Finite Group Databasebeta

← #27180348

Subgroups — #27180348

Order 12928.

764 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108074C211yesmaximal, characteristic, complemented
2C2×#15692275C212yesmaximal, complemented
3#20265337C212yesmaximal, complemented
4#20265332C211yesmaximal, characteristic, complemented
5#20265327C211yesmaximal, characteristic, complemented
6#20265320C212yesmaximal, complemented
7#20265315C211yesmaximal, characteristic, complemented
8#20265310C211yesmaximal, characteristic, complemented
9#20265171C212yesmaximal, complemented
10C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
11#525×C101C211yesmaximal, characteristic, complemented
12C2×#13108074—28no—
13C2×#13108074—21no—
14C2×#13108074—21no—
15C2×#13108074C2212yes—
16C2×#13108074C2212yes—
17C22×Dic202C2211yescharacteristic
18C22×D404C2211yescharacteristic
19C22×C22⋊D101—21no—
20C22×C4×D101—21no—
21#15692275—28no—
22C2×#13108064C2212yescomplemented
23C2×#13108062C2212yescomplemented
24C2×#13108061C2212yescomplemented
25C2×#13108059C2212yescomplemented
26C22×#11717579C2212yescomplemented
27#15692178C2212yescomplemented
28#15692178C2212yescomplemented
29#15692175C2212yescomplemented
30#15692175C2212yescomplemented
31C22⋊#11717579C2211yescharacteristic
32C22⋊#11717579C2211yescharacteristic
33C22⋊#11717579C2212yescomplemented
34C2×#41×C101C2211yescharacteristic, complemented
35C23×C4×C101C2211yescharacteristic
36#123×C101—21no—
37C22⋊Q8×C101C2211yescharacteristic
38C22⋊Q8×C101C2211yescharacteristic, complemented
39#121×C101C2211yescharacteristic
40#121×C101C2211yescharacteristic, complemented
41#120×C101—21no—
42C4×D4×C101—21no—
43C4×D4×C101—21no—
44#117×C101C2211yescharacteristic, complemented
45C23×D101—21no—
46#13108074—28no—
47#13108074—28no—
48#13108074—24no—
49#13108074—24no—
50#13108074—24no—
51#13108074D418yescomplemented
52C2×Dic202—21no—
53C2×Dic202—24no—
54C2×Dic202C2312yes—
55C2×D404—21no—
56C2×D404—24no—
57C2×D404C2312yes—
58C2×C22⋊D101—28no—
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61C22×Dic101—21no—
62C2×C4×D101—28no—
63C2×C4×D101—22no—
64C2×C4×D101—22no—
65#13108064—24no—
66#13108062—24no—
67#13108061—24no—
68#13108059—24no—
69C2×#11717579—24no—
70C2×#11717579C2312yes—
71C2×#11717579C2312yes—
72C24×C101D411yescharacteristic
73#41×C101—24no—
74C2×Q8×C101C2311yescharacteristic
75C2×D4×C101—21no—
76C2×D4×C101—21no—
77C2×D4×C101—21no—
78C2×D4×C101C2311yescharacteristic
79C22×C4×C101—24no—
80C22×C4×C101—21no—
81C22×C4×C101D411yescharacteristic
82C22×C4×C101D411yescharacteristic
83C22×C4×C101D411yescharacteristic
84C22×C4×C101C2311yescharacteristic
85C22×C4×C101C2311yescharacteristic
86C22×C4×C101C2311yescharacteristic
87C4⋊C4×C101—21no—
88C4⋊C4×C101—21no—
89C4⋊C4×C101C2311yescharacteristic
90C4⋊C4×C101C2311yescharacteristic
91C22⋊C4×C101—21no—
92C22⋊C4×C101—21no—
93C22⋊C4×C101—21no—
94C22⋊C4×C101—21no—
95C22⋊C4×C101—21no—
96C42×C101—21no—
97C23×C101—22no—
98C23×C101—22no—
99C23×C101—21no—
100C23×C101—41no—
101C23×C101C2×D411yescharacteristic
102C23×C101C2×D411yescharacteristic
103C23×C101C2×D411yescharacteristic
104C22×D101—24no—
105C22×D101—22no—
106C22×D101—42no—
107C22×D101—42no—
108Q8×C101—22no—
109D4×C101—22no—
110D4×C101—42no—
111D4×C101—42no—
112D4×C101—42no—
113C2×C4×C101—22no—
114C2×C4×C101—22no—
115C2×C4×C101—22no—
116C2×C4×C101—22no—
117C2×C4×C101—22no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—42no—
122C2×C4×C101C2×D414yes—
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C22⋊D101—28no—
128C22⋊D101—28no—
129C2×Dic101—24no—
130C2×Dic101—22no—
131D404—24no—
132D404—22no—
133D404C2×D414yes—
134C4×D101—28no—
135C4×D101—28no—
136Dic202—24no—
137Dic202—22no—
138Dic202C2×D414yes—
139#11717579—22no—
140#11717579—22no—
141C22×C101—21no—
142C22×C101—21no—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101—21no—
146C22×C101—21no—
147C22×C101—21no—
148C22×C101—44no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101—41no—
152C22×C101#12012yes—
153C22×C101#13512yescomplemented
154C22×C101C22×D411yescharacteristic
155C2×D101—24no—
156C2×D101—48no—
157C2×D101—42no—
158C2×D101—42no—
159C4×C101—21no—
160C4×C101—21no—
161C4×C101—41no—
162C4×C101—41no—
163C4×C101—41no—
164C4×C101#12012yes—
165C4×C101C22×D412yes—
166Dic101—24no—
167C2×C101—22no—
168C2×C101—22no—
169C2×C101—41no—
170C2×C101—81no—
171C2×C101C2×#12011yescharacteristic
172C2×C101C2×#13511yescharacteristic
173C2×C101#57511yescharacteristic
174D101—44no—
175#3002—1011nomaximal, Sylow
176C101#300211yescharacteristic, complemented, Sylow
177C22×#41—1011no—
178C2×#135—1012no—
179#525D10111yescharacteristic, complemented
180#452—1012no—
181#451—1011no—
182#450—1011no—
183#449—1012no—
184#448—1011no—
185#447—1011no—
186#416—1012no—
187C2×C22⋊C8—1011no—
188C2×#41—1012no—
189C2×#41—1012no—
190C2×#41—2028no—
191C2×#41—2021no—
192C2×#41—2021no—
193C2×#41C2×D10111yescharacteristic, complemented
194C22×Q8—1011no—
195C22×D4—1011no—
196C22×D4—2021no—
197C23×C4—2021no—
198C23×C4C2×D10111yescharacteristic
199#135—2028no—
200C2×Q16—1012no—
201C2×SD16—1012no—
202C2×SD16—1012no—
203C2×D8—1012no—
204C22×C8—1012no—
205#123—21no—
206C22⋊Q8C2×D10111yescharacteristic
207C22⋊Q8C2×D10111yescharacteristic, complemented
208#121C2×D10111yescharacteristic
209#121C2×D10111yescharacteristic, complemented
210#120—21no—
211C4×D4—21no—
212C4×D4—21no—
213#117C2×D10111yescharacteristic, complemented
214#103—1012no—
215#103—1012no—
216#102—1012no—
217#102—1012no—
218C22⋊C8—1011no—
219C22⋊C8—1011no—
220C22⋊C8—1012no—
221C24—2021no—
222C24C22⋊D10111yescharacteristic
223#41—24no—
224#41—1018no—
225#41—2024no—
226#41—2024no—
227#41—2028no—
228#41—2028no—
229#41—2024no—
230C2×Q8—1012no—
231C2×Q8—2024no—
232C2×Q8—2021no—
233C2×Q8C22×D10111yescharacteristic
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—1012no—
238C2×D4—2024no—
239C2×D4—2028no—
240C2×D4—2022no—
241C2×D4—2022no—
242C2×D4—2021no—
243C2×D4C22×D10111yescharacteristic
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—2028no—
247C22×C4—2021no—
248C22×C4—2022no—
249C22×C4—2022no—
250C22×C4C22⋊D10111yescharacteristic
251C22×C4C22⋊D10111yescharacteristic
252C22×C4C22⋊D10111yescharacteristic
253C22×C4C22×D10111yescharacteristic
254C22×C4C22×D10111yescharacteristic
255C22×C4C22×D10111yescharacteristic
256Q16—2024no—
257SD16—2024no—
258SD16—2024no—
259D8—2024no—
260C2×C8—1012no—
261C2×C8—1012no—
262C2×C8—2024no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4C22×D10111yescharacteristic
266C4⋊C4C22×D10111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C42—21no—
273C23—22no—
274C23—22no—
275C23—21no—
276C23—41no—
277C23—4042no—
278C23—4042no—
279C23—2024no—
280C23—2022no—
281C23C2×C22⋊D10111yescharacteristic
282C23C2×C22⋊D10111yescharacteristic
283C23C2×C22⋊D10111yescharacteristic
284Q8—22no—
285Q8—1014no—
286Q8—2024no—
287Q8—2022no—
288D4—22no—
289D4—42no—
290D4—42no—
291D4—42no—
292D4—1014no—
293D4—2024no—
294D4—2028no—
295D4—2028no—
296D4—2022no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—2028no—
307C2×C4—2024no—
308C2×C4—2022no—
309C2×C4—2028no—
310C2×C4D4×D10114yes—
311C2×C4C2×C22⋊D10111yescharacteristic
312C2×C4C2×C22⋊D10111yescharacteristic
313C2×C4C2×C22⋊D10111yescharacteristic
314C2×C4C23×D10111yescharacteristic
315C8—2022no—
316C8—2022no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—44no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—4048no—
329C22—4042no—
330C22—4042no—
331C22—2024no—
332C22#1569222312yes—
333C22#1569227512yescomplemented
334C22C22×C22⋊D10111yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—2024no—
341C4#1569222312yes—
342C4C2×D4×D10112yes—
343C2—22no—
344C2—22no—
345C2—41no—
346C2—81no—
347C2—4044no—
348C2C2×#1569222311yescharacteristic
349C2C2×#1569227511yescharacteristic
350C2#2026597511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#525×C101C2
socleC22×C101C22×D4
hypercenter#525D101
hypocenterC101#3002
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#13108074C2semidirect
2. C2×#15692275C2semidirect
3. #20265337C2semidirect
4. #20265332C2semidirect
5. #20265327C2semidirect
6. #20265320C2semidirect
7. #20265315C2semidirect
8. #20265310C2semidirect
9. #20265171C2semidirect
10. C2×C22⋊#11717579C2semidirect
11. #525×C101C2semidirect
22. C2×#13108064C22semidirect
23. C2×#13108062C22semidirect
24. C2×#13108061C22semidirect
25. C2×#13108059C22semidirect
26. C22×#11717579C22semidirect
27. #15692178C22semidirect
28. #15692178C22semidirect
29. #15692175C22semidirect
30. #15692175C22semidirect
33. C22⋊#11717579C22semidirect
34. C2×#41×C101C22semidirect
38. C22⋊Q8×C101C22semidirect
40. #121×C101C22semidirect
44. #117×C101C22semidirect
51. #13108074D4semidirect
153. C22×C101#135semidirect
176. C101#3002semidirect
179. #525D101semidirect
193. C2×#41C2×D101semidirect
207. C22⋊Q8C2×D101semidirect
209. #121C2×D101semidirect
213. #117C2×D101semidirect
333. C22#15692275semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922752
#202653372
#202653322
#202653272
#202653202
#202653152
#202653102
#202651712
C2×C22⋊#117175792
#525×C1012
#3002101