Finite Group Databasebeta

← #31784184

Subgroups — #31784184

Order 17536.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984070C211yesmaximal, characteristic, complemented
2C2×#17984056C211yesmaximal, characteristic
3C22×#14555487C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D137C211yesmaximal, characteristic, complemented
5#22577332C211yesmaximal, characteristic, complemented
6#22577330C211yesmaximal, characteristic, complemented
7#22577319C211yesmaximal, characteristic, complemented
8#22577318C211yesmaximal, characteristic, complemented
9#22577283C211yesmaximal, characteristic, complemented
10#22577279C211yesmaximal, characteristic, complemented
11#22577277C211yesmaximal, characteristic, complemented
12#22577265C211yesmaximal, characteristic, complemented
13#22577263C211yesmaximal, characteristic, complemented
14#22577261C211yesmaximal, characteristic, complemented
15#386×C137C211yesmaximal, characteristic, complemented
16C24×D137C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C22×C22⋊D137C2211yescharacteristic
19C23×Dic137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21C22×C4×D137C2211yescharacteristic
22#17984070—24no—
23#17984056—24no—
24C2×C4⋊Dic137C2211yescharacteristic, complemented
25C2×C4⋊Dic137C2211yescharacteristic, complemented
26C2×#14555489C2211yescharacteristic, complemented
27C2×#14555489C2211yescharacteristic, complemented
28C2×C22⋊Dic137C2211yescharacteristic
29C2×C22⋊Dic137C2211yescharacteristic
30C2×C22⋊Dic137C2211yescharacteristic, complemented
31C2×C22⋊Dic137C2211yescharacteristic, complemented
32C2×#14555487—24no—
33C2×#14555487C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×#14555487C2211yescharacteristic
36C2×#14555487C2211yescharacteristic
37C2×#14555487C2211yescharacteristic
38C2×#14555487C2211yescharacteristic
39C2×#14555487C2211yescharacteristic
40C2×#14555487C2211yescharacteristic
41C22⋊C4×D137—24no—
42#17983995C2211yescharacteristic, complemented
43#17983995C2211yescharacteristic, complemented
44#17983995C2211yescharacteristic, complemented
45#17983995C2211yescharacteristic, complemented
46#17983995C2211yescharacteristic, complemented
47#17983995C2211yescharacteristic, complemented
48C22×D4×C137C2211yescharacteristic
49C23×C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C22⋊C4×C137C2211yescharacteristic, complemented
52C2×C22⋊C4×C137C2211yescharacteristic, complemented
53#95×C137C2211yescharacteristic, complemented
54#95×C137C2211yescharacteristic, complemented
55C23×D137—22no—
56C23×D137—22no—
57C23×D137—22no—
58C23×D137—22no—
59C23×D137—41no—
60C23×D137—41no—
61C23×D137D414yes—
62C23×D137C2311yescharacteristic
63C23×D137C2311yescharacteristic
64C2×C22⋊D137—22no—
65C2×C22⋊D137—22no—
66C2×C22⋊D137—22no—
67C2×C22⋊D137—22no—
68C2×C22⋊D137—41no—
69C2×C22⋊D137—41no—
70C22×Dic137—22no—
71C22×Dic137—22no—
72C22×Dic137—41no—
73C22×Dic137C2311yescharacteristic
74C22×Dic137C2311yescharacteristic
75C22×Dic137C2311yescharacteristic
76C22×Dic137C2311yescharacteristic
77C22×Dic137C2311yescharacteristic
78C22×Dic137C2311yescharacteristic
79C2×C4×D137—22no—
80C2×C4×D137—22no—
81C2×C4×D137—22no—
82C2×C4×D137—22no—
83C2×C4×D137—41no—
84C2×C4×D137—41no—
85C4⋊Dic137—22no—
86C4⋊Dic137—41no—
87#14555489—22no—
88#14555489—41no—
89C22⋊Dic137—22no—
90C22⋊Dic137—22no—
91C22⋊Dic137—22no—
92C22⋊Dic137—41no—
93#14555487—22no—
94#14555487—22no—
95#14555487—22no—
96#14555487—22no—
97#14555487—22no—
98#14555487—22no—
99#14555487—22no—
100#14555487D414yescomplemented
101C24×C137C2311yescharacteristic
102C24×C137C2311yescharacteristic
103C2×D4×C137—22no—
104C2×D4×C137—22no—
105C2×D4×C137—22no—
106C22×C4×C137—22no—
107C22×C4×C137—22no—
108C22×C4×C137—41no—
109C22×C4×C137C2311yescharacteristic
110C22×C4×C137C2311yescharacteristic
111C22×C4×C137C2311yescharacteristic
112C22×C4×C137C2311yescharacteristic
113C22×C4×C137C2311yescharacteristic
114C22⋊C4×C137—22no—
115C22⋊C4×C137—22no—
116C22⋊C4×C137—41no—
117C22×D137—21no—
118C22×D137—21no—
119C22×D137—21no—
120C22×D137—21no—
121C22×D137—28no—
122C22×D137—21no—
123C22×D137—21no—
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—21no—
127C22×D137—21no—
128C22×D137—28no—
129C22×D137—44no—
130C22×D137—44no—
131C22×D137—44no—
132C22×D137—44no—
133C22×D137—44no—
134C22×D137C2×D412yes—
135C22×D137C2×D412yes—
136C22×D137C2×D412yes—
137C22×D137#4112yes—
138C22⋊D137—44no—
139C22⋊D137—44no—
140C2×Dic137—21no—
141C2×Dic137—21no—
142C2×Dic137—21no—
143C2×Dic137—21no—
144C2×Dic137—21no—
145C2×Dic137—21no—
146C2×Dic137—21no—
147C2×Dic137—21no—
148C2×Dic137—21no—
149C2×Dic137—21no—
150C2×Dic137—21no—
151C2×Dic137—21no—
152C2×Dic137—21no—
153C2×Dic137—21no—
154C2×Dic137—21no—
155C2×Dic137—21no—
156C2×Dic137—21no—
157C2×Dic137—44no—
158C2×Dic137C2×D412yes—
159C4×D137—44no—
160C4×D137—44no—
161C23×C137—21no—
162C23×C137—21no—
163C23×C137—21no—
164C23×C137—21no—
165C23×C137—21no—
166C23×C137—21no—
167C23×C137—21no—
168C23×C137—21no—
169C23×C137—21no—
170C23×C137—21no—
171C23×C137—21no—
172C23×C137C2×D412yes—
173C23×C137#4112yes—
174C23×C137#4112yes—
175C23×C137C2411yescharacteristic
176D4×C137—44no—
177C2×C4×C137—21no—
178C2×C4×C137—21no—
179C2×C4×C137—21no—
180C2×C4×C137—21no—
181C2×C4×C137—21no—
182C2×C4×C137—21no—
183C2×C4×C137—21no—
184C2×C4×C137—21no—
185C2×C4×C137—21no—
186C2×C4×C137—21no—
187C2×C4×C137—21no—
188C2×C4×C137—21no—
189C2×C4×C137—21no—
190C2×C4×C137—21no—
191C2×C4×C137—44no—
192C2×C4×C137C2×D412yes—
193C22×C137—22no—
194C22×C137—22no—
195C22×C137—22no—
196C22×C137—22no—
197C22×C137—22no—
198C22×C137—22no—
199C22×C137—41no—
200C22×C137—41no—
201C22×C137—41no—
202C22×C137—41no—
203C22×C137—41no—
204C22×C137—41no—
205C22×C137—41no—
206C22×C137#12314yescomplemented
207C22×C137C22×D411yescharacteristic
208C22×C137C22×D411yescharacteristic
209C22×C137C2×#4111yescharacteristic
210C22×C137C2×#4111yescharacteristic
211C22×C137C2×#4111yescharacteristic
212C22×C137H(5,𝔽2)11yescharacteristic
213C22×C137H(5,𝔽2)11yescharacteristic
214C2×D137—22no—
215C2×D137—22no—
216C2×D137—22no—
217C2×D137—22no—
218C2×D137—22no—
219C2×D137—22no—
220C2×D137—22no—
221C2×D137—22no—
222C2×D137—22no—
223C2×D137—416no—
224C2×D137—41no—
225C2×D137—41no—
226C2×D137—41no—
227C2×D137—41no—
228C2×D137#12114yes—
229C4×C137—22no—
230C4×C137—41no—
231C4×C137—41no—
232C4×C137—41no—
233C4×C137—41no—
234Dic137—22no—
235Dic137—41no—
236Dic137—41no—
237Dic137—41no—
238Dic137—41no—
239Dic137—41no—
240C2×C137—24no—
241C2×C137—81no—
242C2×C137C2×#12111yescharacteristic
243C2×C137C2×#12311yescharacteristic
244C2×C137#53411yescharacteristic
245C2×C137#54611yescharacteristic
246C2×C137#54611yescharacteristic
247C2×C137#55111yescharacteristic
248C2×C137#55111yescharacteristic
249D137—24no—
250D137—42no—
251C137#241211yescharacteristic, complemented, Sylow
252#2412—1371nomaximal, Sylow
253C2×#123—1371no—
254C2×#120—1371no—
255C22×C22⋊C4—1371no—
256C22×C22⋊C4—1371no—
257#397—1371no—
258#397—1371no—
259#394—1371no—
260#394—1371no—
261#386—1371no—
262#386—1371no—
263#386D13711yescharacteristic, complemented
264#385—1371no—
265#381—1371no—
266#380—1371no—
267#380—1371no—
268C25—1371no—
269C22×D4—1371no—
270C22×D4—1371no—
271C22×D4C2×D13711yescharacteristic
272C23×C4—1371no—
273C23×C4—1371no—
274C23×C4—1371no—
275C23×C4C2×D13711yescharacteristic
276#123—2744no—
277#120—2744no—
278C2×C4⋊C4—1371no—
279C2×C4⋊C4—1371no—
280C2×C4⋊C4—1371no—
281C2×C4⋊C4—1371no—
282C2×C22⋊C4—2744no—
283C2×C22⋊C4—2744no—
284C2×C22⋊C4—1371no—
285C2×C22⋊C4—1371no—
286C2×C22⋊C4—1371no—
287C2×C22⋊C4—1371no—
288C2×C22⋊C4—1371no—
289C2×C22⋊C4—1371no—
290C2×C22⋊C4—1371no—
291C2×C22⋊C4—1371no—
292C2×C22⋊C4—1371no—
293C2×C22⋊C4—1371no—
294C2×C22⋊C4—1371no—
295C2×C22⋊C4—1371no—
296C2×C22⋊C4C2×D13711yescharacteristic
297C2×C22⋊C4C2×D13711yescharacteristic, complemented
298C2×C22⋊C4C2×D13711yescharacteristic, complemented
299#95—1371no—
300#95—1371no—
301#95—1371no—
302#95—1371no—
303#95—1371no—
304#95—1371no—
305#95C2×D13711yescharacteristic, complemented
306#95C2×D13711yescharacteristic, complemented
307C24—2742no—
308C24—2742no—
309C24—2742no—
310C24—2742no—
311C24—5481no—
312C24—5481no—
313C24—1371no—
314C24—1374no—
315C24—1371no—
316C24C22×D13711yescharacteristic
317C24C22×D13711yescharacteristic
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—2742no—
322C2×D4—2742no—
323C2×D4—2742no—
324C2×D4—2742no—
325C2×D4—5481no—
326C2×D4—5481no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—2742no—
331C22×C4—2742no—
332C22×C4—2742no—
333C22×C4—2742no—
334C22×C4—2742no—
335C22×C4—2742no—
336C22×C4—5481no—
337C22×C4—5481no—
338C22×C4—5481no—
339C22×C4—1371no—
340C22×C4—1371no—
341C22×C4—1371no—
342C22×C4—1371no—
343C22×C4—1371no—
344C22×C4—1371no—
345C22×C4C22×D13711yescharacteristic
346C22×C4C22×D13711yescharacteristic
347C22×C4C22×D13711yescharacteristic
348C22×C4C22×D13711yescharacteristic
349C22×C4C22×D13711yescharacteristic
350C4⋊C4—2742no—
351C4⋊C4—2742no—
352C4⋊C4—5481no—
353C4⋊C4—5481no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—41no—
357C22⋊C4—2742no—
358C22⋊C4—2742no—
359C22⋊C4—2742no—
360C22⋊C4—2742no—
361C22⋊C4—2742no—
362C22⋊C4—2742no—
363C22⋊C4—2742no—
364C22⋊C4—2742no—
365C22⋊C4—2742no—
366C22⋊C4—2742no—
367C22⋊C4—5481no—
368C22⋊C4—1374no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—2748no—
381C23—2741no—
382C23—2741no—
383C23—2741no—
384C23—2741no—
385C23—2748no—
386C23—2741no—
387C23—2741no—
388C23—2741no—
389C23—2741no—
390C23—2741no—
391C23—2741no—
392C23—5484no—
393C23—5484no—
394C23—5484no—
395C23—5484no—
396C23—5484no—
397C23—1372no—
398C23—1372no—
399C23—1372no—
400C23—1372no—
401C23D4×D13712yes—
402C23#1455550912yes—
403C23#1455551012yes—
404C23C23×D13711yescharacteristic
405D4—44no—
406D4—5484no—
407D4—5484no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—44no—
423C2×C4—2741no—
424C2×C4—2741no—
425C2×C4—2741no—
426C2×C4—2741no—
427C2×C4—2741no—
428C2×C4—2741no—
429C2×C4—2741no—
430C2×C4—2741no—
431C2×C4—2741no—
432C2×C4—2741no—
433C2×C4—2741no—
434C2×C4—2741no—
435C2×C4—2741no—
436C2×C4—2741no—
437C2×C4—2741no—
438C2×C4—2741no—
439C2×C4—2741no—
440C2×C4—5484no—
441C2×C4—5484no—
442C2×C4—5484no—
443C2×C4—1372no—
444C2×C4D4×D13712yes—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—2742no—
459C22—2742no—
460C22—2742no—
461C22—2742no—
462C22—2742no—
463C22—2742no—
464C22—2742no—
465C22—2742no—
466C22—2742no—
467C22—54816no—
468C22—5481no—
469C22—5481no—
470C22—5481no—
471C22—5481no—
472C22—1374no—
473C22#1798407014yescomplemented
474C22C2×D4×D13711yescharacteristic
475C22C2×D4×D13711yescharacteristic
476C22#41×D13711yescharacteristic
477C22C2×#1455550911yescharacteristic
478C22C2×#1455551011yescharacteristic
479C22#1798413311yescharacteristic
480C22#1798413411yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—2742no—
487C4—5481no—
488C4—5481no—
489C4—5481no—
490C4—5481no—
491C4—5481no—
492C2—24no—
493C2—81no—
494C2—2744no—
495C2—5482no—
496C2#121×D13711yescharacteristic
497C2C2×#1798407011yescharacteristic
498C2#2257796111yescharacteristic
499C2#2257804011yescharacteristic
500C2#2257804911yescharacteristic
501C2#2257807611yescharacteristic
502C2#2257808211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#386×C137C2
prime radicalC2×#17984056C2
hypercenter#386D137
hypocenterC137#2412
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#17984070C2semidirect
3. C22×#14555487C2semidirect
4. C2×C22⋊C4×D137C2semidirect
5. #22577332C2semidirect
6. #22577330C2semidirect
7. #22577319C2semidirect
8. #22577318C2semidirect
9. #22577283C2semidirect
10. #22577279C2semidirect
11. #22577277C2semidirect
12. #22577265C2semidirect
13. #22577263C2semidirect
14. #22577261C2semidirect
15. #386×C137C2semidirect
24. C2×C4⋊Dic137C22semidirect
25. C2×C4⋊Dic137C22semidirect
26. C2×#14555489C22semidirect
27. C2×#14555489C22semidirect
30. C2×C22⋊Dic137C22semidirect
31. C2×C22⋊Dic137C22semidirect
42. #17983995C22semidirect
43. #17983995C22semidirect
44. #17983995C22semidirect
45. #17983995C22semidirect
46. #17983995C22semidirect
47. #17983995C22semidirect
51. C2×C22⋊C4×C137C22semidirect
52. C2×C22⋊C4×C137C22semidirect
53. #95×C137C22semidirect
54. #95×C137C22semidirect
100. #14555487D4semidirect
206. C22×C137#123semidirect
251. C137#2412semidirect
263. #386D137semidirect
297. C2×C22⋊C4C2×D137semidirect
298. C2×C22⋊C4C2×D137semidirect
305. #95C2×D137semidirect
306. #95C2×D137semidirect
473. C22#17984070semidirect

Maximal subgroups

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

subgroupindex
C2×#179840702
C2×#179840562
C22×#145554872
C2×C22⋊C4×D1372
#225773322
#225773302
#225773192
#225773182
#225772832
#225772792
#225772772
#225772652
#225772632
#225772612
#386×C1372
#2412137