Finite Group Databasebeta

← #30638705

Subgroups — #30638705

Order 16256.

536 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 121 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#22549893C212yesmaximal, complemented
2#22549885C212yesmaximal, complemented
3#22549860C211yesmaximal, characteristic, complemented
4#22549850C211yesmaximal, characteristic, complemented
5C2×2−1+4×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7#16861168C2212yes—
8#16861136—24no—
9#16861133—24no—
10#16861128—21no—
11#16861126—21no—
12#16861124—22no—
132−1+4×C127—24no—
142−1+4×C127—24no—
15C2×#41×C127—21no—
16C2×#41×C127—22no—
17C2×#41×C127C412yescomplemented
18C2×#41×C127C2211yescharacteristic
19C2×#41×C127C2211yescharacteristic
20C22×Q8×C127—21no—
21C22×Q8×C127C412yescomplemented
22C22×Q8×C127C2211yescharacteristic
23#14523232—22no—
24#14523232—22no—
25C2×C127⋊C8—22no—
26C4⋊Dic127—22no—
27C22⋊Dic127—22no—
28C4×Dic127—22no—
29#41×C127—216no—
30#41×C127—28no—
31#41×C127—24no—
32#41×C127—24no—
33#41×C127—22no—
34#41×C127—22no—
35#41×C127—22no—
36#41×C127D414yes—
37C2×Q8×C127—28no—
38C2×Q8×C127—24no—
39C2×Q8×C127—24no—
40C2×Q8×C127—22no—
41C2×Q8×C127—21no—
42C2×Q8×C127—22no—
43C2×Q8×C127C2×C411yescharacteristic
44C2×Q8×C127C2×C412yes—
45C2×Q8×C127D414yes—
46C2×Q8×C127D411yescharacteristic
47C2×D4×C127—22no—
48C2×D4×C127—21no—
49C2×D4×C127C2×C411yescharacteristic
50C2×D4×C127D412yes—
51C2×D4×C127D411yescharacteristic
52C22×C4×C127—22no—
53C22×C4×C127—21no—
54C22×C4×C127—22no—
55C22×C4×C127—21no—
56C22×C4×C127C2×C412yes—
57C22×C4×C127C2311yescharacteristic
58C2×Dic127—42no—
59C127⋊C8—42no—
60C23×C127—21no—
61C23×C127C22⋊C412yes—
62C23×C127C2×D411yescharacteristic
63Q8×C127—28no—
64Q8×C127—21no—
65Q8×C127—22no—
66Q8×C127—24no—
67Q8×C127—22no—
68Q8×C127C22⋊C412yes—
69Q8×C127C22⋊C414yes—
70D4×C127—28no—
71D4×C127—24no—
72D4×C127—24no—
73D4×C127—21no—
74D4×C127—22no—
75D4×C127C22⋊C412yes—
76C2×C4×C127—28no—
77C2×C4×C127—28no—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—24no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—24no—
84C2×C4×C127C22⋊C412yes—
85C2×C4×C127C22×C411yescharacteristic
86C2×C4×C127C2×D412yes—
87C2×C4×C127C2×D412yes—
88C2×C4×C127C2×D411yescharacteristic
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—41no—
92C22×C127—42no—
93C22×C127—42no—
94C22×C127C2×C22⋊C411yescharacteristic
95C22×C127#12012yes—
96C4×C127—24no—
97C4×C127—22no—
98C4×C127—22no—
99C4×C127C2×C22⋊C411yescharacteristic
100C4×C127C2×C22⋊C411yescharacteristic
101C4×C127#12012yes—
102Dic127—82no—
103C2×C127—22no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127—42no—
107C2×C127#37911yescharacteristic
108#1796—1271nomaximal, Sylow
109C127#179611yescharacteristic, complemented, Sylow
110C2×2−1+4D12711yescharacteristic
111#421—1272no—
112#419—1272no—
113#413—1271no—
114#410—1271no—
1152−1+4—24no—
1162−1+4—24no—
117C2×#41—21no—
118C2×#41—22no—
119C2×#41Dic12712yescomplemented
120C2×#41C2×D12711yescharacteristic
121C2×#41C2×D12711yescharacteristic
122C22×Q8—21no—
123C22×Q8Dic12712yescomplemented
124C22×Q8C2×D12711yescharacteristic
125C2×M16—1272no—
126#117—1272no—
127#104—2544no—
128#103—2544no—
129#101—2541no—
130#100—2541no—
131#99—2542no—
132#41—216no—
133#41—28no—
134#41—24no—
135#41—24no—
136#41—22no—
137#41—22no—
138#41—22no—
139#41C22⋊D12714yes—
140C2×Q8—28no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8C2×Dic12711yescharacteristic
147C2×Q8C2×Dic12712yes—
148C2×Q8C22⋊D12714yes—
149C2×Q8C22⋊D12711yescharacteristic
150C2×D4—22no—
151C2×D4—21no—
152C2×D4C2×Dic12711yescharacteristic
153C2×D4C22⋊D12712yes—
154C2×D4C22⋊D12711yescharacteristic
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4C2×Dic12712yes—
160C22×C4C22×D12711yescharacteristic
161M16—2542no—
162M16—2542no—
163C2×C8—2542no—
164C4⋊C4—2542no—
165C22⋊C4—2542no—
166C42—2542no—
167C23—21no—
168C23C22⋊Dic12712yes—
169C23C2×C22⋊D12711yescharacteristic
170Q8—28no—
171Q8—21no—
172Q8—22no—
173Q8—24no—
174Q8—22no—
175Q8C22⋊Dic12712yes—
176Q8C22⋊Dic12714yes—
177D4—28no—
178D4—24no—
179D4—24no—
180D4—21no—
181D4—22no—
182D4C22⋊Dic12712yes—
183C2×C4—28no—
184C2×C4—28no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—5082no—
192C2×C4C22⋊Dic12712yes—
193C2×C4C22×Dic12711yescharacteristic
194C2×C4C2×C22⋊D12712yes—
195C2×C4C2×C22⋊D12712yes—
196C2×C4C2×C22⋊D12711yescharacteristic
197C8—5082no—
198C22—22no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22C2×C22⋊Dic12711yescharacteristic
204C22#1686117812yes—
205C4—24no—
206C4—22no—
207C4—22no—
208C4—10162no—
209C4C2×C22⋊Dic12711yescharacteristic
210C4C2×C22⋊Dic12711yescharacteristic
211C4#1686117812yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—42no—
216C2#2254973211yescharacteristic
subgroupG/H
centerC2#22549732
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×2−1+4×C127C2
socleC2×C127#379
cosocleC22×C4×C127C23
hypercenterC2×2−1+4D127
hypocenterC127#1796
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #22549893C2semidirect
2. #22549885C2semidirect
3. #22549860C2semidirect
4. #22549850C2semidirect
17. C2×#41×C127C4semidirect
21. C22×Q8×C127C4semidirect
109. C127#1796semidirect
119. C2×#41Dic127semidirect
123. C22×Q8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498932
#225498852
#225498602
#225498502
C2×2−1+4×C1272
#1796127