Finite Group Databasebeta

← #37508118

Subgroups — #37508118

Order 22912.

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×D716C211yesmaximal, characteristic
2#26038812C211yesmaximal, characteristic, complemented
3#26038730C211yesmaximal, characteristic, complemented
4#26038729C211yesmaximal, characteristic, complemented
5#26038182C211yesmaximal, characteristic, complemented
6#26038168C211yesmaximal, characteristic, complemented
7D4⋊C8×C179C211yesmaximal, characteristic, complemented
8C22×D716—21no—
9C2×D4×D179—21no—
10C2×#15668248—22no—
11#19153113C2211yescharacteristic, complemented
12#19153110C2211yescharacteristic
13#19153088—21no—
14#19153087—21no—
15#19153083—22no—
16C4×D716C2211yescharacteristic
17#19153045—21no—
18#19153044—21no—
19C4⋊(C179⋊C8)C2211yescharacteristic, complemented
20#19153035—21no—
21#19153034—21no—
22C4×D4×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic, complemented
24C4×C8×C179C2211yescharacteristic, complemented
25C23×D179—22no—
26C2×Dic358—21no—
27C2×D716—22no—
28C2×D716—21no—
29C2×D716—21no—
30C2×D716—42no—
31C2×D716—42no—
32C2×D716D411yescharacteristic
33C2×D716D412yes—
34C2×C22⋊D179—22no—
35D4×D179—42no—
36D4×D179—42no—
37C2×C4×D179—21no—
38#15668248—44no—
39C2×C179⋊C8—21no—
40C4⋊Dic179—21no—
41C4⋊Dic179—21no—
42C4⋊Dic179D411yescharacteristic
43(C2×D179)⋊C4—22no—
44(C2×D179)⋊C4—21no—
45C2×D4×C179D411yescharacteristic
46C22×C4×C179—21no—
47C2×C8×C179—21no—
48C2×C8×C179—21no—
49C4⋊C4×C179D411yescharacteristic
50C22⋊C4×C179—21no—
51C42×C179C2311yescharacteristic
52C22×D179—21no—
53C22×D179—22no—
54C22×D179—44no—
55C22×D179—44no—
56C22×D179—82no—
57Dic358—42no—
58D716—21no—
59D716—42no—
60D716—44no—
61D716—41no—
62D716—41no—
63D716—41no—
64D716—41no—
65D716SD1612yescomplemented
66C22⋊D179—44no—
67C2×Dic179—21no—
68C2×Dic179—41no—
69C4×D179—41no—
70C4×D179—41no—
71C179⋊C8—41no—
72C23×C179—21no—
73D4×C179—21no—
74D4×C179SD1612yescomplemented
75C2×C4×C179—21no—
76C2×C4×C179—41no—
77C2×C4×C179—41no—
78C2×C4×C179C2×D411yescharacteristic
79C2×C4×C179C2×D411yescharacteristic
80C2×C4×C179C2×D411yescharacteristic
81C8×C179—22no—
82C8×C179—41no—
83C22×C179—22no—
84C22×C179—41no—
85C22×C179—41no—
86C22×C179#12011yescharacteristic
87C2×D179—22no—
88C2×D179—41no—
89C2×D179—41no—
90C2×D179—42no—
91C2×D179—42no—
92C2×D179—42no—
93C2×D179—88no—
94C4×C179—21no—
95C4×C179—41no—
96C4×C179C2×SD1611yescharacteristic
97C4×C179C2×SD1611yescharacteristic
98C4×C179AGL(1,ℤ/8)11yescharacteristic
99C4×C179AGL(1,ℤ/8)11yescharacteristic
100Dic179—41no—
101Dic179—81no—
102C2×C179—42no—
103C2×C179#45011yescharacteristic
104C2×C179#45011yescharacteristic
105C2×C179#45311yescharacteristic
106D179—42no—
107D179—82no—
108C179#162911yescharacteristic, complemented, Sylow
109#1629—1791nomaximal, Sylow
110D42—1791no—
111#492—1791no—
112#476—1791no—
113#476—1791no—
114#331—1791no—
115D4⋊C8—1791no—
116D4⋊C8D17911yescharacteristic, complemented
117C22×D4—3581no—
118C22×D4—3581no—
119C2×SD16—3582no—
120C4⋊Q8—1791no—
121#127—1791no—
122#121—3581no—
123#121—3581no—
124#120—3582no—
125C4×D4—1791no—
126C4×D4C2×D17911yescharacteristic
127#106—3581no—
128#106—3581no—
129C4⋊C8—1791no—
130C4⋊C8C2×D17911yescharacteristic, complemented
131#102—3581no—
132#102—3581no—
133C4×C8C2×D17911yescharacteristic, complemented
134C24—3582no—
135C2×Q8—3581no—
136C2×D4—3582no—
137C2×D4—3582no—
138C2×D4—3581no—
139C2×D4—3581no—
140C2×D4—1791no—
141C2×D4—1792no—
142C2×D4—7162no—
143C2×D4—7162no—
144C2×D4—7162no—
145C2×D4—7162no—
146C2×D4D71611yescharacteristic
147C22×C4—21no—
148C22×C4—3581no—
149SD16—7164no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—3581no—
153C4⋊C4—3581no—
154C4⋊C4—3581no—
155C4⋊C4—1791no—
156C4⋊C4D71611yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3582no—
159C22⋊C4—3581no—
160C42C22×D17911yescharacteristic
161C23—21no—
162C23—3581no—
163C23—3582no—
164C23—14322no—
165C23—7164no—
166C23—7164no—
167Q8—7162no—
168D4—21no—
169D4—3581no—
170D4—1792no—
171D4—7162no—
172D4—7164no—
173D4—7164no—
174D4—7161no—
175D4—7161no—
176D4—7161no—
177D4—7161no—
178D4#1566824812yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—3581no—
183C2×C4—7161no—
184C2×C4—7161no—
185C2×C4—7161no—
186C2×C4D4×D17911yescharacteristic
187C2×C4D4×D17911yescharacteristic
188C2×C4C2×D71611yescharacteristic
189C8—22no—
190C8—41no—
191C8—7161no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—3582no—
196C22—14328no—
197C22—7162no—
198C22—7162no—
199C22—7162no—
200C22—7161no—
201C22—7161no—
202C22#1915308311yescharacteristic
203C4—21no—
204C4—41no—
205C4—14321no—
206C4—7161no—
207C4SD16×D17911yescharacteristic
208C4C2×#1566824811yescharacteristic
209C4#1915313711yescharacteristic
210C4#1915314111yescharacteristic
211C2—42no—
212C2—14322no—
213C2—7162no—
214C2#2603859111yescharacteristic
215C2#2603859211yescharacteristic
216C2#2603860811yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
FittingD4⋊C8×C179C2
socleC22×C179#120
prime radicalD4×D716C2
hypercenterD4⋊C8D179
hypocenterC179#1629
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #26038812C2semidirect
3. #26038730C2semidirect
4. #26038729C2semidirect
5. #26038182C2semidirect
6. #26038168C2semidirect
7. D4⋊C8×C179C2semidirect
11. #19153113C22semidirect
19. C4⋊(C179⋊C8)C22semidirect
23. C4⋊C8×C179C22semidirect
24. C4×C8×C179C22semidirect
65. D716SD16semidirect
74. D4×C179SD16semidirect
108. C179#1629semidirect
116. D4⋊C8D179semidirect
130. C4⋊C8C2×D179semidirect
133. C4×C8C2×D179semidirect
178. D4#15668248semidirect

Maximal subgroups

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

subgroupindex
D4×D7162
#260388122
#260387302
#260387292
#260381822
#260381682
D4⋊C8×C1792
#1629179