Finite Group Databasebeta

← #35224252

Subgroups — #35224252

Order 20864.

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#23782463C212yesmaximal, complemented
2C2×#19104741C211yesmaximal, characteristic, complemented
3#23781937C212yesmaximal, complemented
4#23781932C212yesmaximal, complemented
5#23781925C214yesmaximal, complemented
6C2×#19104671C211yesmaximal, characteristic, complemented
7#23781649C212yesmaximal, complemented
8C2×#135×C163C211yesmaximal, characteristic, complemented
9C2×#14579699C2211yescharacteristic
10#19104766C2214yes—
11#19104765—22no—
12#19104761C2214yes—
13#19104760—22no—
14#19104755—24no—
15#19104741—21no—
16#19104741—21no—
17#19104741C2212yes—
18#19104741C2212yes—
19C2×#14579683C2211yescharacteristic
20C2×#14579683C2212yes—
21C2×#14579682—21no—
22C2×C8×D163—21no—
23#19104671C2212yescomplemented
24#19104671C2212yescomplemented
25#19104654C2214yescomplemented
26#19104652C2214yescomplemented
27C2×#41×C163C2212yes—
28#135×C163—21no—
29#135×C163—21no—
30#135×C163—21no—
31#135×C163—21no—
32C2×Q16×C163C2211yescharacteristic, complemented
33C2×SD16×C163C2212yescomplemented
34C2×D8×C163C2211yescharacteristic, complemented
35C22×C8×C163C2211yescharacteristic
36#14579699—21no—
37#14579699—21no—
38#14579699D412yes—
39#14579699C2312yes—
40C2×Dic326D411yescharacteristic
41C2×D652D411yescharacteristic
42C2×C22⋊D163—21no—
43C2×C4×D163—21no—
44#14579689—24no—
45#14579687—24no—
46#14579686—24no—
47#14579684—24no—
48#14579683—21no—
49#14579683—22no—
50#14579683C2312yes—
51#14579683C2314yes—
52#14579682—22no—
53#14579682—22no—
54C2×C163⋊C8—21no—
55C2×C163⋊C8—22no—
56C8×D163—22no—
57C8×D163—22no—
58#41×C163—22no—
59#41×C163—22no—
60#41×C163—22no—
61#41×C163—22no—
62C2×Q8×C163C2312yes—
63C2×D4×C163—22no—
64C2×D4×C163C2312yes—
65C22×C4×C163—22no—
66C22×C4×C163C2311yescharacteristic
67Q16×C163—21no—
68Q16×C163—21no—
69SD16×C163—22no—
70SD16×C163—22no—
71D8×C163—21no—
72D8×C163—21no—
73C2×C8×C163D411yescharacteristic
74C2×C8×C163D411yescharacteristic
75C2×C8×C163D411yescharacteristic
76C2×C8×C163D411yescharacteristic
77C2×C8×C163C2311yescharacteristic
78C2×C8×C163C2311yescharacteristic
79C22×D163—21no—
80Dic326—21no—
81Dic326C2×D412yes—
82D652—21no—
83D652C2×D412yes—
84C22⋊D163—22no—
85C22⋊D163—22no—
86C2×Dic163—21no—
87C4×D163—22no—
88C4×D163—22no—
89C163⋊C8—22no—
90C163⋊C8—24no—
91C23×C163—22no—
92C23×C163#4111yescharacteristic
93Q8×C163—22no—
94Q8×C163—22no—
95D4×C163—22no—
96D4×C163—22no—
97D4×C163—42no—
98D4×C163—42no—
99C2×C4×C163—22no—
100C2×C4×C163—42no—
101C2×C4×C163—42no—
102C2×C4×C163C2×D411yescharacteristic
103C2×C4×C163C2×D411yescharacteristic
104C2×C4×C163C2×D411yescharacteristic
105C2×C4×C163C2×D411yescharacteristic
106C2×C4×C163#4111yescharacteristic
107C2×C4×C163C2411yescharacteristic
108C8×C163C2×D411yescharacteristic
109C8×C163C2×D411yescharacteristic
110C8×C163C2×D411yescharacteristic
111C8×C163C2×D411yescharacteristic
112C22×C163—41no—
113C22×C163—42no—
114C22×C163—82no—
115C22×C163#12111yescharacteristic
116C22×C163#12111yescharacteristic
117C22×C163C2×#4111yescharacteristic
118C2×D163—22no—
119C2×D163—42no—
120C4×C163—42no—
121C4×C163#12111yescharacteristic
122C4×C163#12111yescharacteristic
123C4×C163C22×D411yescharacteristic
124C4×C163C22×D411yescharacteristic
125Dic163—22no—
126C2×C163—21no—
127C2×C163—21no—
128C2×C163—21no—
129C2×C163—82no—
130C2×C163C2×#12111yescharacteristic
131D163—42no—
132C163#306711yescharacteristic, complemented, Sylow
133#3067—1631nomaximal, Sylow
134#575—1632no—
135C2×#135D16311yescharacteristic, complemented
136C2×#131—1631no—
137#473—1632no—
138#472—1632no—
139#471—1634no—
140C2×#108—1631no—
141#413—1632no—
142C2×#41—1631no—
143C2×#41C2×D16312yes—
144#137—3262no—
145#137—1634no—
146AGL(1,ℤ/8)—3262no—
147AGL(1,ℤ/8)—1634no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—3264no—
153C2×Q16C2×D16311yescharacteristic, complemented
154C2×SD16C2×D16312yescomplemented
155C2×D8C2×D16311yescharacteristic, complemented
156#131—3261no—
157#131—3261no—
158#131—1632no—
159#131—1632no—
160C2×M16—3261no—
161C2×M16—1631no—
162C2×M16—1632no—
163C22×C8—3261no—
164C22×C8C2×D16311yescharacteristic
165#108—1632no—
166#108—1632no—
167#101—1634no—
168#100—1634no—
169#41—22no—
170#41—22no—
171#41—22no—
172#41—22no—
173#41—3261no—
174#41—3261no—
175#41—1632no—
176#41—1632no—
177C2×Q8—1631no—
178C2×Q8C22×D16312yes—
179C2×D4—22no—
180C2×D4—3261no—
181C2×D4—1631no—
182C2×D4C22×D16312yes—
183C22×C4—22no—
184C22×C4—3261no—
185C22×C4C22×D16311yescharacteristic
186Q16—21no—
187Q16—21no—
188Q16—3264no—
189SD16—22no—
190SD16—22no—
191SD16—3264no—
192SD16—3264no—
193D8—21no—
194D8—21no—
195D8—3264no—
196M16—3262no—
197M16—3261no—
198M16—3262no—
199M16—3262no—
200M16—1632no—
201M16—1634no—
202C2×C8—3262no—
203C2×C8—3261no—
204C2×C8—3262no—
205C2×C8—3262no—
206C2×C8C22⋊D16311yescharacteristic
207C2×C8C22⋊D16311yescharacteristic
208C2×C8C22⋊D16311yescharacteristic
209C2×C8C22⋊D16311yescharacteristic
210C2×C8C22×D16311yescharacteristic
211C2×C8C22×D16311yescharacteristic
212C23—22no—
213C23—3261no—
214C23#1457969811yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—3261no—
218Q8—1632no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—42no—
223D4—3262no—
224D4—3262no—
225D4—3261no—
226D4—1632no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—3261no—
231C2×C4—3262no—
232C2×C4—3262no—
233C2×C4D4×D16311yescharacteristic
234C2×C4D4×D16311yescharacteristic
235C2×C4C2×C22⋊D16311yescharacteristic
236C2×C4C2×C22⋊D16311yescharacteristic
237C2×C4#1457969811yescharacteristic
238C2×C4C23×D16311yescharacteristic
239C8—3264no—
240C8—3262no—
241C8C2×C22⋊D16311yescharacteristic
242C8C2×C22⋊D16311yescharacteristic
243C8C2×C22⋊D16311yescharacteristic
244C8C2×C22⋊D16311yescharacteristic
245C22—41no—
246C22—42no—
247C22—82no—
248C22—3262no—
249C22—6522no—
250C22#1910470611yescharacteristic
251C22#1910470611yescharacteristic
252C22C2×#1457969811yescharacteristic
253C4—42no—
254C4—3262no—
255C4#1910470611yescharacteristic
256C4#1910470611yescharacteristic
257C4C2×D4×D16311yescharacteristic
258C4C22×C22⋊D16311yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2—6522no—
264C2C2×#1910470611yescharacteristic
subgroupG/H
centerC4#19104706
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#135×C163C2
socleC2×C163C2×#121
hypercenterC2×#135D163
hypocenterC163#3067
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #23782463C2semidirect
2. C2×#19104741C2semidirect
3. #23781937C2semidirect
4. #23781932C2semidirect
5. #23781925C2semidirect
6. C2×#19104671C2semidirect
7. #23781649C2semidirect
8. C2×#135×C163C2semidirect
23. #19104671C22semidirect
24. #19104671C22semidirect
25. #19104654C22semidirect
26. #19104652C22semidirect
32. C2×Q16×C163C22semidirect
33. C2×SD16×C163C22semidirect
34. C2×D8×C163C22semidirect
132. C163#3067semidirect
135. C2×#135D163semidirect
153. C2×Q16C2×D163semidirect
154. C2×SD16C2×D163semidirect
155. C2×D8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824632
C2×#191047412
#237819372
#237819322
#237819252
C2×#191046712
#237816492
C2×#135×C1632
#3067163