Finite Group Databasebeta

← #27170367

Subgroups — #27170367

Order 12928.

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

normal subgroups only

This group has 240 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#20265580C211yesmaximal, characteristic, complemented
2#20265457C211yesmaximal, characteristic, complemented
3#20265382C211yesmaximal, characteristic, complemented
4#20264923C211yesmaximal, characteristic, complemented
5D4⋊#11717579C211yesmaximal, characteristic, complemented
6D4⋊#11717579C211yesmaximal, characteristic, complemented
7#547×C101C211yesmaximal, characteristic
8C2×#13108062—21no—
9#15692255C2211yescharacteristic
10#15692188—22no—
11#15692188—21no—
12C4⋊#11717579C2211yescharacteristic
13C4⋊#11717579C2211yescharacteristic
14#15692178—21no—
15#15692176—21no—
16C4×#11717579C2211yescharacteristic
17C2×#41×C101—21no—
18C4⋊Q8×C101C2211yescharacteristic
19#123×C101—22no—
20C22⋊Q8×C101—21no—
21#121×C101—21no—
22C4×D4×C101C2211yescharacteristic
23C4×D4×C101C2211yescharacteristic
24C2×C4⋊C4×C101—21no—
25C2×Dic202—21no—
26#13108062—42no—
27C2×#11717579—21no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C4⋊Dic101—21no—
31C4⋊Dic101—21no—
32#41×C101—42no—
33#41×C101—42no—
34C2×Q8×C101—21no—
35C2×D4×C101—22no—
36C2×D4×C101D411yescharacteristic
37C2×D4×C101D411yescharacteristic
38C22×C4×C101—22no—
39C22×C4×C101—21no—
40C22×C4×C101—21no—
41C4⋊C4×C101—22no—
42C4⋊C4×C101—21no—
43C4⋊C4×C101D411yescharacteristic
44C4⋊C4×C101D411yescharacteristic
45C4⋊C4×C101D412yes—
46C22⋊C4×C101—22no—
47C22⋊C4×C101—21no—
48C22⋊C4×C101—21no—
49C42×C101C2311yescharacteristic
50C23×C101—21no—
51C23×C101—21no—
52Q8×C101—41no—
53Q8×C101—41no—
54D4×C101—21no—
55D4×C101—21no—
56D4×C101—21no—
57D4×C101—44no—
58D4×C101D812yes—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—22no—
62C2×C4×C101—44no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C2×Dic101—41no—
71Dic202—41no—
72Dic202—41no—
73#11717579—22no—
74#11717579—41no—
75#11717579—41no—
76C22×C101—22no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101—41no—
80C22×C101—41no—
81C22×C101—41no—
82C22×C101#12011yescharacteristic
83C4×C101—21no—
84C4×C101—41no—
85C4×C101—42no—
86C4×C101—41no—
87C4×C101C2×D811yescharacteristic
88C4×C101#13511yescharacteristic
89C4×C101AGL(1,ℤ/8)11yescharacteristic
90C4×C101#13711yescharacteristic
91Dic101—81no—
92C2×C101—42no—
93C2×C101—81no—
94C2×C101#44711yescharacteristic
95C2×C101#45211yescharacteristic
96C2×C101#45611yescharacteristic
97#1647—1011nomaximal, Sylow
98C101#164711yescharacteristic, complemented, Sylow
99#547D10111yescharacteristic
100#500—1011no—
101#474—1011no—
102#461—1011no—
103#332—1011no—
104D4⋊C8—1011no—
105D4⋊C8—1011no—
106C2×#41—21no—
107C2×SD16—2021no—
108C4⋊Q8—1011no—
109C4⋊Q8C2×D10111yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C2×D10111yescharacteristic
114C4×D4C2×D10111yescharacteristic
115C2×C4⋊C4—21no—
116#107—2021no—
117#107—2022no—
118C4⋊C8—1011no—
119C4⋊C8—1011no—
120#103—2021no—
121#102—2021no—
122C4×C8—1011no—
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—2021no—
127C2×D4—22no—
128C2×D4C22⋊D10111yescharacteristic
129C2×D4C22⋊D10111yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—4042no—
134C2×C8—2021no—
135C2×C8—2021no—
136C2×C8—2021no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—2021no—
140C4⋊C4—2021no—
141C4⋊C4C22⋊D10111yescharacteristic
142C4⋊C4C22⋊D10111yescharacteristic
143C4⋊C4C22⋊D10112yes—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C22×D10111yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—4041no—
153Q8—4041no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4#1310805912yes—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—4041no—
168C2×C4C2×C22⋊D10111yescharacteristic
169C2×C4C2×C22⋊D10111yescharacteristic
170C2×C4C2×C22⋊D10111yescharacteristic
171C8—4041no—
172C8—4041no—
173C8—2022no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#1569222411yescharacteristic
181C4—21no—
182C4—41no—
183C4—42no—
184C4—41no—
185C4—8081no—
186C4C2×#1310805911yescharacteristic
187C4#1569227511yescharacteristic
188C4#1569228111yescharacteristic
189C4#1569228511yescharacteristic
190C2—42no—
191C2—81no—
192C2#2026531111yescharacteristic
193C2#2026533911yescharacteristic
194C2#2026536011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radical#547×C101C2
socleC22×C101#120
hypercenter#547D101
hypocenterC101#1647
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20265580C2semidirect
2. #20265457C2semidirect
3. #20265382C2semidirect
4. #20264923C2semidirect
5. D4⋊#11717579C2semidirect
6. D4⋊#11717579C2semidirect
98. C101#1647semidirect

Maximal subgroups

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

subgroupindex
#202655802
#202654572
#202653822
#202649232
D4⋊#117175792
#547×C1012
#1647101