Finite Group Databasebeta

← #30639201

Subgroups — #30639201

Order 16256.

304 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16861239C211yesmaximal, characteristic
2#22550643C211yesmaximal, characteristic
3#22549884C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5#22549646C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239—22no—
11C2×#14523238C412yescomplemented
12C2×#14523236C412yescomplemented
13#16861216—22no—
14#16861216—22no—
15C2×#14523232—22no—
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2211yescharacteristic
19#16861132—22no—
20#16861127—22no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#104×C127—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254C2×C412yes—
31C22×Dic127—21no—
32#14523238—24no—
33#14523236—24no—
34#14523232—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232—21no—
40#14523232D412yes—
41C2×C127⋊C8—24no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8C2×C412yes—
44C2×C127⋊C8D412yes—
45#41×C127—21no—
46#41×C127—21no—
47#41×C127D412yes—
48#41×C127D412yes—
49C2×Q8×C127C2×C411yescharacteristic
50C2×D4×C127—21no—
51C2×D4×C127C2×C411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127—21no—
60Dic254—22no—
61Dic254—22no—
62Dic254—41no—
63Dic254—41no—
64C2×Dic127—22no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C127⋊C8—22no—
68C127⋊C8—22no—
69C127⋊C8—22no—
70C23×C127—21no—
71C23×C127#4111yescharacteristic
72Q8×C127—21no—
73Q8×C127C22⋊C412yes—
74D4×C127—22no—
75D4×C127—22no—
76D4×C127—21no—
77D4×C127C22⋊C412yes—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—41no—
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—81no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12111yescharacteristic
94C22×C127#12311yescharacteristic
95C4×C127—22no—
96C4×C127—81no—
97C4×C127C2×C22⋊C411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12011yescharacteristic
100C4×C127#12111yescharacteristic
101Dic127—42no—
102C2×C127—21no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127#38611yescharacteristic
107#1902—1271nomaximal, Sylow
108C127#190211yescharacteristic, complemented, Sylow
109C2×#137—1271no—
110#568—1271no—
111#421D12711yescharacteristic
112#419—1271no—
113C2×#101—1271no—
114#344—1271no—
115#338—1271no—
116C2×#41C2×D12711yescharacteristic
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—1271no—
127C2×M16—2542no—
128C2×M16C2×D12711yescharacteristic
129#117C2×D12711yescharacteristic
130#104—22no—
131#103—2542no—
132#101—2542no—
133#41—21no—
134#41—21no—
135#41C22⋊D12712yes—
136#41D50812yes—
137C2×Q8—1272no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×Q8C4×D12711yescharacteristic
144C2×D4—21no—
145C2×D4C4×D12711yescharacteristic
146C22×C4—21no—
147C22×C4—2541no—
148C22×C4C22×D12711yescharacteristic
149Q16—2544no—
150SD16—2544no—
151M16—21no—
152M16—21no—
153M16—1272no—
154M16—2544no—
155M16—2541no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16—2541no—
160C2×C8—21no—
161C2×C8—1272no—
162C2×C8—1272no—
163C2×C8—2544no—
164C2×C8—2541no—
165C4⋊C4—21no—
166C22⋊C4—21no—
167C42—21no—
168C23—21no—
169C23#1452324711yescharacteristic
170Q8—21no—
171Q8—5081no—
172Q8—5081no—
173Q8—2542no—
174Q8—2542no—
175Q8(C2×D127)⋊C412yes—
176D4—22no—
177D4—22no—
178D4—21no—
179D4(C2×D127)⋊C412yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—5081no—
185C2×C4—5081no—
186C2×C4—2542no—
187C2×C4C2×C4×D12711yescharacteristic
188C2×C4D4×D12711yescharacteristic
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×D50811yescharacteristic
192C2×C4#1452324711yescharacteristic
193C8—41no—
194C8—2542no—
195C8—2542no—
196C8—2542no—
197C22—22no—
198C22—41no—
199C22—81no—
200C22(C22⋊D127)⋊C411yescharacteristic
201C22#1686118111yescharacteristic
202C22#1686119311yescharacteristic
203C4—22no—
204C4—81no—
205C4—5082no—
206C4C2×(C2×D127)⋊C411yescharacteristic
207C4(C22⋊D127)⋊C411yescharacteristic
208C4#1686117911yescharacteristic
209C4#1686118111yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—42no—
214C2#2254975311yescharacteristic
subgroupG/H
centerC2#22549753
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1902
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
3. #22549884C2semidirect
4. C2×#16861127C2semidirect
5. #22549646C2semidirect
6. C127⋊#338C2semidirect
11. C2×#14523238C4semidirect
12. C2×#14523236C4semidirect
108. C127#1902semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225506432
#225498842
C2×#168611272
#225496462
C127⋊#3382
#421×C1272
#1902127