Finite Group Databasebeta

← #30639202

Subgroups — #30639202

Order 16256.

304 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16861242C211yesmaximal, characteristic
2#22550642C211yesmaximal, characteristic
3#22549894C211yesmaximal, characteristic
4#22549884C211yesmaximal, characteristic, complemented
5C2×#16861127C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7#344×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861242—22no—
11#16861242—22no—
12C2×Dic508C412yescomplemented
13C2×#14523237C412yescomplemented
14#16861217—22no—
15#16861215—22no—
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18M16×D127—22no—
19#16861137—22no—
20#16861132—22no—
21#16861127—22no—
22C2×M16×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25#14523248—21no—
26#14523248—21no—
27#14523248D412yes—
28#14523248D412yes—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254C2×C412yes—
35C2×Dic254C2×C411yescharacteristic
36C2×D508C2×C411yescharacteristic
37C2×C22⋊D127—21no—
38C22×Dic127—21no—
39C2×C4×D127—21no—
40Dic508—24no—
41#14523237—24no—
42#14523232—21no—
43#14523232—21no—
44#14523232—21no—
45#14523232—21no—
46#14523231—24no—
47C2×C127⋊C8—21no—
48C2×C127⋊C8D412yes—
49C8×D127—24no—
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53M16×C127—21no—
54M16×C127D412yes—
55C2×C8×C127—21no—
56C2×C8×C127C2×C412yes—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127—21no—
60C22×D127—21no—
61Dic254—21no—
62Dic254—22no—
63Dic254—22no—
64Dic254—41no—
65Dic254—41no—
66Dic254C22⋊C412yes—
67D508—21no—
68D508C22⋊C412yes—
69C22⋊D127—22no—
70C22⋊D127—22no—
71C2×Dic127—21no—
72C2×Dic127—22no—
73C2×Dic127—41no—
74C2×Dic127—41no—
75C4×D127—22no—
76C4×D127—22no—
77C127⋊C8—22no—
78C127⋊C8—41no—
79C23×C127#4111yescharacteristic
80C2×C4×C127—41no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—22no—
88C8×C127—22no—
89C22×C127—41no—
90C22×C127C4×D411yescharacteristic
91C22×C127#12111yescharacteristic
92C22×C127#12311yescharacteristic
93C2×D127—22no—
94C2×D127—81no—
95C4×C127—81no—
96C4×C127C2×C22⋊C411yescharacteristic
97C4×C127C4×D411yescharacteristic
98C4×C127#12011yescharacteristic
99C4×C127#12111yescharacteristic
100Dic127—22no—
101Dic127—42no—
102C2×C127—21no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127#38611yescharacteristic
106D127—42no—
107#1902—1271nomaximal, Sylow
108C127#190211yescharacteristic, complemented, Sylow
109C2×#137—1271no—
110#568—1271no—
111#421—1271no—
112#419—1271no—
113C2×#101—1271no—
114#344D12711yescharacteristic, complemented
115#338—1271no—
116C2×#41—1271no—
117C22×Q8—1271no—
118#137—2542no—
119#137—2542no—
120C2×Q16—1272no—
121C2×SD16—1272no—
122#131—2542no—
123#131—2542no—
124C2×M16—1271no—
125C2×M16—1271no—
126C2×M16—2542no—
127C2×M16C2×D12711yescharacteristic
128C2×M16C2×D12711yescharacteristic
129#117C2×D12711yescharacteristic
130#104—2542no—
131#103—2542no—
132#101—2542no—
133#41—1272no—
134#41—1272no—
135#41—2541no—
136#41—2541no—
137C2×Q8—1271no—
138C2×Q8—1272no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×Q8—2541no—
144C2×D4—1271no—
145C2×D4—2541no—
146C22×C4—2541no—
147C22×C4—2541no—
148C22×C4C22×D12711yescharacteristic
149Q16—2544no—
150SD16—2544no—
151M16—21no—
152M16—21no—
153M16—21no—
154M16—2544no—
155M16—2541no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16D50812yes—
160C2×C8—21no—
161C2×C8—1272no—
162C2×C8—2544no—
163C2×C8—2541no—
164C2×C8C4×D12712yes—
165C4⋊C4—21no—
166C22⋊C4—21no—
167C42—21no—
168C23—2541no—
169C23#1452324811yescharacteristic
170Q8—1272no—
171Q8—5081no—
172Q8—5081no—
173Q8—2542no—
174Q8—2542no—
175Q8—2541no—
176D4—1272no—
177D4—2542no—
178D4—2542no—
179D4—2541no—
180C2×C4—41no—
181C2×C4—5081no—
182C2×C4—5081no—
183C2×C4—2542no—
184C2×C4—2541no—
185C2×C4—2542no—
186C2×C4—2542no—
187C2×C4C2×C4×D12711yescharacteristic
188C2×C4D4×D12711yescharacteristic
189C2×C4D4×D12711yescharacteristic
190C2×C4D4×D12711yescharacteristic
191C2×C4C2×D50811yescharacteristic
192C2×C4#1452324911yescharacteristic
193C8—22no—
194C8—22no—
195C8—5081no—
196C8—2542no—
197C22—41no—
198C22—10161no—
199C22—2542no—
200C22C4×D50811yescharacteristic
201C22#1686118211yescharacteristic
202C22#1686119411yescharacteristic
203C4—81no—
204C4—5082no—
205C4—2542no—
206C4C22⋊C4×D12711yescharacteristic
207C4#1686117111yescharacteristic
208C4#1686117911yescharacteristic
209C4#1686118411yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—5082no—
214C2#2254975411yescharacteristic
subgroupG/H
centerC2#22549754
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#344×C127C2
socleC2×C127#386
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#344D127
hypocenterC127#1902
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. #22549884C2semidirect
5. C2×#16861127C2semidirect
6. C127⋊#338C2semidirect
7. #344×C127C2semidirect
12. C2×Dic508C4semidirect
13. C2×#14523237C4semidirect
108. C127#1902semidirect
114. #344D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
#225506422
#225498942
#225498842
C2×#168611272
C127⋊#3382
#344×C1272
#1902127