Finite Group Databasebeta

← #36352631

Subgroups — #36352631

Order 22144.

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

normal subgroups only

This group has 278 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#26012162C211yesmaximal, characteristic, complemented
2#26011868C211yesmaximal, characteristic, complemented
3#26011692C211yesmaximal, characteristic, complemented
4#26011686C211yesmaximal, characteristic, complemented
5#26011234C211yesmaximal, characteristic, complemented
6#26011223C211yesmaximal, characteristic, complemented
7D4⋊C8×C173C211yesmaximal, characteristic, complemented
8C2×#14583826—21no—
9C2×#14583815—21no—
10C2×D1384—22no—
11C2×#14583811—21no—
12#19149676C2211yescharacteristic
13#19149676C2211yescharacteristic
14#19149675—21no—
15#19149653—21no—
16#19149652—22no—
17C4×Dic346C2211yescharacteristic
18#19149641—21no—
19C4⋊#13063436C2211yescharacteristic
20#19149601—21no—
21#19149599—21no—
22C4×D4×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4×C8×C173C2211yescharacteristic
25#14583826—42no—
26#14583826—42no—
27C2×Dic346D411yescharacteristic
28C2×D692—21no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692—21no—
32C2×D692—21no—
33C2×D692D412yes—
34C2×C22⋊D173—22no—
35C2×C22⋊D173—22no—
36C2×C4×D173—22no—
37#14583815—42no—
38D1384—44no—
39#14583811—42no—
40C2×#13063436—21no—
41C4⋊Dic173D411yescharacteristic
42#14583805—21no—
43#14583803—22no—
44C4×Dic173—21no—
45C2×D4×C173D411yescharacteristic
46C22×C4×C173—21no—
47C2×C8×C173—21no—
48C2×C8×C173—21no—
49C4⋊C4×C173D411yescharacteristic
50C22⋊C4×C173—21no—
51C42×C173C2311yescharacteristic
52C22×D173—22no—
53C22×D173—41no—
54Dic346—21no—
55Dic346D812yescomplemented
56D692—42no—
57D692—42no—
58D692—41no—
59D692—41no—
60D692—41no—
61D692—41no—
62D692—41no—
63D692—41no—
64D692—41no—
65D692—41no—
66C22⋊D173—44no—
67C22⋊D173—82no—
68C2×Dic173—21no—
69C2×Dic173—21no—
70C4×D173—44no—
71#13063436—41no—
72C23×C173—21no—
73D4×C173—21no—
74D4×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—41no—
77C2×C4×C173—41no—
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C8×C173—22no—
82C8×C173—41no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173#12011yescharacteristic
87C2×D173—42no—
88C2×D173—42no—
89C2×D173—42no—
90C2×D173—81no—
91C2×D173—81no—
92C2×D173—81no—
93C4×C173—21no—
94C4×C173—41no—
95C4×C173C2×D811yescharacteristic
96C4×C173#13511yescharacteristic
97C4×C173AGL(1,ℤ/8)11yescharacteristic
98C4×C173AGL(1,ℤ/8)11yescharacteristic
99Dic173—22no—
100Dic173—41no—
101C2×C173—81no—
102C2×C173#44711yescharacteristic
103C2×C173#44911yescharacteristic
104C2×C173#45311yescharacteristic
105D173—82no—
106D173—161no—
107C173#162111yescharacteristic, complemented, Sylow
108#1621—1731nomaximal, Sylow
109#550—1731no—
110#493—1731no—
111#460—1731no—
112#459—1731no—
113#331—1731no—
114Q8⋊C8—1731no—
115D4⋊C8D17311yescharacteristic, complemented
116C2×#41—3461no—
117C2×SD16—3461no—
118C2×D8—3462no—
119C2×D8—3461no—
120#127—3461no—
121#127—1731no—
122#127—1731no—
123#121—3462no—
124#121—3461no—
125C4×Q8—1731no—
126C4×D4—3461no—
127C4×D4C2×D17311yescharacteristic
128C4⋊C8—1731no—
129C4⋊C8C2×D17311yescharacteristic
130#102—3461no—
131#102—3461no—
132C4×C8C2×D17311yescharacteristic
133#41—6922no—
134#41—6922no—
135C2×Q8—1731no—
136C2×D4—3462no—
137C2×D4—3462no—
138C2×D4—3461no—
139C2×D4—3461no—
140C2×D4—3461no—
141C2×D4—3461no—
142C2×D4—3461no—
143C2×D4—1732no—
144C2×D4D69211yescharacteristic
145C22×C4—21no—
146C22×C4—3462no—
147SD16—6922no—
148D8—6924no—
149D8—6922no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—3461no—
153C4⋊C4—3461no—
154C4⋊C4—1731no—
155C4⋊C4D69211yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—3462no—
158C42—3461no—
159C42C22×D17311yescharacteristic
160C23—21no—
161C23—3462no—
162C23—6921no—
163Q8—3461no—
164Q8—1732no—
165D4—21no—
166D4—21no—
167D4—13842no—
168D4—6922no—
169D4—6922no—
170D4—6924no—
171D4—6921no—
172D4—6921no—
173D4—6921no—
174D4—6921no—
175D4—6921no—
176D4—6921no—
177D4—6921no—
178D4—6921no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—3461no—
183C2×C4—3461no—
184C2×C4—6924no—
185C2×C4D4×D17311yescharacteristic
186C2×C4D4×D17311yescharacteristic
187C2×C4C2×D69211yescharacteristic
188C8—22no—
189C8—41no—
190C8—6921no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—13841no—
195C22—13841no—
196C22—13841no—
197C22—6922no—
198C22—6922no—
199C22—6922no—
200C22#1914964911yescharacteristic
201C4—21no—
202C4—41no—
203C4—3462no—
204C4—6921no—
205C4D8×D17311yescharacteristic
206C4#1914970111yescharacteristic
207C4#1914970611yescharacteristic
208C4#1914970711yescharacteristic
209C2—81no—
210C2—13842no—
211C2—27681no—
212C2#2601162611yescharacteristic
213C2#2601163811yescharacteristic
214C2#2601166011yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
FittingD4⋊C8×C173C2
socleC22×C173#120
hypercenterD4⋊C8D173
hypocenterC173#1621
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. #26012162C2semidirect
2. #26011868C2semidirect
3. #26011692C2semidirect
4. #26011686C2semidirect
5. #26011234C2semidirect
6. #26011223C2semidirect
7. D4⋊C8×C173C2semidirect
55. Dic346D8semidirect
107. C173#1621semidirect
115. D4⋊C8D173semidirect

Maximal subgroups

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

subgroupindex
#260121622
#260118682
#260116922
#260116862
#260112342
#260112232
D4⋊C8×C1732
#1621173