Finite Group Databasebeta

← #30639637

Subgroups — #30639637

Order 16256.

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

normal subgroups only

This group has 316 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
1C22×#14523233C211yesmaximal, characteristic
2C2×#16861144C211yesmaximal, characteristic, complemented
3#22549882C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549834C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523233—21no—
10C2×#14523233—21no—
11C2×#14523233—22no—
12C2×#14523233C414yescomplemented
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861144—22no—
17#16861131—22no—
18#16861130—22no—
19#16861125—22no—
20#16861119—21no—
21C8×Dic127—21no—
22C22×D4×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#100×C127—22no—
26C23×D127—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508C2×C412yes—
33#14523233—24no—
34#14523233—24no—
35#14523232—21no—
36#14523232—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8C2×C412yes—
39C2×C127⋊C8D411yescharacteristic
40C2×C127⋊C8D411yescharacteristic
41C2×C127⋊C8D412yes—
42C4⋊Dic127D412yes—
43C22⋊Dic127—21no—
44C4×Dic127—21no—
45C24×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127C2×C412yes—
52C22×C4×C127C2311yescharacteristic
53M16×C127—21no—
54M16×C127—21no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—21no—
58C2×C8×C127—21no—
59C22×D127—22no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127—82no—
63D508—22no—
64D508—22no—
65D508—41no—
66D508—41no—
67C2×Dic127—22no—
68C127⋊C8—41no—
69C127⋊C8C22⋊C412yes—
70C127⋊C8C22⋊C412yes—
71C23×C127—22no—
72C23×C127—41no—
73C23×C127—41no—
74C23×C127—82no—
75C23×C127#4111yescharacteristic
76D4×C127—22no—
77D4×C127—22no—
78D4×C127—41no—
79D4×C127—41no—
80C2×C4×C127C22×C411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—22no—
87C8×C127—41no—
88C22×C127—22no—
89C22×C127—42no—
90C22×C127—82no—
91C22×C127—82no—
92C22×C127—82no—
93C22×C127C4×D411yescharacteristic
94C22×C127#12111yescharacteristic
95C22×C127#12411yescharacteristic
96C2×D127—42no—
97C2×D127—82no—
98C2×D127—82no—
99C2×D127—82no—
100C4×C127C2×C22⋊C411yescharacteristic
101C4×C127C4×D411yescharacteristic
102C4×C127#12111yescharacteristic
103C4×C127#12711yescharacteristic
104Dic127—42no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—82no—
108C2×C127#39011yescharacteristic
109C2×C127#47212yes—
110D127—82no—
111#1978—1271nomaximal, Sylow
112C127#197811yescharacteristic, complemented, Sylow
113C22×D8—1271no—
114C2×#108—1271no—
115#418—1271no—
116#418—1271no—
117C2×#100—1271no—
118C2×#100D12711yescharacteristic, complemented
119#405—1271no—
120C22×D4—1271no—
121C22×D4C2×D12711yescharacteristic
122C2×D8—1274no—
123C2×D8—2541no—
124C2×D8—2541no—
125C2×D8—2542no—
126C2×M16—1271no—
127C2×M16C2×D12711yescharacteristic
128C2×M16C2×D12711yescharacteristic
129C22×C8—1271no—
130#117—1271no—
131#108—2542no—
132#102—2542no—
133#102—2542no—
134#100—22no—
135#100—2542no—
136#97—2541no—
137C4×C8—2541no—
138C24—21no—
139C24—2541no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—1272no—
146C2×D4—2541no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4—2541no—
150C2×D4—2541no—
151C2×D4C4×D12712yes—
152C22×C4C22×D12711yescharacteristic
153D8—2544no—
154D8—2544no—
155M16—21no—
156M16—21no—
157M16—21no—
158M16—2541no—
159M16—2541no—
160M16C22⋊D12712yes—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—1272no—
164C2×C8—1271no—
165C2×C8—1271no—
166C2×C8—1272no—
167C2×C8—2541no—
168C4⋊C4—1272no—
169C22⋊C4—2541no—
170C42—2541no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—82no—
175C23—10162no—
176C23—5081no—
177C23—5081no—
178C23—2542no—
179C23#1452324811yescharacteristic
180D4—22no—
181D4—22no—
182D4—41no—
183D4—41no—
184D4—5081no—
185D4—5081no—
186D4—2542no—
187D4—2542no—
188C2×C4—2542no—
189C2×C4C2×C4×D12711yescharacteristic
190C2×C4D4×D12711yescharacteristic
191C2×C4D4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4C2×C22⋊D12711yescharacteristic
194C2×C4#1452324711yescharacteristic
195C8—22no—
196C8—41no—
197C8—1272no—
198C8—1272no—
199C8—5081no—
200C22—22no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22—10162no—
206C22—10162no—
207C22—10162no—
208C22—5082no—
209C22C4×C22⋊D12711yescharacteristic
210C22#1686118211yescharacteristic
211C22#1686119511yescharacteristic
212C4—5082no—
213C4C22⋊C4×D12711yescharacteristic
214C4(C22⋊D127)⋊C411yescharacteristic
215C4#1686118111yescharacteristic
216C4#1686120511yescharacteristic
217C2—21no—
218C2—21no—
219C2—82no—
220C2—10162no—
221C2#2254977611yescharacteristic
222C2#2255014512yes—
subgroupG/H
centerC22#16861182
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#121
prime radicalC22×#14523233C2
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1978
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16861144C2semidirect
3. #22549882C2semidirect
4. #22549881C2semidirect
5. C2×#16861125C2semidirect
6. #22549834C2semidirect
7. C2×#100×C127C2semidirect
12. C2×#14523233C4semidirect
112. C127#1978semidirect
118. C2×#100D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232332
C2×#168611442
#225498822
#225498812
C2×#168611252
#225498342
C2×#100×C1272
#1978127