Finite Group Databasebeta

← #30648534

Subgroups — #30648534

Order 16256.

376 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 135 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#22550429C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic
3#22550186C212yesmaximal, complemented
4#22550182C212yesmaximal, complemented
5#22550170C212yesmaximal, complemented
6#22550160C212yesmaximal, complemented
7#22549916C211yesmaximal, characteristic
8#22549899C211yesmaximal, characteristic
9#22549886C211yesmaximal, characteristic
10#22549884C211yesmaximal, characteristic
11#428×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14#16861208C2211yescharacteristic
15#16861208C2211yescharacteristic
16#16861200C2211yescharacteristic
17#16861200C2211yescharacteristic
18#16861186—21no—
19#16861186—21no—
20C4×Dic254—21no—
21C4×Dic254—21no—
22#16861174—21no—
23#16861174—21no—
24#16861174C2212yes—
25#16861174C2212yes—
26#16861168C2211yescharacteristic
27C2×Dic127⋊C4—21no—
28C2×C4×Dic127—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861142C2212yes—
31#16861140C2212yes—
32Dic127⋊C8C2212yes—
33Dic127⋊C8C2212yes—
34#16861134C2212yes—
35#16861134C2212yes—
36#16861132C2212yes—
37#16861132C2212yes—
38C2×M16×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#107×C127C2212yes—
42#106×C127C2212yes—
43C2×Dic254—21no—
44C2×Dic254—21no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254Q814yescomplemented
48C2×Dic254C2312yes—
49C22×Dic127—21no—
50C22×Dic127D411yescharacteristic
51C22×Dic127D411yescharacteristic
52#14523232—22no—
53C2×C127⋊C8C2312yes—
54C4⋊Dic127—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127C2311yescharacteristic
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—21no—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—22no—
63Dic127⋊C4—22no—
64C22⋊Dic127—21no—
65C4×Dic127—22no—
66C4×Dic127—22no—
67C4×Dic127—21no—
68C4×Dic127D411yescharacteristic
69C4×Dic127D411yescharacteristic
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72C22×C4×C127—21no—
73C22×C4×C127C2311yescharacteristic
74M16×C127—22no—
75C2×C8×C127C2312yes—
76C4⋊C4×C127—21no—
77C4⋊C4×C127C2311yescharacteristic
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C42×C127—21no—
82Dic254—24no—
83Dic254—22no—
84Dic254—42no—
85Dic254—42no—
86Dic254C2×Q814yes—
87C2×Dic127—24no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C2×Dic127—41no—
93C2×Dic127C2×D411yescharacteristic
94C2×Dic127C2×D411yescharacteristic
95C2×Dic127C2×D411yescharacteristic
96C2×Dic127C2×D411yescharacteristic
97C127⋊C8—22no—
98C23×C127C2×D411yescharacteristic
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—22no—
102C2×C4×C127—42no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×Q812yes—
105C2×C4×C127#4112yes—
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127C22⋊Q812yes—
111C22×C127C22×D411yescharacteristic
112C4×C127—41no—
113C4×C127—42no—
114C4×C127—41no—
115C4×C127C22⋊Q812yes—
116C4×C127C22×Q811yescharacteristic
117C4×C127C2×#4111yescharacteristic
118Dic127—24no—
119Dic127—21no—
120Dic127—21no—
121Dic127—21no—
122Dic127—41no—
123Dic127—41no—
124Dic127#13712yes—
125C2×C127—22no—
126C2×C127C2×C22⋊Q811yescharacteristic
127C2×C127C2×#13711yescharacteristic
128C2×C127#57511yescharacteristic
129#3078—1271nomaximal, Sylow
130C127#307811yescharacteristic, complemented, Sylow
131#533—1271no—
132C2×C4×Q8—1271no—
133#479—1272no—
134#479—1272no—
135#477—1272no—
136#475—1272no—
137#428—1271no—
138#428D12711yescharacteristic
139#423—1271no—
140#419—1271no—
141#419—1271no—
142C22×Q8—1271no—
143C2×M16—1271no—
144C2×M16C2×D12711yescharacteristic
145C4⋊Q8—1271no—
146C4⋊Q8—1271no—
147#125—1271no—
148#125—1271no—
149C22⋊Q8—2541no—
150C22⋊Q8—2541no—
151C4×Q8—1272no—
152C4×Q8—1272no—
153C4×Q8—2541no—
154C4×Q8—2541no—
155C4×Q8—2541no—
156C4×Q8—2541no—
157#117—1271no—
158#117C2×D12711yescharacteristic
159C2×C4⋊C4—2541no—
160C2×C4⋊C4C2×D12711yescharacteristic
161C2×C42—1271no—
162C2×C42—2541no—
163#107—1272no—
164#107C2×D12712yes—
165#106—1272no—
166#106C2×D12712yes—
167C4⋊C8—1272no—
168C4⋊C8—1272no—
169#103—1272no—
170#103—1272no—
171#103—1272no—
172#103—1272no—
173C2×Q8—1274no—
174C2×Q8—1272no—
175C2×Q8—2541no—
176C2×Q8—2541no—
177C2×Q8—2541no—
178C2×Q8—2542no—
179C22×C4—21no—
180C22×C4—1271no—
181C22×C4—1271no—
182C22×C4—2541no—
183C22×C4C22×D12711yescharacteristic
184M16—22no—
185M16—2542no—
186C2×C8—1272no—
187C2×C8C22×D12712yes—
188C4⋊C4—21no—
189C4⋊C4—1271no—
190C4⋊C4—1271no—
191C4⋊C4—2542no—
192C4⋊C4—2542no—
193C4⋊C4—2541no—
194C4⋊C4—2541no—
195C4⋊C4—2541no—
196C4⋊C4—2541no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4C22×D12711yescharacteristic
200C4⋊C4C22×D12711yescharacteristic
201C4⋊C4C22×D12712yes—
202C22⋊C4—21no—
203C22⋊C4—2541no—
204C42—21no—
205C42—1271no—
206C42—1271no—
207C42—1271no—
208C42—1271no—
209C42—2542no—
210C42—2542no—
211C42—2541no—
212C23D4×D12711yescharacteristic
213Q8—1274no—
214Q8—5082no—
215Q8—5082no—
216Q8—2544no—
217Q8—2542no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—1271no—
223C2×C4—1271no—
224C2×C4—1271no—
225C2×C4—1271no—
226C2×C4—5081no—
227C2×C4—2542no—
228C2×C4—2541no—
229C2×C4—2541no—
230C2×C4—2544no—
231C2×C4—2542no—
232C2×C4D4×D12711yescharacteristic
233C2×C4Q8×D12712yes—
234C2×C4#1452324812yes—
235C2×C4C23×D12711yescharacteristic
236C8—22no—
237C8—2542no—
238C22—21no—
239C22—21no—
240C22#1686118712yes—
241C22C2×D4×D12711yescharacteristic
242C4—41no—
243C4—42no—
244C4—41no—
245C4—1272no—
246C4—5081no—
247C4—5081no—
248C4—2544no—
249C4—2541no—
250C4—2541no—
251C4—2541no—
252C4#1686118712yes—
253C4C2×Q8×D12711yescharacteristic
254C4C2×#1452324811yescharacteristic
255C2—22no—
256C2C2×#1686118711yescharacteristic
257C2#137×D12711yescharacteristic
258C2#2255067211yescharacteristic
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#3078
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. #22550186C2semidirect
4. #22550182C2semidirect
5. #22550170C2semidirect
6. #22550160C2semidirect
47. C2×Dic254Q8semidirect
130. C127#3078semidirect

Maximal subgroups

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

subgroupindex
#225504292
C2×#168611742
#225501862
#225501822
#225501702
#225501602
#225499162
#225498992
#225498862
#225498842
#428×C1272
#3078127