Finite Group Databasebeta

← #27174111

Subgroups — #27174111

Order 12928.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692224C211yesmaximal, characteristic
2C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
3C2×#15692215C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic101C211yesmaximal, characteristic, complemented
5#20265077C211yesmaximal, characteristic, complemented
6#20265071C211yesmaximal, characteristic, complemented
7#20265065C211yesmaximal, characteristic, complemented
8#20265056C211yesmaximal, characteristic, complemented
9#20265054C211yesmaximal, characteristic, complemented
10#20265053C211yesmaximal, characteristic, complemented
11#20265050C211yesmaximal, characteristic, complemented
12#20265049C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C23×Dic101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic, complemented
22#15692224C418yescomplemented
23C4×C22⋊D101—24no—
24#15692215—24no—
25C2×#13108053C2211yescharacteristic, complemented
26C2×#13108053C2211yescharacteristic, complemented
27C2×#13108053C2211yescharacteristic, complemented
28C2×C22⋊Dic101—24no—
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×C22⋊Dic101C2211yescharacteristic
33C2×C22⋊Dic101C2211yescharacteristic
34C2×C22⋊Dic101C2211yescharacteristic
35C2×C22⋊Dic101C2211yescharacteristic
36C2×#13108051C2211yescharacteristic, complemented
37C2×#13108051C2211yescharacteristic, complemented
38C2×#13108051C2211yescharacteristic, complemented
39C2×C4×Dic101C2211yescharacteristic, complemented
40C2×C4×Dic101C2211yescharacteristic, complemented
41C2×C4×Dic101C2211yescharacteristic, complemented
42#15692162C2211yescharacteristic, complemented
43#15692162C2211yescharacteristic, complemented
44#15692162C2211yescharacteristic, complemented
45#15692162C2211yescharacteristic, complemented
46#15692162C2211yescharacteristic, complemented
47#15692162C2211yescharacteristic, complemented
48C25×C101C2211yescharacteristic
49C23×C4×C101C2211yescharacteristic
50C23×C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101—24no—
52C2×C22⋊C4×C101C2211yescharacteristic
53C2×C22⋊C4×C101C2211yescharacteristic
54C2×C22⋊C4×C101C2211yescharacteristic
55C2×C22⋊C4×C101C2211yescharacteristic
56C23×D101C2311yescharacteristic
57C2×C22⋊D101—22no—
58C2×C22⋊D101—22no—
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61C2×C22⋊D101—41no—
62C2×C22⋊D101—41no—
63C2×C22⋊D101C2×C414yescomplemented
64C2×C22⋊D101C2×C414yescomplemented
65C2×C22⋊D101C2×C414yescomplemented
66C22×Dic101—22no—
67C22×Dic101—22no—
68C22×Dic101—22no—
69C22×Dic101—22no—
70C22×Dic101—41no—
71C22×Dic101—41no—
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C22×Dic101C2311yescharacteristic
79C2×C4×D101—22no—
80C2×C4×D101—22no—
81C2×C4×D101—41no—
82#13108053—22no—
83#13108053—22no—
84#13108053—41no—
85C22⋊Dic101—22no—
86C22⋊Dic101—22no—
87C22⋊Dic101—22no—
88C22⋊Dic101C2×C414yescomplemented
89C22⋊Dic101C2×C414yescomplemented
90C22⋊Dic101C2×C414yescomplemented
91C22⋊Dic101D414yescomplemented
92#13108051—22no—
93#13108051—22no—
94#13108051—41no—
95C4×Dic101—22no—
96C4×Dic101—22no—
97C4×Dic101—41no—
98C24×C101—22no—
99C24×C101—22no—
100C24×C101—22no—
101C24×C101—22no—
102C24×C101—41no—
103C24×C101—41no—
104C24×C101C2×C414yescomplemented
105C24×C101C2311yescharacteristic
106C24×C101C2311yescharacteristic
107C24×C101C2311yescharacteristic
108C22×C4×C101—22no—
109C22×C4×C101—22no—
110C22×C4×C101—22no—
111C22×C4×C101—22no—
112C22×C4×C101—41no—
113C22×C4×C101—41no—
114C22×C4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116C22×C4×C101C2311yescharacteristic
117C22×C4×C101C2311yescharacteristic
118C22⋊C4×C101—22no—
119C22⋊C4×C101—22no—
120C22⋊C4×C101—22no—
121C22⋊C4×C101D414yescomplemented
122C23×C101—21no—
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—21no—
128C23×C101—21no—
129C23×C101—21no—
130C23×C101—28no—
131C23×C101—28no—
132C23×C101—21no—
133C23×C101—21no—
134C23×C101—21no—
135C23×C101—21no—
136C23×C101—21no—
137C23×C101—21no—
138C23×C101—44no—
139C23×C101—44no—
140C23×C101—44no—
141C23×C101—44no—
142C23×C101—44no—
143C23×C101C22×C412yes—
144C23×C101C22×C412yes—
145C23×C101C22×C412yes—
146C23×C101C2×D412yes—
147C23×C101C2×D412yes—
148C23×C101#4112yes—
149C23×C101#4112yes—
150C23×C101C2411yescharacteristic
151C22×D101—21no—
152C22×D101—21no—
153C22×D101—21no—
154C22×D101—21no—
155C22×D101—21no—
156C22×D101—21no—
157C22×D101C22×C412yes—
158C2×C4×C101—21no—
159C2×C4×C101—21no—
160C2×C4×C101—21no—
161C2×C4×C101—21no—
162C2×C4×C101—21no—
163C2×C4×C101—21no—
164C2×C4×C101—21no—
165C2×C4×C101—21no—
166C2×C4×C101—21no—
167C2×C4×C101—21no—
168C2×C4×C101—44no—
169C2×C4×C101—44no—
170C2×C4×C101C2×D412yes—
171C2×C4×C101C2×D412yes—
172C22⋊D101—28no—
173C22⋊D101—28no—
174C22⋊D101—44no—
175C2×Dic101—21no—
176C2×Dic101—21no—
177C2×Dic101—21no—
178C2×Dic101—21no—
179C2×Dic101—21no—
180C2×Dic101—21no—
181C2×Dic101—21no—
182C2×Dic101—21no—
183C2×Dic101—21no—
184C2×Dic101—21no—
185C2×Dic101—21no—
186C2×Dic101—21no—
187C2×Dic101—21no—
188C2×Dic101—21no—
189C2×Dic101—21no—
190C2×Dic101—21no—
191C2×Dic101—44no—
192C2×Dic101—44no—
193C2×Dic101C22×C412yes—
194C2×Dic101C22×C412yes—
195C2×Dic101C22×C412yes—
196C2×Dic101C2×D412yes—
197C2×Dic101C2×D412yes—
198C4×D101—44no—
199C22×C101—22no—
200C22×C101—22no—
201C22×C101—22no—
202C22×C101—22no—
203C22×C101—22no—
204C22×C101—22no—
205C22×C101—22no—
206C22×C101—22no—
207C22×C101—22no—
208C22×C101—22no—
209C22×C101—22no—
210C22×C101—22no—
211C22×C101—22no—
212C22×C101—22no—
213C22×C101—22no—
214C22×C101—416no—
215C22×C101—41no—
216C22×C101—41no—
217C22×C101—41no—
218C22×C101—41no—
219C22×C101C4×D414yescomplemented
220C22×C101C4×D414yescomplemented
221C22×C101C23×C411yescharacteristic
222C22×C101C22×D411yescharacteristic
223C22×C101C22×D411yescharacteristic
224C22×C101C2×#4111yescharacteristic
225C22×C101C2×#4111yescharacteristic
226C22×C101H(5,𝔽2)11yescharacteristic
227C22×C101H(5,𝔽2)11yescharacteristic
228C2×D101—22no—
229C2×D101—22no—
230C2×D101—22no—
231C2×D101—41no—
232C2×D101—41no—
233C2×D101—41no—
234C2×D101—41no—
235C4×C101—22no—
236C4×C101—22no—
237C4×C101—41no—
238C4×C101—41no—
239Dic101—22no—
240Dic101—22no—
241Dic101—22no—
242Dic101—22no—
243Dic101—22no—
244Dic101—41no—
245Dic101—41no—
246C2×C101—24no—
247C2×C101—24no—
248C2×C101—42no—
249C2×C101C2×C4×D411yescharacteristic
250C2×C101C2×C4×D411yescharacteristic
251C2×C101#51811yescharacteristic
252C2×C101D4211yescharacteristic
253C2×C101#54611yescharacteristic
254C2×C101#54611yescharacteristic
255C2×C101#55111yescharacteristic
256D101—42no—
257#2358—1011nomaximal, Sylow
258C101#235811yescharacteristic, complemented, Sylow
259C2×#120—1011no—
260C2×C4×D4—1011no—
261C2×C4×D4—1011no—
262C22×C22⋊C4—1011no—
263C22×C22⋊C4D10111yescharacteristic, complemented
264#390—1011no—
265#388—1011no—
266#388—1011no—
267#386—1011no—
268#386—1011no—
269#386—1011no—
270#385—1011no—
271#385—1011no—
272C4×C22⋊C4—1011no—
273C4×C22⋊C4—1011no—
274C25C2×D10111yescharacteristic
275C22×D4—1011no—
276C22×D4—1011no—
277C22×D4—1011no—
278C23×C4—1011no—
279C23×C4—1011no—
280C23×C4—1011no—
281C23×C4C2×D10111yescharacteristic
282C23×C4C2×D10111yescharacteristic
283#120—1018no—
284C4×D4—2024no—
285C4×D4—2024no—
286C2×C4⋊C4—1011no—
287C2×C4⋊C4—1011no—
288C2×C4⋊C4—1011no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—1011no—
291C2×C22⋊C4—1011no—
292C2×C22⋊C4—1011no—
293C2×C22⋊C4—1011no—
294C2×C22⋊C4—1011no—
295C2×C22⋊C4—1011no—
296C2×C22⋊C4—1011no—
297C2×C22⋊C4—1011no—
298C2×C22⋊C4—1011no—
299C2×C22⋊C4—1011no—
300C2×C22⋊C4—2024no—
301C2×C22⋊C4C2×D10111yescharacteristic
302C2×C22⋊C4C2×D10111yescharacteristic
303C2×C22⋊C4C2×D10111yescharacteristic
304C2×C22⋊C4C2×D10111yescharacteristic
305C2×C42—1011no—
306C2×C42—1011no—
307C2×C42—1011no—
308#95—1011no—
309#95—1011no—
310#95—1011no—
311#95—1011no—
312#95—1011no—
313#95—1011no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—1011no—
321C24C4×D10114yescomplemented
322C24C22×D10111yescharacteristic
323C24C22×D10111yescharacteristic
324C24C22×D10111yescharacteristic
325C2×D4—1014no—
326C2×D4—1014no—
327C2×D4—1014no—
328C2×D4—4041no—
329C2×D4—4041no—
330C2×D4—2022no—
331C2×D4—2022no—
332C2×D4—2022no—
333C2×D4—2022no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—1011no—
341C22×C4—1011no—
342C22×C4—1011no—
343C22×C4—1011no—
344C22×C4—1011no—
345C22×C4—1011no—
346C22×C4—1011no—
347C22×C4—4041no—
348C22×C4—4041no—
349C22×C4—4041no—
350C22×C4—2022no—
351C22×C4—2022no—
352C22×C4—2022no—
353C22×C4—2022no—
354C22×C4—2022no—
355C22×C4—2022no—
356C22×C4C22×D10111yescharacteristic
357C22×C4C22×D10111yescharacteristic
358C22×C4C22×D10111yescharacteristic
359C22×C4C22×D10111yescharacteristic
360C4⋊C4—4041no—
361C4⋊C4—2022no—
362C4⋊C4—2022no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—1014no—
367C22⋊C4—1014no—
368C22⋊C4—1014no—
369C22⋊C4—1014no—
370C22⋊C4—4041no—
371C22⋊C4—2022no—
372C22⋊C4—2022no—
373C22⋊C4—2022no—
374C22⋊C4—2022no—
375C22⋊C4—2022no—
376C22⋊C4C22⋊D10114yescomplemented
377C42—4041no—
378C42—2022no—
379C42—2022no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—28no—
395C23—28no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—1012no—
402C23—2021no—
403C23—2021no—
404C23—2021no—
405C23—2021no—
406C23—2021no—
407C23—2021no—
408C23C2×C4×D10112yes—
409C23C2×C4×D10112yes—
410C23C2×C4×D10112yes—
411C23D4×D10112yes—
412C23C2×C22⋊D10112yes—
413C23#1310807312yes—
414C23#1310807412yes—
415C23C23×D10111yescharacteristic
416D4—4044no—
417D4—2028no—
418D4—2028no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—1012no—
432C2×C4—1012no—
433C2×C4—1012no—
434C2×C4—1012no—
435C2×C4—1012no—
436C2×C4—4044no—
437C2×C4—4044no—
438C2×C4—4044no—
439C2×C4—2021no—
440C2×C4—2021no—
441C2×C4—2021no—
442C2×C4—2021no—
443C2×C4—2021no—
444C2×C4—2021no—
445C2×C4—2021no—
446C2×C4—2021no—
447C2×C4—2021no—
448C2×C4—2021no—
449C2×C4—2021no—
450C2×C4—2021no—
451C2×C4—2021no—
452C2×C4—2021no—
453C2×C4—2021no—
454C2×C4—2021no—
455C2×C4C2×C22⋊D10112yes—
456C2×C4C2×C22⋊D10112yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—416no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—4041no—
478C22—4041no—
479C22—4041no—
480C22—4041no—
481C22—2022no—
482C22—2022no—
483C22—2022no—
484C22#1569221514yescomplemented
485C22C4×C22⋊D10114yescomplemented
486C22C22×C4×D10111yescharacteristic
487C22C2×D4×D10111yescharacteristic
488C22C22×C22⋊D10111yescharacteristic
489C22C2×#1310807311yescharacteristic
490C22C2×#1310807411yescharacteristic
491C22#1569230011yescharacteristic
492C22#1569230011yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—4041no—
498C4—4041no—
499C4—2022no—
500C4—2022no—
501C4—2022no—
502C4—2022no—
503C4—2022no—
504C2—24no—
505C2—24no—
506C2—42no—
507C2—4042no—
508C2C2×#1569221511yescharacteristic
509C2C2×C4×C22⋊D10111yescharacteristic
510C2#2026565011yescharacteristic
511C2D4×C22⋊D10111yescharacteristic
512C2#2026581711yescharacteristic
513C2#2026582111yescharacteristic
514C2#2026585311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
FittingC22×C22⋊C4×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692224C2
hypercenterC22×C22⋊C4D101
hypocenterC101#2358
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D101C2semidirect
3. C2×#15692215C2semidirect
4. C22×C22⋊Dic101C2semidirect
5. #20265077C2semidirect
6. #20265071C2semidirect
7. #20265065C2semidirect
8. #20265056C2semidirect
9. #20265054C2semidirect
10. #20265053C2semidirect
11. #20265050C2semidirect
12. #20265049C2semidirect
13. C22⋊C4×Dic101C2semidirect
14. C4×C22⋊Dic101C2semidirect
15. C22×C22⋊C4×C101C2semidirect
21. C22×C4×D101C22semidirect
22. #15692224C4semidirect
25. C2×#13108053C22semidirect
26. C2×#13108053C22semidirect
27. C2×#13108053C22semidirect
36. C2×#13108051C22semidirect
37. C2×#13108051C22semidirect
38. C2×#13108051C22semidirect
39. C2×C4×Dic101C22semidirect
40. C2×C4×Dic101C22semidirect
41. C2×C4×Dic101C22semidirect
42. #15692162C22semidirect
43. #15692162C22semidirect
44. #15692162C22semidirect
45. #15692162C22semidirect
46. #15692162C22semidirect
47. #15692162C22semidirect
63. C2×C22⋊D101C2×C4semidirect
64. C2×C22⋊D101C2×C4semidirect
65. C2×C22⋊D101C2×C4semidirect
88. C22⋊Dic101C2×C4semidirect
89. C22⋊Dic101C2×C4semidirect
90. C22⋊Dic101C2×C4semidirect
91. C22⋊Dic101D4semidirect
104. C24×C101C2×C4semidirect
121. C22⋊C4×C101D4semidirect
219. C22×C101C4×D4semidirect
220. C22×C101C4×D4semidirect
258. C101#2358semidirect
263. C22×C22⋊C4D101semidirect
321. C24C4×D101semidirect
376. C22⋊C4C22⋊D101semidirect
484. C22#15692215semidirect
485. C22C4×C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922242
C2×C4×C22⋊D1012
C2×#156922152
C22×C22⋊Dic1012
#202650772
#202650712
#202650652
#202650562
#202650542
#202650532
#202650502
#202650492
C22⋊C4×Dic1012
C4×C22⋊Dic1012
C22×C22⋊C4×C1012
#2358101