Finite Group Databasebeta

← #27174109

Subgroups — #27174109

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×#15692223C211yesmaximal, characteristic
2C2×C4×C22⋊D101C211yesmaximal, characteristic, complemented
3C22×#13108051C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
5#20265077C211yesmaximal, characteristic, complemented
6#20265068C211yesmaximal, characteristic, complemented
7#20265067C211yesmaximal, characteristic, complemented
8#20265057C211yesmaximal, characteristic, complemented
9#20265056C211yesmaximal, characteristic, complemented
10#20265053C211yesmaximal, characteristic, complemented
11#20265050C211yesmaximal, characteristic, complemented
12#20265048C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
14C4×#13108051C211yesmaximal, characteristic, complemented
15C2×C4×D4×C101C211yesmaximal, characteristic, complemented
16C24×D101C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22#15692223C418yescomplemented
23C4×C22⋊D101—24no—
24C2×#13108053C2211yescharacteristic, complemented
25C2×#13108053C2211yescharacteristic, complemented
26C2×C22⋊Dic101C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic, complemented
29C2×C22⋊Dic101C2211yescharacteristic, complemented
30C2×#13108051—24no—
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C2×#13108051C2211yescharacteristic
39C22⋊C4×D101—24no—
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
48C22×D4×C101C2211yescharacteristic
49C23×C4×C101C2211yescharacteristic
50C23×C4×C101C2211yescharacteristic, complemented
51C4×D4×C101—24no—
52C2×C4⋊C4×C101C2211yescharacteristic, complemented
53C2×C22⋊C4×C101C2211yescharacteristic
54C2×C22⋊C4×C101C2211yescharacteristic, complemented
55C2×C42×C101C2211yescharacteristic, complemented
56C23×D101—22no—
57C23×D101—22no—
58C23×D101—22no—
59C23×D101—22no—
60C23×D101—41no—
61C23×D101—41no—
62C23×D101C2×C414yescomplemented
63C23×D101C2311yescharacteristic
64C23×D101C2311yescharacteristic
65C2×C22⋊D101—22no—
66C2×C22⋊D101—22no—
67C2×C22⋊D101—41no—
68C2×C22⋊D101—41no—
69C2×C22⋊D101C2×C414yescomplemented
70C2×C22⋊D101C2×C414yescomplemented
71C22×Dic101—22no—
72C22×Dic101—22no—
73C22×Dic101—41no—
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C22×Dic101C2311yescharacteristic
79C22×Dic101C2311yescharacteristic
80C2×C4×D101—22no—
81C2×C4×D101—22no—
82C2×C4×D101—22no—
83C2×C4×D101—22no—
84C2×C4×D101—41no—
85C2×C4×D101—41no—
86#13108053—22no—
87#13108053—41no—
88C22⋊Dic101—22no—
89C22⋊Dic101—22no—
90C22⋊Dic101—41no—
91C22⋊Dic101C2×C414yescomplemented
92#13108051—22no—
93#13108051—22no—
94#13108051—22no—
95#13108051—22no—
96#13108051—22no—
97#13108051C2×C414yescomplemented
98#13108051C2×C414yescomplemented
99#13108051D414yescomplemented
100C4×Dic101—22no—
101C4×Dic101—41no—
102C24×C101C2311yescharacteristic
103C24×C101C2311yescharacteristic
104C2×D4×C101—22no—
105C2×D4×C101—22no—
106C2×D4×C101C2×C414yescomplemented
107C22×C4×C101—22no—
108C22×C4×C101—22no—
109C22×C4×C101—22no—
110C22×C4×C101—22no—
111C22×C4×C101—41no—
112C22×C4×C101—41no—
113C22×C4×C101C2311yescharacteristic
114C22×C4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116C22×C4×C101C2311yescharacteristic
117C22×C4×C101C2311yescharacteristic
118C4⋊C4×C101—22no—
119C22⋊C4×C101—22no—
120C22⋊C4×C101D414yescomplemented
121C42×C101—22no—
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—21no—
131C23×C101—21no—
132C23×C101C22×C412yes—
133C23×C101C22×C412yes—
134C23×C101C2×D412yes—
135C23×C101#4112yes—
136C23×C101C2411yescharacteristic
137C22×D101—21no—
138C22×D101—21no—
139C22×D101—21no—
140C22×D101—21no—
141C22×D101—28no—
142C22×D101—21no—
143C22×D101—21no—
144C22×D101—21no—
145C22×D101—21no—
146C22×D101—21no—
147C22×D101—21no—
148C22×D101—28no—
149C22×D101—44no—
150C22×D101—44no—
151C22×D101—44no—
152C22×D101—44no—
153C22×D101—44no—
154C22×D101C22×C412yes—
155C22×D101C22×C412yes—
156C22×D101C2×D412yes—
157C22×D101#4112yes—
158D4×C101—28no—
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—21no—
169C2×C4×C101—21no—
170C2×C4×C101—44no—
171C2×C4×C101—44no—
172C2×C4×C101C22×C412yes—
173C2×C4×C101C2×D412yes—
174C2×C4×C101C2×D412yes—
175C2×C4×C101C2×D412yes—
176C22⋊D101—28no—
177C22⋊D101—44no—
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—21no—
192C2×Dic101—21no—
193C2×Dic101—44no—
194C2×Dic101C22×C412yes—
195C2×Dic101C22×C412yes—
196C2×Dic101C2×D412yes—
197C4×D101—44no—
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—41no—
209C22×C101—41no—
210C22×C101—41no—
211C22×C101—41no—
212C22×C101C4×D414yescomplemented
213C22×C101C23×C411yescharacteristic
214C22×C101C22×D411yescharacteristic
215C22×C101C22×D411yescharacteristic
216C22×C101C2×#4111yescharacteristic
217C22×C101C2×#4111yescharacteristic
218C22×C101H(5,𝔽2)11yescharacteristic
219C22×C101H(5,𝔽2)11yescharacteristic
220C2×D101—22no—
221C2×D101—22no—
222C2×D101—22no—
223C2×D101—22no—
224C2×D101—22no—
225C2×D101—22no—
226C2×D101—22no—
227C2×D101—22no—
228C2×D101—22no—
229C2×D101—416no—
230C2×D101—41no—
231C2×D101—41no—
232C2×D101—41no—
233C2×D101—41no—
234C2×D101C4×D414yescomplemented
235C4×C101—22no—
236C4×C101—22no—
237C4×C101—22no—
238C4×C101—22no—
239C4×C101—41no—
240Dic101—22no—
241Dic101—22no—
242Dic101—22no—
243Dic101—41no—
244Dic101—41no—
245Dic101—41no—
246C2×C101—24no—
247C2×C101—42no—
248C2×C101C2×C4×D411yescharacteristic
249C2×C101C2×C4×D411yescharacteristic
250C2×C101#51811yescharacteristic
251C2×C101D4211yescharacteristic
252C2×C101#54611yescharacteristic
253C2×C101#54611yescharacteristic
254C2×C101#55111yescharacteristic
255D101—24no—
256D101—42no—
257#2358—1011nomaximal, Sylow
258C101#235811yescharacteristic, complemented, Sylow
259C2×#120—1011no—
260C2×C4×D4—1011no—
261C2×C4×D4D10111yescharacteristic, complemented
262C22×C22⋊C4—1011no—
263C22×C22⋊C4—1011no—
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—
274C25—1011no—
275C22×D4—1011no—
276C22×D4—1011no—
277C22×D4C2×D10111yescharacteristic
278C23×C4—1011no—
279C23×C4—1011no—
280C23×C4—1011no—
281C23×C4C2×D10111yescharacteristic
282C23×C4C2×D10111yescharacteristic, complemented
283#120—1018no—
284C4×D4—24no—
285C4×D4—2024no—
286C2×C4⋊C4—1011no—
287C2×C4⋊C4—1011no—
288C2×C4⋊C4C2×D10111yescharacteristic, complemented
289C2×C22⋊C4—1011no—
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—1011no—
301C2×C22⋊C4—2024no—
302C2×C22⋊C4—2024no—
303C2×C22⋊C4C2×D10111yescharacteristic
304C2×C22⋊C4C2×D10111yescharacteristic, complemented
305C2×C42—1011no—
306C2×C42—1011no—
307C2×C42C2×D10111yescharacteristic, complemented
308#95—1011no—
309#95—1011no—
310#95—1011no—
311#95—1011no—
312#95—1011no—
313#95—1011no—
314C24—1011no—
315C24—1014no—
316C24—1011no—
317C24—4041no—
318C24—4041no—
319C24—2022no—
320C24—2022no—
321C24—2022no—
322C24—2022no—
323C24C22×D10111yescharacteristic
324C24C22×D10111yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—1014no—
328C2×D4—1014no—
329C2×D4—4041no—
330C2×D4—4041no—
331C2×D4—2022no—
332C2×D4—2022no—
333C2×D4C4×D10114yescomplemented
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—4041no—
347C22×C4—4041no—
348C22×C4—4041no—
349C22×C4—2022no—
350C22×C4—2022no—
351C22×C4—2022no—
352C22×C4—2022no—
353C22×C4—2022no—
354C22×C4—2022no—
355C22×C4C22×D10111yescharacteristic
356C22×C4C22×D10111yescharacteristic
357C22×C4C22×D10111yescharacteristic
358C22×C4C22×D10111yescharacteristic
359C22×C4C22×D10111yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—4041no—
362C4⋊C4—2022no—
363C22⋊C4—22no—
364C22⋊C4—1014no—
365C22⋊C4—1014no—
366C22⋊C4—1014no—
367C22⋊C4—1014no—
368C22⋊C4—4041no—
369C22⋊C4—2022no—
370C22⋊C4—2022no—
371C22⋊C4—2022no—
372C22⋊C4—2022no—
373C22⋊C4—2022no—
374C22⋊C4—2022no—
375C22⋊C4—2022no—
376C22⋊C4C22⋊D10114yescomplemented
377C42—22no—
378C42—4041no—
379C42—2022no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—1012no—
391C23—1012no—
392C23—1012no—
393C23—1012no—
394C23—4044no—
395C23—4044no—
396C23—4044no—
397C23—4044no—
398C23—4044no—
399C23—2028no—
400C23—2021no—
401C23—2021no—
402C23—2021no—
403C23—2021no—
404C23—2021no—
405C23—2021no—
406C23—2028no—
407C23—2021no—
408C23—2021no—
409C23—2021no—
410C23—2021no—
411C23C2×C4×D10112yes—
412C23C2×C4×D10112yes—
413C23C2×C22⋊D10112yes—
414C23#1310807412yes—
415C23C23×D10111yescharacteristic
416D4—28no—
417D4—4044no—
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—21no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—1012no—
433C2×C4—1012no—
434C2×C4—1012no—
435C2×C4—4044no—
436C2×C4—4044no—
437C2×C4—4044no—
438C2×C4—2021no—
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×C4C2×C4×D10112yes—
454C2×C4D4×D10112yes—
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—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—1014no—
471C22—40416no—
472C22—4041no—
473C22—4041no—
474C22—4041no—
475C22—4041no—
476C22—2022no—
477C22—2022no—
478C22—2022no—
479C22—2022no—
480C22—2022no—
481C22—2022no—
482C22—2022no—
483C22—2022no—
484C22—2022no—
485C22C4×C22⋊D10114yescomplemented
486C22C22×C4×D10111yescharacteristic
487C22C2×D4×D10111yescharacteristic
488C22C22×C22⋊D10111yescharacteristic
489C22#41×D10111yescharacteristic
490C22C2×#1310807411yescharacteristic
491C22#1569230011yescharacteristic
492C22#1569230111yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—4041no—
499C4—4041no—
500C4—4041no—
501C4—2022no—
502C4—2022no—
503C4—2022no—
504C2—24no—
505C2—42no—
506C2—4042no—
507C2—2024no—
508C2C4×D4×D10111yescharacteristic
509C2C2×C4×C22⋊D10111yescharacteristic
510C2#2026564911yescharacteristic
511C2D4×C22⋊D10111yescharacteristic
512C2#2026581511yescharacteristic
513C2#2026582311yescharacteristic
514C2#2026585611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
FittingC2×C4×D4×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692223C2
hypercenterC2×C4×D4D101
hypocenterC101#2358
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D101C2semidirect
3. C22×#13108051C2semidirect
4. C2×C22⋊C4×D101C2semidirect
5. #20265077C2semidirect
6. #20265068C2semidirect
7. #20265067C2semidirect
8. #20265057C2semidirect
9. #20265056C2semidirect
10. #20265053C2semidirect
11. #20265050C2semidirect
12. #20265048C2semidirect
13. C22⋊C4×Dic101C2semidirect
14. C4×#13108051C2semidirect
15. C2×C4×D4×C101C2semidirect
22. #15692223C4semidirect
24. C2×#13108053C22semidirect
25. C2×#13108053C22semidirect
28. C2×C22⋊Dic101C22semidirect
29. C2×C22⋊Dic101C22semidirect
40. C2×C4×Dic101C22semidirect
41. C2×C4×Dic101C22semidirect
42. #15692162C22semidirect
43. #15692162C22semidirect
44. #15692162C22semidirect
45. #15692162C22semidirect
46. #15692162C22semidirect
47. #15692162C22semidirect
50. C23×C4×C101C22semidirect
52. C2×C4⋊C4×C101C22semidirect
54. C2×C22⋊C4×C101C22semidirect
55. C2×C42×C101C22semidirect
62. C23×D101C2×C4semidirect
69. C2×C22⋊D101C2×C4semidirect
70. C2×C22⋊D101C2×C4semidirect
91. C22⋊Dic101C2×C4semidirect
97. #13108051C2×C4semidirect
98. #13108051C2×C4semidirect
99. #13108051D4semidirect
106. C2×D4×C101C2×C4semidirect
120. C22⋊C4×C101D4semidirect
212. C22×C101C4×D4semidirect
234. C2×D101C4×D4semidirect
258. C101#2358semidirect
261. C2×C4×D4D101semidirect
282. C23×C4C2×D101semidirect
288. C2×C4⋊C4C2×D101semidirect
304. C2×C22⋊C4C2×D101semidirect
307. C2×C42C2×D101semidirect
333. C2×D4C4×D101semidirect
376. C22⋊C4C22⋊D101semidirect
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×#156922232
C2×C4×C22⋊D1012
C22×#131080512
C2×C22⋊C4×D1012
#202650772
#202650682
#202650672
#202650572
#202650562
#202650532
#202650502
#202650482
C22⋊C4×Dic1012
C4×#131080512
C2×C4×D4×C1012
#2358101