Finite Group Databasebeta

← #27173794

Subgroups — #27173794

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×#15692225C211yesmaximal, characteristic
2C2×C4×D404C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265078C211yesmaximal, characteristic, complemented
5#20265077C211yesmaximal, characteristic, complemented
6#20265070C211yesmaximal, characteristic, complemented
7#20265069C211yesmaximal, characteristic, complemented
8#20265068C211yesmaximal, characteristic, complemented
9#20265055C211yesmaximal, characteristic, complemented
10#20265055C211yesmaximal, characteristic, complemented
11#20265053C211yesmaximal, characteristic, complemented
12#20265048C211yesmaximal, characteristic, complemented
13#20265032C211yesmaximal, characteristic, complemented
14#20265024C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C24×D101C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C22×D404C2211yescharacteristic
19C22×C22⋊D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22C22×C4×D101C2211yescharacteristic, complemented
23#15692225C418yescomplemented
24C4×D404—24no—
25C2×C4⋊Dic101C2211yescharacteristic, complemented
26C2×#13108053C2211yescharacteristic, complemented
27C2×C22⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic, complemented
38C2×#13108051C2211yescharacteristic, complemented
39C2×#13108051C2211yescharacteristic, complemented
40C22⋊C4×D101—24no—
41C2×C4×Dic101C2211yescharacteristic, complemented
42C2×C4×Dic101C2211yescharacteristic, complemented
43#15692162C2211yescharacteristic, complemented
44#15692162C2211yescharacteristic, complemented
45#15692162C2211yescharacteristic, complemented
46#15692162C2211yescharacteristic, complemented
47C2×C4⋊C4×C101C2211yescharacteristic, complemented
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C42×C101C2211yescharacteristic, complemented
52#95×C101C2211yescharacteristic, complemented
53#95×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
63C23×D101C2311yescharacteristic
64C2×D404—22no—
65C2×D404—22no—
66C2×D404—41no—
67C2×D404—41no—
68C2×D404C2×C414yes—
69C2×D404C2×C414yescomplemented
70C2×C22⋊D101—41no—
71C2×C22⋊D101—41no—
72C2×C22⋊D101C2×C414yescomplemented
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C22×Dic101C2311yescharacteristic
77C22×Dic101C2311yescharacteristic
78C2×C4×D101—22no—
79C2×C4×D101—22no—
80C2×C4×D101—41no—
81C2×C4×D101—41no—
82C2×C4×D101—41no—
83C2×C4×D101—41no—
84C2×C4×D101—41no—
85C2×C4×D101—41no—
86C2×C4×D101D414yescomplemented
87C4⋊Dic101—22no—
88#13108053—41no—
89C22⋊Dic101—22no—
90C22⋊Dic101—41no—
91#13108051—22no—
92#13108051—22no—
93#13108051—22no—
94#13108051—22no—
95#13108051—22no—
96#13108051—41no—
97#13108051—41no—
98#13108051—41no—
99#13108051—41no—
100#13108051C2×C414yescomplemented
101#13108051C2×C414yescomplemented
102C4×Dic101—41no—
103C4×Dic101—41no—
104C24×C101C2311yescharacteristic
105C22×C4×C101C2311yescharacteristic
106C22×C4×C101C2311yescharacteristic
107C22×C4×C101C2311yescharacteristic
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C22×C4×C101C2311yescharacteristic
111C4⋊C4×C101—41no—
112C22⋊C4×C101—22no—
113C22⋊C4×C101—41no—
114C22⋊C4×C101C2×C414yescomplemented
115C42×C101—22no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101—21no—
119C23×C101—21no—
120C23×C101—21no—
121C23×C101—21no—
122C23×C101C22×C412yes—
123C23×C101C2411yescharacteristic
124C22×D101—21no—
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101—21no—
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—28no—
141C22×D101—44no—
142C22×D101—44no—
143C22×D101—44no—
144C22×D101—44no—
145C22×D101—44no—
146C22×D101—44no—
147C22×D101C22×C412yes—
148C22×D101C22×C412yes—
149C22×D101C22×C412yes—
150C22×D101C2×D412yes—
151C22×D101#4112yes—
152C2×C4×C101—21no—
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×C101—21no—
167C2×C4×C101C22×C412yes—
168C2×C4×C101C22×C412yes—
169C2×C4×C101C2×D412yes—
170C22⋊D101—44no—
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×Dic101C22×C412yes—
185C2×Dic101C2×D412yes—
186D404—28no—
187D404—44no—
188C4×D101—44no—
189C4×D101—44no—
190C4×D101—44no—
191C22×C101—22no—
192C22×C101—22no—
193C22×C101—22no—
194C22×C101—41no—
195C22×C101—41no—
196C22×C101—41no—
197C22×C101—41no—
198C22×C101C23×C411yescharacteristic
199C22×C101C22×D411yescharacteristic
200C22×C101C2×#4111yescharacteristic
201C22×C101H(5,𝔽2)11yescharacteristic
202C22×C101H(5,𝔽2)11yescharacteristic
203C22×C101H(5,𝔽2)11yescharacteristic
204C22×C101H(5,𝔽2)11yescharacteristic
205C2×D101—22no—
206C2×D101—22no—
207C2×D101—22no—
208C2×D101—22no—
209C2×D101—22no—
210C2×D101—22no—
211C2×D101—22no—
212C2×D101—22no—
213C2×D101—22no—
214C2×D101—22no—
215C2×D101—22no—
216C2×D101—22no—
217C2×D101—416no—
218C2×D101—41no—
219C2×D101—41no—
220C2×D101—41no—
221C2×D101—41no—
222C2×D101—41no—
223C2×D101—41no—
224C2×D101—41no—
225C2×D101—41no—
226C2×D101C4×D414yes—
227C4×C101—22no—
228C4×C101—22no—
229C4×C101—22no—
230C4×C101—41no—
231C4×C101—41no—
232C4×C101—41no—
233Dic101—22no—
234Dic101—22no—
235Dic101—41no—
236Dic101—41no—
237Dic101—41no—
238C2×C101—42no—
239C2×C101C2×C4×D411yescharacteristic
240C2×C101#51811yescharacteristic
241C2×C101#51811yescharacteristic
242C2×C101#53411yescharacteristic
243C2×C101#53511yescharacteristic
244C2×C101#53811yescharacteristic
245C2×C101#53811yescharacteristic
246D101—24no—
247D101—42no—
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#390—1011no—
256#388—1011no—
257#388—1011no—
258#388—1011no—
259#388D10111yescharacteristic, complemented
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×D4—1011no—
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—1011no—
291C2×C22⋊C4—2024no—
292C2×C22⋊C4C2×D10111yescharacteristic
293C2×C22⋊C4C2×D10111yescharacteristic
294C2×C22⋊C4C2×D10111yescharacteristic
295C2×C42—1011no—
296C2×C42—1011no—
297C2×C42C2×D10111yescharacteristic, complemented
298#95—1011no—
299#95—1011no—
300#95—1011no—
301#95—1011no—
302#95C2×D10111yescharacteristic, complemented
303#95C2×D10111yescharacteristic, complemented
304C24—1011no—
305C24—1014no—
306C24—1011no—
307C24—1011no—
308C24—4041no—
309C24—4041no—
310C24—4041no—
311C24—4041no—
312C24—2022no—
313C24—2022no—
314C24C22×D10111yescharacteristic
315C2×D4—1014no—
316C2×D4—1014no—
317C2×D4—1014no—
318C2×D4—4041no—
319C2×D4—4041no—
320C2×D4—4041no—
321C2×D4—4041no—
322C2×D4—2022no—
323C2×D4—2022no—
324C22×C4—1011no—
325C22×C4—1011no—
326C22×C4—1014no—
327C22×C4—1011no—
328C22×C4—1011no—
329C22×C4—1011no—
330C22×C4—4041no—
331C22×C4—4041no—
332C22×C4—4041no—
333C22×C4—4041no—
334C22×C4—4041no—
335C22×C4—4041no—
336C22×C4—2022no—
337C22×C4—2022no—
338C22×C4C22×D10111yescharacteristic
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—41no—
349C22⋊C4—1014no—
350C22⋊C4—1014no—
351C22⋊C4—4041no—
352C22⋊C4—4041no—
353C22⋊C4—4041no—
354C22⋊C4—4041no—
355C22⋊C4—4041no—
356C22⋊C4—2022no—
357C22⋊C4—2022no—
358C22⋊C4—2022no—
359C22⋊C4—2022no—
360C22⋊C4—2022no—
361C22⋊C4—2022no—
362C22⋊C4C4×D10114yescomplemented
363C42—22no—
364C42—4041no—
365C42—4041no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—1012no—
373C23—1012no—
374C23—1012no—
375C23—1012no—
376C23—1012no—
377C23—4044no—
378C23—4044no—
379C23—4044no—
380C23—4044no—
381C23—4044no—
382C23—4044no—
383C23—2028no—
384C23—2021no—
385C23—2021no—
386C23—2021no—
387C23—2021no—
388C23—2021no—
389C23—2021no—
390C23—2021no—
391C23—2021no—
392C23—2021no—
393C23—2021no—
394C23—2021no—
395C23—2021no—
396C23—2021no—
397C23—2021no—
398C23—2021no—
399C23—2021no—
400C23C2×C4×D10112yes—
401C23C23×D10111yescharacteristic
402D4—4044no—
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—21no—
419C2×C4—21no—
420C2×C4—1012no—
421C2×C4—1012no—
422C2×C4—4044no—
423C2×C4—4044no—
424C2×C4—4044no—
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×C4C2×C4×D10112yes—
439C2×C4C2×C4×D10112yes—
440C2×C4D4×D10112yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—1014no—
449C22—40416no—
450C22—4041no—
451C22—4041no—
452C22—4041no—
453C22—4041no—
454C22—4041no—
455C22—4041no—
456C22—4041no—
457C22—4041no—
458C22—2022no—
459C22—2022no—
460C22—2022no—
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#1569230111yescharacteristic
475C22#1569230111yescharacteristic
476C22#1569230111yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—4041no—
484C4—4041no—
485C4—4041no—
486C4—2022no—
487C4—2022no—
488C2—42no—
489C2—4042no—
490C2—4042no—
491C2—2024no—
492C2C4×D4×D10111yescharacteristic
493C2#2026564911yescharacteristic
494C2#2026565111yescharacteristic
495C2#2026573811yescharacteristic
496C2#2026574311yescharacteristic
497C2#2026576111yescharacteristic
498C2#2026576611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692225C2
hypercenter#388D101
hypocenterC101#2321
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D404C2semidirect
3. C2×C22⋊C4×D101C2semidirect
4. #20265078C2semidirect
5. #20265077C2semidirect
6. #20265070C2semidirect
7. #20265069C2semidirect
8. #20265068C2semidirect
9. #20265055C2semidirect
10. #20265055C2semidirect
11. #20265053C2semidirect
12. #20265048C2semidirect
13. #20265032C2semidirect
14. #20265024C2semidirect
15. #388×C101C2semidirect
22. C22×C4×D101C22semidirect
23. #15692225C4semidirect
25. C2×C4⋊Dic101C22semidirect
26. C2×#13108053C22semidirect
37. C2×#13108051C22semidirect
38. C2×#13108051C22semidirect
39. C2×#13108051C22semidirect
41. C2×C4×Dic101C22semidirect
42. C2×C4×Dic101C22semidirect
43. #15692162C22semidirect
44. #15692162C22semidirect
45. #15692162C22semidirect
46. #15692162C22semidirect
47. C2×C4⋊C4×C101C22semidirect
51. C2×C42×C101C22semidirect
52. #95×C101C22semidirect
53. #95×C101C22semidirect
60. C23×D101C2×C4semidirect
69. C2×D404C2×C4semidirect
72. C2×C22⋊D101C2×C4semidirect
86. C2×C4×D101D4semidirect
100. #13108051C2×C4semidirect
101. #13108051C2×C4semidirect
114. C22⋊C4×C101C2×C4semidirect
250. C101#2321semidirect
259. #388D101semidirect
277. C2×C4⋊C4C2×D101semidirect
297. C2×C42C2×D101semidirect
302. #95C2×D101semidirect
303. #95C2×D101semidirect
362. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922252
C2×C4×D4042
C2×C22⋊C4×D1012
#202650782
#202650772
#202650702
#202650692
#202650682
#202650552
#202650532
#202650482
#202650322
#202650242
#388×C1012
#2321101