Finite Group Databasebeta

← #30638744

Subgroups — #30638744

Order 16256.

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

normal subgroups only

This group has 292 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
1C2×#16861174C211yesmaximal, characteristic
2C2×Dic127⋊C8C211yesmaximal, characteristic
3C2×#16861134C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5#22549741C211yesmaximal, characteristic
6#22549638C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861174—22no—
11#16861174C414yescomplemented
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4—21no—
14C2×C4×Dic127—21no—
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17#16861134—22no—
18#16861132—22no—
19C127⋊#95—21no—
20C127⋊#95—21no—
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254C2×C412yes—
28C2×Dic254D412yes—
29C2×Dic254Q812yes—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C22×Dic127D411yescharacteristic
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39Dic127⋊C4—24no—
40C4×Dic127—24no—
41C4×Dic127C2×C412yes—
42C4×Dic127D412yes—
43C22×C4×C127—21no—
44C22×C4×C127—21no—
45C22×C4×C127C2311yescharacteristic
46C2×C8×C127—21no—
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127C2×C412yes—
53Dic254—22no—
54Dic254—22no—
55Dic254—22no—
56Dic254C4⋊C414yes—
57C2×Dic127—22no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127C22⋊C412yes—
68C2×Dic127C22⋊C412yes—
69C127⋊C8—41no—
70C23×C127C2×D411yescharacteristic
71C2×C4×C127—22no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127C22×C411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×Q811yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C8×C127—41no—
84C22×C127C2×C22⋊C411yescharacteristic
85C22×C127#12111yescharacteristic
86C22×C127C22⋊Q811yescharacteristic
87C22×C127#13511yescharacteristic
88C22×C127#13511yescharacteristic
89C22×C127#13711yescharacteristic
90C22×C127#13711yescharacteristic
91C4×C127—42no—
92C4×C127—81no—
93C4×C127C2×C4⋊C411yescharacteristic
94C4×C127#11711yescharacteristic
95C4×C127#12111yescharacteristic
96C4×C127C22⋊Q811yescharacteristic
97Dic127—24no—
98Dic127—22no—
99Dic127—22no—
100Dic127—81no—
101C2×C127#38011yescharacteristic
102C2×C127#41611yescharacteristic
103C2×C127#41911yescharacteristic
104C2×C127#46411yescharacteristic
105C2×C127#46411yescharacteristic
106C2×C127#47911yescharacteristic
107C2×C127#47911yescharacteristic
108#1805—1271nomaximal, Sylow
109C127#180511yescharacteristic, complemented, Sylow
110C2×C4×Q8—1271no—
111C2×C4⋊C8—1271no—
112C2×#103—1271no—
113C2×#103—1271no—
114#382—1271no—
115#340—1271no—
116#340D12711yescharacteristic
117C22×Q8—1271no—
118C22×C8—1271no—
119C22×C8C2×D12711yescharacteristic
120C4×Q8—1274no—
121C4×Q8—2542no—
122C2×C4⋊C4—1271no—
123C2×C4⋊C4—2541no—
124C2×C4⋊C4C2×D12711yescharacteristic
125C2×C4⋊C4C2×D12711yescharacteristic
126C2×C42—1271no—
127C2×C42—2541no—
128C4⋊C8—2542no—
129#103—2542no—
130#103—2542no—
131#95—2541no—
132#95—2541no—
133C2×Q8—1272no—
134C2×Q8—1272no—
135C2×Q8—1272no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—1271no—
142C22×C4—1271no—
143C22×C4—2541no—
144C22×C4—2541no—
145C22×C4C22×D12711yescharacteristic
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—2541no—
150C2×C8—2541no—
151C2×C8—2541no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—2544no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4C4×D12712yes—
159C42—1272no—
160C42—1272no—
161C42—2544no—
162C23D4×D12711yescharacteristic
163Q8—1274no—
164Q8—2542no—
165Q8—2542no—
166Q8—2542no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—1272no—
174C2×C4—1272no—
175C2×C4—5081no—
176C2×C4—5081no—
177C2×C4—5081no—
178C2×C4—2542no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4—2541no—
182C2×C4—2541no—
183C2×C4—2542no—
184C2×C4—2542no—
185C2×C4C2×C4×D12711yescharacteristic
186C2×C4D4×D12711yescharacteristic
187C2×C4D4×D12711yescharacteristic
188C2×C4Q8×D12711yescharacteristic
189C2×C4#1452324811yescharacteristic
190C2×C4#1452324811yescharacteristic
191C8—41no—
192C8—5081no—
193C22C22⋊C4×D12711yescharacteristic
194C22#1686118211yescharacteristic
195C22#1686118711yescharacteristic
196C22#1686122811yescharacteristic
197C22#1686123011yescharacteristic
198C22#1686123811yescharacteristic
199C22#1686124111yescharacteristic
200C4—42no—
201C4—81no—
202C4—10161no—
203C4—2542no—
204C4—2542no—
205C4—2544no—
206C4C4⋊C4×D12711yescharacteristic
207C4#1686116611yescharacteristic
208C4#1686118211yescharacteristic
209C4#1686118711yescharacteristic
210C2#2254973511yescharacteristic
211C2#2254986911yescharacteristic
212C2#2254988311yescharacteristic
213C2#2255009711yescharacteristic
214C2#2255010111yescharacteristic
215C2#2255018211yescharacteristic
216C2#2255018611yescharacteristic
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#1805
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
11. #16861174C4semidirect
109. C127#1805semidirect

Maximal subgroups

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

subgroupindex
C2×#168611742
C2×Dic127⋊C82
C2×#168611342
C2×#168611322
#225497412
#225496382
#340×C1272
#1805127