Finite Group Databasebeta

← #27173791

Subgroups — #27173791

Order 12928.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692223C211yesmaximal, characteristic
2C2×#15692215C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265077C211yesmaximal, characteristic, complemented
5#20265071C211yesmaximal, characteristic, complemented
6#20265071C211yesmaximal, characteristic, complemented
7#20265068C211yesmaximal, characteristic, complemented
8#20265067C211yesmaximal, characteristic, complemented
9#20265057C211yesmaximal, characteristic, complemented
10#20265054C211yesmaximal, characteristic, complemented
11#20265053C211yesmaximal, characteristic, complemented
12#20265048C211yesmaximal, characteristic, complemented
13#20265032C211yesmaximal, characteristic, complemented
14C4×#13108051C211yesmaximal, characteristic, complemented
15#390×C101C211yesmaximal, characteristic, complemented
16C24×D101C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic, complemented
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22#15692223C418yescomplemented
23#15692215—24no—
24C2×#13108053C2211yescharacteristic, complemented
25C2×#13108053C2211yescharacteristic, complemented
26C2×C22⋊Dic101C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic, complemented
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C22⋊C4×D101—24no—
39C2×C4×Dic101C2211yescharacteristic, complemented
40C2×C4×Dic101C2211yescharacteristic, complemented
41#15692162C2211yescharacteristic, complemented
42#15692162C2211yescharacteristic, complemented
43#15692162C2211yescharacteristic, complemented
44#15692162C2211yescharacteristic, complemented
45#15692162C2211yescharacteristic, complemented
46#15692162C2211yescharacteristic, complemented
47C22×D4×C101C2211yescharacteristic
48C2×C4⋊C4×C101C2211yescharacteristic, complemented
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic, complemented
52C2×C22⋊C4×C101C2211yescharacteristic, complemented
53C2×C42×C101C2211yescharacteristic, complemented
54C23×D101—22no—
55C23×D101—22no—
56C23×D101—41no—
57C23×D101—41no—
58C23×D101—41no—
59C23×D101—41no—
60C23×D101C2×C414yescomplemented
61C23×D101C2311yescharacteristic
62C23×D101C2311yescharacteristic
63C2×C22⋊D101—22no—
64C2×C22⋊D101—22no—
65C2×C22⋊D101—41no—
66C2×C22⋊D101—41no—
67C2×C22⋊D101C2×C414yes—
68C2×C22⋊D101C2×C414yescomplemented
69C22×Dic101—22no—
70C22×Dic101—41no—
71C22×Dic101—41no—
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C2×C4×D101—22no—
79C2×C4×D101—41no—
80C2×C4×D101—41no—
81C2×C4×D101—41no—
82C2×C4×D101—41no—
83C2×C4×D101D414yescomplemented
84#13108053—22no—
85#13108053—41no—
86C22⋊Dic101—22no—
87C22⋊Dic101—41no—
88C22⋊Dic101—41no—
89C22⋊Dic101C2×C414yescomplemented
90#13108051—22no—
91#13108051—22no—
92#13108051—22no—
93#13108051—22no—
94#13108051—41no—
95#13108051—41no—
96#13108051C2×C414yescomplemented
97#13108051C2×C414yescomplemented
98C4×Dic101—22no—
99C4×Dic101—41no—
100C24×C101C2311yescharacteristic
101C24×C101C2311yescharacteristic
102C2×D4×C101—41no—
103C2×D4×C101—41no—
104C2×D4×C101C2×C414yescomplemented
105C22×C4×C101C2311yescharacteristic
106C22×C4×C101C2311yescharacteristic
107C22×C4×C101C2311yescharacteristic
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C4⋊C4×C101—41no—
111C22⋊C4×C101—22no—
112C22⋊C4×C101—22no—
113C22⋊C4×C101—41no—
114C22⋊C4×C101—41no—
115C42×C101—41no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101—21no—
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—21no—
127C23×C101—21no—
128C23×C101C22×C412yes—
129C23×C101C22×C412yes—
130C23×C101C2411yescharacteristic
131C22×D101—21no—
132C22×D101—21no—
133C22×D101—21no—
134C22×D101—21no—
135C22×D101—21no—
136C22×D101—21no—
137C22×D101—21no—
138C22×D101—21no—
139C22×D101—21no—
140C22×D101—21no—
141C22×D101—28no—
142C22×D101—44no—
143C22×D101—44no—
144C22×D101—44no—
145C22×D101—44no—
146C22×D101—44no—
147C22×D101—44no—
148C22×D101C22×C412yes—
149C22×D101C22×C412yes—
150C22×D101C2×D412yes—
151C22×D101#4112yes—
152D4×C101—44no—
153C2×C4×C101—21no—
154C2×C4×C101—21no—
155C2×C4×C101—21no—
156C2×C4×C101—21no—
157C2×C4×C101—21no—
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×C101C22×C412yes—
167C2×C4×C101C2×D412yes—
168C22⋊D101—28no—
169C22⋊D101—44no—
170C2×Dic101—21no—
171C2×Dic101—21no—
172C2×Dic101—21no—
173C2×Dic101—21no—
174C2×Dic101—21no—
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—44no—
186C2×Dic101C22×C412yes—
187C2×Dic101C22×C412yes—
188C2×Dic101C2×D412yes—
189C4×D101—44no—
190C4×D101—44no—
191C22×C101—22no—
192C22×C101—22no—
193C22×C101—22no—
194C22×C101—22no—
195C22×C101—22no—
196C22×C101—22no—
197C22×C101—41no—
198C22×C101—41no—
199C22×C101—41no—
200C22×C101—41no—
201C22×C101—41no—
202C22×C101—41no—
203C22×C101—41no—
204C22×C101—41no—
205C22×C101C23×C411yescharacteristic
206C22×C101C22×D411yescharacteristic
207C22×C101C2×#4111yescharacteristic
208C22×C101H(5,𝔽2)11yescharacteristic
209C22×C101H(5,𝔽2)11yescharacteristic
210C22×C101H(5,𝔽2)11yescharacteristic
211C22×C101H(5,𝔽2)11yescharacteristic
212C2×D101—22no—
213C2×D101—22no—
214C2×D101—22no—
215C2×D101—22no—
216C2×D101—22no—
217C2×D101—22no—
218C2×D101—22no—
219C2×D101—22no—
220C2×D101—22no—
221C2×D101—416no—
222C2×D101—41no—
223C2×D101—41no—
224C2×D101—41no—
225C2×D101—41no—
226C2×D101C4×D414yes—
227C4×C101—22no—
228C4×C101—22no—
229C4×C101—41no—
230C4×C101—41no—
231C4×C101—41no—
232Dic101—22no—
233Dic101—22no—
234Dic101—22no—
235Dic101—41no—
236Dic101—41no—
237Dic101—41no—
238C2×C101—42no—
239C2×C101—42no—
240C2×C101C2×C4×D411yescharacteristic
241C2×C101#51811yescharacteristic
242C2×C101#51811yescharacteristic
243C2×C101#53411yescharacteristic
244C2×C101#53511yescharacteristic
245C2×C101#53811yescharacteristic
246C2×C101#53811yescharacteristic
247D101—24no—
248D101—42no—
249#2321—1011nomaximal, Sylow
250C101#232111yescharacteristic, complemented, Sylow
251C2×#120—1011no—
252C2×C4×D4—1011no—
253C22×C22⋊C4—1011no—
254#390—1011no—
255#390D10111yescharacteristic, complemented
256#388—1011no—
257#388—1011no—
258#388—1011no—
259#388—1011no—
260#386—1011no—
261#386—1011no—
262#386—1011no—
263#385—1011no—
264#379—1011no—
265C4×C22⋊C4—1011no—
266C25—1011no—
267C22×D4—1011no—
268C22×D4—1011no—
269C22×D4C2×D10111yescharacteristic
270C23×C4—1011no—
271C23×C4—1011no—
272C23×C4—1011no—
273#120—1018no—
274C4×D4—2024no—
275C2×C4⋊C4—1011no—
276C2×C4⋊C4—1011no—
277C2×C4⋊C4C2×D10111yescharacteristic, complemented
278C2×C22⋊C4—1011no—
279C2×C22⋊C4—1011no—
280C2×C22⋊C4—1011no—
281C2×C22⋊C4—1011no—
282C2×C22⋊C4—1011no—
283C2×C22⋊C4—1011no—
284C2×C22⋊C4—1011no—
285C2×C22⋊C4—1011no—
286C2×C22⋊C4—1011no—
287C2×C22⋊C4—1011no—
288C2×C22⋊C4—1011no—
289C2×C22⋊C4—1011no—
290C2×C22⋊C4—2024no—
291C2×C22⋊C4C2×D10111yescharacteristic
292C2×C22⋊C4C2×D10111yescharacteristic
293C2×C22⋊C4C2×D10111yescharacteristic, complemented
294C2×C22⋊C4C2×D10111yescharacteristic, complemented
295C2×C42—1011no—
296C2×C42—1011no—
297C2×C42C2×D10111yescharacteristic, complemented
298#95—1011no—
299#95—1011no—
300#95—1011no—
301#95—1011no—
302#95—1011no—
303#95—1011no—
304C24—1011no—
305C24—1014no—
306C24—1011no—
307C24—4041no—
308C24—4041no—
309C24—4041no—
310C24—4041no—
311C24—2022no—
312C24—2022no—
313C24C22×D10111yescharacteristic
314C24C22×D10111yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—1014no—
318C2×D4—1014no—
319C2×D4—4041no—
320C2×D4—4041no—
321C2×D4—2022no—
322C2×D4—2022no—
323C2×D4C4×D10114yescomplemented
324C22×C4—1011no—
325C22×C4—1011no—
326C22×C4—1011no—
327C22×C4—1014no—
328C22×C4—1011no—
329C22×C4—1011no—
330C22×C4—1011no—
331C22×C4—4041no—
332C22×C4—4041no—
333C22×C4—4041no—
334C22×C4—4041no—
335C22×C4—4041no—
336C22×C4—4041no—
337C22×C4—2022no—
338C22×C4—2022no—
339C22×C4C22×D10111yescharacteristic
340C22×C4C22×D10111yescharacteristic
341C22×C4C22×D10111yescharacteristic
342C22×C4C22×D10111yescharacteristic
343C22×C4C22×D10111yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—4041no—
346C4⋊C4—2022no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—1014no—
352C22⋊C4—1014no—
353C22⋊C4—1014no—
354C22⋊C4—4041no—
355C22⋊C4—4041no—
356C22⋊C4—4041no—
357C22⋊C4—4041no—
358C22⋊C4—2022no—
359C22⋊C4—2022no—
360C22⋊C4—2022no—
361C22⋊C4—2022no—
362C22⋊C4—2022no—
363C42—41no—
364C42—4041no—
365C42—2022no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—1012no—
379C23—1012no—
380C23—1012no—
381C23—1012no—
382C23—4044no—
383C23—4044no—
384C23—4044no—
385C23—4044no—
386C23—4044no—
387C23—4044no—
388C23—2028no—
389C23—2021no—
390C23—2021no—
391C23—2021no—
392C23—2021no—
393C23—2021no—
394C23—2021no—
395C23—2021no—
396C23—2021no—
397C23—2021no—
398C23—2021no—
399C23C2×C4×D10112yes—
400C23C2×C4×D10112yes—
401C23C23×D10111yescharacteristic
402D4—44no—
403D4—4044no—
404D4—2028no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
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—1012no—
419C2×C4—1012no—
420C2×C4—1012no—
421C2×C4—4044no—
422C2×C4—4044no—
423C2×C4—4044no—
424C2×C4—2021no—
425C2×C4—2021no—
426C2×C4—2021no—
427C2×C4—2021no—
428C2×C4—2021no—
429C2×C4—2021no—
430C2×C4—2021no—
431C2×C4—2021no—
432C2×C4—2021no—
433C2×C4—2021no—
434C2×C4—2021no—
435C2×C4—2021no—
436C2×C4—2021no—
437C2×C4—2021no—
438C2×C4—2021no—
439C2×C4C2×C4×D10112yes—
440C2×C4D4×D10112yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—1014no—
456C22—40416no—
457C22—4041no—
458C22—4041no—
459C22—4041no—
460C22—4041no—
461C22—2022no—
462C22—2022no—
463C22—2022no—
464C22—2022no—
465C22—2022no—
466C22—2022no—
467C22—2022no—
468C22—2022no—
469C22—2022no—
470C22C22×C4×D10111yescharacteristic
471C22C2×D4×D10111yescharacteristic
472C22#41×D10111yescharacteristic
473C22#1569230011yescharacteristic
474C22#1569230011yescharacteristic
475C22#1569230011yescharacteristic
476C22#1569230111yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—4041no—
483C4—4041no—
484C4—4041no—
485C4—2022no—
486C4—2022no—
487C4—2022no—
488C2—42no—
489C2—42no—
490C2—4042no—
491C2—2024no—
492C2C4×D4×D10111yescharacteristic
493C2#2026564911yescharacteristic
494C2#2026565011yescharacteristic
495C2#2026573511yescharacteristic
496C2#2026574011yescharacteristic
497C2#2026575911yescharacteristic
498C2#2026576111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#390×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692223C2
hypercenter#390D101
hypocenterC101#2321
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#15692215C2semidirect
3. C2×C22⋊C4×D101C2semidirect
4. #20265077C2semidirect
5. #20265071C2semidirect
6. #20265071C2semidirect
7. #20265068C2semidirect
8. #20265067C2semidirect
9. #20265057C2semidirect
10. #20265054C2semidirect
11. #20265053C2semidirect
12. #20265048C2semidirect
13. #20265032C2semidirect
14. C4×#13108051C2semidirect
15. #390×C101C2semidirect
19. C23×Dic101C22semidirect
22. #15692223C4semidirect
24. C2×#13108053C22semidirect
25. C2×#13108053C22semidirect
29. C2×C22⋊Dic101C22semidirect
39. C2×C4×Dic101C22semidirect
40. C2×C4×Dic101C22semidirect
41. #15692162C22semidirect
42. #15692162C22semidirect
43. #15692162C22semidirect
44. #15692162C22semidirect
45. #15692162C22semidirect
46. #15692162C22semidirect
48. C2×C4⋊C4×C101C22semidirect
51. C2×C22⋊C4×C101C22semidirect
52. C2×C22⋊C4×C101C22semidirect
53. C2×C42×C101C22semidirect
60. C23×D101C2×C4semidirect
68. C2×C22⋊D101C2×C4semidirect
83. C2×C4×D101D4semidirect
89. C22⋊Dic101C2×C4semidirect
96. #13108051C2×C4semidirect
97. #13108051C2×C4semidirect
104. C2×D4×C101C2×C4semidirect
250. C101#2321semidirect
255. #390D101semidirect
277. C2×C4⋊C4C2×D101semidirect
293. C2×C22⋊C4C2×D101semidirect
294. C2×C22⋊C4C2×D101semidirect
297. C2×C42C2×D101semidirect
323. C2×D4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922232
C2×#156922152
C2×C22⋊C4×D1012
#202650772
#202650712
#202650682
#202650672
#202650572
#202650542
#202650532
#202650482
#202650322
C4×#131080512
#390×C1012
#2321101