Finite Group Databasebeta

← #27182938

Subgroups — #27182938

Order 12928.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265827C211yesmaximal, characteristic, complemented
2D4×C22⋊D101C211yesmaximal, characteristic, complemented
3#20265522C211yesmaximal, characteristic, complemented
4#20265491C211yesmaximal, characteristic, complemented
5#20265489C211yesmaximal, characteristic, complemented
6#20265455C211yesmaximal, characteristic, complemented
7#20265411C211yesmaximal, characteristic, complemented
8#20265410C211yesmaximal, characteristic, complemented
9#20265370C211yesmaximal, characteristic, complemented
10#20265309C211yesmaximal, characteristic, complemented
11D8×Dic101C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic, complemented
13C2×#15692176C211yesmaximal, characteristic, complemented
14#102×D101C211yesmaximal, characteristic, complemented
15#447×C101C211yesmaximal, characteristic, complemented
16C2×#13108073—21no—
17C22×C22⋊D101—21no—
18C2×D4×D101C2211yescharacteristic
19C2×#13108059C2211yescharacteristic, complemented
20C22×#11717579C2211yescharacteristic, complemented
21C2×C8×D101C2211yescharacteristic, complemented
22#15692254C2211yescharacteristic, complemented
23#15692251C2211yescharacteristic
24#15692237—21no—
25#15692236—21no—
26#15692232—21no—
27#15692229C2211yescharacteristic
28#15692227—21no—
29#15692226C2211yescharacteristic
30#15692226C2211yescharacteristic, complemented
31#15692224—21no—
32#15692223—21no—
33D4×Dic101C2211yescharacteristic
34D4×Dic101C2211yescharacteristic, complemented
35C4×C22⋊D101C2211yescharacteristic
36C2×C4⋊Dic101C2211yescharacteristic, complemented
37C4⋊C4×D101C2211yescharacteristic, complemented
38#15692189C2211yescharacteristic, complemented
39#15692189C2211yescharacteristic, complemented
40#15692188C2211yescharacteristic, complemented
41#15692184C2211yescharacteristic, complemented
42#15692177C2211yescharacteristic, complemented
43#15692177C2211yescharacteristic, complemented
44#15692176C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692176C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic, complemented
48#15692176C2211yescharacteristic, complemented
49#15692175C2211yescharacteristic, complemented
50#15692167C2211yescharacteristic, complemented
51C8×Dic101C2211yescharacteristic, complemented
52C22×D4×C101C2211yescharacteristic
53C2×D8×C101C2211yescharacteristic
54C2×D8×C101C2211yescharacteristic, complemented
55#121×C101C2211yescharacteristic, complemented
56#102×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic, complemented
59C23×D101—21no—
60#13108073—44no—
61C2×Dic202—21no—
62C2×D404C2311yescharacteristic
63C2×C22⋊D101—21no—
64C2×C22⋊D101—21no—
65C2×C22⋊D101—21no—
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—44no—
70C2×C22⋊D101D411yescharacteristic
71D4×D101—22no—
72D4×D101—41no—
73C22×Dic101—21no—
74C22×Dic101—21no—
75C22×Dic101—21no—
76C2×C4×D101—21no—
77C2×C4×D101C2311yescharacteristic
78#13108059—41no—
79C2×#11717579—41no—
80C2×#11717579C2311yescharacteristic
81C2×#11717579C2311yescharacteristic
82C8×D101—41no—
83C4⋊Dic101—21no—
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86C4⋊Dic101C2311yescharacteristic
87#13108053—21no—
88#13108053—21no—
89#13108053D411yescharacteristic
90C22⋊Dic101—21no—
91C22⋊Dic101—21no—
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101D411yescharacteristic
95#13108051—21no—
96#13108051D411yescharacteristic
97C4×Dic101C2311yescharacteristic
98C24×C101—21no—
99C2×D4×C101—22no—
100C2×D4×C101—21no—
101C2×D4×C101—21no—
102C2×D4×C101—41no—
103C2×D4×C101C2311yescharacteristic
104C2×D4×C101C2311yescharacteristic
105C2×D4×C101C2311yescharacteristic
106C22×C4×C101C2311yescharacteristic
107D8×C101—22no—
108D8×C101—41no—
109C2×C8×C101C2311yescharacteristic
110C2×C8×C101C2311yescharacteristic
111C4⋊C4×C101C2311yescharacteristic
112C22⋊C4×C101—21no—
113C23×C101—21no—
114C23×C101—21no—
115C23×C101—21no—
116C23×C101—42no—
117C23×C101—42no—
118C23×C101—81no—
119C23×C101C2×D411yescharacteristic
120C22×D101—21no—
121C22×D101—42no—
122C22×D101—42no—
123C22×D101—81no—
124C22×D101C2×D411yescharacteristic
125D4×C101—21no—
126D4×C101—21no—
127D4×C101—21no—
128D4×C101—21no—
129D4×C101—21no—
130D4×C101—42no—
131D4×C101—42no—
132D4×C101#4112yescomplemented
133C2×C4×C101—21no—
134C2×C4×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101C2×D411yescharacteristic
137C2×C4×C101C2411yescharacteristic
138C22⋊D101—42no—
139C22⋊D101—42no—
140C22⋊D101—42no—
141C22⋊D101—42no—
142C22⋊D101—42no—
143C22⋊D101—81no—
144C22⋊D101D814yescomplemented
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—42no—
149C2×Dic101—42no—
150C2×Dic101—42no—
151C2×Dic101C2×D411yescharacteristic
152C2×Dic101C2×D411yescharacteristic
153C2×Dic101C2×D411yescharacteristic
154D404—21no—
155D404—21no—
156C4×D101—21no—
157C4×D101—21no—
158C4×D101—42no—
159Dic202—42no—
160C8×C101—21no—
161C8×C101—21no—
162#11717579—21no—
163#11717579—21no—
164C22×C101—21no—
165C22×C101—21no—
166C22×C101—22no—
167C22×C101—41no—
168C22×C101—41no—
169C22×C101—41no—
170C22×C101—41no—
171C22×C101—41no—
172C22×C101—41no—
173C22×C101—41no—
174C22×C101—41no—
175C22×C101—84no—
176C22×C101C2×D812yes—
177C22×C101C22×D411yescharacteristic
178C2×D101—21no—
179C2×D101—21no—
180C2×D101—41no—
181C2×D101—41no—
182C2×D101—41no—
183C2×D101—84no—
184C2×D101C2×D812yes—
185C4×C101—21no—
186C4×C101—41no—
187C4×C101C2×#4111yescharacteristic
188C4×C101H(5,𝔽2)11yescharacteristic
189Dic101—21no—
190Dic101—21no—
191Dic101—41no—
192Dic101—41no—
193Dic101—41no—
194Dic101C2×D812yes—
195C2×C101—22no—
196C2×C101—42no—
197C2×C101—81no—
198C2×C101—81no—
199C2×C101#54611yescharacteristic
200C2×C101C22×D811yescharacteristic
201C2×C101#57711yescharacteristic
202D101—22no—
203D101—81no—
204#3294—1011nomaximal, Sylow
205C101#329411yescharacteristic, complemented, Sylow
206#547—1011no—
207D42—1011no—
208#486—1011no—
209#480—1011no—
210#480—1011no—
211#474—1011no—
212#466—1011no—
213#466—1011no—
214#459—1011no—
215#447—1011no—
216#447D10111yescharacteristic, complemented
217C4×D8—1011no—
218C8×D4—1011no—
219C2×#102—1011no—
220C2×#102—1011no—
221C2×#41—2021no—
222C22×D4—1011no—
223C22×D4—2021no—
224C22×D4C2×D10111yescharacteristic
225C2×D8—1011no—
226C2×D8C2×D10111yescharacteristic
227C2×D8C2×D10111yescharacteristic, complemented
228C22×C8—1011no—
229C22×C8—1011no—
230C4⋊Q8—1011no—
231#127—1011no—
232#123—2021no—
233#123—2021no—
234C22⋊Q8—2021no—
235#121—1011no—
236#121—1011no—
237#121—1011no—
238#121—2021no—
239#121C2×D10111yescharacteristic, complemented
240#120—2021no—
241#120—2021no—
242C4×D4—1011no—
243C4×D4—1011no—
244C4×D4—1011no—
245C2×C4⋊C4—1011no—
246C2×C4⋊C4—1011no—
247#107—1011no—
248#107—1011no—
249#107—1011no—
250C4⋊C8—1011no—
251#102—1011no—
252#102—1011no—
253#102—1011no—
254#102—1011no—
255#102—1011no—
256#102—1011no—
257#102—1011no—
258#102—1011no—
259#102C2×D10111yescharacteristic
260#102C2×D10111yescharacteristic, complemented
261C22⋊C8—1011no—
262C22⋊C8C2×D10111yescharacteristic, complemented
263C4×C8—1011no—
264C24—21no—
265C24—2021no—
266#41—4044no—
267C2×Q8—2021no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—1011no—
273C2×D4—1011no—
274C2×D4—4044no—
275C2×D4—4041no—
276C2×D4—2021no—
277C2×D4—2021no—
278C2×D4—2021no—
279C2×D4—2021no—
280C2×D4—2021no—
281C2×D4—2021no—
282C2×D4—2022no—
283C2×D4C22×D10111yescharacteristic
284C2×D4C22×D10111yescharacteristic
285C2×D4C22×D10111yescharacteristic
286C22×C4—1011no—
287C22×C4—2021no—
288C22×C4—2021no—
289C22×C4—2021no—
290C22×C4—2021no—
291C22×C4C22×D10111yescharacteristic
292D8—22no—
293D8—41no—
294D8—4041no—
295C2×C8—1011no—
296C2×C8—1011no—
297C2×C8—4041no—
298C2×C8—4041no—
299C2×C8C22×D10111yescharacteristic
300C2×C8C22×D10111yescharacteristic
301C4⋊C4—1011no—
302C4⋊C4—1011no—
303C4⋊C4—1011no—
304C4⋊C4—1011no—
305C4⋊C4—2021no—
306C4⋊C4—2021no—
307C4⋊C4—2021no—
308C4⋊C4C22×D10111yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—1011no—
311C22⋊C4—1011no—
312C22⋊C4—2021no—
313C22⋊C4—2021no—
314C22⋊C4—2021no—
315C22⋊C4—2021no—
316C22⋊C4—2021no—
317C42—1011no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—8081no—
325C23—1011no—
326C23—4042no—
327C23—4042no—
328C23—2021no—
329C23D4×D10111yescharacteristic
330Q8—4042no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—8081no—
339D4—1014no—
340D4—4042no—
341D4—4042no—
342D4—4042no—
343D4—4042no—
344D4—4042no—
345D4—2021no—
346D4—2021no—
347D4#1310807312yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—1011no—
352C2×C4—1011no—
353C2×C4—1011no—
354C2×C4—4042no—
355C2×C4—4042no—
356C2×C4—4042no—
357C2×C4—4042no—
358C2×C4—2021no—
359C2×C4—2021no—
360C2×C4—2021no—
361C2×C4—2021no—
362C2×C4—2021no—
363C2×C4D4×D10111yescharacteristic
364C2×C4C23×D10111yescharacteristic
365C8—21no—
366C8—21no—
367C8—2021no—
368C8—2021no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—8084no—
382C22—1012no—
383C22—4041no—
384C22—4041no—
385C22—4041no—
386C22—2021no—
387C22—2021no—
388C22D8×D10112yes—
389C22C2×D4×D10111yescharacteristic
390C4—21no—
391C4—41no—
392C4—1012no—
393C4—4041no—
394C4—4041no—
395C4—4041no—
396C4—2021no—
397C4—2021no—
398C4C2×#1310807311yescharacteristic
399C4#1569230011yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—8081no—
405C2—2022no—
406C2#2026581411yescharacteristic
407C2C2×D8×D10111yescharacteristic
408C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#447×C101C2
socleC22×C101C22×D4
hypercenter#447D101
hypocenterC101#3294
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20265827C2semidirect
2. D4×C22⋊D101C2semidirect
3. #20265522C2semidirect
4. #20265491C2semidirect
5. #20265489C2semidirect
6. #20265455C2semidirect
7. #20265411C2semidirect
8. #20265410C2semidirect
9. #20265370C2semidirect
10. #20265309C2semidirect
11. D8×Dic101C2semidirect
12. #20265240C2semidirect
13. C2×#15692176C2semidirect
14. #102×D101C2semidirect
15. #447×C101C2semidirect
19. C2×#13108059C22semidirect
20. C22×#11717579C22semidirect
21. C2×C8×D101C22semidirect
22. #15692254C22semidirect
30. #15692226C22semidirect
34. D4×Dic101C22semidirect
36. C2×C4⋊Dic101C22semidirect
37. C4⋊C4×D101C22semidirect
38. #15692189C22semidirect
39. #15692189C22semidirect
40. #15692188C22semidirect
41. #15692184C22semidirect
42. #15692177C22semidirect
43. #15692177C22semidirect
46. #15692176C22semidirect
47. #15692176C22semidirect
48. #15692176C22semidirect
49. #15692175C22semidirect
50. #15692167C22semidirect
51. C8×Dic101C22semidirect
54. C2×D8×C101C22semidirect
55. #121×C101C22semidirect
57. #102×C101C22semidirect
58. C22⋊C8×C101C22semidirect
132. D4×C101#41semidirect
144. C22⋊D101D8semidirect
205. C101#3294semidirect
216. #447D101semidirect
227. C2×D8C2×D101semidirect
239. #121C2×D101semidirect
260. #102C2×D101semidirect
262. C22⋊C8C2×D101semidirect
347. D4#13108073semidirect

Maximal subgroups

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

subgroupindex
#202658272
D4×C22⋊D1012
#202655222
#202654912
#202654892
#202654552
#202654112
#202654102
#202653702
#202653092
D8×Dic1012
#202652402
C2×#156921762
#102×D1012
#447×C1012
#3294101