Finite Group Databasebeta

← #41003639

Subgroups — #41003639

Order 26944.

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

normal subgroups only

This group has 262 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
1C2×#20269772C211yesmaximal, characteristic, complemented
2#41×D421C211yesmaximal, characteristic, complemented
3#27212398C212yesmaximal, complemented
4#27212397C211yesmaximal, characteristic, complemented
5#27212394C211yesmaximal, characteristic, complemented
6#27212393C212yesmaximal, complemented
7#27212391C211yesmaximal, characteristic, complemented
8#27212389C212yesmaximal, complemented
9#27212387C212yesmaximal, complemented
10M16×D421C211yesmaximal, characteristic, complemented
11#137×C421C211yesmaximal, characteristic, complemented
12#20269772—21no—
13#20269772—21no—
14#20269772C2212yes—
15#20269772C2212yescomplemented
16#20269772C2211yescharacteristic, complemented
17#20269771—21no—
18#20269771C2211yescharacteristic
19#20269770—21no—
20#20269770C2211yescharacteristic, complemented
21Q8×D421C2211yescharacteristic, complemented
22C2×D1684—21no—
23C2×D1684C2211yescharacteristic, complemented
24D4×D421—21no—
25D4×D421C2211yescharacteristic, complemented
26C2×C4×D421—21no—
27C2×C4×D421—21no—
28C2×C4×D421C2211yescharacteristic
29#20269761C2212yescomplemented
30#20269760C2212yescomplemented
31#20269758C2212yescomplemented
32#20269758C2212yescomplemented
33D3368C2212yescomplemented
34#20269756C2212yescomplemented
35#20269755C2211yescharacteristic, complemented
36#20269754C2212yescomplemented
37C8×D421C2212yescomplemented
38#41×C421C2211yescharacteristic
39C2×Q8×C421C2211yescharacteristic, complemented
40Q16×C421C2212yescomplemented
41SD16×C421C2212yescomplemented
42M16×C421C2211yescharacteristic, complemented
43C22×D421—21no—
44C22×D421—21no—
45C22×D421D411yescharacteristic
46Dic842—21no—
47Dic842C2311yescharacteristic
48D1684—21no—
49D1684—22no—
50D1684—22no—
51D1684—21no—
52D1684C2312yes—
53D1684C2311yescharacteristic
54C22⋊D421—21no—
55C22⋊D421—21no—
56C22⋊D421—21no—
57C2×Dic421—21no—
58C2×Dic421D411yescharacteristic
59C4×D421—21no—
60C4×D421—21no—
61C4×D421—21no—
62C4×D421—22no—
63C4×D421—22no—
64C4×D421D411yescharacteristic
65C4×D421D411yescharacteristic
66C4×D421C2311yescharacteristic
67C4×D421C2311yescharacteristic
68#16809890C2312yes—
69Q8×C421—21no—
70Q8×C421C2312yes—
71Q8×C421C2311yescharacteristic
72D4×C421—21no—
73D4×C421C2311yescharacteristic
74C2×C4×C421—21no—
75C2×C4×C421—21no—
76C2×C4×C421C2311yescharacteristic
77C8×C421C2312yes—
78C22×C421—21no—
79C22×C421C2×D411yescharacteristic
80C2×D421—21no—
81C2×D421—22no—
82C2×D421—41no—
83C2×D421—42no—
84C2×D421—42no—
85C2×D421C2×D411yescharacteristic
86C2×D421C2×D411yescharacteristic
87C4×C421—21no—
88C4×C421—22no—
89C4×C421C2×D411yescharacteristic
90C4×C421C2411yescharacteristic
91Dic421—21no—
92Dic421C2×D411yescharacteristic
93Dic421C2×D411yescharacteristic
94C2×C421—21no—
95C2×C421—41no—
96C2×C421C22×D411yescharacteristic
97D421—21no—
98D421—21no—
99D421—41no—
100D421—42no—
101C421#57511yescharacteristic, complemented, Sylow
102#575—4211nomaximal, Sylow
103C2×#41—4211no—
104C2×#41—4211no—
105#137—4211no—
106#137—4212no—
107#137D42111yescharacteristic, complemented
108AGL(1,ℤ/8)—4211no—
109AGL(1,ℤ/8)—4212no—
110AGL(1,ℤ/8)—4211no—
111#135—4212no—
112#135—4212no—
113C2×M16—4211no—
114#41—8421no—
115#41—8421no—
116#41—8421no—
117#41—8421no—
118#41—4211no—
119#41—4212no—
120#41—4212no—
121#41—4211no—
122#41—4211no—
123#41C2×D42111yescharacteristic
124C2×Q8—4211no—
125C2×Q8C2×D42111yescharacteristic, complemented
126C2×D4—8421no—
127C2×D4—8421no—
128C2×D4—4211no—
129C2×D4—4211no—
130C22×C4—8421no—
131C22×C4—8421no—
132C22×C4—4211no—
133Q16—4212no—
134Q16C2×D42112yescomplemented
135SD16—4212no—
136SD16—4212no—
137SD16—4212no—
138SD16C2×D42112yescomplemented
139D8—4212no—
140D8—4212no—
141M16—4211no—
142M16—4212no—
143M16C2×D42111yescharacteristic, complemented
144C2×C8—4212no—
145C23—8421no—
146C23—8421no—
147C23—4211no—
148Q8—21no—
149Q8—8421no—
150Q8—4211no—
151Q8C22×D42111yescharacteristic
152Q8C22×D42112yes—
153D4—21no—
154D4—8421no—
155D4—8422no—
156D4—8422no—
157D4—8421no—
158D4—8421no—
159D4—8421no—
160D4—8421no—
161D4—4212no—
162D4—4211no—
163D4C22×D42111yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—8421no—
167C2×C4—8421no—
168C2×C4—8422no—
169C2×C4—8422no—
170C2×C4—8421no—
171C2×C4—8421no—
172C2×C4—4211no—
173C2×C4—4211no—
174C2×C4—4211no—
175C2×C4—4211no—
176C2×C4—4211no—
177C2×C4C22×D42111yescharacteristic
178C8—4212no—
179C8C22×D42112yes—
180C22—21no—
181C22—8422no—
182C22—8421no—
183C22—16841no—
184C22—16842no—
185C22—16842no—
186C22—4211no—
187C22—4211no—
188C22D4×D42111yescharacteristic
189C4—21no—
190C4—22no—
191C4—8421no—
192C4—4211no—
193C4—4211no—
194C4D4×D42111yescharacteristic
195C4C23×D42111yescharacteristic
196C2—21no—
197C2—41no—
198C2—8421no—
199C2—8421no—
200C2—16842no—
201C2—16841no—
202C2C2×D4×D42111yescharacteristic
subgroupG/H
centerC2C2×D4×D421
commutator, cosocleC4×C421C24
FrattiniC4C23×D421
Fitting#137×C421C2
socleC2×C421C22×D4
hypercenter#137D421
hypocenterC421#575
LCSG ▷ C4×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#20269772C2semidirect
2. #41×D421C2semidirect
3. #27212398C2semidirect
4. #27212397C2semidirect
5. #27212394C2semidirect
6. #27212393C2semidirect
7. #27212391C2semidirect
8. #27212389C2semidirect
9. #27212387C2semidirect
10. M16×D421C2semidirect
11. #137×C421C2semidirect
15. #20269772C22semidirect
16. #20269772C22semidirect
20. #20269770C22semidirect
21. Q8×D421C22semidirect
23. C2×D1684C22semidirect
25. D4×D421C22semidirect
29. #20269761C22semidirect
30. #20269760C22semidirect
31. #20269758C22semidirect
32. #20269758C22semidirect
33. D3368C22semidirect
34. #20269756C22semidirect
35. #20269755C22semidirect
36. #20269754C22semidirect
37. C8×D421C22semidirect
39. C2×Q8×C421C22semidirect
40. Q16×C421C22semidirect
41. SD16×C421C22semidirect
42. M16×C421C22semidirect
101. C421#575semidirect
107. #137D421semidirect
125. C2×Q8C2×D421semidirect
134. Q16C2×D421semidirect
138. SD16C2×D421semidirect
143. M16C2×D421semidirect

Maximal subgroups

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

subgroupindex
C2×#202697722
#41×D4212
#272123982
#272123972
#272123942
#272123932
#272123912
#272123892
#272123872
M16×D4212
#137×C4212
#575421