Finite Group Databasebeta

← #30639870

Subgroups — #30639870

Order 16256.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16861208C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic
3C2×#16861132C211yesmaximal, characteristic
4#22549781C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×Dic254—21no—
9C22×Dic254C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11#16861208—22no—
12#16861208—22no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×C4×Dic127C2211yescharacteristic
16#16861134—22no—
17#16861132—22no—
18C127⋊#95—21no—
19C127⋊#95—21no—
20C22×C8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×Dic254—24no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—42no—
30C2×Dic254—42no—
31C2×Dic254Q812yescomplemented
32C22×Dic127—21no—
33C22×Dic127—21no—
34C22×Dic127D411yescharacteristic
35C22×Dic127D411yescharacteristic
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127D412yes—
41Dic127⋊C4—24no—
42C4×Dic127—21no—
43C4×Dic127—21no—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C22×C4×C127C2311yescharacteristic
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127D412yes—
54Dic254—22no—
55Dic254—22no—
56Dic254—48no—
57Dic254—41no—
58Dic254—41no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127SD1612yes—
70C2×Dic127Q1612yes—
71C127⋊C8—41no—
72C23×C127C2×D411yescharacteristic
73C2×C4×C127—22no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×Q811yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127—41no—
86C22×C127#12011yescharacteristic
87C22×C127C22⋊Q811yescharacteristic
88C22×C127#12411yescharacteristic
89C22×C127C2×SD1611yescharacteristic
90C22×C127C2×Q1611yescharacteristic
91C22×C127AGL(1,ℤ/8)11yescharacteristic
92C22×C127#13711yescharacteristic
93C4×C127—42no—
94C4×C127—81no—
95C4×C127C22⋊Q811yescharacteristic
96C4×C127C22⋊Q811yescharacteristic
97C4×C127#12411yescharacteristic
98C4×C127#12411yescharacteristic
99Dic127—22no—
100Dic127—22no—
101Dic127—42no—
102Dic127—42no—
103C2×C127#39311yescharacteristic
104C2×C127#45011yescharacteristic
105C2×C127#45111yescharacteristic
106C2×C127#47511yescharacteristic
107C2×C127#47711yescharacteristic
108C2×C127#48711yescharacteristic
109C2×C127#48911yescharacteristic
110#2024—1271nomaximal, Sylow
111C127#202411yescharacteristic, complemented, Sylow
112C2×C4⋊Q8—1271no—
113C2×#103—1271no—
114C2×#103—1271no—
115#391—1271no—
116#340—1271no—
117#340D12711yescharacteristic
118#336—1271no—
119C22×Q8—1271no—
120C22×Q8—2541no—
121C22×C8—1271no—
122C22×C8C2×D12711yescharacteristic
123C4⋊Q8—2542no—
124C4⋊Q8—2542no—
125C2×C4⋊C4—1271no—
126C2×C4⋊C4—2541no—
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C4⋊C4C2×D12711yescharacteristic
129C2×C42—1271no—
130#103—2542no—
131#103—2542no—
132#95—2541no—
133#95—2541no—
134C2×Q8—1272no—
135C2×Q8—5082no—
136C2×Q8—5082no—
137C2×Q8—2544no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—1271no—
146C22×C4—1271no—
147C22×C4—2541no—
148C22×C4—2541no—
149C22×C4C22×D12711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—2541no—
154C2×C8—2541no—
155C2×C8—2541no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—1272no—
160C4⋊C4—2544no—
161C4⋊C4—2541no—
162C4⋊C4D50812yes—
163C42—2541no—
164C42—2541no—
165C23D4×D12711yescharacteristic
166Q8—5088no—
167Q8—5081no—
168Q8—5081no—
169Q8—2542no—
170Q8—2542no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—1272no—
178C2×C4—1272no—
179C2×C4—5081no—
180C2×C4—5081no—
181C2×C4—5081no—
182C2×C4—5081no—
183C2×C4—2542no—
184C2×C4—2542no—
185C2×C4—2541no—
186C2×C4—2541no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×D50811yescharacteristic
191C2×C4Q8×D12711yescharacteristic
192C2×C4#1452324711yescharacteristic
193C2×C4#1452324811yescharacteristic
194C2×C4#1452324811yescharacteristic
195C8—41no—
196C8—5081no—
197C22#1686117911yescharacteristic
198C22#1686118711yescharacteristic
199C22#1686119511yescharacteristic
200C22SD16×D12711yescharacteristic
201C22Q16×D12711yescharacteristic
202C22#1686123311yescharacteristic
203C22#1686123811yescharacteristic
204C4—42no—
205C4—81no—
206C4—5082no—
207C4—5082no—
208C4—2542no—
209C4—2542no—
210C4#1686118711yescharacteristic
211C4#1686118911yescharacteristic
212C4#1686119511yescharacteristic
213C4#1686119711yescharacteristic
214C2#2254978911yescharacteristic
215C2#2255002811yescharacteristic
216C2#2255003311yescharacteristic
217C2#2255016011yescharacteristic
218C2#2255017011yescharacteristic
219C2#2255022511yescharacteristic
220C2#2255023411yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socle, prime radicalC23×C127C2×D4
hypercenter#340D127
hypocenterC127#2024
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic254Q8semidirect
111. C127#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#168612082
C2×#168611342
C2×#168611322
#225497812
#225496382
#225496302
#340×C1272
#2024127