Finite Group Databasebeta

← #30639072

Subgroups — #30639072

Order 16256.

338 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×Dic508C211yesmaximal, characteristic
2C2×#16861174C211yesmaximal, characteristic
3C2×#16861132C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5#22549781C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10C2×Dic508—22no—
11C2×Dic508—22no—
12C2×Dic508C414yescomplemented
13C22×C127⋊C8C2211yescharacteristic
14#16861174—22no—
15#16861174—22no—
16C2×Dic127⋊C4—21no—
17C2×C4×Dic127—21no—
18C2×C4×Dic127C2211yescharacteristic
19#16861132—22no—
20#16861132—22no—
21C127⋊#95—21no—
22C127⋊#95—21no—
23C22×C8×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254C2×C412yes—
35C2×Dic254C2×C412yes—
36C2×Dic254D412yes—
37C2×Dic254D412yes—
38C22×Dic127—21no—
39C22×Dic127—21no—
40C22×Dic127D411yescharacteristic
41C22×Dic127D411yescharacteristic
42Dic508—28no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46Dic127⋊C4—24no—
47C4×Dic127—24no—
48C4×Dic127—21no—
49C4×Dic127—21no—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C2×C8×C127C2×C412yes—
56C4⋊C4×C127—21no—
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59C4⋊C4×C127D412yes—
60Dic254—22no—
61Dic254—22no—
62Dic254—22no—
63Dic254—22no—
64Dic254—22no—
65Dic254—41no—
66Dic254—41no—
67Dic254C22⋊C414yes—
68C2×Dic127—22no—
69C2×Dic127—21no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—21no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127Q1612yescomplemented
80C127⋊C8—41no—
81C23×C127C2×D411yescharacteristic
82C2×C4×C127—22no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127C22×C411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127#4111yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C8×C127—22no—
95C22×C127C4×D411yescharacteristic
96C22×C127#12011yescharacteristic
97C22×C127#12111yescharacteristic
98C22×C127C2×Q1611yescharacteristic
99C22×C127#13511yescharacteristic
100C22×C127#13711yescharacteristic
101C22×C127#13711yescharacteristic
102C4×C127—42no—
103C4×C127—81no—
104C4×C127C2×C22⋊C411yescharacteristic
105C4×C127C4×D411yescharacteristic
106C4×C127#12111yescharacteristic
107C4×C127#12311yescharacteristic
108Dic127—24no—
109Dic127—22no—
110Dic127—42no—
111Dic127—41no—
112C2×C127#38611yescharacteristic
113C2×C127C4×Q1611yescharacteristic
114C2×C127#44111yescharacteristic
115C2×C127#45111yescharacteristic
116C2×C127#45211yescharacteristic
117C2×C127#46211yescharacteristic
118C2×C127#46411yescharacteristic
119#1878—1271nomaximal, Sylow
120C127#187811yescharacteristic, complemented, Sylow
121C22×Q16—1271no—
122C2×C4×Q8—1271no—
123C2×#103—1271no—
124C2×#103—1271no—
125#391—1271no—
126#340D12711yescharacteristic
127#336—1271no—
128C22×Q8—1271no—
129C22×Q8—1271no—
130C2×Q16—1274no—
131C2×Q16—2542no—
132C2×Q16—2542no—
133C22×C8—1271no—
134C22×C8C2×D12711yescharacteristic
135C4×Q8—2542no—
136C4×Q8—2542no—
137C2×C4⋊C4—2541no—
138C2×C4⋊C4C2×D12711yescharacteristic
139C2×C4⋊C4C2×D12711yescharacteristic
140C2×C42—1271no—
141C2×C42—2541no—
142#103—2542no—
143#103—2542no—
144#95—2541no—
145#95—2541no—
146C2×Q8—1272no—
147C2×Q8—1272no—
148C2×Q8—1272no—
149C2×Q8—1272no—
150C2×Q8—2541no—
151C2×Q8—2541no—
152C2×Q8—2541no—
153C2×Q8—2541no—
154C2×Q8—2541no—
155C2×Q8—2541no—
156C2×Q8—2541no—
157C2×Q8—2541no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1271no—
161C22×C4—1271no—
162C22×C4—2541no—
163C22×C4—2541no—
164C22×C4C22×D12711yescharacteristic
165Q16—2548no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—2541no—
169C2×C8—2541no—
170C2×C8—2541no—
171C2×C8C4×D12712yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—2544no—
176C4⋊C4D50812yes—
177C42—2544no—
178C42—2541no—
179C42—2541no—
180C23D4×D12711yescharacteristic
181Q8—1274no—
182Q8—5081no—
183Q8—5081no—
184Q8—2542no—
185Q8—2542no—
186Q8—2542no—
187Q8—2542no—
188Q8—2542no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—1272no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—2541no—
199C2×C4—2541no—
200C2×C4—2542no—
201C2×C4—2541no—
202C2×C4—2541no—
203C2×C4—2542no—
204C2×C4—2541no—
205C2×C4—2542no—
206C2×C4—2542no—
207C2×C4C2×C4×D12711yescharacteristic
208C2×C4D4×D12711yescharacteristic
209C2×C4D4×D12711yescharacteristic
210C2×C4C2×D50811yescharacteristic
211C2×C4#1452324811yescharacteristic
212C2×C4#1452324911yescharacteristic
213C8—22no—
214C8—5081no—
215C22#1686117111yescharacteristic
216C22#1686117911yescharacteristic
217C22#1686118211yescharacteristic
218C22Q16×D12711yescharacteristic
219C22#1686122811yescharacteristic
220C22#1686123811yescharacteristic
221C22#1686123811yescharacteristic
222C4—42no—
223C4—81no—
224C4—5081no—
225C4—5082no—
226C4—2542no—
227C4—2544no—
228C4C22⋊C4×D12711yescharacteristic
229C4C4×D50811yescharacteristic
230C4#1686118411yescharacteristic
231C4#1686119411yescharacteristic
232C2#2254975411yescharacteristic
233C2#2254997011yescharacteristic
234C2#2254998211yescharacteristic
235C2#2255003311yescharacteristic
236C2#2255003911yescharacteristic
237C2#2255008511yescharacteristic
238C2#2255009711yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#340D127
hypocenterC127#1878
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
12. C2×Dic508C4semidirect
79. C2×Dic127Q16semidirect
120. C127#1878semidirect

Maximal subgroups

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

subgroupindex
C22×Dic5082
C2×#168611742
C2×#168611322
#225497812
#225496302
#340×C1272
#1878127