Finite Group Databasebeta

← #30639203

Subgroups — #30639203

Order 16256.

304 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 89 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#22549893C211yesmaximal, characteristic
3#22549886C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5#22549648C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7#568×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239—22no—
11C2×#14523238C412yescomplemented
12C2×#14523236C412yescomplemented
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861136—22no—
17#16861134—22no—
18#16861127—22no—
19C2×#41×C127C2211yescharacteristic
20#131×C127—22no—
21#131×C127—22no—
22C2×M16×C127—22no—
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
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—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8C2×C412yes—
40C4⋊Dic127—21no—
41C22⋊Dic127—21no—
42C4×Dic127—21no—
43#41×C127—21no—
44#41×C127—21no—
45#41×C127D412yes—
46#41×C127D412yes—
47C2×Q8×C127C2×C411yescharacteristic
48C2×D4×C127—21no—
49C2×D4×C127C2×C411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52M16×C127—24no—
53M16×C127—21no—
54M16×C127—21no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—24no—
58C2×C8×C127—21no—
59C2×C8×C127D412yes—
60Dic254—22no—
61Dic254—22no—
62Dic254—41no—
63Dic254—41no—
64C2×Dic127—22no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C127⋊C8—22no—
69C127⋊C8—41no—
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×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—22no—
90C22×C127—41no—
91C22×C127—81no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12111yescharacteristic
94C22×C127#12311yescharacteristic
95C4×C127—22no—
96C4×C127C2×C22⋊C411yescharacteristic
97C4×C127C4×D411yescharacteristic
98C4×C127#12011yescharacteristic
99C4×C127#12111yescharacteristic
100Dic127—42no—
101Dic127—81no—
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#568D12711yescharacteristic
111#421—1271no—
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—22no—
123#131—22no—
124C2×M16—22no—
125C2×M16—1271no—
126C2×M16—1271no—
127C2×M16C2×D12711yescharacteristic
128C2×M16C2×D12711yescharacteristic
129#117—1271no—
130#104—2542no—
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—24no—
152M16—21no—
153M16—21no—
154M16—21no—
155M16—2541no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16C22⋊D12712yes—
160C2×C8—24no—
161C2×C8—21no—
162C2×C8—1272no—
163C2×C8—2541no—
164C2×C8C22⋊D12712yes—
165C4⋊C4—2541no—
166C22⋊C4—2541no—
167C42—2541no—
168C23—21no—
169C23#1452324811yescharacteristic
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—5081no—
184C2×C4—5081no—
185C2×C4—5081no—
186C2×C4—2542no—
187C2×C4C2×C4×D12711yescharacteristic
188C2×C4C2×C22⋊D12711yescharacteristic
189C2×C4C2×C22⋊D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×D50811yescharacteristic
192C2×C4#1452324811yescharacteristic
193C8—22no—
194C8—22no—
195C8—5081no—
196C8—2542no—
197C22—22no—
198C22—41no—
199C22—81no—
200C22C4×C22⋊D12711yescharacteristic
201C22#1686118311yescharacteristic
202C22#1686119211yescharacteristic
203C4—22no—
204C4—10161no—
205C4—5082no—
206C4C2×(C2×D127)⋊C411yescharacteristic
207C4C4×C22⋊D12711yescharacteristic
208C4#1686117811yescharacteristic
209C4#1686118311yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—42no—
214C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1902
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
3. #22549886C2semidirect
4. C2×#16861127C2semidirect
5. #22549648C2semidirect
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
#225498932
#225498862
C2×#168611272
#225496482
C127⋊#3382
#568×C1272
#1902127