Finite Group Databasebeta

← #27170397

Subgroups — #27170397

Order 12928.

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×Dic202C211yesmaximal, characteristic
2#20265575C211yesmaximal, characteristic, complemented
3#20265463C211yesmaximal, characteristic, complemented
4#20265382C211yesmaximal, characteristic
5#20264924C211yesmaximal, characteristic, complemented
6#20264909C211yesmaximal, characteristic, complemented
7D4⋊C8×C101C211yesmaximal, characteristic
8C22×Dic202—21no—
9C2×#13108062—21no—
10#15692255C2211yescharacteristic
11#15692255C2211yescharacteristic
12#15692254—21no—
13#15692234—21no—
14#15692232—22no—
15C4×Dic202C2211yescharacteristic
16D4×Dic101—21no—
17#15692187—22no—
18#15692187—21no—
19C4⋊#11717579C2211yescharacteristic
20#15692180—21no—
21#15692178—21no—
22C4×D4×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic
24C4×C8×C101C2211yescharacteristic
25C2×Dic202—22no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202—42no—
29C2×Dic202—42no—
30C2×Dic202D411yescharacteristic
31C22×Dic101—22no—
32#13108062—42no—
33C2×#11717579—21no—
34C4⋊Dic101—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101D411yescharacteristic
38C4⋊Dic101D412yes—
39#13108053—22no—
40#13108053—21no—
41C22⋊Dic101—22no—
42C4×Dic101—21no—
43C2×D4×C101D411yescharacteristic
44C22×C4×C101—21no—
45C2×C8×C101—21no—
46C2×C8×C101—21no—
47C4⋊C4×C101D411yescharacteristic
48C22⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51D4×C101—21no—
52D4×C101SD1612yes—
53C2×C4×C101—21no—
54C2×C4×C101—41no—
55C2×C4×C101—41no—
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101C2×D411yescharacteristic
58C2×C4×C101C2×D411yescharacteristic
59C2×Dic101—21no—
60C2×Dic101—21no—
61C2×Dic101—22no—
62C2×Dic101—44no—
63C2×Dic101—41no—
64Dic202—21no—
65Dic202—44no—
66Dic202—41no—
67Dic202—41no—
68Dic202—41no—
69Dic202—41no—
70Dic202SD1612yescomplemented
71C8×C101—22no—
72C8×C101—41no—
73#11717579—41no—
74C22×C101—22no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101#12011yescharacteristic
78C4×C101—21no—
79C4×C101—41no—
80C4×C101C2×SD1611yescharacteristic
81C4×C101C2×SD1611yescharacteristic
82C4×C101AGL(1,ℤ/8)11yescharacteristic
83C4×C101#13711yescharacteristic
84Dic101—22no—
85Dic101—42no—
86Dic101—41no—
87Dic101—81no—
88C2×C101—42no—
89C2×C101#44811yescharacteristic
90C2×C101#45011yescharacteristic
91C2×C101#45611yescharacteristic
92#1651—1011nomaximal, Sylow
93C101#165111yescharacteristic, complemented, Sylow
94D4×Q8—1011no—
95#498—1011no—
96#475—1011no—
97#461—1011no—
98#332—1011no—
99Q8⋊C8—1011no—
100D4⋊C8D10111yescharacteristic
101C22×Q8—2021no—
102C2×SD16—2021no—
103C4⋊Q8—1011no—
104C4⋊Q8—1011no—
105C4⋊Q8—2021no—
106C22⋊Q8—2022no—
107C22⋊Q8—2021no—
108C4×Q8—1011no—
109C4×D4—2021no—
110C4×D4C2×D10111yescharacteristic
111#106—2021no—
112#106—2022no—
113C4⋊C8—1011no—
114C4⋊C8C2×D10111yescharacteristic
115#103—2021no—
116#103—2021no—
117C4×C8C2×D10111yescharacteristic
118C2×Q8—1011no—
119C2×Q8—4042no—
120C2×Q8—4042no—
121C2×Q8—2022no—
122C2×Q8—2021no—
123C2×Q8—2021no—
124C2×D4D40411yescharacteristic
125C22×C4—21no—
126C22×C4—2022no—
127SD16—4042no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2021no—
131C4⋊C4—1011no—
132C4⋊C4—1012no—
133C4⋊C4—2022no—
134C4⋊C4—2021no—
135C4⋊C4—2021no—
136C4⋊C4—2021no—
137C4⋊C4—2021no—
138C4⋊C4D40411yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—2022no—
141C42—2021no—
142C42C22×D10111yescharacteristic
143C23—21no—
144Q8—1012no—
145Q8—4044no—
146Q8—4041no—
147Q8—4041no—
148Q8—4041no—
149Q8—4041no—
150Q8—2021no—
151D4—21no—
152D4#1310806312yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—4041no—
157C2×C4—4044no—
158C2×C4—2021no—
159C2×C4—2022no—
160C2×C4—2021no—
161C2×C4D4×D10111yescharacteristic
162C2×C4D4×D10111yescharacteristic
163C2×C4C2×D40411yescharacteristic
164C8—22no—
165C8—41no—
166C8—4041no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1569222511yescharacteristic
171C4—21no—
172C4—41no—
173C4—8081no—
174C4—4042no—
175C4—4041no—
176C4—2022no—
177C4SD16×D10111yescharacteristic
178C4C2×#1310806311yescharacteristic
179C4#1569227911yescharacteristic
180C4#1569228811yescharacteristic
181C2—42no—
182C2#2026531811yescharacteristic
183C2#2026532911yescharacteristic
184C2#2026536111yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
FittingD4⋊C8×C101C2
socleC22×C101#120
prime radicalC2×D4×C101D4
hypercenterD4⋊C8D101
hypocenterC101#1651
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20265575C2semidirect
3. #20265463C2semidirect
5. #20264924C2semidirect
6. #20264909C2semidirect
70. Dic202SD16semidirect
93. C101#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2022
#202655752
#202654632
#202653822
#202649242
#202649092
D4⋊C8×C1012
#1651101