Finite Group Databasebeta

← #27173975

Subgroups — #27173975

Order 12928.

792 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692212C211yesmaximal, characteristic, complemented
2C22×#13108051C211yesmaximal, characteristic
3C2×C22⋊C4×D101C211yesmaximal, characteristic
4#20265068C211yesmaximal, characteristic, complemented
5#20265068C211yesmaximal, characteristic, complemented
6#20265066C211yesmaximal, characteristic, complemented
7#20265066C211yesmaximal, characteristic, complemented
8#20265050C211yesmaximal, characteristic, complemented
9#20265049C211yesmaximal, characteristic, complemented
10#20265039C211yesmaximal, characteristic, complemented
11#20265036C211yesmaximal, characteristic, complemented
12#20265032C211yesmaximal, characteristic
13C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
14C4×#13108051C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C101C211yesmaximal, characteristic, complemented
16C24×D101C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20#15692212—24no—
21C2×#13108053C2211yescharacteristic, complemented
22C2×#13108053C2211yescharacteristic, complemented
23C2×#13108053C2211yescharacteristic, complemented
24C2×#13108053C2211yescharacteristic, complemented
25C2×C22⋊Dic101C2211yescharacteristic
26C2×C22⋊Dic101C2211yescharacteristic
27C2×#13108051—24no—
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×#13108051C2211yescharacteristic
38C22⋊C4×D101C418yescomplemented
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
47C23×C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C42×C101C2211yescharacteristic, complemented
51C2×C42×C101C2211yescharacteristic, complemented
52#95×C101C2211yescharacteristic, complemented
53#95×C101C2211yescharacteristic, complemented
54C23×D101—22no—
55C23×D101—22no—
56C23×D101—22no—
57C23×D101—22no—
58C23×D101—41no—
59C23×D101—41no—
60C23×D101C2×C414yes—
61C23×D101C2311yescharacteristic
62C23×D101C2311yescharacteristic
63C22×Dic101—22no—
64C22×Dic101—22no—
65C22×Dic101—41no—
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C22×Dic101C2311yescharacteristic
72C2×C4×D101—22no—
73C2×C4×D101—22no—
74C2×C4×D101—41no—
75C2×C4×D101—41no—
76C2×C4×D101C2×C414yescomplemented
77C2×C4×D101C2×C414yescomplemented
78#13108053—22no—
79#13108053—22no—
80#13108053—41no—
81#13108053—41no—
82C22⋊Dic101—22no—
83C22⋊Dic101C2×C414yes—
84#13108051—22no—
85#13108051—22no—
86#13108051—22no—
87#13108051—22no—
88#13108051—22no—
89#13108051—41no—
90#13108051—41no—
91#13108051C2×C414yescomplemented
92#13108051C2×C414yescomplemented
93#13108051D414yes—
94C4×Dic101—22no—
95C4×Dic101—41no—
96C24×C101C2311yescharacteristic
97C22×C4×C101—22no—
98C22×C4×C101—22no—
99C22×C4×C101—41no—
100C22×C4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C22×C4×C101C2311yescharacteristic
105C22×C4×C101C2311yescharacteristic
106C22⋊C4×C101—22no—
107C22⋊C4×C101C2×C414yes—
108C42×C101—22no—
109C42×C101—41no—
110C23×C101—21no—
111C23×C101—21no—
112C23×C101—21no—
113C23×C101—21no—
114C23×C101C22×C412yes—
115C23×C101#4112yes—
116C23×C101#4112yes—
117C23×C101C2411yescharacteristic
118C22×D101—21no—
119C22×D101—21no—
120C22×D101—21no—
121C22×D101—21no—
122C22×D101—28no—
123C22×D101—21no—
124C22×D101—21no—
125C22×D101—21no—
126C22×D101—21no—
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—28no—
130C22×D101—44no—
131C22×D101—44no—
132C22×D101—44no—
133C22×D101—44no—
134C22×D101—44no—
135C22×D101C22×C412yes—
136C22×D101C22×C412yes—
137C22×D101C2×D412yes—
138C22×D101#4112yes—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—21no—
145C2×C4×C101—21no—
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—44no—
155C2×C4×C101C22×C412yes—
156C2×C4×C101C22×C412yes—
157C2×C4×C101C2×D412yes—
158C2×Dic101—21no—
159C2×Dic101—21no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162C2×Dic101—21no—
163C2×Dic101—21no—
164C2×Dic101—21no—
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—44no—
174C2×Dic101C22×C412yes—
175C2×Dic101C22×C412yes—
176C2×Dic101C2×D412yes—
177C4×D101—28no—
178C4×D101—44no—
179C22×C101—22no—
180C22×C101—22no—
181C22×C101—22no—
182C22×C101—22no—
183C22×C101—22no—
184C22×C101—22no—
185C22×C101#11714yescomplemented
186C22×C101C23×C411yescharacteristic
187C22×C101C22×D411yescharacteristic
188C22×C101C2×#4111yescharacteristic
189C22×C101C2×#4111yescharacteristic
190C22×C101C2×#4111yescharacteristic
191C22×C101H(5,𝔽2)11yescharacteristic
192C22×C101H(5,𝔽2)11yescharacteristic
193C2×D101—22no—
194C2×D101—22no—
195C2×D101—22no—
196C2×D101—22no—
197C2×D101—22no—
198C2×D101—22no—
199C2×D101—22no—
200C2×D101—22no—
201C2×D101—22no—
202C2×D101—416no—
203C2×D101—41no—
204C2×D101—41no—
205C2×D101—41no—
206C2×D101—41no—
207C2×D101C4×D414yes—
208C4×C101—22no—
209C4×C101—22no—
210C4×C101—22no—
211C4×C101—41no—
212C4×C101—41no—
213C4×C101—41no—
214Dic101—22no—
215Dic101—22no—
216Dic101—22no—
217Dic101—41no—
218Dic101—41no—
219Dic101—41no—
220C2×C101—24no—
221C2×C101C2×#11711yescharacteristic
222C2×C101C2×C4×D411yescharacteristic
223C2×C101#51811yescharacteristic
224C2×C101#54611yescharacteristic
225C2×C101#54611yescharacteristic
226C2×C101#55111yescharacteristic
227C2×C101#55111yescharacteristic
228D101—24no—
229D101—42no—
230#2342—1011nomaximal, Sylow
231C101#234211yescharacteristic, complemented, Sylow
232C2×#117—1011no—
233C22×C22⋊C4—1011no—
234C22×C22⋊C4—1011no—
235#388—1011no—
236#388—1011no—
237#388—1011no—
238#388—1011no—
239#385—1011no—
240#385—1011no—
241#381—1011no—
242#380—1011no—
243#379—1011no—
244C4×C22⋊C4—1011no—
245C4×C22⋊C4—1011no—
246C4×C22⋊C4D10111yescharacteristic, complemented
247C25—1011no—
248C23×C4—1011no—
249C23×C4—1011no—
250C23×C4—1011no—
251C23×C4C2×D10111yescharacteristic
252#117—2024no—
253C2×C4⋊C4—1011no—
254C2×C4⋊C4—1011no—
255C2×C4⋊C4—1011no—
256C2×C4⋊C4—1011no—
257C2×C22⋊C4—1011no—
258C2×C22⋊C4—1011no—
259C2×C22⋊C4—1011no—
260C2×C22⋊C4—1011no—
261C2×C22⋊C4—1018no—
262C2×C22⋊C4—1011no—
263C2×C22⋊C4—1011no—
264C2×C22⋊C4—1011no—
265C2×C22⋊C4—1011no—
266C2×C22⋊C4—1011no—
267C2×C22⋊C4—1011no—
268C2×C22⋊C4—1011no—
269C2×C22⋊C4—1011no—
270C2×C22⋊C4—2024no—
271C2×C22⋊C4C2×D10111yescharacteristic
272C2×C22⋊C4C2×D10111yescharacteristic
273C2×C42—1011no—
274C2×C42—1011no—
275C2×C42C2×D10111yescharacteristic, complemented
276C2×C42C2×D10111yescharacteristic, complemented
277#95—1011no—
278#95—1011no—
279#95—1011no—
280#95—1011no—
281#95—1011no—
282#95—1011no—
283#95C2×D10111yescharacteristic, complemented
284#95C2×D10111yescharacteristic, complemented
285C24—1011no—
286C24—1014no—
287C24—1011no—
288C24—4041no—
289C24—4041no—
290C24—2022no—
291C24—2022no—
292C24—2022no—
293C24—2022no—
294C24C22×D10111yescharacteristic
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—1011no—
299C22×C4—1011no—
300C22×C4—1011no—
301C22×C4—1014no—
302C22×C4—1014no—
303C22×C4—1011no—
304C22×C4—1011no—
305C22×C4—1011no—
306C22×C4—4041no—
307C22×C4—4041no—
308C22×C4—4041no—
309C22×C4—2022no—
310C22×C4—2022no—
311C22×C4—2022no—
312C22×C4—2022no—
313C22×C4C22×D10111yescharacteristic
314C22×C4C22×D10111yescharacteristic
315C22×C4C22×D10111yescharacteristic
316C22×C4C22×D10111yescharacteristic
317C22×C4C22×D10111yescharacteristic
318C22×C4C22×D10111yescharacteristic
319C4⋊C4—4041no—
320C4⋊C4—4041no—
321C4⋊C4—2022no—
322C4⋊C4—2022no—
323C22⋊C4—22no—
324C22⋊C4—1014no—
325C22⋊C4—1014no—
326C22⋊C4—1014no—
327C22⋊C4—1014no—
328C22⋊C4—4041no—
329C22⋊C4—4041no—
330C22⋊C4—2022no—
331C22⋊C4—2022no—
332C22⋊C4—2022no—
333C22⋊C4—2022no—
334C22⋊C4—2022no—
335C22⋊C4—2022no—
336C22⋊C4C4×D10114yes—
337C42—22no—
338C42—41no—
339C42—4041no—
340C42—2022no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—1012no—
346C23—1012no—
347C23—1012no—
348C23—1012no—
349C23—4044no—
350C23—4044no—
351C23—4044no—
352C23—4044no—
353C23—4044no—
354C23—2028no—
355C23—2021no—
356C23—2021no—
357C23—2021no—
358C23—2021no—
359C23—2021no—
360C23—2021no—
361C23—2028no—
362C23—2021no—
363C23—2021no—
364C23—2021no—
365C23—2021no—
366C23C2×C4×D10112yes—
367C23#1310807412yes—
368C23#1310807412yes—
369C23C23×D10111yescharacteristic
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—44no—
386C2×C4—1012no—
387C2×C4—1012no—
388C2×C4—1012no—
389C2×C4—4044no—
390C2×C4—4044no—
391C2×C4—2028no—
392C2×C4—2021no—
393C2×C4—2021no—
394C2×C4—2021no—
395C2×C4—2021no—
396C2×C4—2021no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4—2021no—
401C2×C4—2021no—
402C2×C4—2021no—
403C2×C4—2021no—
404C2×C4—2021no—
405C2×C4—2021no—
406C2×C4—2021no—
407C2×C4C2×C4×D10112yes—
408C2×C4C2×C4×D10112yes—
409C2×C4D4×D10112yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—1014no—
417C22—40416no—
418C22—4041no—
419C22—4041no—
420C22—4041no—
421C22—4041no—
422C22—2022no—
423C22—2022no—
424C22—2022no—
425C22—2022no—
426C22—2022no—
427C22—2022no—
428C22—2022no—
429C22—2022no—
430C22—2022no—
431C22#1569221214yescomplemented
432C22C22×C4×D10111yescharacteristic
433C22C2×D4×D10111yescharacteristic
434C22#41×D10111yescharacteristic
435C22C2×#1310807411yescharacteristic
436C22C2×#1310807411yescharacteristic
437C22#1569230011yescharacteristic
438C22#1569230111yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—4041no—
446C4—4041no—
447C4—4041no—
448C4—2022no—
449C4—2022no—
450C4—2022no—
451C2—24no—
452C2—4042no—
453C2—2024no—
454C2C2×#1569221211yescharacteristic
455C2C4×D4×D10111yescharacteristic
456C2#2026564911yescharacteristic
457C2#2026581711yescharacteristic
458C2#2026582311yescharacteristic
459C2#2026585111yescharacteristic
460C2#2026585611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
FittingC4×C22⋊C4×C101C2
socle, cosocleC23×C101C24
prime radicalC24×D101C22
hypercenterC4×C22⋊C4D101
hypocenterC101#2342
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15692212C2semidirect
4. #20265068C2semidirect
5. #20265068C2semidirect
6. #20265066C2semidirect
7. #20265066C2semidirect
8. #20265050C2semidirect
9. #20265049C2semidirect
10. #20265039C2semidirect
11. #20265036C2semidirect
13. C22⋊C4×Dic101C2semidirect
14. C4×#13108051C2semidirect
15. C4×C22⋊C4×C101C2semidirect
21. C2×#13108053C22semidirect
22. C2×#13108053C22semidirect
23. C2×#13108053C22semidirect
24. C2×#13108053C22semidirect
38. C22⋊C4×D101C4semidirect
39. C2×C4×Dic101C22semidirect
40. C2×C4×Dic101C22semidirect
41. #15692162C22semidirect
42. #15692162C22semidirect
43. #15692162C22semidirect
44. #15692162C22semidirect
45. #15692162C22semidirect
46. #15692162C22semidirect
50. C2×C42×C101C22semidirect
51. C2×C42×C101C22semidirect
52. #95×C101C22semidirect
53. #95×C101C22semidirect
76. C2×C4×D101C2×C4semidirect
77. C2×C4×D101C2×C4semidirect
91. #13108051C2×C4semidirect
92. #13108051C2×C4semidirect
185. C22×C101#117semidirect
231. C101#2342semidirect
246. C4×C22⋊C4D101semidirect
275. C2×C42C2×D101semidirect
276. C2×C42C2×D101semidirect
283. #95C2×D101semidirect
284. #95C2×D101semidirect
431. C22#15692212semidirect

Maximal subgroups

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

subgroupindex
C2×#156922122
C22×#131080512
C2×C22⋊C4×D1012
#202650682
#202650662
#202650502
#202650492
#202650392
#202650362
#202650322
C22⋊C4×Dic1012
C4×#131080512
C4×C22⋊C4×C1012
#2342101