Finite Group Databasebeta

← #30639908

Subgroups — #30639908

Order 16256.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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#22550502C211yesmaximal, characteristic, complemented
2C2×#16861123C211yesmaximal, characteristic
3C2×#16861123C211yesmaximal, characteristic
4C2×#16861123C211yesmaximal, characteristic
5#22549643C211yesmaximal, characteristic, complemented
6#22549642C211yesmaximal, characteristic, complemented
7#342×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2211yescharacteristic
9#16861188—22no—
10#16861186—22no—
11#16861186—22no—
12C2×C22⋊Dic127C2211yescharacteristic
13C2×C22⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127C2211yescharacteristic
15#16861123—22no—
16#16861123—22no—
17#16861123—22no—
18C2×C22⋊C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2211yescharacteristic
20C2×C22⋊C4×C127C2211yescharacteristic
21C23×D127Q811yescharacteristic, complemented
22C2×Dic254—22no—
23C2×C22⋊D127—21no—
24C2×C22⋊D127—21no—
25C2×C22⋊D127—21no—
26C2×C22⋊D127—21no—
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C22×Dic127—21no—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127Q811yescharacteristic, complemented
33C4⋊Dic127—22no—
34Dic127⋊C4—24no—
35C22⋊Dic127—21no—
36C22⋊Dic127—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127D412yes—
40C22⋊Dic127D412yes—
41C24×C127C2311yescharacteristic
42C22×C4×C127—21no—
43C22×C4×C127—21no—
44C22×C4×C127—21no—
45C22⋊C4×C127—21no—
46C22⋊C4×C127—21no—
47C22⋊C4×C127—21no—
48C22⋊C4×C127—21no—
49C22⋊C4×C127—21no—
50C22⋊C4×C127D412yes—
51C22×D127—21no—
52C22×D127—21no—
53C22×D127—21no—
54C22×D127—81no—
55Dic254—82no—
56C22⋊D127—41no—
57C22⋊D127—41no—
58C22⋊D127—41no—
59C22⋊D127—41no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—41no—
66C2×Dic127—81no—
67C2×Dic127—81no—
68C2×Dic127—81no—
69C23×C127—81no—
70C23×C127C2×D411yescharacteristic
71C23×C127C2×D411yescharacteristic
72C23×C127C2×D411yescharacteristic
73C23×C127C2×Q811yescharacteristic
74C23×C127#4111yescharacteristic
75C23×C127#4111yescharacteristic
76C23×C127#4111yescharacteristic
77C2×C4×C127—22no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—81no—
81C2×C4×C127—81no—
82C2×C4×C127—81no—
83C22×C127—41no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127—41no—
87C22×C127—41no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127#12011yescharacteristic
91C22×C127C22⋊Q811yescharacteristic
92C22×C127C22⋊Q811yescharacteristic
93C22×C127C22⋊Q811yescharacteristic
94C22×C127#12411yescharacteristic
95C22×C127#12411yescharacteristic
96C22×C127#12411yescharacteristic
97C2×D127—41no—
98C2×D127—81no—
99C2×D127—81no—
100C2×D127—81no—
101C4×C127—42no—
102C4×C127—81no—
103C4×C127—81no—
104Dic127—42no—
105Dic127—42no—
106Dic127—41no—
107Dic127—81no—
108C2×C127—21no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127#39311yescharacteristic
116D127—81no—
117#2032—1271nomaximal, Sylow
118C127#203211yescharacteristic, complemented, Sylow
119#543—1271no—
120C2×#99—1271no—
121C2×#99—1271no—
122C2×#99—1271no—
123#342—1271no—
124#342—1271no—
125#342D12711yescharacteristic, complemented
126C22×D4—1271no—
127C22⋊Q8—2542no—
128C22⋊Q8—2542no—
129C22⋊Q8—2542no—
130C2×C22⋊C4—1271no—
131C2×C22⋊C4—1271no—
132C2×C22⋊C4—1271no—
133C2×C22⋊C4C2×D12711yescharacteristic
134C2×C22⋊C4C2×D12711yescharacteristic
135C2×C22⋊C4C2×D12711yescharacteristic
136#99—2542no—
137#99—2542no—
138#99—2542no—
139C24—1271no—
140C24C22×D12711yescharacteristic
141C2×Q8—2542no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1271no—
152C22×C4—2541no—
153C22×C4—2541no—
154C22×C4—2541no—
155C4⋊C4—2544no—
156C4⋊C4—2542no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—1272no—
163C22⋊C4—1272no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4D50812yes—
169C23—81no—
170C23—10161no—
171C23—2541no—
172C23—2541no—
173C23—2541no—
174C23D4×D12711yescharacteristic
175C23D4×D12711yescharacteristic
176C23C2×D50811yescharacteristic
177C23Q8×D12711yescharacteristic
178C23#1452324711yescharacteristic
179C23#1452324811yescharacteristic
180C23#1452324811yescharacteristic
181Q8—10162no—
182D4—5081no—
183D4—5081no—
184D4—5081no—
185D4—5081no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—10161no—
193C2×C4—10161no—
194C2×C4—10161no—
195C2×C4—5081no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4—2541no—
199C2×C4—2542no—
200C2×C4—2542no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—10161no—
209C22—10161no—
210C22—10161no—
211C22—5081no—
212C22#1686117911yescharacteristic
213C22#1686118711yescharacteristic
214C22#1686118711yescharacteristic
215C22#1686118911yescharacteristic
216C22#1686119511yescharacteristic
217C22#1686119511yescharacteristic
218C22#1686119711yescharacteristic
219C4—42no—
220C4—81no—
221C4—81no—
222C4—10161no—
223C4—5082no—
224C4—5082no—
225C4—5081no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—10161no—
234C2#2254978911yescharacteristic
subgroupG/H
centerC2#22549789
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#342×C127C2
socleC2×C127#393
prime radicalC22×C22⋊D127C22
hypercenter#342D127
hypocenterC127#2032
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #22550502C2semidirect
5. #22549643C2semidirect
6. #22549642C2semidirect
7. #342×C127C2semidirect
21. C23×D127Q8semidirect
32. C22×Dic127Q8semidirect
118. C127#2032semidirect
125. #342D127semidirect

Maximal subgroups

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

subgroupindex
#225505022
C2×#168611232
#225496432
#225496422
#342×C1272
#2032127