Finite Group Databasebeta

← #41012516

Subgroups — #41012516

Order 27008.

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×D844C211yesmaximal, characteristic
2#27213519C211yesmaximal, characteristic, complemented
3#27213437C211yesmaximal, characteristic, complemented
4#27213436C211yesmaximal, characteristic, complemented
5#27212889C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic, complemented
8C22×D844—21no—
9C2×D4×D211—21no—
10C2×#16809951—22no—
11#20269977C2211yescharacteristic, complemented
12#20269974C2211yescharacteristic
13#20269952—21no—
14#20269951—21no—
15#20269947—22no—
16C4×D844C2211yescharacteristic
17#20269909—21no—
18#20269908—21no—
19C4⋊(C211⋊C8)C2211yescharacteristic, complemented
20#20269899—21no—
21#20269898—21no—
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic, complemented
24C4×C8×C211C2211yescharacteristic, complemented
25C23×D211—22no—
26C2×Dic422—21no—
27C2×D844—22no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—42no—
31C2×D844—42no—
32C2×D844D411yescharacteristic
33C2×D844D412yes—
34C2×C22⋊D211—22no—
35D4×D211—42no—
36D4×D211—42no—
37C2×C4×D211—21no—
38#16809951—44no—
39C2×C211⋊C8—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42C4⋊Dic211D411yescharacteristic
43(C2×D211)⋊C4—22no—
44(C2×D211)⋊C4—21no—
45C2×D4×C211D411yescharacteristic
46C22×C4×C211—21no—
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C4⋊C4×C211D411yescharacteristic
50C22⋊C4×C211—21no—
51C42×C211C2311yescharacteristic
52C22×D211—21no—
53C22×D211—22no—
54C22×D211—44no—
55C22×D211—44no—
56C22×D211—82no—
57Dic422—42no—
58D844—21no—
59D844—42no—
60D844—44no—
61D844—41no—
62D844—41no—
63D844—41no—
64D844—41no—
65D844SD1612yescomplemented
66C22⋊D211—44no—
67C2×Dic211—21no—
68C2×Dic211—41no—
69C4×D211—41no—
70C4×D211—41no—
71C211⋊C8—41no—
72C23×C211—21no—
73D4×C211—21no—
74D4×C211SD1612yescomplemented
75C2×C4×C211—21no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C8×C211—22no—
82C8×C211—41no—
83C22×C211—22no—
84C22×C211—41no—
85C22×C211—41no—
86C22×C211#12011yescharacteristic
87C2×D211—22no—
88C2×D211—41no—
89C2×D211—41no—
90C2×D211—42no—
91C2×D211—42no—
92C2×D211—42no—
93C2×D211—88no—
94C4×C211—21no—
95C4×C211—41no—
96C4×C211C2×SD1611yescharacteristic
97C4×C211C2×SD1611yescharacteristic
98C4×C211AGL(1,ℤ/8)11yescharacteristic
99C4×C211AGL(1,ℤ/8)11yescharacteristic
100Dic211—41no—
101Dic211—81no—
102C2×C211—42no—
103C2×C211#45011yescharacteristic
104C2×C211#45011yescharacteristic
105C2×C211#45311yescharacteristic
106D211—42no—
107D211—82no—
108C211#162911yescharacteristic, complemented, Sylow
109#1629—2111nomaximal, Sylow
110D42—2111no—
111#492—2111no—
112#476—2111no—
113#476—2111no—
114#331—2111no—
115D4⋊C8—2111no—
116D4⋊C8D21111yescharacteristic, complemented
117C22×D4—4221no—
118C22×D4—4221no—
119C2×SD16—4222no—
120C4⋊Q8—2111no—
121#127—2111no—
122#121—4221no—
123#121—4221no—
124#120—4222no—
125C4×D4—2111no—
126C4×D4C2×D21111yescharacteristic
127#106—4221no—
128#106—4221no—
129C4⋊C8—2111no—
130C4⋊C8C2×D21111yescharacteristic, complemented
131#102—4221no—
132#102—4221no—
133C4×C8C2×D21111yescharacteristic, complemented
134C24—4222no—
135C2×Q8—4221no—
136C2×D4—8442no—
137C2×D4—8442no—
138C2×D4—8442no—
139C2×D4—8442no—
140C2×D4—4222no—
141C2×D4—4222no—
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—2112no—
145C2×D4—2111no—
146C2×D4D84411yescharacteristic
147C22×C4—21no—
148C22×C4—4221no—
149SD16—8444no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—4221no—
153C4⋊C4—4221no—
154C4⋊C4—4221no—
155C4⋊C4—2111no—
156C4⋊C4D84411yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—4222no—
159C22⋊C4—4221no—
160C42C22×D21111yescharacteristic
161C23—21no—
162C23—8444no—
163C23—8444no—
164C23—16882no—
165C23—4222no—
166C23—4221no—
167Q8—8442no—
168D4—21no—
169D4—8442no—
170D4—8444no—
171D4—8444no—
172D4—8441no—
173D4—8441no—
174D4—8441no—
175D4—8441no—
176D4—4221no—
177D4—2112no—
178D4#1680995112yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—8441no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—4221no—
186C2×C4D4×D21111yescharacteristic
187C2×C4D4×D21111yescharacteristic
188C2×C4C2×D84411yescharacteristic
189C8—22no—
190C8—41no—
191C8—8441no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—8441no—
196C22—8441no—
197C22—8442no—
198C22—8442no—
199C22—8442no—
200C22—16888no—
201C22—4222no—
202C22#2026994711yescharacteristic
203C4—21no—
204C4—41no—
205C4—8441no—
206C4—16881no—
207C4SD16×D21111yescharacteristic
208C4C2×#1680995111yescharacteristic
209C4#2027000111yescharacteristic
210C4#2027000511yescharacteristic
211C2—42no—
212C2—8442no—
213C2—16882no—
214C2#2721329811yescharacteristic
215C2#2721329911yescharacteristic
216C2#2721331511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingD4⋊C8×C211C2
socleC22×C211#120
prime radicalD4×D844C2
hypercenterD4⋊C8D211
hypocenterC211#1629
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27213519C2semidirect
3. #27213437C2semidirect
4. #27213436C2semidirect
5. #27212889C2semidirect
6. #27212875C2semidirect
7. D4⋊C8×C211C2semidirect
11. #20269977C22semidirect
19. C4⋊(C211⋊C8)C22semidirect
23. C4⋊C8×C211C22semidirect
24. C4×C8×C211C22semidirect
65. D844SD16semidirect
74. D4×C211SD16semidirect
108. C211#1629semidirect
116. D4⋊C8D211semidirect
130. C4⋊C8C2×D211semidirect
133. C4×C8C2×D211semidirect
178. D4#16809951semidirect

Maximal subgroups

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

subgroupindex
D4×D8442
#272135192
#272134372
#272134362
#272128892
#272128752
D4⋊C8×C2112
#1629211