Finite Group Databasebeta

← #27206632

Subgroups — #27206632

Order 13184.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15693048C211yesmaximal, characteristic, complemented
2C22×#13110092C211yesmaximal, characteristic, complemented
3#20268452C211yesmaximal, characteristic, complemented
4#20268425C211yesmaximal, characteristic, complemented
5#20268177C211yesmaximal, characteristic, complemented
6#20268040C211yesmaximal, characteristic, complemented
7#20268025C211yesmaximal, characteristic, complemented
8#20268009C211yesmaximal, characteristic, complemented
9#20267989C211yesmaximal, characteristic, complemented
10#20267940C211yesmaximal, characteristic, complemented
11#20267939C211yesmaximal, characteristic, complemented
12#20267927C211yesmaximal, characteristic, complemented
13#20267894C211yesmaximal, characteristic, complemented
14#20267859C211yesmaximal, characteristic, complemented
15#448×C103C211yesmaximal, characteristic, complemented
16C2×#13110106C2211yescharacteristic, complemented
17C22×D412C2211yescharacteristic
18C2×D4×D103—21no—
19C2×D4×D103C2211yescharacteristic
20#15693048—24no—
21C2×#13110092—24no—
22C2×#13110092C2211yescharacteristic
23C2×#13110092C2211yescharacteristic
24C2×#13110092C2211yescharacteristic, complemented
25C2×#13110092C2211yescharacteristic, complemented
26C2×D824C2211yescharacteristic
27C2×D824C2211yescharacteristic, complemented
28C2×#13110090C2211yescharacteristic, complemented
29C2×#13110088C2211yescharacteristic, complemented
30C22×C103⋊C8C2211yescharacteristic, complemented
31#15693017C2211yescharacteristic
32#15693010—21no—
33#15693010C2211yescharacteristic, complemented
34#15692997—21no—
35#15692997C2211yescharacteristic, complemented
36#15692996C2211yescharacteristic
37#15692995—21no—
38#15692995C2211yescharacteristic, complemented
39#15692994—21no—
40#15692991—21no—
41#15692989—21no—
42Q8×Dic103C2211yescharacteristic, complemented
43#15692983C2211yescharacteristic
44C4×C22⋊D103—21no—
45C4×C22⋊D103C2211yescharacteristic
46#15692954C2211yescharacteristic, complemented
47Dic103⋊C8C2211yescharacteristic, complemented
48#15692943C2211yescharacteristic
49#15692943C2211yescharacteristic, complemented
50#15692942C2211yescharacteristic, complemented
51#15692941C2211yescharacteristic, complemented
52(C2×D103)⋊C8C2211yescharacteristic, complemented
53#15692931C2211yescharacteristic, complemented
54C22×Q8×C103C2211yescharacteristic, complemented
55C2×SD16×C103C2211yescharacteristic, complemented
56C2×SD16×C103C2211yescharacteristic, complemented
57#121×C103C2211yescharacteristic
58#103×C103C2211yescharacteristic, complemented
59#103×C103C2211yescharacteristic, complemented
60C22⋊C8×C103C2211yescharacteristic, complemented
61C23×D103—21no—
62C23×D103—21no—
63#13110106—22no—
64#13110106—41no—
65C2×D412—22no—
66C2×D412—21no—
67C2×D412—21no—
68C2×D412—21no—
69C2×D412—41no—
70C2×D412C2311yescharacteristic
71C2×D412C2311yescharacteristic
72C2×D412C2311yescharacteristic
73C2×C22⋊D103—21no—
74C2×C22⋊D103—21no—
75C2×C22⋊D103—21no—
76C2×C22⋊D103—21no—
77C2×C22⋊D103D411yescharacteristic
78D4×D103—22no—
79D4×D103—41no—
80D4×D103—44no—
81C2×C4×D103—21no—
82C2×C4×D103—21no—
83C2×C4×D103C2311yescharacteristic
84#13110092—22no—
85#13110092—22no—
86#13110092—22no—
87#13110092D414yescomplemented
88D824—22no—
89D824—41no—
90#13110090—22no—
91#13110088—22no—
92C2×C103⋊C8—22no—
93C2×C103⋊C8C2311yescharacteristic
94C2×C103⋊C8C2311yescharacteristic
95C4⋊Dic103—21no—
96C4⋊Dic103C2311yescharacteristic
97Dic103⋊C4—21no—
98Dic103⋊C4D411yescharacteristic
99C22⋊Dic103D411yescharacteristic
100(C2×D103)⋊C4—21no—
101(C2×D103)⋊C4—21no—
102(C2×D103)⋊C4—21no—
103(C2×D103)⋊C4—21no—
104(C2×D103)⋊C4—21no—
105(C2×D103)⋊C4—21no—
106(C2×D103)⋊C4D411yescharacteristic
107C4×Dic103—21no—
108C4×Dic103C2311yescharacteristic
109C2×Q8×C103—21no—
110C2×Q8×C103—22no—
111C2×Q8×C103—41no—
112C2×Q8×C103C2311yescharacteristic
113C2×Q8×C103C2311yescharacteristic
114C2×D4×C103—21no—
115C2×D4×C103C2311yescharacteristic
116C22×C4×C103—21no—
117C22×C4×C103C2311yescharacteristic
118SD16×C103—22no—
119SD16×C103—41no—
120C2×C8×C103C2311yescharacteristic
121C2×C8×C103C2311yescharacteristic
122C4⋊C4×C103C2311yescharacteristic
123C22⋊C4×C103—21no—
124C23×C103—21no—
125C23×C103C2×D411yescharacteristic
126C22×D103—21no—
127C22×D103—21no—
128C22×D103—21no—
129C22×D103—42no—
130C22×D103—42no—
131C22×D103—42no—
132C22×D103—42no—
133C22×D103—81no—
134C22×D103—81no—
135C22×D103C2×D411yescharacteristic
136Q8×C103—21no—
137Q8×C103—21no—
138Q8×C103—21no—
139Q8×C103—42no—
140Q8×C103C2×D412yescomplemented
141D4×C103—21no—
142D4×C103—21no—
143D4×C103—42no—
144C2×C4×C103—21no—
145C2×C4×C103—21no—
146C2×C4×C103—21no—
147C2×C4×C103—21no—
148C2×C4×C103—21no—
149C2×C4×C103—42no—
150C2×C4×C103C2×D411yescharacteristic
151C2×C4×C103C2411yescharacteristic
152C22⋊D103—22no—
153C22⋊D103—42no—
154C22⋊D103—42no—
155C22⋊D103—42no—
156C22⋊D103—81no—
157C2×Dic103—21no—
158C2×Dic103C2×D411yescharacteristic
159C2×Dic103C2×D411yescharacteristic
160C2×Dic103C2×D411yescharacteristic
161D412—21no—
162D412—21no—
163D412—21no—
164D412—21no—
165D412—21no—
166D412—42no—
167D412—42no—
168D412—42no—
169D412C2×D412yes—
170C4×D103—21no—
171C4×D103—21no—
172C4×D103—42no—
173C4×D103—42no—
174C8×C103—21no—
175C8×C103—21no—
176C103⋊C8—21no—
177C103⋊C8C2×D412yescomplemented
178C22×C103—21no—
179C22×C103—21no—
180C22×C103—41no—
181C22×C103—41no—
182C22×C103—41no—
183C22×C103AGL(1,ℤ/8)12yescomplemented
184C22×C103C22×D411yescharacteristic
185C2×D103—21no—
186C2×D103—22no—
187C2×D103—21no—
188C2×D103—21no—
189C2×D103—41no—
190C2×D103—41no—
191C2×D103—41no—
192C2×D103—41no—
193C2×D103—41no—
194C2×D103—41no—
195C2×D103—41no—
196C2×D103—41no—
197C2×D103—84no—
198C2×D103—84no—
199C4×C103—22no—
200C4×C103—21no—
201C4×C103—41no—
202C4×C103—41no—
203C4×C103C22×D411yescharacteristic
204C4×C103H(5,𝔽2)11yescharacteristic
205Dic103—21no—
206Dic103—21no—
207Dic103—21no—
208Dic103—41no—
209C2×C103—22no—
210C2×C103—81no—
211C2×C103D4211yescharacteristic
212C2×C103C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C103#57711yescharacteristic
214D103—41no—
215D103—42no—
216D103—81no—
217D103—81no—
218#3265—1031nomaximal, Sylow
219C103#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1031no—
221C22×SD16—1031no—
222#548—1031no—
223D42—1031no—
224#496—1031no—
225#468—1031no—
226#466—1031no—
227#463—1031no—
228#460—1031no—
229#450—1031no—
230#449—1031no—
231#448D10311yescharacteristic, complemented
232#447—1031no—
233#440—1031no—
234#435—1031no—
235C2×#41—1031no—
236C22×Q8C2×D10311yescharacteristic, complemented
237C22×D4—1031no—
238C22×D4—1031no—
239C22×D4—2061no—
240AGL(1,ℤ/8)—2064no—
241C2×SD16—1031no—
242C2×SD16—1031no—
243C2×SD16—1031no—
244C2×SD16—1031no—
245C2×SD16—2064no—
246C2×SD16C2×D10311yescharacteristic, complemented
247C2×SD16C2×D10311yescharacteristic, complemented
248C2×D8—1031no—
249C2×D8—1031no—
250C2×D8—1031no—
251C2×M16—1031no—
252C22×C8—1031no—
253#127—1031no—
254#124—1031no—
255#124—2061no—
256C22⋊Q8—1031no—
257C22⋊Q8—2061no—
258#121—1031no—
259#121—1031no—
260#121—2061no—
261#121—2061no—
262#121C2×D10311yescharacteristic
263#120—2061no—
264#120—2061no—
265C4×Q8—1031no—
266C4×D4—1031no—
267C4×D4—1031no—
268C4×D4—2061no—
269#107—1031no—
270C4⋊C8—1031no—
271#103C2×D10311yescharacteristic, complemented
272#103C2×D10311yescharacteristic, complemented
273#102—1031no—
274#102—1031no—
275#102—1031no—
276#102—1031no—
277C22⋊C8—1031no—
278C22⋊C8C2×D10311yescharacteristic, complemented
279#97—1031no—
280C24—2061no—
281C24—2061no—
282#41—4121no—
283#41—2062no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D10311yescharacteristic
288C2×Q8C22×D10311yescharacteristic
289C2×D4—21no—
290C2×D4—1031no—
291C2×D4—1031no—
292C2×D4—1031no—
293C2×D4—1031no—
294C2×D4—4121no—
295C2×D4—4124no—
296C2×D4—4121no—
297C2×D4—2062no—
298C2×D4—2061no—
299C2×D4—2061no—
300C2×D4—2061no—
301C2×D4—2061no—
302C2×D4—2061no—
303C2×D4—2062no—
304C2×D4—2061no—
305C2×D4—2061no—
306C2×D4C22×D10311yescharacteristic
307C22×C4—21no—
308C22×C4—1031no—
309C22×C4—2061no—
310C22×C4—2061no—
311C22×C4C22×D10311yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—1034no—
315SD16—2062no—
316SD16—2062no—
317SD16—2062no—
318D8—4121no—
319D8—2062no—
320D8—2062no—
321M16—2062no—
322C2×C8—1031no—
323C2×C8—1031no—
324C2×C8—2062no—
325C2×C8C22×D10311yescharacteristic
326C2×C8C22×D10311yescharacteristic
327C4⋊C4—1031no—
328C4⋊C4—1031no—
329C4⋊C4—2061no—
330C4⋊C4—2061no—
331C4⋊C4C22×D10311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—1031no—
334C22⋊C4—1031no—
335C22⋊C4—2061no—
336C22⋊C4—2061no—
337C22⋊C4—2061no—
338C22⋊C4—2061no—
339C22⋊C4—2061no—
340C22⋊C4—2061no—
341C42—1031no—
342C42—2061no—
343C23—21no—
344C23—8241no—
345C23—8241no—
346C23—1031no—
347C23—4122no—
348C23—4122no—
349C23—4122no—
350C23—4122no—
351C23—2061no—
352C23—2061no—
353C23—2061no—
354C23D4×D10311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D10312yescomplemented
360D4—21no—
361D4—21no—
362D4—42no—
363D4—8241no—
364D4—1032no—
365D4—4122no—
366D4—4122no—
367D4—4122no—
368D4—4122no—
369D4—4122no—
370D4—4122no—
371D4—2062no—
372D4—2061no—
373D4—2061no—
374D4—2061no—
375D4—2061no—
376D4—2061no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—1031no—
384C2×C4—1031no—
385C2×C4—1031no—
386C2×C4—4122no—
387C2×C4—4122no—
388C2×C4—2061no—
389C2×C4—2061no—
390C2×C4—2061no—
391C2×C4D4×D10311yescharacteristic
392C2×C4C23×D10311yescharacteristic
393C8—21no—
394C8—21no—
395C8—1032no—
396C8—2061no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—8244no—
403C22—8244no—
404C22—4121no—
405C22—4121no—
406C22—4121no—
407C22—4121no—
408C22—4121no—
409C22—4121no—
410C22—4121no—
411C22—4121no—
412C22—2061no—
413C22—2061no—
414C22—2061no—
415C22—2062no—
416C22#1569304812yescomplemented
417C22C2×D4×D10311yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—4121no—
423C4—2061no—
424C4—2061no—
425C4—2061no—
426C4C2×D4×D10311yescharacteristic
427C4#1569306611yescharacteristic
428C2—22no—
429C2—81no—
430C2—8241no—
431C2—8241no—
432C2—4122no—
433C2—4121no—
434C2#2026842311yescharacteristic
435C2C2×#1569304811yescharacteristic
436C2#2026860011yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#448×C103C2
socleC22×C103C22×D4
hypercenter#448D103
hypocenterC103#3265
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#15693048C2semidirect
2. C22×#13110092C2semidirect
3. #20268452C2semidirect
4. #20268425C2semidirect
5. #20268177C2semidirect
6. #20268040C2semidirect
7. #20268025C2semidirect
8. #20268009C2semidirect
9. #20267989C2semidirect
10. #20267940C2semidirect
11. #20267939C2semidirect
12. #20267927C2semidirect
13. #20267894C2semidirect
14. #20267859C2semidirect
15. #448×C103C2semidirect
16. C2×#13110106C22semidirect
24. C2×#13110092C22semidirect
25. C2×#13110092C22semidirect
27. C2×D824C22semidirect
28. C2×#13110090C22semidirect
29. C2×#13110088C22semidirect
30. C22×C103⋊C8C22semidirect
33. #15693010C22semidirect
35. #15692997C22semidirect
38. #15692995C22semidirect
42. Q8×Dic103C22semidirect
46. #15692954C22semidirect
47. Dic103⋊C8C22semidirect
49. #15692943C22semidirect
50. #15692942C22semidirect
51. #15692941C22semidirect
52. (C2×D103)⋊C8C22semidirect
53. #15692931C22semidirect
54. C22×Q8×C103C22semidirect
55. C2×SD16×C103C22semidirect
56. C2×SD16×C103C22semidirect
58. #103×C103C22semidirect
59. #103×C103C22semidirect
60. C22⋊C8×C103C22semidirect
87. #13110092D4semidirect
140. Q8×C103C2×D4semidirect
177. C103⋊C8C2×D4semidirect
183. C22×C103AGL(1,ℤ/8)semidirect
219. C103#3265semidirect
231. #448D103semidirect
236. C22×Q8C2×D103semidirect
246. C2×SD16C2×D103semidirect
247. C2×SD16C2×D103semidirect
271. #103C2×D103semidirect
272. #103C2×D103semidirect
278. C22⋊C8C2×D103semidirect
359. Q8D4×D103semidirect
416. C22#15693048semidirect

Maximal subgroups

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

subgroupindex
C2×#156930482
C22×#131100922
#202684522
#202684252
#202681772
#202680402
#202680252
#202680092
#202679892
#202679402
#202679392
#202679272
#202678942
#202678592
#448×C1032
#3265103