Finite Group Databasebeta

← #37508212

Subgroups — #37508212

Order 22912.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039149C211yesmaximal, characteristic
2#26038791C211yesmaximal, characteristic, complemented
3#26038716C211yesmaximal, characteristic
4#26038187C211yesmaximal, characteristic, complemented
5Q8⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic
7#462×C179C211yesmaximal, characteristic, complemented
8#19153113C2211yescharacteristic
9#19153105—21no—
10#19153091—22no—
11#19153089—21no—
12C4×D716—21no—
13C4×D716C2211yescharacteristic
14C4⋊C4×D179—21no—
15#19153046—21no—
16C4⋊(C179⋊C8)C2211yescharacteristic
17#19153036—22no—
18#19153033—21no—
19C4×C179⋊C8C2211yescharacteristic
20C2×Q16×C179—21no—
21C4⋊Q8×C179C2211yescharacteristic
22C4×Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24#103×C179—21no—
25C2×Dic358D412yes—
26C2×D716D411yescharacteristic
27C2×C4×D179—22no—
28C2×C4×D179—21no—
29C2×C179⋊C8—21no—
30C2×C179⋊C8—21no—
31C4⋊Dic179—21no—
32C4⋊Dic179—21no—
33C4⋊Dic179D411yescharacteristic
34Dic179⋊C4—22no—
35Dic179⋊C4—22no—
36(C2×D179)⋊C4—22no—
37(C2×D179)⋊C4—21no—
38C2×Q8×C179—21no—
39C2×Q8×C179D411yescharacteristic
40Q16×C179—42no—
41C2×C8×C179—21no—
42C4⋊C4×C179—21no—
43C4⋊C4×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179D411yescharacteristic
46C42×C179—21no—
47C42×C179C2311yescharacteristic
48C22×D179—21no—
49Dic358—42no—
50D716—21no—
51D716Q1612yescomplemented
52C2×Dic179—21no—
53C2×Dic179—22no—
54C4×D179—41no—
55C4×D179—41no—
56C4×D179—82no—
57C179⋊C8—41no—
58C179⋊C8—41no—
59Q8×C179—21no—
60Q8×C179—21no—
61Q8×C179—41no—
62Q8×C179—41no—
63C2×C4×C179—21no—
64C2×C4×C179—21no—
65C2×C4×C179—41no—
66C2×C4×C179C2×D411yescharacteristic
67C2×C4×C179C2×D411yescharacteristic
68C2×C4×C179C2×D411yescharacteristic
69C8×C179—41no—
70C22×C179#12011yescharacteristic
71C2×D179—22no—
72C2×D179—41no—
73C2×D179—41no—
74C4×C179—21no—
75C4×C179—41no—
76C4×C179—41no—
77C4×C179—81no—
78C4×C179C2×Q1611yescharacteristic
79C4×C179#13511yescharacteristic
80C4×C179AGL(1,ℤ/8)11yescharacteristic
81C4×C179#13711yescharacteristic
82Dic179—41no—
83Dic179—42no—
84C2×C179#44911yescharacteristic
85C2×C179#45111yescharacteristic
86C2×C179#45511yescharacteristic
87D179—42no—
88C179#164411yescharacteristic, complemented, Sylow
89#1644—1791nomaximal, Sylow
90#554—1791no—
91#487—1791no—
92#474—1791no—
93#462D17911yescharacteristic, complemented
94#333—1791no—
95Q8⋊C8—1791no—
96D4⋊C8—1791no—
97C2×Q16—21no—
98C4⋊Q8—1791no—
99C4⋊Q8C2×D17911yescharacteristic
100#125—3581no—
101C22⋊Q8—3582no—
102C22⋊Q8—3581no—
103C4×Q8C2×D17911yescharacteristic
104C4×D4—3581no—
105C4×D4—1791no—
106C2×C4⋊C4—3581no—
107#107—3581no—
108C4⋊C8—1791no—
109C4⋊C8C2×D17911yescharacteristic
110#103—21no—
111#103—3582no—
112#102—3581no—
113C4×C8—1791no—
114C2×Q8—21no—
115C2×Q8—1792no—
116C2×Q8C22⋊D17911yescharacteristic
117C2×D4—1791no—
118C22×C4—3582no—
119C22×C4—3581no—
120Q16—42no—
121C2×C8—21no—
122C2×C8—3581no—
123C2×C8—3581no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3582no—
128C4⋊C4—3582no—
129C4⋊C4—3581no—
130C4⋊C4—3581no—
131C4⋊C4—1791no—
132C4⋊C4C22⋊D17911yescharacteristic
133C22⋊C4—3582no—
134C22⋊C4—3581no—
135C42—21no—
136C42C22×D17911yescharacteristic
137C23—3581no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—7162no—
143D4—3581no—
144D4—1792no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—3581no—
149C2×C4—3582no—
150C2×C4—14322no—
151C2×C4—7161no—
152C2×C4—7161no—
153C2×C4D4×D17911yescharacteristic
154C2×C4D4×D17911yescharacteristic
155C2×C4C2×C22⋊D17911yescharacteristic
156C8—41no—
157C8—7161no—
158C8—7161no—
159C22—3582no—
160C22—7161no—
161C22—7161no—
162C22#1915308111yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—7161no—
168C4—7162no—
169C4Q16×D17911yescharacteristic
170C4#1915313311yescharacteristic
171C4#1915313911yescharacteristic
172C4#1915314511yescharacteristic
173C2—7162no—
174C2#2603858211yescharacteristic
175C2#2603859311yescharacteristic
176C2#2603861511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#462×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#462D179
hypocenterC179#1644
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #26038791C2semidirect
4. #26038187C2semidirect
5. Q8⋊(C179⋊C8)C2semidirect
7. #462×C179C2semidirect
51. D716Q16semidirect
88. C179#1644semidirect
93. #462D179semidirect

Maximal subgroups

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

subgroupindex
#260391492
#260387912
#260387162
#260381872
Q8⋊(C179⋊C8)2
#260381662
#462×C1792
#1644179