Finite Group Databasebeta

← #27179580

Subgroups — #27179580

Order 12928.

774 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D101C211yesmaximal, characteristic
2C2×#15692263C211yesmaximal, characteristic, complemented
3C2×#15692261C211yesmaximal, characteristic, complemented
4#20265144C211yesmaximal, characteristic, complemented
5#20265142C212yesmaximal, complemented
6#20265142C212yesmaximal, complemented
7#20265140C211yesmaximal, characteristic, complemented
8#20265139C212yesmaximal, complemented
9C2×#15692167C211yesmaximal, characteristic, complemented
10C22⋊C8×D101C212yesmaximal, complemented
11#408×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C411yescharacteristic
13C2×#13108074C2212yes—
14C2×#13108074C2211yescharacteristic
15C2×#13108073C411yescharacteristic
16C2×#13108073C412yes—
17#41×D101—24no—
18#41×D101—24no—
19C2×Q8×D101C411yescharacteristic
20C2×D4×D101C412yes—
21C2×D4×D101C411yescharacteristic
22C22×C4×D101C2212yes—
23C22×C4×D101C2211yescharacteristic
24#15692263—24no—
25#15692261—24no—
26C2×#13108058C2211yescharacteristic, complemented
27C2×#13108057C2212yescomplemented
28C2×#13108057C2212yescomplemented
29C22×#11717579C2211yescharacteristic, complemented
30C2×C8×D101C2212yescomplemented
31C2×C8×D101C2212yescomplemented
32C22⋊#11717579C2212yescomplemented
33C22⋊#11717579C2212yescomplemented
34#15692167C2211yescharacteristic
35#15692167C2212yes—
36#15692167C2211yescharacteristic
37#15692167C2211yescharacteristic, complemented
38#15692167C2211yescharacteristic, complemented
39#15692167C2212yescomplemented
40C2×#41×C101C2211yescharacteristic
41C2×M16×C101C2211yescharacteristic, complemented
42C22×C8×C101C2211yescharacteristic, complemented
43C22⋊C8×C101C2212yescomplemented
44C22⋊C8×C101C2212yescomplemented
45C23×D101C2×C411yescharacteristic
46C23×D101C2×C412yes—
47#13108075—24no—
48#13108074—24no—
49#13108074—24no—
50#13108074D414yes—
51#13108074D414yes—
52#13108073—28no—
53#13108073—24no—
54C2×Dic202C2×C412yes—
55C2×Dic202C2×C411yescharacteristic
56Q8×D101—24no—
57C2×D404C2×C412yes—
58C2×D404C2×C411yescharacteristic
59C2×C22⋊D101C2×C412yes—
60C2×C22⋊D101C2×C412yes—
61C2×C22⋊D101C2×C412yes—
62D4×D101—28no—
63D4×D101—24no—
64C22×Dic101C2×C411yescharacteristic
65C22×Dic101C2×C412yes—
66C2×C4×D101—22no—
67C2×C4×D101—22no—
68C2×C4×D101—24no—
69C2×C4×D101—24no—
70C2×C4×D101C2×C411yescharacteristic
71C2×C4×D101C2×C412yes—
72C2×C4×D101C2×C412yes—
73C2×C4×D101C2×C411yescharacteristic
74C2×C4×D101C2311yescharacteristic
75C2×C4×D101C2311yescharacteristic
76C2×C4×D101C2312yes—
77#13108058—22no—
78#13108057—24no—
79#13108057—24no—
80C2×#11717579—22no—
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2312yes—
83C2×#11717579C2311yescharacteristic
84C8×D101—24no—
85C8×D101—24no—
86#41×C101—22no—
87#41×C101—22no—
88C2×Q8×C101C2×C411yescharacteristic
89C2×D4×C101C2×C412yes—
90C2×D4×C101C2×C411yescharacteristic
91C22×C4×C101C2312yes—
92C22×C4×C101C2311yescharacteristic
93M16×C101—22no—
94C2×C8×C101—22no—
95C2×C8×C101C2311yescharacteristic
96C2×C8×C101C2311yescharacteristic
97C2×C8×C101C2312yes—
98C23×C101C22×C412yes—
99C23×C101C22×C411yescharacteristic
100C22×D101—22no—
101C22×D101—24no—
102C22×D101—24no—
103C22×D101—22no—
104C22×D101—42no—
105C22×D101—41no—
106C22×D101C22×C411yescharacteristic
107C22×D101C22×C412yes—
108C22×D101C22×C411yescharacteristic
109Q8×C101—22no—
110D4×C101—24no—
111D4×C101—22no—
112C2×C4×C101—22no—
113C2×C4×C101—22no—
114C2×C4×C101C22×C412yes—
115C2×C4×C101C22×C411yescharacteristic
116C2×C4×C101C2×D412yes—
117C2×C4×C101C2×D412yes—
118C2×C4×C101C2411yescharacteristic
119C22⋊D101—24no—
120C22⋊D101—24no—
121C22⋊D101C22⋊C418yes—
122C2×Dic101—22no—
123C2×Dic101—24no—
124C2×Dic101C22×C411yescharacteristic
125C2×Dic101C22×C411yescharacteristic
126C2×Dic101C22×C412yes—
127D404—24no—
128D404C22⋊C414yes—
129C4×D101—24no—
130C4×D101—22no—
131C4×D101—22no—
132C4×D101—21no—
133C4×D101—21no—
134C4×D101—21no—
135C4×D101—21no—
136C4×D101—24no—
137C4×D101C2×D414yes—
138C4×D101C2×D414yes—
139Dic202—24no—
140Dic202C22⋊C414yes—
141C8×C101—21no—
142C8×C101—21no—
143C8×C101—22no—
144#11717579—21no—
145#11717579—22no—
146#11717579—21no—
147C22×C101—21no—
148C22×C101—21no—
149C22×C101—22no—
150C22×C101—22no—
151C22×C101—22no—
152C22×C101C2×C22⋊C412yes—
153C22×C101C23×C411yescharacteristic
154C2×D101—21no—
155C2×D101—22no—
156C2×D101—21no—
157C2×D101—22no—
158C2×D101—21no—
159C2×D101—21no—
160C2×D101—21no—
161C2×D101—48no—
162C2×D101—44no—
163C2×D101C2×C22⋊C414yes—
164C2×D101#13112yescomplemented
165C4×C101—22no—
166C4×C101C2×C22⋊C412yes—
167C4×C101C22×D411yescharacteristic
168C4×C101C22×D411yescharacteristic
169Dic101—21no—
170Dic101—21no—
171Dic101C2×C22⋊C414yes—
172C2×C101—22no—
173C2×C101—42no—
174C2×C101C22×C22⋊C411yescharacteristic
175C2×C101C2×#13111yescharacteristic
176C2×C101#56811yescharacteristic
177D101—22no—
178D101—24no—
179D101—41no—
180#2880—1011nomaximal, Sylow
181C101#288011yescharacteristic, complemented, Sylow
182C22×#41—1011no—
183C2×#131—1011no—
184C2×#131—1011no—
185#408—1011no—
186#408—1012no—
187#408—1012no—
188#408D10111yescharacteristic, complemented
189#407—1011no—
190#407—1012no—
191C2×C22⋊C8—1011no—
192C2×C22⋊C8—1012no—
193C2×#41—1011no—
194C2×#41—1012no—
195C2×#41—1011no—
196C2×#41—1011no—
197C2×#41—1012no—
198C2×#41—2024no—
199C2×#41—2024no—
200C2×#41C2×D10111yescharacteristic
201C22×Q8—1011no—
202C22×D4—1012no—
203C22×D4—1011no—
204C23×C4—1012no—
205C23×C4—1011no—
206#131—2024no—
207#131—2024no—
208C2×M16—1011no—
209C2×M16—1012no—
210C2×M16—1012no—
211C2×M16C2×D10111yescharacteristic, complemented
212C22×C8—1011no—
213C22×C8—1012no—
214C22×C8—1012no—
215C22×C8C2×D10111yescharacteristic, complemented
216C22⋊C8—1012no—
217C22⋊C8—1012no—
218C22⋊C8—1011no—
219C22⋊C8—1012no—
220C22⋊C8—1011no—
221C22⋊C8—1011no—
222C22⋊C8—1011no—
223C22⋊C8—1012no—
224C22⋊C8C2×D10112yescomplemented
225C22⋊C8C2×D10112yescomplemented
226C24—1012no—
227C24—1011no—
228#41—22no—
229#41—22no—
230#41—1014no—
231#41—1014no—
232#41—2024no—
233#41—2024no—
234#41—2028no—
235#41—2024no—
236#41—2024no—
237C2×Q8—1011no—
238C2×Q8—1012no—
239C2×Q8—2024no—
240C2×Q8C4×D10111yescharacteristic
241C2×D4—1012no—
242C2×D4—1012no—
243C2×D4—1012no—
244C2×D4—1011no—
245C2×D4—1012no—
246C2×D4—2028no—
247C2×D4—2024no—
248C2×D4C4×D10112yes—
249C2×D4C4×D10111yescharacteristic
250C22×C4—1012no—
251C22×C4—1012no—
252C22×C4—1011no—
253C22×C4—1012no—
254C22×C4—1011no—
255C22×C4—1012no—
256C22×C4—1011no—
257C22×C4—1011no—
258C22×C4—1011no—
259C22×C4—2024no—
260C22×C4—2024no—
261C22×C4—2022no—
262C22×C4—2022no—
263C22×C4C22×D10112yes—
264C22×C4C22×D10111yescharacteristic
265M16—22no—
266M16—2024no—
267M16—2024no—
268M16—2022no—
269C2×C8—22no—
270C2×C8—1011no—
271C2×C8—1012no—
272C2×C8—1011no—
273C2×C8—2024no—
274C2×C8—2024no—
275C2×C8—2022no—
276C2×C8C22×D10111yescharacteristic
277C2×C8C22×D10112yes—
278C2×C8C22×D10111yescharacteristic
279C23—1011no—
280C23—1012no—
281C23—1011no—
282C23—4041no—
283C23—4042no—
284C23—2024no—
285C23—2022no—
286C23—2024no—
287C23—2022no—
288C23C2×C4×D10112yes—
289C23C2×C4×D10111yescharacteristic
290Q8—22no—
291Q8—1014no—
292Q8—2024no—
293D4—24no—
294D4—22no—
295D4—1018no—
296D4—1014no—
297D4—2024no—
298D4—2024no—
299D4—2024no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—1012no—
303C2×C4—1014no—
304C2×C4—1014no—
305C2×C4—1011no—
306C2×C4—1011no—
307C2×C4—2024no—
308C2×C4—2024no—
309C2×C4—2024no—
310C2×C4—2022no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4—2022no—
316C2×C4—2022no—
317C2×C4C2×C4×D10112yes—
318C2×C4C2×C4×D10111yescharacteristic
319C2×C4D4×D10112yes—
320C2×C4D4×D10112yes—
321C2×C4C23×D10111yescharacteristic
322C8—21no—
323C8—21no—
324C8—22no—
325C8—2021no—
326C8—2022no—
327C8—2021no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—1014no—
334C22—1012no—
335C22—4044no—
336C22—4048no—
337C22—2021no—
338C22—2021no—
339C22—2021no—
340C22—2021no—
341C22—2022no—
342C22—2021no—
343C22—2022no—
344C22C22⋊C4×D10112yes—
345C22C22×C4×D10111yescharacteristic
346C4—22no—
347C4—1014no—
348C4—2021no—
349C4—2021no—
350C4C22⋊C4×D10112yes—
351C4C2×D4×D10111yescharacteristic
352C4C2×D4×D10111yescharacteristic
353C2—22no—
354C2—42no—
355C2—4041no—
356C2—2022no—
357C2—2024no—
358C2C2×C22⋊C4×D10111yescharacteristic
359C2#131×D10111yescharacteristic
360C2#2026594211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D101
commutator, socleC22×C101C23×C4
Fitting#408×C101C2
prime radicalC2×#41×D101C2
cosocleC2×C4×C101C24
hypercenter#408D101
hypocenterC101#2880
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#15692263C2semidirect
3. C2×#15692261C2semidirect
4. #20265144C2semidirect
5. #20265142C2semidirect
6. #20265142C2semidirect
7. #20265140C2semidirect
8. #20265139C2semidirect
9. C2×#15692167C2semidirect
10. C22⋊C8×D101C2semidirect
11. #408×C101C2semidirect
26. C2×#13108058C22semidirect
27. C2×#13108057C22semidirect
28. C2×#13108057C22semidirect
29. C22×#11717579C22semidirect
30. C2×C8×D101C22semidirect
31. C2×C8×D101C22semidirect
32. C22⋊#11717579C22semidirect
33. C22⋊#11717579C22semidirect
37. #15692167C22semidirect
38. #15692167C22semidirect
39. #15692167C22semidirect
41. C2×M16×C101C22semidirect
42. C22×C8×C101C22semidirect
43. C22⋊C8×C101C22semidirect
44. C22⋊C8×C101C22semidirect
164. C2×D101#131semidirect
181. C101#2880semidirect
188. #408D101semidirect
211. C2×M16C2×D101semidirect
215. C22×C8C2×D101semidirect
224. C22⋊C8C2×D101semidirect
225. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1012
C2×#156922632
C2×#156922612
#202651442
#202651422
#202651402
#202651392
C2×#156921672
C22⋊C8×D1012
#408×C1012
#2880101