Finite Group Databasebeta

← #27180593

Subgroups — #27180593

Order 12928.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692281C211yesmaximal, characteristic, complemented
2#20265723C211yesmaximal, characteristic, complemented
3C2×D4×Dic101C211yesmaximal, characteristic, complemented
4#20265394C212yesmaximal, complemented
5#20265390C212yesmaximal, complemented
6#20265380C212yesmaximal, complemented
7#20265370C212yesmaximal, complemented
8#20265200C211yesmaximal, characteristic, complemented
9#20265179C211yesmaximal, characteristic, complemented
10#20265178C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C23×Dic101—21no—
14#15692281—24no—
15C2×#13108062C2212yes—
16C2×#13108059C2212yes—
17C2×#13108058C2211yescharacteristic
18#15692253C2211yescharacteristic, complemented
19#15692251C2211yescharacteristic, complemented
20#15692244C2211yescharacteristic, complemented
21#15692242C2211yescharacteristic, complemented
22#15692227—21no—
23#15692227—21no—
24D4×Dic101—21no—
25D4×Dic101—21no—
26D4×Dic101C2212yes—
27D4×Dic101C2212yes—
28C4×C22⋊D101—21no—
29C4×C22⋊D101—21no—
30C2×C4⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101—21no—
32C2×C4×Dic101C2211yescharacteristic
33#15692184C2212yescomplemented
34#15692184C2212yescomplemented
35#15692180C2212yescomplemented
36#15692179C2212yescomplemented
37#15692177C2212yescomplemented
38#15692176C2212yescomplemented
39C2×#41×C101C2211yescharacteristic
40C22×D4×C101C2211yescharacteristic
41AGL(1,ℤ/8)×C101—24no—
42C2×SD16×C101C2212yes—
43C2×D8×C101C2212yes—
44C2×M16×C101C2211yescharacteristic
45#13108074—22no—
46#13108074—22no—
47C2×Dic202C2311yescharacteristic
48C2×D404C2311yescharacteristic
49C2×C22⋊D101—21no—
50C2×C22⋊D101—21no—
51C2×C22⋊D101—21no—
52C22×Dic101—21no—
53C22×Dic101—22no—
54C22×Dic101—22no—
55C22×Dic101—44no—
56C22×Dic101D411yescharacteristic
57C22×Dic101D411yescharacteristic
58C2×C4×D101—21no—
59#13108062—24no—
60#13108059—24no—
61#13108058—22no—
62C2×#11717579C2312yes—
63C4⋊Dic101—21no—
64C4⋊Dic101C2312yes—
65#13108053—21no—
66#13108053—21no—
67C22⋊Dic101—22no—
68C22⋊Dic101—22no—
69C22⋊Dic101—21no—
70C22⋊Dic101—21no—
71#13108051—21no—
72#13108051—21no—
73C4×Dic101D411yescharacteristic
74C4×Dic101D411yescharacteristic
75C4×Dic101C2311yescharacteristic
76C4×Dic101C2311yescharacteristic
77C24×C101—21no—
78#41×C101—22no—
79#41×C101—22no—
80C2×Q8×C101C2311yescharacteristic
81C2×D4×C101—21no—
82C2×D4×C101—21no—
83C2×D4×C101—22no—
84C2×D4×C101D414yescomplemented
85C2×D4×C101C2311yescharacteristic
86C2×D4×C101C2312yes—
87C22×C4×C101—21no—
88C22×C4×C101C2311yescharacteristic
89SD16×C101—24no—
90D8×C101—24no—
91M16×C101—22no—
92C2×C8×C101C2312yes—
93C23×C101—24no—
94C23×C101—22no—
95C23×C101—21no—
96C23×C101—42no—
97C23×C101—42no—
98C23×C101C2×D411yescharacteristic
99C22×D101—21no—
100Q8×C101—21no—
101Q8×C101—21no—
102D4×C101—24no—
103D4×C101—21no—
104D4×C101—22no—
105D4×C101—21no—
106D4×C101—42no—
107D4×C101C2×D414yes—
108C2×C4×C101—21no—
109C2×C4×C101—42no—
110C2×C4×C101C2×D411yescharacteristic
111C2×C4×C101C2×D412yes—
112C2×C4×C101#4112yes—
113C2×C4×C101C2411yescharacteristic
114C22⋊D101—42no—
115C22⋊D101—42no—
116C22⋊D101—42no—
117C2×Dic101—22no—
118C2×Dic101—21no—
119C2×Dic101—22no—
120C2×Dic101—41no—
121C2×Dic101—44no—
122C2×Dic101—42no—
123C2×Dic101—44no—
124C2×Dic101C2×D411yescharacteristic
125C2×Dic101C2×D411yescharacteristic
126C2×Dic101C2×D411yescharacteristic
127C2×Dic101C2×D411yescharacteristic
128D404—21no—
129D404—21no—
130C4×D101—42no—
131Dic202—21no—
132Dic202—21no—
133C8×C101—22no—
134#11717579—22no—
135C22×C101—21no—
136C22×C101—21no—
137C22×C101—24no—
138C22×C101—48no—
139C22×C101—42no—
140C22×C101—42no—
141C22×C101—41no—
142C22×C101—41no—
143C22×C101—41no—
144C22×C101#12112yes—
145C22×C101C22×D411yescharacteristic
146C2×D101—41no—
147C2×D101—41no—
148C2×D101—41no—
149C4×C101—41no—
150C4×C101#12112yes—
151C4×C101C22×D411yescharacteristic
152C4×C101C2×#4111yescharacteristic
153Dic101—21no—
154Dic101—21no—
155Dic101—21no—
156Dic101—41no—
157Dic101—42no—
158Dic101AGL(1,ℤ/8)12yescomplemented
159C2×C101—22no—
160C2×C101—44no—
161C2×C101—81no—
162C2×C101C2×#12111yescharacteristic
163C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C101#57511yescharacteristic
165D101—81no—
166#3036—1011nomaximal, Sylow
167C101#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1011no—
169C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
170#532—1011no—
171C2×C4×D4—1011no—
172#463—1012no—
173#463—1012no—
174#461—1012no—
175#459—1012no—
176#423—1011no—
177#417—1011no—
178#417—1011no—
179C2×#41—1011no—
180C2×#41C2×D10111yescharacteristic
181C22×D4C2×D10111yescharacteristic
182C23×C4—2021no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—2024no—
185C2×SD16—1012no—
186C2×SD16C2×D10112yes—
187C2×D8—1012no—
188C2×D8C2×D10112yes—
189C2×M16—1011no—
190C2×M16C2×D10111yescharacteristic
191C4⋊Q8—1011no—
192#127—1011no—
193#124—1011no—
194#124—1011no—
195#121—2021no—
196#121—2021no—
197C4×D4—1012no—
198C4×D4—1012no—
199C4×D4—2021no—
200C4×D4—2021no—
201C4×D4—2021no—
202C4×D4—2021no—
203C2×C4⋊C4—1011no—
204C2×C22⋊C4—2021no—
205C2×C42—1011no—
206C4⋊C8—1012no—
207C4⋊C8—1012no—
208#103—1012no—
209#103—1012no—
210#102—1012no—
211#102—1012no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—2022no—
216#41—2022no—
217C2×Q8—1011no—
218C2×Q8C22×D10111yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—1011no—
223C2×D4—2021no—
224C2×D4—2021no—
225C2×D4—2021no—
226C2×D4C22⋊D10114yescomplemented
227C2×D4C22×D10111yescharacteristic
228C2×D4C22×D10112yes—
229C22×C4—21no—
230C22×C4—1011no—
231C22×C4—1011no—
232C22×C4—4044no—
233C22×C4—2022no—
234C22×C4—2022no—
235C22×C4—2021no—
236C22×C4—2021no—
237C22×C4C22×D10111yescharacteristic
238SD16—24no—
239SD16—2024no—
240D8—24no—
241D8—2024no—
242M16—22no—
243M16—2022no—
244C2×C8—1012no—
245C2×C8C22×D10112yes—
246C4⋊C4—1012no—
247C4⋊C4—2021no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C22⋊C4—2021no—
251C22⋊C4—2021no—
252C22⋊C4—2022no—
253C22⋊C4—2022no—
254C22⋊C4—2021no—
255C22⋊C4—2021no—
256C42—1011no—
257C42—1011no—
258C42—1011no—
259C42—1011no—
260C23—22no—
261C23—24no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—2021no—
266C23D4×D10111yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2021no—
270Q8—2021no—
271D4—21no—
272D4—22no—
273D4—21no—
274D4—24no—
275D4—42no—
276D4—4042no—
277D4—4042no—
278D4—4042no—
279D4—2021no—
280D4—2021no—
281D4C2×C22⋊D10114yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—1011no—
285C2×C4—1011no—
286C2×C4—1011no—
287C2×C4—1011no—
288C2×C4—4044no—
289C2×C4—4042no—
290C2×C4—4044no—
291C2×C4—4042no—
292C2×C4—4041no—
293C2×C4—2022no—
294C2×C4—2022no—
295C2×C4—2021no—
296C2×C4D4×D10111yescharacteristic
297C2×C4C2×C22⋊D10112yes—
298C2×C4#1310807312yes—
299C2×C4C23×D10111yescharacteristic
300C8—22no—
301C8—2022no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—4041no—
312C22—4041no—
313C22—4041no—
314C22#1569222712yes—
315C22C2×D4×D10111yescharacteristic
316C4—41no—
317C4—1012no—
318C4—4041no—
319C4—4042no—
320C4—2021no—
321C4—2021no—
322C4—2021no—
323C4#1569222712yes—
324C4C22×C22⋊D10111yescharacteristic
325C4C2×#1310807311yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—8081no—
330C2C2×#1569222711yescharacteristic
331C2AGL(1,ℤ/8)×D10111yescharacteristic
332C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3036
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. #20265723C2semidirect
3. C2×D4×Dic101C2semidirect
4. #20265394C2semidirect
5. #20265390C2semidirect
6. #20265380C2semidirect
7. #20265370C2semidirect
8. #20265200C2semidirect
9. #20265179C2semidirect
10. #20265178C2semidirect
11. C2×AGL(1,ℤ/8)×C101C2semidirect
18. #15692253C22semidirect
19. #15692251C22semidirect
20. #15692244C22semidirect
21. #15692242C22semidirect
33. #15692184C22semidirect
34. #15692184C22semidirect
35. #15692180C22semidirect
36. #15692179C22semidirect
37. #15692177C22semidirect
38. #15692176C22semidirect
84. C2×D4×C101D4semidirect
158. Dic101AGL(1,ℤ/8)semidirect
167. C101#3036semidirect
169. C2×AGL(1,ℤ/8)D101semidirect
226. C2×D4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
#202657232
C2×D4×Dic1012
#202653942
#202653902
#202653802
#202653702
#202652002
#202651792
#202651782
C2×AGL(1,ℤ/8)×C1012
#3036101