Finite Group Databasebeta

← #30639540

Subgroups — #30639540

Order 16256.

336 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16861196C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic, complemented
3C2×#16861132C211yesmaximal, characteristic, complemented
4C2×#16861130C211yesmaximal, characteristic
5C2×C8×Dic127C211yesmaximal, characteristic, complemented
6#22549741C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861196—22no—
11#16861196C414yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×C22⋊Dic127—22no—
14C2×C4×Dic127C2211yescharacteristic
15#16861134—22no—
16#16861132—22no—
17#16861130—22no—
18C8×Dic127—22no—
19C127⋊#95—22no—
20C22×D4×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23#102×C127—22no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D412yes—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C22⋊Dic127—28no—
39C4×Dic127C2×C412yes—
40C4×Dic127D412yes—
41C24×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127C2×C412yes—
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C2×C8×C127D412yes—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55Dic254—22no—
56Dic254—22no—
57Dic254—41no—
58Dic254—41no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127—42no—
66C2×Dic127C22⋊C414yes—
67C127⋊C8—22no—
68C23×C127—22no—
69C23×C127—41no—
70C23×C127—41no—
71C23×C127—42no—
72C23×C127—42no—
73C23×C127C2×D411yescharacteristic
74D4×C127—22no—
75D4×C127—22no—
76D4×C127—41no—
77D4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—42no—
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—22no—
87C22×C127—42no—
88C22×C127—44no—
89C22×C127—82no—
90C22×C127—81no—
91C22×C127—81no—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#12411yescharacteristic
94C22×C127#12711yescharacteristic
95C22×C127#13512yes—
96C22×C127#13512yes—
97C4×C127—81no—
98C4×C127C4×D411yescharacteristic
99C4×C127C4×D411yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12111yescharacteristic
102Dic127—24no—
103Dic127—42no—
104Dic127—81no—
105C2×C127—82no—
106C2×C127#39011yescharacteristic
107C2×C127#41612yes—
108C2×C127#48812yes—
109C2×C127#49511yescharacteristic
110C2×C127#49511yescharacteristic
111#1958—1271nomaximal, Sylow
112C127#195811yescharacteristic, complemented, Sylow
113C2×#124—1271no—
114C2×#103—1271no—
115C2×#103—1271no—
116C2×#102—1271no—
117C2×#102D12711yescharacteristic
118C2×C4×C8—1271no—
119#382—1271no—
120C22×Q8—1271no—
121C22×D4C2×D12711yescharacteristic
122C22×C8—1271no—
123C22×C8C2×D12711yescharacteristic
124#124—1274no—
125#124—2542no—
126C2×C4⋊C4—1271no—
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C22⋊C4—2542no—
129C2×C42—1271no—
130#103—2542no—
131#103—2542no—
132#102—22no—
133#102—2542no—
134C4×C8—2542no—
135#95—2542no—
136C24—21no—
137C2×Q8—1272no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C4×D12712yes—
149C22×C4—21no—
150C22×C4—1272no—
151C22×C4—2541no—
152C22×C4—2541no—
153C22×C4C22×D12711yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—1272no—
157C2×C8—2541no—
158C2×C8—2541no—
159C2×C8C22⋊D12712yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—2541no—
163C4⋊C4—2541no—
164C22⋊C4—2548no—
165C42—1272no—
166C42—1272no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—42no—
172C23D4×D12711yescharacteristic
173Q8—5081no—
174Q8—5081no—
175Q8—2542no—
176Q8—2542no—
177D4—22no—
178D4—22no—
179D4—41no—
180D4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—1274no—
184C2×C4—5081no—
185C2×C4—5082no—
186C2×C4—5081no—
187C2×C4—5081no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4D4×D12711yescharacteristic
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4#1452324711yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—22no—
198C8—2542no—
199C22—42no—
200C22—44no—
201C22—82no—
202C22—81no—
203C22—81no—
204C22C22⋊C4×D12711yescharacteristic
205C22#1686119511yescharacteristic
206C22#1686120511yescharacteristic
207C22#1686122812yes—
208C22#1686123012yes—
209C4—81no—
210C4—10161no—
211C4—5082no—
212C4—2544no—
213C4(C22⋊D127)⋊C411yescharacteristic
214C4C4×C22⋊D12711yescharacteristic
215C4#1686118111yescharacteristic
216C4#1686118211yescharacteristic
217C2—82no—
218C2#2254977611yescharacteristic
219C2#2254986912yes—
220C2#2255022912yes—
221C2#2255025711yescharacteristic
222C2#2255025911yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#102D127
hypocenterC127#1958
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16861134C2semidirect
3. C2×#16861132C2semidirect
5. C2×C8×Dic127C2semidirect
6. #22549741C2semidirect
11. #16861196C4semidirect
112. C127#1958semidirect

Maximal subgroups

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

subgroupindex
C2×#168611962
C2×#168611342
C2×#168611322
C2×#168611302
C2×C8×Dic1272
#225497412
C2×#102×C1272
#1958127