Finite Group Databasebeta

← #30639880

Subgroups — #30639880

Order 16256.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861200C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic
3C2×#16861132C211yesmaximal, characteristic
4#22549781C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861200—22no—
11#16861200—22no—
12C2×C4⋊Dic127—21no—
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—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254Q812yes—
29C22×Dic127—21no—
30C22×Dic127—21no—
31C22×Dic127D411yescharacteristic
32C22×Dic127D411yescharacteristic
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—24no—
37C4⋊Dic127—21no—
38C4⋊Dic127D412yes—
39Dic127⋊C4—24no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C22×C4×C127—21no—
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C2×C8×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127D412yes—
52Dic254—22no—
53Dic254—22no—
54Dic254—41no—
55Dic254—41no—
56C2×Dic127—21no—
57C2×Dic127—21no—
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×Dic127—41no—
68C127⋊C8—41no—
69C23×C127C2×D411yescharacteristic
70C2×C4×C127—22no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×Q811yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C8×C127—41no—
83C22×C127#12011yescharacteristic
84C22×C127C22⋊Q811yescharacteristic
85C22×C127#12411yescharacteristic
86C22×C127#13511yescharacteristic
87C22×C127#13511yescharacteristic
88C22×C127#13711yescharacteristic
89C22×C127#13711yescharacteristic
90C4×C127—42no—
91C4×C127—81no—
92C4×C127C22⋊Q811yescharacteristic
93C4×C127C22⋊Q811yescharacteristic
94C4×C127#12411yescharacteristic
95C4×C127#12411yescharacteristic
96Dic127—42no—
97Dic127—41no—
98Dic127—41no—
99Dic127—42no—
100C2×C127#39311yescharacteristic
101C2×C127#45211yescharacteristic
102C2×C127#45211yescharacteristic
103C2×C127#47911yescharacteristic
104C2×C127#47911yescharacteristic
105C2×C127#48811yescharacteristic
106C2×C127#49111yescharacteristic
107#2026—1271nomaximal, Sylow
108C127#202611yescharacteristic, complemented, Sylow
109C2×#125—1271no—
110C2×#103—1271no—
111C2×#103—1271no—
112#391—1271no—
113#340—1271no—
114#340D12711yescharacteristic
115#336—1271no—
116C22×Q8—1271no—
117C22×C8—1271no—
118C22×C8C2×D12711yescharacteristic
119#125—2542no—
120#125—2542no—
121C2×C4⋊C4—1271no—
122C2×C4⋊C4—2541no—
123C2×C4⋊C4—2541no—
124C2×C4⋊C4C2×D12711yescharacteristic
125C2×C4⋊C4C2×D12711yescharacteristic
126C2×C42—1271no—
127#103—2542no—
128#103—2542no—
129#95—2541no—
130#95—2541no—
131C2×Q8—1272no—
132C2×Q8—2541no—
133C2×Q8—2541no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—1271no—
140C22×C4—1271no—
141C22×C4—2541no—
142C22×C4—2541no—
143C22×C4C22×D12711yescharacteristic
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—2541no—
148C2×C8—2541no—
149C2×C8—2541no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—1272no—
154C4⋊C4—2544no—
155C4⋊C4—2544no—
156C4⋊C4—2541no—
157C4⋊C4D50812yes—
158C42—2541no—
159C42—2541no—
160C23D4×D12711yescharacteristic
161Q8—5081no—
162Q8—5081no—
163Q8—2542no—
164Q8—2542no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—5081no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—5081no—
175C2×C4—2542no—
176C2×C4—2541no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4—2541no—
182C2×C4—2542no—
183C2×C4D4×D12711yescharacteristic
184C2×C4C2×D50811yescharacteristic
185C2×C4Q8×D12711yescharacteristic
186C2×C4#1452324711yescharacteristic
187C2×C4#1452324811yescharacteristic
188C2×C4#1452324811yescharacteristic
189C8—41no—
190C8—5081no—
191C22#1686117911yescharacteristic
192C22#1686118711yescharacteristic
193C22#1686119511yescharacteristic
194C22#1686122811yescharacteristic
195C22#1686123011yescharacteristic
196C22#1686123811yescharacteristic
197C22#1686124111yescharacteristic
198C4—42no—
199C4—81no—
200C4—5081no—
201C4—5082no—
202C4—5081no—
203C4—5082no—
204C4#1686118711yescharacteristic
205C4#1686118911yescharacteristic
206C4#1686119511yescharacteristic
207C4#1686119711yescharacteristic
208C2#2254978911yescharacteristic
209C2#2255003911yescharacteristic
210C2#2255004111yescharacteristic
211C2#2255018211yescharacteristic
212C2#2255018611yescharacteristic
213C2#2255022911yescharacteristic
214C2#2255024311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socle, prime radicalC23×C127C2×D4
hypercenter#340D127
hypocenterC127#2026
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
108. C127#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#168612002
C2×#168611342
C2×#168611322
#225497812
#225496382
#225496302
#340×C1272
#2026127