Finite Group Databasebeta

← #27180848

Subgroups — #27180848

Order 12928.

452 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#20265975C212yesmaximal, complemented
2C2×#15692261C211yesmaximal, characteristic, complemented
3#20265449C212yesmaximal, complemented
4#20265444C212yesmaximal, complemented
5#20265437C214yesmaximal, complemented
6C2×#15692191C211yesmaximal, characteristic, complemented
7#20265161C212yesmaximal, complemented
8C2×#135×C101C211yesmaximal, characteristic, complemented
9C2×#13108074C2211yescharacteristic
10#15692286C2214yes—
11#15692285—22no—
12#15692281C2214yes—
13#15692280—22no—
14#15692275—24no—
15#15692261—21no—
16#15692261—21no—
17#15692261C2212yes—
18#15692261C2212yes—
19C2×#13108058C2212yes—
20C2×#13108058C2211yescharacteristic
21C2×#13108057—21no—
22C2×C8×D101—21no—
23#15692191C2212yescomplemented
24#15692191C2212yescomplemented
25#15692174C2214yescomplemented
26#15692172C2214yescomplemented
27C2×#41×C101C2212yes—
28#135×C101—21no—
29#135×C101—21no—
30#135×C101—21no—
31#135×C101—21no—
32C2×Q16×C101C2211yescharacteristic, complemented
33C2×SD16×C101C2212yescomplemented
34C2×D8×C101C2211yescharacteristic, complemented
35C22×C8×C101C2211yescharacteristic
36#13108074—21no—
37#13108074—21no—
38#13108074D412yes—
39#13108074C2312yes—
40C2×Dic202D411yescharacteristic
41C2×D404D411yescharacteristic
42C2×C22⋊D101—21no—
43C2×C4×D101—21no—
44#13108064—24no—
45#13108062—24no—
46#13108061—24no—
47#13108059—24no—
48#13108058—21no—
49#13108058—22no—
50#13108058C2312yes—
51#13108058C2314yes—
52#13108057—22no—
53#13108057—22no—
54C2×#11717579—21no—
55C2×#11717579—22no—
56C8×D101—22no—
57C8×D101—22no—
58#41×C101—22no—
59#41×C101—22no—
60#41×C101—22no—
61#41×C101—22no—
62C2×Q8×C101C2312yes—
63C2×D4×C101—22no—
64C2×D4×C101C2312yes—
65C22×C4×C101—22no—
66C22×C4×C101C2311yescharacteristic
67Q16×C101—21no—
68Q16×C101—21no—
69SD16×C101—22no—
70SD16×C101—22no—
71D8×C101—21no—
72D8×C101—21no—
73C2×C8×C101D411yescharacteristic
74C2×C8×C101D411yescharacteristic
75C2×C8×C101D411yescharacteristic
76C2×C8×C101D411yescharacteristic
77C2×C8×C101C2311yescharacteristic
78C2×C8×C101C2311yescharacteristic
79C23×C101—22no—
80C23×C101#4111yescharacteristic
81C22×D101—21no—
82Q8×C101—22no—
83Q8×C101—22no—
84D4×C101—22no—
85D4×C101—22no—
86D4×C101—42no—
87D4×C101—42no—
88C2×C4×C101—22no—
89C2×C4×C101—42no—
90C2×C4×C101—42no—
91C2×C4×C101C2×D411yescharacteristic
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101#4111yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—22no—
98C22⋊D101—22no—
99C2×Dic101—21no—
100D404—21no—
101D404C2×D412yes—
102C4×D101—22no—
103C4×D101—22no—
104Dic202—21no—
105Dic202C2×D412yes—
106C8×C101C2×D411yescharacteristic
107C8×C101C2×D411yescharacteristic
108C8×C101C2×D411yescharacteristic
109C8×C101C2×D411yescharacteristic
110#11717579—22no—
111#11717579—24no—
112C22×C101—41no—
113C22×C101—42no—
114C22×C101—82no—
115C22×C101#12111yescharacteristic
116C22×C101#12111yescharacteristic
117C22×C101C2×#4111yescharacteristic
118C2×D101—22no—
119C2×D101—42no—
120C4×C101—42no—
121C4×C101#12111yescharacteristic
122C4×C101#12111yescharacteristic
123C4×C101C22×D411yescharacteristic
124C4×C101C22×D411yescharacteristic
125Dic101—22no—
126C2×C101—21no—
127C2×C101—21no—
128C2×C101—21no—
129C2×C101—82no—
130C2×C101C2×#12111yescharacteristic
131D101—42no—
132#3067—1011nomaximal, Sylow
133C101#306711yescharacteristic, complemented, Sylow
134#575—1012no—
135C2×#135D10111yescharacteristic, complemented
136C2×#131—1011no—
137#473—1012no—
138#472—1012no—
139#471—1014no—
140C2×#108—1011no—
141#413—1012no—
142C2×#41—1011no—
143C2×#41C2×D10112yes—
144#137—1014no—
145#137—2022no—
146AGL(1,ℤ/8)—1014no—
147AGL(1,ℤ/8)—2022no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—2024no—
153C2×Q16C2×D10111yescharacteristic, complemented
154C2×SD16C2×D10112yescomplemented
155C2×D8C2×D10111yescharacteristic, complemented
156#131—1012no—
157#131—1012no—
158#131—2021no—
159#131—2021no—
160C2×M16—1011no—
161C2×M16—1012no—
162C2×M16—2021no—
163C22×C8—2021no—
164C22×C8C2×D10111yescharacteristic
165#108—1012no—
166#108—1012no—
167#101—1014no—
168#100—1014no—
169#41—22no—
170#41—22no—
171#41—22no—
172#41—22no—
173#41—1012no—
174#41—1012no—
175#41—2021no—
176#41—2021no—
177C2×Q8—1011no—
178C2×Q8C22×D10112yes—
179C2×D4—22no—
180C2×D4—1011no—
181C2×D4—2021no—
182C2×D4C22×D10112yes—
183C22×C4—22no—
184C22×C4—2021no—
185C22×C4C22×D10111yescharacteristic
186Q16—21no—
187Q16—21no—
188Q16—2024no—
189SD16—22no—
190SD16—22no—
191SD16—2024no—
192SD16—2024no—
193D8—21no—
194D8—21no—
195D8—2024no—
196M16—1012no—
197M16—1014no—
198M16—2022no—
199M16—2021no—
200M16—2022no—
201M16—2022no—
202C2×C8—2022no—
203C2×C8—2021no—
204C2×C8—2022no—
205C2×C8—2022no—
206C2×C8C22⋊D10111yescharacteristic
207C2×C8C22⋊D10111yescharacteristic
208C2×C8C22⋊D10111yescharacteristic
209C2×C8C22⋊D10111yescharacteristic
210C2×C8C22×D10111yescharacteristic
211C2×C8C22×D10111yescharacteristic
212C23—22no—
213C23—2021no—
214C23#1310807311yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—1012no—
218Q8—2021no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—42no—
223D4—1012no—
224D4—2022no—
225D4—2022no—
226D4—2021no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—2021no—
231C2×C4—2022no—
232C2×C4—2022no—
233C2×C4D4×D10111yescharacteristic
234C2×C4D4×D10111yescharacteristic
235C2×C4C2×C22⋊D10111yescharacteristic
236C2×C4C2×C22⋊D10111yescharacteristic
237C2×C4#1310807311yescharacteristic
238C2×C4C23×D10111yescharacteristic
239C8—2024no—
240C8—2022no—
241C8C2×C22⋊D10111yescharacteristic
242C8C2×C22⋊D10111yescharacteristic
243C8C2×C22⋊D10111yescharacteristic
244C8C2×C22⋊D10111yescharacteristic
245C22—41no—
246C22—42no—
247C22—82no—
248C22—4042no—
249C22—2022no—
250C22#1569222611yescharacteristic
251C22#1569222611yescharacteristic
252C22C2×#1310807311yescharacteristic
253C4—42no—
254C4—2022no—
255C4#1569222611yescharacteristic
256C4#1569222611yescharacteristic
257C4C2×D4×D10111yescharacteristic
258C4C22×C22⋊D10111yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2—4042no—
264C2C2×#1569222611yescharacteristic
subgroupG/H
centerC4#15692226
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC2×C101C2×#121
hypercenterC2×#135D101
hypocenterC101#3067
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #20265975C2semidirect
2. C2×#15692261C2semidirect
3. #20265449C2semidirect
4. #20265444C2semidirect
5. #20265437C2semidirect
6. C2×#15692191C2semidirect
7. #20265161C2semidirect
8. C2×#135×C101C2semidirect
23. #15692191C22semidirect
24. #15692191C22semidirect
25. #15692174C22semidirect
26. #15692172C22semidirect
32. C2×Q16×C101C22semidirect
33. C2×SD16×C101C22semidirect
34. C2×D8×C101C22semidirect
133. C101#3067semidirect
135. C2×#135D101semidirect
153. C2×Q16C2×D101semidirect
154. C2×SD16C2×D101semidirect
155. C2×D8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202659752
C2×#156922612
#202654492
#202654442
#202654372
C2×#156921912
#202651612
C2×#135×C1012
#3067101