Finite Group Databasebeta

← #30665079

Subgroups — #30665079

Order 16768.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D524C211yesmaximal, characteristic
2#22553685C211yesmaximal, characteristic, complemented
3#22553603C211yesmaximal, characteristic, complemented
4#22553602C211yesmaximal, characteristic, complemented
5#22553055C211yesmaximal, characteristic, complemented
6#22553041C211yesmaximal, characteristic, complemented
7D4⋊C8×C131C211yesmaximal, characteristic, complemented
8C22×D524—21no—
9C2×D4×D131—21no—
10C2×#14523502—22no—
11#16862246C2211yescharacteristic, complemented
12#16862243C2211yescharacteristic
13#16862221—21no—
14#16862220—21no—
15#16862216—22no—
16C4×D524C2211yescharacteristic
17#16862178—21no—
18#16862177—21no—
19C4⋊(C131⋊C8)C2211yescharacteristic, complemented
20#16862168—21no—
21#16862167—21no—
22C4×D4×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic, complemented
24C4×C8×C131C2211yescharacteristic, complemented
25C23×D131—22no—
26C2×Dic262—21no—
27C2×D524—22no—
28C2×D524—21no—
29C2×D524—21no—
30C2×D524—42no—
31C2×D524—42no—
32C2×D524D411yescharacteristic
33C2×D524D412yes—
34C2×C22⋊D131—22no—
35D4×D131—42no—
36D4×D131—42no—
37C2×C4×D131—21no—
38#14523502—44no—
39C2×C131⋊C8—21no—
40C4⋊Dic131—21no—
41C4⋊Dic131—21no—
42C4⋊Dic131D411yescharacteristic
43(C2×D131)⋊C4—22no—
44(C2×D131)⋊C4—21no—
45C2×D4×C131D411yescharacteristic
46C22×C4×C131—21no—
47C2×C8×C131—21no—
48C2×C8×C131—21no—
49C4⋊C4×C131D411yescharacteristic
50C22⋊C4×C131—21no—
51C42×C131C2311yescharacteristic
52C22×D131—21no—
53C22×D131—22no—
54C22×D131—44no—
55C22×D131—44no—
56C22×D131—82no—
57Dic262—42no—
58D524—21no—
59D524—42no—
60D524—44no—
61D524—41no—
62D524—41no—
63D524—41no—
64D524—41no—
65D524SD1612yescomplemented
66C22⋊D131—44no—
67C2×Dic131—21no—
68C2×Dic131—41no—
69C4×D131—41no—
70C4×D131—41no—
71C131⋊C8—41no—
72C23×C131—21no—
73D4×C131—21no—
74D4×C131SD1612yescomplemented
75C2×C4×C131—21no—
76C2×C4×C131—41no—
77C2×C4×C131—41no—
78C2×C4×C131C2×D411yescharacteristic
79C2×C4×C131C2×D411yescharacteristic
80C2×C4×C131C2×D411yescharacteristic
81C8×C131—22no—
82C8×C131—41no—
83C22×C131—22no—
84C22×C131—41no—
85C22×C131—41no—
86C22×C131#12011yescharacteristic
87C2×D131—22no—
88C2×D131—41no—
89C2×D131—41no—
90C2×D131—42no—
91C2×D131—42no—
92C2×D131—42no—
93C2×D131—88no—
94C4×C131—21no—
95C4×C131—41no—
96C4×C131C2×SD1611yescharacteristic
97C4×C131C2×SD1611yescharacteristic
98C4×C131AGL(1,ℤ/8)11yescharacteristic
99C4×C131AGL(1,ℤ/8)11yescharacteristic
100Dic131—41no—
101Dic131—81no—
102C2×C131—42no—
103C2×C131#45011yescharacteristic
104C2×C131#45011yescharacteristic
105C2×C131#45311yescharacteristic
106D131—42no—
107D131—82no—
108C131#162911yescharacteristic, complemented, Sylow
109#1629—1311nomaximal, Sylow
110D42—1311no—
111#492—1311no—
112#476—1311no—
113#476—1311no—
114#331—1311no—
115D4⋊C8—1311no—
116D4⋊C8D13111yescharacteristic, complemented
117C22×D4—2621no—
118C22×D4—2621no—
119C2×SD16—2622no—
120C4⋊Q8—1311no—
121#127—1311no—
122#121—2621no—
123#121—2621no—
124#120—2622no—
125C4×D4—1311no—
126C4×D4C2×D13111yescharacteristic
127#106—2621no—
128#106—2621no—
129C4⋊C8—1311no—
130C4⋊C8C2×D13111yescharacteristic, complemented
131#102—2621no—
132#102—2621no—
133C4×C8C2×D13111yescharacteristic, complemented
134C24—2622no—
135C2×Q8—2621no—
136C2×D4—2622no—
137C2×D4—2622no—
138C2×D4—2621no—
139C2×D4—2621no—
140C2×D4—5242no—
141C2×D4—5242no—
142C2×D4—5242no—
143C2×D4—5242no—
144C2×D4—1311no—
145C2×D4—1312no—
146C2×D4D52411yescharacteristic
147C22×C4—21no—
148C22×C4—2621no—
149SD16—5244no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—2621no—
153C4⋊C4—2621no—
154C4⋊C4—2621no—
155C4⋊C4—1311no—
156C4⋊C4D52411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—2622no—
159C22⋊C4—2621no—
160C42C22×D13111yescharacteristic
161C23—21no—
162C23—2621no—
163C23—2622no—
164C23—5244no—
165C23—5244no—
166C23—10482no—
167Q8—5242no—
168D4—21no—
169D4—2621no—
170D4—5242no—
171D4—5244no—
172D4—5244no—
173D4—5241no—
174D4—5241no—
175D4—5241no—
176D4—5241no—
177D4—1312no—
178D4#1452350212yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—2621no—
183C2×C4—5241no—
184C2×C4—5241no—
185C2×C4—5241no—
186C2×C4D4×D13111yescharacteristic
187C2×C4D4×D13111yescharacteristic
188C2×C4C2×D52411yescharacteristic
189C8—22no—
190C8—41no—
191C8—5241no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—2622no—
196C22—5241no—
197C22—5241no—
198C22—5242no—
199C22—5242no—
200C22—5242no—
201C22—10488no—
202C22#1686221611yescharacteristic
203C4—21no—
204C4—41no—
205C4—5241no—
206C4—10481no—
207C4SD16×D13111yescharacteristic
208C4C2×#1452350211yescharacteristic
209C4#1686227011yescharacteristic
210C4#1686227411yescharacteristic
211C2—42no—
212C2—5242no—
213C2—10482no—
214C2#2255346411yescharacteristic
215C2#2255346511yescharacteristic
216C2#2255348111yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
FittingD4⋊C8×C131C2
socleC22×C131#120
prime radicalD4×D524C2
hypercenterD4⋊C8D131
hypocenterC131#1629
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22553685C2semidirect
3. #22553603C2semidirect
4. #22553602C2semidirect
5. #22553055C2semidirect
6. #22553041C2semidirect
7. D4⋊C8×C131C2semidirect
11. #16862246C22semidirect
19. C4⋊(C131⋊C8)C22semidirect
23. C4⋊C8×C131C22semidirect
24. C4×C8×C131C22semidirect
65. D524SD16semidirect
74. D4×C131SD16semidirect
108. C131#1629semidirect
116. D4⋊C8D131semidirect
130. C4⋊C8C2×D131semidirect
133. C4×C8C2×D131semidirect
178. D4#14523502semidirect

Maximal subgroups

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

subgroupindex
D4×D5242
#225536852
#225536032
#225536022
#225530552
#225530412
D4⋊C8×C1312
#1629131