Finite Group Databasebeta

← #30640104

Subgroups — #30640104

Order 16256.

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

normal subgroups only

This group has 284 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×#16861140C211yesmaximal, characteristic, complemented
2C2×C4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic
4C2×#16861130C211yesmaximal, characteristic
5#22549774C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7#390×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12C2×Dic127⋊C4—21no—
13#16861140—22no—
14C4⋊(C127⋊C8)—22no—
15#16861130—22no—
16#16861130—22no—
17C127⋊#95—21no—
18C22×D4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C22⋊C4×C127—21no—
21C2×C22⋊C4×C127—21no—
22C2×C42×C127C2211yescharacteristic
23C22×Dic127—21no—
24C22×Dic127—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8D412yes—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D412yes—
35Dic127⋊C4—42no—
36C24×C127—21no—
37C2×D4×C127—21no—
38C2×D4×C127—21no—
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41C2×D4×C127—21no—
42C2×D4×C127Q812yes—
43C22×C4×C127—21no—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127D411yescharacteristic
46C22×C4×C127C2311yescharacteristic
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C22⋊C4×C127—42no—
50C22⋊C4×C127—42no—
51C42×C127—21no—
52C42×C127—21no—
53C2×Dic127—22no—
54C2×Dic127—41no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C127⋊C8—22no—
60C127⋊C8—41no—
61C23×C127—22no—
62C23×C127—41no—
63C23×C127—41no—
64C23×C127—41no—
65C23×C127—41no—
66C23×C127—41no—
67C23×C127—41no—
68C23×C127C2×D411yescharacteristic
69D4×C127—22no—
70D4×C127—22no—
71D4×C127—41no—
72D4×C127—41no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127SD1612yes—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×Q811yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127—81no—
92C22×C127—81no—
93C22×C127—81no—
94C22×C127—81no—
95C22×C127#12111yescharacteristic
96C22×C127#12111yescharacteristic
97C22×C127C22⋊Q811yescharacteristic
98C22×C127C2×SD1611yescharacteristic
99C22×C127#13511yescharacteristic
100C22×C127AGL(1,ℤ/8)11yescharacteristic
101C22×C127AGL(1,ℤ/8)11yescharacteristic
102C4×C127—22no—
103C4×C127—41no—
104C4×C127—81no—
105C4×C127#12111yescharacteristic
106C4×C127C22⋊Q811yescharacteristic
107C4×C127C22⋊Q811yescharacteristic
108C4×C127#12611yescharacteristic
109Dic127—42no—
110Dic127—81no—
111C2×C127—82no—
112C2×C127#39611yescharacteristic
113C2×C127#46011yescharacteristic
114C2×C127#46311yescharacteristic
115C2×C127#46511yescharacteristic
116C2×C127#46911yescharacteristic
117C2×C127#47611yescharacteristic
118C2×C127#47811yescharacteristic
119#2065—1271nomaximal, Sylow
120C127#206511yescharacteristic, complemented, Sylow
121C2×#106—1271no—
122C2×C4⋊C8—1271no—
123C2×#102—1271no—
124C2×#102—1271no—
125#390D12711yescharacteristic
126#389—1271no—
127#340—1271no—
128C22×D4C2×D12711yescharacteristic
129C22×C8—1271no—
130C22×C8—1271no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4—1271no—
133C2×C4⋊C4—2541no—
134C2×C4⋊C4C2×D12711yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C42C2×D12711yescharacteristic
138#106—2542no—
139C4⋊C8—2542no—
140#102—2542no—
141#102—2542no—
142#95—2541no—
143C24—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4Dic25412yes—
150C22×C4—21no—
151C22×C4—2541no—
152C22×C4—2541no—
153C22×C4C22⋊D12711yescharacteristic
154C22×C4C22⋊D12711yescharacteristic
155C22×C4C22×D12711yescharacteristic
156C2×C8—1272no—
157C2×C8—2541no—
158C2×C8—2541no—
159C2×C8—2541no—
160C2×C8—2541no—
161C2×C8—2541no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—1272no—
165C4⋊C4—5082no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C42—21no—
172C42—21no—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23C2×C22⋊D12711yescharacteristic
181D4—22no—
182D4—22no—
183D4—41no—
184D4—41no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—5081no—
194C2×C4—5081no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—2542no—
199C2×C4#1452323612yes—
200C2×C4D4×D12711yescharacteristic
201C2×C4D4×D12711yescharacteristic
202C2×C4C2×Dic25411yescharacteristic
203C2×C4#1452324711yescharacteristic
204C2×C4#1452324711yescharacteristic
205C2×C4#1452324811yescharacteristic
206C8—5081no—
207C8—2542no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#1686118011yescharacteristic
216C22#1686118111yescharacteristic
217C22#1686118811yescharacteristic
218C22C2×#1452323611yescharacteristic
219C22#1686122911yescharacteristic
220C22#1686123411yescharacteristic
221C22#1686123511yescharacteristic
222C4—22no—
223C4—41no—
224C4—81no—
225C4—10161no—
226C4—5082no—
227C4#1686118211yescharacteristic
228C4#1686118611yescharacteristic
229C4#1686118611yescharacteristic
230C4#1686120211yescharacteristic
231C2—82no—
232C2#2254979911yescharacteristic
233C2#2255007311yescharacteristic
234C2#2255008811yescharacteristic
235C2#2255010311yescharacteristic
236C2#2255012711yescharacteristic
237C2#2255016611yescharacteristic
238C2#2255017611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenter#390D127
hypocenterC127#2065
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16861140C2semidirect
2. C2×C4⋊(C127⋊C8)C2semidirect
5. #22549774C2semidirect
6. #22549638C2semidirect
120. C127#2065semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
C2×C4⋊(C127⋊C8)2
C2×#168611302
#225497742
#225496382
#390×C1272
#2065127