Finite Group Databasebeta

← #30649260

Subgroups — #30649260

Order 16256.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#22550507C211yesmaximal, characteristic
2#22550504C211yesmaximal, characteristic
3#22550284C212yesmaximal, complemented
4#22550283C212yesmaximal, complemented
5#22550277C212yesmaximal, complemented
6#22550273C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic
8#22549912C211yesmaximal, characteristic
9C2×#16861140C211yesmaximal, characteristic
10#22549834C211yesmaximal, characteristic
11#428×C127C211yesmaximal, characteristic
12C22×C127⋊C8C2211yescharacteristic
13#16861209—21no—
14#16861208—21no—
15#16861208C2212yes—
16#16861208C2212yes—
17#16861201—21no—
18#16861200—21no—
19#16861200C2212yes—
20#16861200C2212yes—
21#16861188—22no—
22#16861186—22no—
23#16861168C2211yescharacteristic
24#16861168C2211yescharacteristic
25C2×C4⋊Dic127C2211yescharacteristic
26#16861143C2212yes—
27#16861142C2211yescharacteristic
28#16861142C2211yescharacteristic
29#16861141C2212yes—
30#16861140C2211yescharacteristic
31#16861140C2212yes—
32#16861140C2211yescharacteristic
33#16861140C2212yes—
34#16861119C2212yes—
35C8×Dic127C2212yes—
36C2×M16×C127C2211yescharacteristic
37#117×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#107×C127C2212yes—
40#106×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254—22no—
43C22×Dic127—21no—
44C2×C127⋊C8Q814yescomplemented
45C2×C127⋊C8C2311yescharacteristic
46C2×C127⋊C8C2311yescharacteristic
47C4⋊Dic127—22no—
48C4⋊Dic127—22no—
49C4⋊Dic127—21no—
50C4⋊Dic127D411yescharacteristic
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53C4⋊Dic127C2312yes—
54C4⋊Dic127C2311yescharacteristic
55Dic127⋊C4—22no—
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59C22⋊Dic127—21no—
60C22⋊Dic127D412yes—
61C4×Dic127—21no—
62C4×Dic127C2312yes—
63C22×C4×C127—21no—
64C22×C4×C127C2311yescharacteristic
65M16×C127Q814yescomplemented
66C2×C8×C127C2312yes—
67C4⋊C4×C127—21no—
68C4⋊C4×C127C2311yescharacteristic
69C4⋊C4×C127C2312yes—
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C42×C127—21no—
73Dic254—82no—
74Dic254—82no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—81no—
79C2×Dic127C2×D412yes—
80C2×Dic127C2×D412yes—
81C127⋊C8C2×Q812yes—
82C127⋊C8C2×Q812yes—
83C23×C127C2×D411yescharacteristic
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127—81no—
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×Q812yes—
90C2×C4×C127C2×Q812yes—
91C2×C4×C127C2411yescharacteristic
92C8×C127C2×Q814yes—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127C4⋊Q812yes—
96C22×C127C22×D411yescharacteristic
97C4×C127—41no—
98C4×C127—42no—
99C4×C127—41no—
100C4×C127C4⋊Q812yes—
101C4×C127C22×Q811yescharacteristic
102C4×C127C22×Q811yescharacteristic
103Dic127—22no—
104Dic127—22no—
105Dic127—41no—
106Dic127—42no—
107Dic127—41no—
108C2×C127—22no—
109C2×C127C2×C4⋊Q811yescharacteristic
110C2×C127#57711yescharacteristic
111C2×C127#57711yescharacteristic
112#3163—1271nomaximal, Sylow
113C127#316311yescharacteristic, complemented, Sylow
114#544—1271no—
115#544—1271no—
116#501—1272no—
117#501—1272no—
118#499—1272no—
119#498—1272no—
120#428—1271no—
121#428D12711yescharacteristic
122#427—1271no—
123C2×#106—1271no—
124#405—1271no—
125C2×M16C2×D12711yescharacteristic
126C22×C8—1271no—
127C4⋊Q8—1272no—
128C4⋊Q8—1272no—
129C4⋊Q8—2541no—
130C4⋊Q8—2541no—
131#125—1272no—
132#125—1272no—
133#125—2541no—
134#125—2541no—
135C22⋊Q8—2542no—
136C22⋊Q8—2542no—
137#117—1271no—
138#117—1271no—
139#117C2×D12711yescharacteristic
140C2×C4⋊C4—1271no—
141C2×C4⋊C4C2×D12711yescharacteristic
142#107—1272no—
143#107—1271no—
144#107—1271no—
145#107C2×D12712yes—
146#106—1272no—
147#106—1272no—
148#106—1272no—
149#106—1271no—
150#106—1271no—
151#106C2×D12712yes—
152#97—1272no—
153C4×C8—1272no—
154C2×Q8—2542no—
155C2×Q8—2542no—
156C22×C4—21no—
157C22×C4—2541no—
158C22×C4C22×D12711yescharacteristic
159M16Dic25414yescomplemented
160C2×C8—1274no—
161C2×C8—1271no—
162C2×C8—1271no—
163C2×C8C22×D12712yes—
164C4⋊C4—21no—
165C4⋊C4—1271no—
166C4⋊C4—1271no—
167C4⋊C4—1272no—
168C4⋊C4—1271no—
169C4⋊C4—1271no—
170C4⋊C4—2542no—
171C4⋊C4—2542no—
172C4⋊C4—2542no—
173C4⋊C4—2542no—
174C4⋊C4—2542no—
175C4⋊C4—2542no—
176C4⋊C4—2541no—
177C4⋊C4C22×D12712yes—
178C4⋊C4C22×D12711yescharacteristic
179C4⋊C4C22×D12711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—1272no—
182C22⋊C4—2541no—
183C42—21no—
184C42—1272no—
185C42—2541no—
186C23D4×D12711yescharacteristic
187Q8—10162no—
188Q8—10162no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—1272no—
194C2×C4—1272no—
195C2×C4—10161no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4—2542no—
199C2×C4D4×D12711yescharacteristic
200C2×C4Q8×D12712yes—
201C2×C4C2×Dic25412yes—
202C2×C4C23×D12711yescharacteristic
203C8—1272no—
204C8—1272no—
205C8C2×Dic25414yes—
206C22—21no—
207C22—21no—
208C22#1686120812yes—
209C22C2×D4×D12711yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—5082no—
214C4—5081no—
215C4—5081no—
216C4—2542no—
217C4—2542no—
218C4#1686120812yes—
219C4C2×Q8×D12711yescharacteristic
220C4C22×Dic25411yescharacteristic
221C2—22no—
222C2C2×#1686120811yescharacteristic
223C2#2255068511yescharacteristic
224C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#428D127
hypocenterC127#3163
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #22550284C2semidirect
4. #22550283C2semidirect
5. #22550277C2semidirect
6. #22550273C2semidirect
44. C2×C127⋊C8Q8semidirect
65. M16×C127Q8semidirect
113. C127#3163semidirect
159. M16Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225505042
#225502842
#225502832
#225502772
#225502732
#225499182
#225499122
C2×#168611402
#225498342
#428×C1272
#3163127