Finite Group Databasebeta

← #31780041

Subgroups — #31780041

Order 17536.

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

normal subgroups only

This group has 234 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×Dic274C211yesmaximal, characteristic
2#22577808C211yesmaximal, characteristic, complemented
3#22577683C211yesmaximal, characteristic
4#22577614C211yesmaximal, characteristic, complemented
5#22577150C211yesmaximal, characteristic, complemented
6#22577135C211yesmaximal, characteristic, complemented
7D4⋊C8×C137C211yesmaximal, characteristic
8C22×Dic274—21no—
9C2×Dic548—21no—
10#17984088C2211yescharacteristic
11#17984088C2211yescharacteristic
12#17984087—21no—
13#17984067—21no—
14#17984065—22no—
15C4×Dic274C2211yescharacteristic
16D4×Dic137—21no—
17#17984022—22no—
18#17984021—21no—
19C4⋊#11764040C2211yescharacteristic
20#17984013—21no—
21#17984009—21no—
22C4×D4×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25C2×Dic274—22no—
26C2×Dic274—21no—
27C2×Dic274—21no—
28C2×Dic274—42no—
29C2×Dic274—42no—
30C2×Dic274D411yescharacteristic
31C22×Dic137—22no—
32Dic548—42no—
33C2×#11764040—21no—
34C4⋊Dic137—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137D411yescharacteristic
38C4⋊Dic137D412yes—
39#14555489—22no—
40#14555489—21no—
41C22⋊Dic137—22no—
42C4×Dic137—21no—
43C2×D4×C137D411yescharacteristic
44C22×C4×C137—21no—
45C2×C8×C137—21no—
46C2×C8×C137—21no—
47C4⋊C4×C137D411yescharacteristic
48C22⋊C4×C137—21no—
49C42×C137C2311yescharacteristic
50Dic274—21no—
51Dic274—44no—
52Dic274—41no—
53Dic274—41no—
54Dic274—41no—
55Dic274—41no—
56Dic274D812yes—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137—22no—
60C2×Dic137—44no—
61C2×Dic137—41no—
62#11764040—41no—
63C23×C137—21no—
64D4×C137—21no—
65D4×C137Q1612yescomplemented
66C2×C4×C137—21no—
67C2×C4×C137—41no—
68C2×C4×C137—41no—
69C2×C4×C137C2×D411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C8×C137—22no—
73C8×C137—41no—
74C22×C137—22no—
75C22×C137—41no—
76C22×C137—41no—
77C22×C137#12011yescharacteristic
78C4×C137—21no—
79C4×C137—41no—
80C4×C137C2×D811yescharacteristic
81C4×C137C2×Q1611yescharacteristic
82C4×C137AGL(1,ℤ/8)11yescharacteristic
83C4×C137#13711yescharacteristic
84Dic137—22no—
85Dic137—42no—
86Dic137—41no—
87Dic137—81no—
88C2×C137—42no—
89C2×C137#44711yescharacteristic
90C2×C137#45111yescharacteristic
91C2×C137#45611yescharacteristic
92C137#164911yescharacteristic, complemented, Sylow
93#1649—1371nomaximal, Sylow
94D4×Q8—1371no—
95#500—1371no—
96#474—1371no—
97#462—1371no—
98#332—1371no—
99Q8⋊C8—1371no—
100D4⋊C8D13711yescharacteristic
101C22×Q8—2741no—
102C2×Q16—2741no—
103C4⋊Q8—2741no—
104C4⋊Q8—1371no—
105C4⋊Q8—1371no—
106C22⋊Q8—2742no—
107C22⋊Q8—2741no—
108C4×Q8—1371no—
109C4×D4—2741no—
110C4×D4C2×D13711yescharacteristic
111#107—2742no—
112#107—2741no—
113C4⋊C8—1371no—
114C4⋊C8C2×D13711yescharacteristic
115#103—2741no—
116#102—2741no—
117C4×C8C2×D13711yescharacteristic
118C2×Q8—2742no—
119C2×Q8—2741no—
120C2×Q8—2741no—
121C2×Q8—5482no—
122C2×Q8—5482no—
123C2×Q8—1371no—
124C2×D4D54811yescharacteristic
125C22×C4—21no—
126C22×C4—2742no—
127Q16—5482no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2741no—
131C4⋊C4—2742no—
132C4⋊C4—2741no—
133C4⋊C4—2741no—
134C4⋊C4—2741no—
135C4⋊C4—2741no—
136C4⋊C4—1371no—
137C4⋊C4—1372no—
138C4⋊C4D54811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2742no—
141C42—2741no—
142C42C22×D13711yescharacteristic
143C23—21no—
144Q8—2741no—
145Q8—5484no—
146Q8—5481no—
147Q8—5481no—
148Q8—5481no—
149Q8—5481no—
150Q8—1372no—
151D4—21no—
152D4Dic54812yescomplemented
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—2742no—
159C2×C4—5484no—
160C2×C4—5481no—
161C2×C4D4×D13711yescharacteristic
162C2×C4D4×D13711yescharacteristic
163C2×C4C2×D54811yescharacteristic
164C8—22no—
165C8—41no—
166C8—5481no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1798405811yescharacteristic
171C4—21no—
172C4—41no—
173C4—2742no—
174C4—5482no—
175C4—5481no—
176C4—10961no—
177C4D8×D13711yescharacteristic
178C4C2×Dic54811yescharacteristic
179C4#1798411511yescharacteristic
180C4#1798412111yescharacteristic
181C2—42no—
182C2#2257753811yescharacteristic
183C2#2257756111yescharacteristic
184C2#2257758711yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingD4⋊C8×C137C2
socleC22×C137#120
prime radicalC2×D4×C137D4
hypercenterD4⋊C8D137
hypocenterC137#1649
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22577808C2semidirect
4. #22577614C2semidirect
5. #22577150C2semidirect
6. #22577135C2semidirect
65. D4×C137Q16semidirect
92. C137#1649semidirect
152. D4Dic548semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2742
#225778082
#225776832
#225776142
#225771502
#225771352
D4⋊C8×C1372
#1649137