Finite Group Databasebeta

← #27174272

Subgroups — #27174272

Order 12928.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692225C211yesmaximal, characteristic
2#20265079C211yesmaximal, characteristic, complemented
3#20265078C211yesmaximal, characteristic, complemented
4#20265077C211yesmaximal, characteristic, complemented
5#20265070C211yesmaximal, characteristic, complemented
6#20265069C211yesmaximal, characteristic, complemented
7#20265068C211yesmaximal, characteristic, complemented
8#20265067C211yesmaximal, characteristic, complemented
9#20265066C211yesmaximal, characteristic, complemented
10#20265055C211yesmaximal, characteristic, complemented
11#20265055C211yesmaximal, characteristic, complemented
12#20265053C211yesmaximal, characteristic, complemented
13#20265032C211yesmaximal, characteristic, complemented
14#20265032C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C24×D101C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C22×D404C2211yescharacteristic
19C22×C22⋊D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic, complemented
21#15692225C418yescomplemented
22C2×C4⋊Dic101C2211yescharacteristic, complemented
23C2×#13108053C2211yescharacteristic, complemented
24C2×C22⋊Dic101C2211yescharacteristic
25C2×C22⋊Dic101C2211yescharacteristic
26C2×#13108051C2211yescharacteristic
27C2×#13108051C2211yescharacteristic
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic, complemented
37C2×#13108051C2211yescharacteristic, complemented
38C2×#13108051C2211yescharacteristic, complemented
39C2×C4×Dic101C2211yescharacteristic, complemented
40C2×C4×Dic101C2211yescharacteristic, complemented
41#15692162C2211yescharacteristic, complemented
42#15692162C2211yescharacteristic, complemented
43#15692162C2211yescharacteristic, complemented
44#15692162C2211yescharacteristic, complemented
45C2×C4⋊C4×C101C2211yescharacteristic, complemented
46C2×C22⋊C4×C101C2211yescharacteristic
47C2×C22⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C42×C101C2211yescharacteristic, complemented
50#95×C101C2211yescharacteristic, complemented
51#95×C101C2211yescharacteristic, complemented
52C23×D101—41no—
53C23×D101—41no—
54C23×D101—41no—
55C23×D101—41no—
56C23×D101—41no—
57C23×D101—41no—
58C23×D101C2×C414yescomplemented
59C23×D101C2311yescharacteristic
60C23×D101C2311yescharacteristic
61C23×D101C2311yescharacteristic
62C2×D404—41no—
63C2×D404—41no—
64C2×D404—41no—
65C2×D404—41no—
66C2×D404C2×C414yes—
67C2×D404C2×C414yes—
68C2×C22⋊D101—41no—
69C2×C22⋊D101—41no—
70C2×C22⋊D101C2×C414yes—
71C22×Dic101C2311yescharacteristic
72C22×Dic101C2311yescharacteristic
73C22×Dic101C2311yescharacteristic
74C22×Dic101C2311yescharacteristic
75C22×Dic101C2311yescharacteristic
76C2×C4×D101—41no—
77C2×C4×D101—41no—
78C2×C4×D101—41no—
79C4⋊Dic101—41no—
80#13108053—41no—
81C22⋊Dic101—41no—
82C22⋊Dic101—41no—
83#13108051—41no—
84#13108051—41no—
85#13108051—41no—
86#13108051—41no—
87#13108051—41no—
88#13108051—41no—
89#13108051—41no—
90#13108051—41no—
91#13108051—41no—
92#13108051—41no—
93#13108051—41no—
94#13108051C2×C414yescomplemented
95#13108051C2×C414yescomplemented
96C4×Dic101—41no—
97C4×Dic101—41no—
98C24×C101C2311yescharacteristic
99C22×C4×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105C4⋊C4×C101—41no—
106C22⋊C4×C101—41no—
107C22⋊C4×C101—41no—
108C22⋊C4×C101C2×C414yescomplemented
109C42×C101—41no—
110C23×C101—21no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101—21no—
115C23×C101—21no—
116C23×C101C22×C412yes—
117C23×C101C2411yescharacteristic
118C22×D101—21no—
119C22×D101—21no—
120C22×D101—21no—
121C22×D101—21no—
122C22×D101—21no—
123C22×D101—21no—
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—44no—
137C22×D101—44no—
138C22×D101—44no—
139C22×D101—44no—
140C22×D101—44no—
141C22×D101—44no—
142C22×D101—44no—
143C22×D101C22×C412yes—
144C22×D101C22×C412yes—
145C22×D101C22×C412yes—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—21no—
150C2×C4×C101—21no—
151C2×C4×C101—21no—
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×C101C22×C412yes—
163C2×C4×C101C22×C412yes—
164C22⋊D101—44no—
165C2×Dic101—21no—
166C2×Dic101—21no—
167C2×Dic101—21no—
168C2×Dic101—21no—
169C2×Dic101—21no—
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×Dic101C22×C412yes—
180D404—44no—
181D404—44no—
182C4×D101—44no—
183C22×C101—22no—
184C22×C101—22no—
185C22×C101—22no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101C23×C411yescharacteristic
191C22×C101H(5,𝔽2)11yescharacteristic
192C22×C101H(5,𝔽2)11yescharacteristic
193C22×C101H(5,𝔽2)11yescharacteristic
194C22×C101H(5,𝔽2)11yescharacteristic
195C22×C101H(5,𝔽2)11yescharacteristic
196C22×C101H(5,𝔽2)11yescharacteristic
197C2×D101—22no—
198C2×D101—22no—
199C2×D101—22no—
200C2×D101—22no—
201C2×D101—22no—
202C2×D101—22no—
203C2×D101—22no—
204C2×D101—22no—
205C2×D101—22no—
206C2×D101—416no—
207C2×D101—41no—
208C2×D101—41no—
209C2×D101—41no—
210C2×D101—41no—
211C2×D101—41no—
212C2×D101—41no—
213C2×D101—41no—
214C2×D101—41no—
215C2×D101—41no—
216C2×D101—41no—
217C2×D101—41no—
218C2×D101—41no—
219C4×C101—22no—
220C4×C101—22no—
221C4×C101—41no—
222C4×C101—41no—
223C4×C101—41no—
224C4×C101—41no—
225Dic101—22no—
226Dic101—41no—
227Dic101—41no—
228Dic101—41no—
229Dic101—41no—
230C2×C101—42no—
231C2×C101#51811yescharacteristic
232C2×C101#51811yescharacteristic
233C2×C101#51811yescharacteristic
234C2×C101#56011yescharacteristic
235C2×C101#56011yescharacteristic
236C2×C101#56011yescharacteristic
237C2×C101H(3,𝔽4)11yescharacteristic
238D101—42no—
239D101—42no—
240D101—42no—
241#2375—1011nomaximal, Sylow
242C101#237511yescharacteristic, complemented, Sylow
243C2×#120—1011no—
244#390—1011no—
245#390—1011no—
246#390—1011no—
247#388—1011no—
248#388—1011no—
249#388—1011no—
250#388—1011no—
251#388—1011no—
252#388D10111yescharacteristic, complemented
253#386—1011no—
254#386—1011no—
255#386—1011no—
256#379—1011no—
257#379—1011no—
258C25—1011no—
259C22×D4—1011no—
260C22×D4—1011no—
261C22×D4—1011no—
262C23×C4—1011no—
263#120—1018no—
264C2×C4⋊C4—1011no—
265C2×C4⋊C4—1011no—
266C2×C4⋊C4C2×D10111yescharacteristic, complemented
267C2×C22⋊C4—1011no—
268C2×C22⋊C4—1011no—
269C2×C22⋊C4—1011no—
270C2×C22⋊C4—1011no—
271C2×C22⋊C4—1011no—
272C2×C22⋊C4—1011no—
273C2×C22⋊C4—1011no—
274C2×C22⋊C4—1011no—
275C2×C22⋊C4—1011no—
276C2×C22⋊C4—1011no—
277C2×C22⋊C4—1011no—
278C2×C22⋊C4—1011no—
279C2×C22⋊C4—1011no—
280C2×C22⋊C4—1011no—
281C2×C22⋊C4—1011no—
282C2×C22⋊C4C2×D10111yescharacteristic
283C2×C22⋊C4C2×D10111yescharacteristic
284C2×C22⋊C4C2×D10111yescharacteristic
285C2×C42—1011no—
286C2×C42—1011no—
287C2×C42C2×D10111yescharacteristic, complemented
288#95—1011no—
289#95—1011no—
290#95—1011no—
291#95—1011no—
292#95C2×D10111yescharacteristic, complemented
293#95C2×D10111yescharacteristic, complemented
294C24—1014no—
295C24—1011no—
296C24—1011no—
297C24—1011no—
298C24—4041no—
299C24—4041no—
300C24—4041no—
301C24—4041no—
302C24—4041no—
303C24—4041no—
304C24C22×D10111yescharacteristic
305C2×D4—1014no—
306C2×D4—1014no—
307C2×D4—1014no—
308C2×D4—4041no—
309C2×D4—4041no—
310C2×D4—4041no—
311C2×D4—4041no—
312C2×D4—4041no—
313C2×D4—4041no—
314C22×C4—1011no—
315C22×C4—1011no—
316C22×C4—1011no—
317C22×C4—1011no—
318C22×C4—1011no—
319C22×C4—4041no—
320C22×C4—4041no—
321C22×C4—4041no—
322C22×C4C22×D10111yescharacteristic
323C22×C4C22×D10111yescharacteristic
324C22×C4C22×D10111yescharacteristic
325C22×C4C22×D10111yescharacteristic
326C22×C4C22×D10111yescharacteristic
327C22×C4C22×D10111yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—4041no—
330C4⋊C4—4041no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—1014no—
334C22⋊C4—1014no—
335C22⋊C4—4041no—
336C22⋊C4—4041no—
337C22⋊C4—4041no—
338C22⋊C4—4041no—
339C22⋊C4—4041no—
340C22⋊C4—4041no—
341C22⋊C4—4041no—
342C22⋊C4—4041no—
343C22⋊C4—4041no—
344C22⋊C4—4041no—
345C22⋊C4—4041no—
346C22⋊C4—4041no—
347C22⋊C4—4041no—
348C22⋊C4C4×D10114yescomplemented
349C42—41no—
350C42—4041no—
351C42—4041no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—1012no—
359C23—1012no—
360C23—1012no—
361C23—4044no—
362C23—4044no—
363C23—4044no—
364C23—4044no—
365C23—4044no—
366C23—4044no—
367C23—4044no—
368C23—2021no—
369C23—2021no—
370C23—2021no—
371C23—2021no—
372C23—2021no—
373C23—2021no—
374C23—2021no—
375C23—2021no—
376C23—2021no—
377C23—2021no—
378C23—2021no—
379C23—2021no—
380C23—2021no—
381C23—2021no—
382C23—2021no—
383C23—2021no—
384C23—2021no—
385C23—2021no—
386C23C2×C4×D10112yes—
387C23C23×D10111yescharacteristic
388D4—4044no—
389D4—4044no—
390D4—4044no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—1012no—
408C2×C4—4044no—
409C2×C4—2021no—
410C2×C4—2021no—
411C2×C4—2021no—
412C2×C4—2021no—
413C2×C4—2021no—
414C2×C4—2021no—
415C2×C4—2021no—
416C2×C4—2021no—
417C2×C4—2021no—
418C2×C4—2021no—
419C2×C4—2021no—
420C2×C4—2021no—
421C2×C4—2021no—
422C2×C4—2021no—
423C2×C4C2×C4×D10112yes—
424C2×C4C2×C4×D10112yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—40416no—
433C22—4041no—
434C22—4041no—
435C22—4041no—
436C22—4041no—
437C22—4041no—
438C22—4041no—
439C22—4041no—
440C22—4041no—
441C22—4041no—
442C22—4041no—
443C22—4041no—
444C22—4041no—
445C22—2022no—
446C22—2022no—
447C22—2022no—
448C22—2022no—
449C22—2022no—
450C22—2022no—
451C22—2022no—
452C22—2022no—
453C22—2022no—
454C22C22×C4×D10111yescharacteristic
455C22#1569230011yescharacteristic
456C22#1569230011yescharacteristic
457C22#1569230111yescharacteristic
458C22#1569230111yescharacteristic
459C22#1569230111yescharacteristic
460C22#1569230111yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—4041no—
468C4—4041no—
469C4—4041no—
470C4—4041no—
471C4—2022no—
472C2—42no—
473C2—4042no—
474C2—4042no—
475C2—4042no—
476C2#2026564911yescharacteristic
477C2#2026564911yescharacteristic
478C2#2026565111yescharacteristic
479C2#2026591511yescharacteristic
480C2#2026591711yescharacteristic
481C2#2026591811yescharacteristic
482C2#2026591911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692225C2
hypercenter#388D101
hypocenterC101#2375
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20265079C2semidirect
3. #20265078C2semidirect
4. #20265077C2semidirect
5. #20265070C2semidirect
6. #20265069C2semidirect
7. #20265068C2semidirect
8. #20265067C2semidirect
9. #20265066C2semidirect
10. #20265055C2semidirect
11. #20265055C2semidirect
12. #20265053C2semidirect
13. #20265032C2semidirect
14. #20265032C2semidirect
15. #388×C101C2semidirect
20. C22×C4×D101C22semidirect
21. #15692225C4semidirect
22. C2×C4⋊Dic101C22semidirect
23. C2×#13108053C22semidirect
36. C2×#13108051C22semidirect
37. C2×#13108051C22semidirect
38. C2×#13108051C22semidirect
39. C2×C4×Dic101C22semidirect
40. C2×C4×Dic101C22semidirect
41. #15692162C22semidirect
42. #15692162C22semidirect
43. #15692162C22semidirect
44. #15692162C22semidirect
45. C2×C4⋊C4×C101C22semidirect
49. C2×C42×C101C22semidirect
50. #95×C101C22semidirect
51. #95×C101C22semidirect
58. C23×D101C2×C4semidirect
94. #13108051C2×C4semidirect
95. #13108051C2×C4semidirect
108. C22⋊C4×C101C2×C4semidirect
242. C101#2375semidirect
252. #388D101semidirect
266. C2×C4⋊C4C2×D101semidirect
287. C2×C42C2×D101semidirect
292. #95C2×D101semidirect
293. #95C2×D101semidirect
348. 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
#202650792
#202650782
#202650772
#202650702
#202650692
#202650682
#202650672
#202650662
#202650552
#202650532
#202650322
#388×C1012
#2375101