Finite Group Databasebeta

← #36352813

Subgroups — #36352813

Order 22144.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#26012156C211yesmaximal, characteristic
2#26011894C211yesmaximal, characteristic, complemented
3#26011772C211yesmaximal, characteristic
4#26011702C211yesmaximal, characteristic, complemented
5#26011221C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic
7#332×C173C211yesmaximal, characteristic, complemented
8C2×Q8×D173—21no—
9C2×Dic692—21no—
10#19149679C2211yescharacteristic
11#19149678—21no—
12#19149657—22no—
13#19149655—21no—
14C4×Dic346C2211yescharacteristic
15C4×D692C2211yescharacteristic
16#19149641—21no—
17#19149613—21no—
18#19149612—22no—
19#19149604—21no—
20#19149601—21no—
21C4×#13063436C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4⋊C8×C173C2211yescharacteristic
25C2×Dic346—22no—
26C2×Dic346—21no—
27C2×Dic346D411yescharacteristic
28Q8×D173—42no—
29Q8×D173—42no—
30C2×D692D411yescharacteristic
31C2×C4×D173—22no—
32C2×C4×D173—21no—
33Dic692—42no—
34C2×#13063436—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173D411yescharacteristic
38C4⋊Dic173D411yescharacteristic
39#14583805—22no—
40#14583805—21no—
41#14583803—22no—
42#14583803—21no—
43C4×Dic173—21no—
44C2×Q8×C173—21no—
45C2×C8×C173—21no—
46C2×C8×C173—21no—
47C4⋊C4×C173—21no—
48C4⋊C4×C173D412yes—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51Dic346—21no—
52Dic346—44no—
53Dic346—41no—
54Dic346—41no—
55Dic346D812yes—
56D692—21no—
57D692Q1612yescomplemented
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—41no—
62C4×D173—44no—
63C4×D173—41no—
64C4×D173—41no—
65#13063436—22no—
66Q8×C173—41no—
67Q8×C173—41no—
68C2×C4×C173—22no—
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173C2×D411yescharacteristic
72C8×C173—41no—
73C8×C173—41no—
74C22×C173#12011yescharacteristic
75C2×D173—22no—
76C2×D173—41no—
77C2×D173—41no—
78C4×C173—21no—
79C4×C173—42no—
80C4×C173C2×D811yescharacteristic
81C4×C173C2×Q1611yescharacteristic
82C4×C173AGL(1,ℤ/8)11yescharacteristic
83C4×C173#13711yescharacteristic
84Dic173—22no—
85Dic173—41no—
86Dic173—41no—
87Dic173—81no—
88C2×C173#44711yescharacteristic
89C2×C173#45111yescharacteristic
90C2×C173#45611yescharacteristic
91D173—42no—
92C173#164911yescharacteristic, complemented, Sylow
93#1649—1731nomaximal, Sylow
94D4×Q8—1731no—
95#500—1731no—
96#474—1731no—
97#462—1731no—
98#332D17311yescharacteristic, complemented
99Q8⋊C8—1731no—
100D4⋊C8—1731no—
101C22×Q8—3461no—
102C2×Q16—3461no—
103C4⋊Q8—3461no—
104C4⋊Q8—1731no—
105C4⋊Q8C2×D17311yescharacteristic
106C22⋊Q8—3461no—
107C22⋊Q8—3462no—
108C4×Q8—1731no—
109C4×D4—3461no—
110C4×D4—1731no—
111#107—3461no—
112#107—3462no—
113C4⋊C8C2×D17311yescharacteristic
114C4⋊C8C2×D17311yescharacteristic
115#103—3461no—
116#102—3461no—
117C4×C8—1731no—
118C2×Q8—21no—
119C2×Q8—3462no—
120C2×Q8—3461no—
121C2×Q8—1731no—
122C2×Q8—6922no—
123C2×Q8—6922no—
124C2×D4—1731no—
125C22×C4—3462no—
126C22×C4—3461no—
127Q16—6922no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3461no—
131C4⋊C4—21no—
132C4⋊C4—3462no—
133C4⋊C4—3461no—
134C4⋊C4—3461no—
135C4⋊C4—3461no—
136C4⋊C4—1731no—
137C4⋊C4—1731no—
138C4⋊C4D69212yes—
139C22⋊C4—3462no—
140C22⋊C4—3461no—
141C42—3461no—
142C42C22×D17311yescharacteristic
143C23—3461no—
144Q8—41no—
145Q8—41no—
146Q8—3461no—
147Q8—1732no—
148Q8—6924no—
149Q8—6921no—
150Q8—6921no—
151D4—3461no—
152D4—1732no—
153C2×C4—22no—
154C2×C4—3461no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4—6924no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4—6921no—
161C2×C4D4×D17311yescharacteristic
162C2×C4D4×D17311yescharacteristic
163C2×C4C2×D69211yescharacteristic
164C8—41no—
165C8—41no—
166C8—3462no—
167C22—3462no—
168C22—6921no—
169C22—6921no—
170C22#1914964911yescharacteristic
171C4—21no—
172C4—42no—
173C4—3462no—
174C4—13841no—
175C4—6921no—
176C4—6921no—
177C4D8×D17311yescharacteristic
178C4Q16×D17311yescharacteristic
179C4#1914970611yescharacteristic
180C4#1914970811yescharacteristic
181C2—6922no—
182C2#2601162611yescharacteristic
183C2#2601164811yescharacteristic
184C2#2601167611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#332×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#332D173
hypocenterC173#1649
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #26011894C2semidirect
4. #26011702C2semidirect
5. #26011221C2semidirect
7. #332×C173C2semidirect
57. D692Q16semidirect
92. C173#1649semidirect
98. #332D173semidirect

Maximal subgroups

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

subgroupindex
#260121562
#260118942
#260117722
#260117022
#260112212
#260112182
#332×C1732
#1649173