Finite Group Databasebeta

← #27170197

Subgroups — #27170197

Order 12928.

278 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 61 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#20265553C211yesmaximal, characteristic, complemented
2#20265377C211yesmaximal, characteristic, complemented
3#20265372C211yesmaximal, characteristic, complemented
4#20264921C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#550×C101C211yesmaximal, characteristic, complemented
8C2×#13108061—21no—
9C2×#13108059—22no—
10C2×#13108059—21no—
11#15692252C2211yescharacteristic
12C4⋊#11717579C2211yescharacteristic
13C4⋊#11717579C2211yescharacteristic
14#15692175—21no—
15#15692175—21no—
16C4×#11717579C2211yescharacteristic
17C2×#41×C101—21no—
18#127×C101—21no—
19#127×C101C2211yescharacteristic
20#121×C101—22no—
21#121×C101—21no—
22C4×Q8×C101C2211yescharacteristic
23C4×D4×C101—21no—
24C4×D4×C101C2211yescharacteristic
25C2×D404—21no—
26C2×D404—21no—
27C2×D404—21no—
28#13108061—42no—
29#13108059—44no—
30#13108059—42no—
31C2×#11717579—21no—
32C2×#11717579—21no—
33C2×#11717579—21no—
34#41×C101—42no—
35#41×C101—42no—
36C2×Q8×C101D411yescharacteristic
37C2×D4×C101—22no—
38C2×D4×C101—22no—
39C2×D4×C101—21no—
40C2×D4×C101—21no—
41C2×D4×C101D411yescharacteristic
42C2×D4×C101D412yes—
43C22×C4×C101—22no—
44C22×C4×C101—21no—
45C4⋊C4×C101—21no—
46C4⋊C4×C101D411yescharacteristic
47C4⋊C4×C101D411yescharacteristic
48C22⋊C4×C101—22no—
49C22⋊C4×C101—21no—
50C42×C101—21no—
51C42×C101C2311yescharacteristic
52C23×C101—21no—
53C23×C101—22no—
54C22×D101—41no—
55Q8×C101—21no—
56Q8×C101D812yescomplemented
57D4×C101—21no—
58D4×C101—21no—
59D4×C101—42no—
60D4×C101—44no—
61D4×C101—41no—
62D4×C101—41no—
63D4×C101—41no—
64D4×C101—41no—
65D4×C101—82no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—44no—
70C2×C4×C101—41no—
71C2×C4×C101—41no—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×D411yescharacteristic
74C2×C4×C101C2×D411yescharacteristic
75D404—42no—
76D404—41no—
77D404—41no—
78D404—41no—
79D404—41no—
80#11717579—22no—
81#11717579—41no—
82#11717579—41no—
83C22×C101—42no—
84C22×C101—41no—
85C22×C101—41no—
86C22×C101—42no—
87C22×C101—42no—
88C22×C101—41no—
89C22×C101#12011yescharacteristic
90C2×D101—81no—
91C2×D101—81no—
92C2×D101—81no—
93C4×C101—22no—
94C4×C101—21no—
95C4×C101—41no—
96C4×C101—41no—
97C4×C101C2×D811yescharacteristic
98C4×C101#13511yescharacteristic
99C4×C101AGL(1,ℤ/8)11yescharacteristic
100C4×C101AGL(1,ℤ/8)11yescharacteristic
101C2×C101—81no—
102C2×C101—82no—
103C2×C101#44711yescharacteristic
104C2×C101#44911yescharacteristic
105C2×C101#45311yescharacteristic
106D101—161no—
107#1621—1011nomaximal, Sylow
108C101#162111yescharacteristic, complemented, Sylow
109#550D10111yescharacteristic, complemented
110#493—1011no—
111#460—1011no—
112#459—1011no—
113#331—1011no—
114Q8⋊C8—1011no—
115D4⋊C8—1011no—
116C2×#41—21no—
117C2×SD16—2021no—
118C2×D8—2021no—
119C2×D8—2022no—
120#127—21no—
121#127—1011no—
122#127C2×D10111yescharacteristic
123#121—22no—
124#121—21no—
125C4×Q8C2×D10111yescharacteristic
126C4×D4—21no—
127C4×D4C2×D10111yescharacteristic
128C4⋊C8—1011no—
129C4⋊C8—1011no—
130#102—2021no—
131#102—2021no—
132C4×C8—1011no—
133#41—42no—
134#41—42no—
135C2×Q8C22⋊D10111yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—2021no—
141C2×D4—2021no—
142C2×D4—2021no—
143C2×D4C22⋊D10111yescharacteristic
144C2×D4C22⋊D10112yes—
145C22×C4—22no—
146C22×C4—21no—
147SD16—4042no—
148D8—4042no—
149D8—4044no—
150C2×C8—2021no—
151C2×C8—2021no—
152C2×C8—2021no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D10111yescharacteristic
155C4⋊C4C22⋊D10111yescharacteristic
156C22⋊C4—22no—
157C22⋊C4—21no—
158C42—21no—
159C42C22×D10111yescharacteristic
160C23—21no—
161C23—22no—
162C23—4041no—
163Q8—21no—
164Q8#1310805912yescomplemented
165D4—21no—
166D4—21no—
167D4—42no—
168D4—44no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—82no—
174D4—4042no—
175D4—4041no—
176D4—4041no—
177D4—4041no—
178D4—4041no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×C22⋊D10111yescharacteristic
186C2×C4C2×C22⋊D10111yescharacteristic
187C2×C4C2×C22⋊D10111yescharacteristic
188C8—4041no—
189C8—4041no—
190C8—2022no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—41no—
197C22—8081no—
198C22—8081no—
199C22—8081no—
200C22#1569222411yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4C2×#1310805911yescharacteristic
206C4#1569227511yescharacteristic
207C4#1569228011yescharacteristic
208C4#1569228111yescharacteristic
209C2—81no—
210C2—82no—
211C2—16161no—
212C2#2026531111yescharacteristic
213C2#2026532211yescharacteristic
214C2#2026534511yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#550×C101C2
socleC22×C101#120
hypercenter#550D101
hypocenterC101#1621
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #20265553C2semidirect
2. #20265377C2semidirect
3. #20265372C2semidirect
4. #20264921C2semidirect
5. Q8⋊#11717579C2semidirect
6. D4⋊#11717579C2semidirect
7. #550×C101C2semidirect
56. Q8×C101D8semidirect
108. C101#1621semidirect
109. #550D101semidirect
164. Q8#13108059semidirect

Maximal subgroups

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

subgroupindex
#202655532
#202653772
#202653722
#202649212
Q8⋊#117175792
D4⋊#117175792
#550×C1012
#1621101