Finite Group Databasebeta

← #30639281

Subgroups — #30639281

Order 16256.

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

normal subgroups only

This group has 308 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
1#22550642C211yesmaximal, characteristic
2#22550452C211yesmaximal, characteristic
3#22549858C211yesmaximal, characteristic
4C2×#16861127C211yesmaximal, characteristic, complemented
5C2×#16861127C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#344×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861217—22no—
11#16861215—22no—
12C2×#14523232C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14M16×D127—22no—
15#16861209—21no—
16#16861197—21no—
17#16861193C414yes—
18#16861189—22no—
19#16861127—21no—
20#16861127—22no—
21#16861127—22no—
22#16861125—21no—
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#117×C127C2211yescharacteristic
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254D411yescharacteristic
37C2×D508D411yescharacteristic
38C2×C22⋊D127C2×C412yes—
39C22×Dic127C2×C412yes—
40C2×C4×D127D412yes—
41#14523232—21no—
42#14523232—21no—
43#14523232—21no—
44#14523232D412yes—
45#14523231—24no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8—21no—
48C8×D127—24no—
49C4⋊Dic127—24no—
50(C2×D127)⋊C4—24no—
51C22×C4×C127C2311yescharacteristic
52M16×C127—21no—
53M16×C127—21no—
54M16×C127—21no—
55M16×C127D412yes—
56C2×C8×C127—21no—
57C2×C8×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127C2×C412yes—
60C42×C127—21no—
61C22×D127C22⋊C412yes—
62Dic254—21no—
63Dic254—21no—
64Dic254—41no—
65Dic254—41no—
66Dic254—41no—
67Dic254—41no—
68D508—21no—
69D508—21no—
70C22⋊D127—22no—
71C22⋊D127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127C22⋊C412yes—
76C4×D127—22no—
77C4×D127—22no—
78C127⋊C8—22no—
79C127⋊C8—41no—
80C23×C127C22×C411yescharacteristic
81C2×C4×C127—22no—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—22no—
89C8×C127—41no—
90C22×C127—41no—
91C22×C127C2×C22⋊C411yescharacteristic
92C22×C127C4×D411yescharacteristic
93C22×C127C4×D411yescharacteristic
94C2×D127—22no—
95C2×D127—42no—
96C4×C127—42no—
97C4×C127#12011yescharacteristic
98C4×C127#12111yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12311yescharacteristic
101Dic127—22no—
102Dic127—42no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127#38611yescharacteristic
107D127—42no—
108#1915—1271nomaximal, Sylow
109C127#191511yescharacteristic, complemented, Sylow
110#568—1271no—
111#536—1271no—
112#413—1271no—
113C2×#101—1271no—
114C2×#101—1271no—
115#344—1271no—
116#344D12711yescharacteristic, complemented
117C2×#41—1271no—
118C22×Q8—1271no—
119#131—2542no—
120#131—2542no—
121C2×M16—1271no—
122C2×M16—1271no—
123C2×M16—2542no—
124C2×M16C2×D12711yescharacteristic
125C2×M16C2×D12711yescharacteristic
126C4⋊Q8—2541no—
127#124—2541no—
128#123—1274no—
129C22⋊Q8—2542no—
130#117C2×D12711yescharacteristic
131#101—2542no—
132#101—2541no—
133#101—2542no—
134#100—2541no—
135#41—2541no—
136#41—2541no—
137#41—2541no—
138#41—2541no—
139C2×Q8—1271no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×Q8—2541no—
144C2×Q8—2541no—
145C2×Q8—2541no—
146C2×D4—1272no—
147C2×D4—1271no—
148C22×C4—1272no—
149C22×C4—1272no—
150C22×C4C22×D12711yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—1272no—
155M16—2544no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16D50812yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—2544no—
163C2×C8—2541no—
164C2×C8—2541no—
165C4⋊C4—21no—
166C4⋊C4—2544no—
167C22⋊C4—2544no—
168C22⋊C4C4×D12712yes—
169C42—21no—
170C23—1272no—
171C23C2×C4×D12711yescharacteristic
172Q8—5081no—
173Q8—5081no—
174Q8—5081no—
175Q8—5081no—
176Q8—2541no—
177Q8—2541no—
178D4—2542no—
179D4—2542no—
180D4—2541no—
181D4—2541no—
182C2×C4—22no—
183C2×C4—1272no—
184C2×C4—2542no—
185C2×C4—2542no—
186C2×C4—2542no—
187C2×C4—2542no—
188C2×C4—2542no—
189C2×C4D4×D12711yescharacteristic
190C2×C4D4×D12711yescharacteristic
191C2×C4D4×D12711yescharacteristic
192C2×C4C2×D50811yescharacteristic
193C2×C4#1452324811yescharacteristic
194C2×C4#1452324911yescharacteristic
195C8—22no—
196C8—41no—
197C8—5081no—
198C8—2542no—
199C22—41no—
200C22—5082no—
201C22—2542no—
202C22C22⋊C4×D12711yescharacteristic
203C22#1686117111yescharacteristic
204C22C4×D50811yescharacteristic
205C4—42no—
206C4—5082no—
207C4—2542no—
208C4#1686117911yescharacteristic
209C4#1686118211yescharacteristic
210C4#1686118411yescharacteristic
211C4#1686119411yescharacteristic
212C2—21no—
213C2—21no—
214C2—21no—
215C2—5082no—
216C2#2254975411yescharacteristic
subgroupG/H
centerC2#22549754
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#344×C127C2
socleC2×C127#386
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#344D127
hypocenterC127#1915
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. C2×#16861127C2semidirect
5. C2×#16861127C2semidirect
6. #22549646C2semidirect
7. #344×C127C2semidirect
109. C127#1915semidirect
116. #344D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225504522
#225498582
C2×#168611272
#225496462
#344×C1272
#1915127