Finite Group Databasebeta

← #36374948

Subgroups — #36374948

Order 22336.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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#26013685C211yesmaximal, characteristic, complemented
2#26013684C211yesmaximal, characteristic, complemented
3#26013667C211yesmaximal, characteristic, complemented
4#26013666C212yesmaximal, complemented
5#26013663C212yesmaximal, complemented
6#26013660C212yesmaximal, complemented
7#26013659C211yesmaximal, characteristic, complemented
8C2×#19150218C211yesmaximal, characteristic, complemented
9D8×D349C212yesmaximal, complemented
10#26013647C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C349C211yesmaximal, characteristic, complemented
12#19150234C2212yescomplemented
13#19150233—21no—
14#19150233C2211yescharacteristic
15#19150233C2211yescharacteristic, complemented
16#19150232—21no—
17#19150232C2212yescomplemented
18C2×D1396—21no—
19C2×D1396C2211yescharacteristic, complemented
20C2×C22⋊D349—22no—
21D4×D349—21no—
22D4×D349—22no—
23D4×D349C2212yescomplemented
24D4×D349C2212yescomplemented
25#19150223C2211yescharacteristic, complemented
26#19150222C2212yescomplemented
27#19150221C2211yescharacteristic, complemented
28#19150220C2211yescharacteristic, complemented
29D2792C2212yescomplemented
30#19150218C2211yescharacteristic, complemented
31#19150218C2212yescomplemented
32#19150218C2212yescomplemented
33#19150216C2212yescomplemented
34C2×#14584095C2211yescharacteristic, complemented
35C8×D349C2212yescomplemented
36#41×C349C2211yescharacteristic, complemented
37C2×D4×C349C2211yescharacteristic, complemented
38SD16×C349C2212yescomplemented
39D8×C349C2212yescomplemented
40M16×C349C2211yescharacteristic, complemented
41C22×D349—21no—
42C22×D349—22no—
43C22×D349—22no—
44Dic698D411yescharacteristic
45Dic698C2311yescharacteristic
46D1396—22no—
47D1396—21no—
48D1396D411yescharacteristic
49D1396C2312yes—
50D1396C2311yescharacteristic
51C22⋊D349—21no—
52C22⋊D349—22no—
53C22⋊D349—22no—
54C22⋊D349—22no—
55C22⋊D349D412yes—
56C2×Dic349—22no—
57C4×D349—21no—
58C4×D349—22no—
59C4×D349C2312yes—
60#14584095C2311yescharacteristic
61#14584095C2311yescharacteristic
62C23×C349—21no—
63Q8×C349C2311yescharacteristic
64D4×C349—21no—
65D4×C349—21no—
66D4×C349C2311yescharacteristic
67D4×C349C2312yes—
68C2×C4×C349—21no—
69C2×C4×C349C2311yescharacteristic
70C8×C349C2312yes—
71C22×C349—21no—
72C22×C349—22no—
73C22×C349—81no—
74C22×C349C2×D411yescharacteristic
75C2×D349—21no—
76C2×D349—22no—
77C2×D349—82no—
78C2×D349—82no—
79C2×D349—81no—
80C2×D349C2×D412yes—
81C4×C349—21no—
82C4×C349C2×D411yescharacteristic
83C4×C349C2411yescharacteristic
84Dic349—21no—
85Dic349C2×D412yes—
86C2×C349—21no—
87C2×C349—41no—
88C2×C349—42no—
89C2×C349C22×D411yescharacteristic
90D349—22no—
91D349—41no—
92D349—42no—
93C349#57611yescharacteristic, complemented, Sylow
94#576—3491nomaximal, Sylow
95H(5,𝔽2)—3491no—
96H(5,𝔽2)—3491no—
97AGL(1,ℤ/8)—3492no—
98AGL(1,ℤ/8)—3492no—
99AGL(1,ℤ/8)—3491no—
100AGL(1,ℤ/8)D34911yescharacteristic, complemented
101#135—3491no—
102#135—3492no—
103C2×D8—3491no—
104C2×D8—3492no—
105#131—3491no—
106#41—3491no—
107#41—3492no—
108#41—3492no—
109#41—3491no—
110#41—6981no—
111#41—6981no—
112#41C2×D34911yescharacteristic, complemented
113C2×D4—3492no—
114C2×D4—3492no—
115C2×D4—3491no—
116C2×D4—6981no—
117C2×D4—6982no—
118C2×D4—6981no—
119C2×D4—6982no—
120C2×D4C2×D34911yescharacteristic, complemented
121Q16—3491no—
122SD16—3491no—
123SD16—3491no—
124SD16—3492no—
125SD16C2×D34912yescomplemented
126D8—3492no—
127D8—3491no—
128D8—3492no—
129D8—3492no—
130D8C2×D34912yescomplemented
131M16—3492no—
132M16C2×D34911yescharacteristic, complemented
133C2×C8—3491no—
134C2×C8—3492no—
135C23—21no—
136C23—6982no—
137C23—6982no—
138C23—6981no—
139Q8—3491no—
140Q8—3491no—
141Q8C22×D34911yescharacteristic
142D4—21no—
143D4—21no—
144D4—3492no—
145D4—3491no—
146D4—3492no—
147D4—3491no—
148D4—6982no—
149D4—6982no—
150D4—6982no—
151D4—6982no—
152D4—6981no—
153D4—6981no—
154D4C22×D34912yes—
155D4C22×D34911yescharacteristic
156C2×C4—21no—
157C2×C4—3492no—
158C2×C4—6982no—
159C2×C4—6982no—
160C2×C4—6981no—
161C2×C4C22×D34911yescharacteristic
162C8—3491no—
163C8—3491no—
164C8C22×D34912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—3492no—
169C22—6982no—
170C22—6981no—
171C22—27921no—
172C22—27922no—
173C22—27922no—
174C22D4×D34911yescharacteristic
175C4—21no—
176C4—3492no—
177C4—6981no—
178C4D4×D34911yescharacteristic
179C4C23×D34911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—13962no—
184C2—13961no—
185C2—6982no—
186C2C2×D4×D34911yescharacteristic
subgroupG/H
centerC2C2×D4×D349
commutator, cosocleC4×C349C24
FrattiniC4C23×D349
FittingAGL(1,ℤ/8)×C349C2
socleC2×C349C22×D4
hypercenterAGL(1,ℤ/8)D349
hypocenterC349#576
LCSG ▷ C4×C349 ▷ C2×C349 ▷ C349
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26013685C2semidirect
2. #26013684C2semidirect
3. #26013667C2semidirect
4. #26013666C2semidirect
5. #26013663C2semidirect
6. #26013660C2semidirect
7. #26013659C2semidirect
8. C2×#19150218C2semidirect
9. D8×D349C2semidirect
10. #26013647C2semidirect
11. AGL(1,ℤ/8)×C349C2semidirect
12. #19150234C22semidirect
15. #19150233C22semidirect
17. #19150232C22semidirect
19. C2×D1396C22semidirect
23. D4×D349C22semidirect
24. D4×D349C22semidirect
25. #19150223C22semidirect
26. #19150222C22semidirect
27. #19150221C22semidirect
28. #19150220C22semidirect
29. D2792C22semidirect
30. #19150218C22semidirect
31. #19150218C22semidirect
32. #19150218C22semidirect
33. #19150216C22semidirect
34. C2×#14584095C22semidirect
35. C8×D349C22semidirect
36. #41×C349C22semidirect
37. C2×D4×C349C22semidirect
38. SD16×C349C22semidirect
39. D8×C349C22semidirect
40. M16×C349C22semidirect
93. C349#576semidirect
100. AGL(1,ℤ/8)D349semidirect
112. #41C2×D349semidirect
120. C2×D4C2×D349semidirect
125. SD16C2×D349semidirect
130. D8C2×D349semidirect
132. M16C2×D349semidirect

Maximal subgroups

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

subgroupindex
#260136852
#260136842
#260136672
#260136662
#260136632
#260136602
#260136592
C2×#191502182
D8×D3492
#260136472
AGL(1,ℤ/8)×C3492
#576349