#1568935711yescharacteristic
subgroupG/H
centerC2C2×#15689357
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3067C2
socleC2×C47C2×#466
hypercenter#3067D47
hypocenterC47#33618
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#15689558C2semidirect
2. #20256491C2semidirect
3. C2×#15689393C2semidirect
4. #20245520C2semidirect
5. #20245506C2semidirect
6. #20245182C2semidirect
7. #20245137C2semidirect
8. C2×#15688935C2semidirect
9. C47×#3067C2semidirect
14. C2×#13093967C22semidirect
15. C2×#13093965C22semidirect
17. #15689540C22semidirect
25. #15688952C22semidirect
26. #15688939C22semidirect
27. #15688936C22semidirect
28. #15688935C22semidirect
32. C47×#473C22semidirect
33. C47×#472C22semidirect
213. #3067D47semidirect
241. #473C2×D47semidirect
244. #472C2×D47semidirect
254. C47#33618semidirect

Maximal subgroups

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

subgroupindex
C2×#156895582
#202564912
C2×#156893932
#202455202
#202455062
#202451822
#202451372
C2×#156889352
C47×#30672
#3361847