Finite Group Databasebeta

← #30639944

Subgroups — #30639944

Order 16256.

278 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 71 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#22549919C211yesmaximal, characteristic, complemented
2C2×#16861136C211yesmaximal, characteristic
3#22549877C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic, complemented
5#22549783C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#536×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4×Dic127C2211yescharacteristic
12#16861142—21no—
13#16861140—21no—
14#16861136—22no—
15#16861133—21no—
16#16861130—21no—
17#16861128—22no—
18C127⋊#95—22no—
19C2×#41×C127C2211yescharacteristic
20C22×Q8×C127C2211yescharacteristic
21C4⋊Q8×C127—21no—
22#124×C127—21no—
23#123×C127—22no—
24C22⋊Q8×C127—21no—
25C22⋊Q8×C127—21no—
26#117×C127C2211yescharacteristic
27C22×Dic127—21no—
28C22×Dic127—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232—21no—
32#14523232D412yes—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C4×Dic127—21no—
38C4×Dic127—21no—
39#41×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42#41×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127D411yescharacteristic
50C2×D4×C127—21no—
51C2×D4×C127D411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127D412yes—
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C4⋊C4×C127—22no—
57C4⋊C4×C127D411yescharacteristic
58C4⋊C4×C127D411yescharacteristic
59C22⋊C4×C127—22no—
60C22⋊C4×C127—22no—
61C22⋊C4×C127—21no—
62C42×C127—21no—
63C2×Dic127—22no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C2×Dic127—42no—
67C2×Dic127—41no—
68C127⋊C8—22no—
69C127⋊C8—41no—
70C23×C127—21no—
71C23×C127C2×D411yescharacteristic
72Q8×C127—21no—
73Q8×C127—21no—
74Q8×C127—41no—
75Q8×C127—41no—
76Q8×C127—41no—
77Q8×C127—41no—
78D4×C127—21no—
79D4×C127—21no—
80D4×C127—41no—
81D4×C127—41no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—22no—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127#12011yescharacteristic
102C22×C127#12111yescharacteristic
103C22×C127#12311yescharacteristic
104C4×C127—42no—
105C4×C127—41no—
106C4×C127—41no—
107C4×C127—41no—
108C4×C127#12111yescharacteristic
109C4×C127#12111yescharacteristic
110C4×C127#12311yescharacteristic
111C4×C127#12411yescharacteristic
112Dic127—42no—
113Dic127—81no—
114C2×C127—21no—
115C2×C127—21no—
116C2×C127—81no—
117C2×C127#39411yescharacteristic
118C2×C127#45411yescharacteristic
119C2×C127#45611yescharacteristic
120#2038—1271nomaximal, Sylow
121C127#203811yescharacteristic, complemented, Sylow
122#536D12711yescharacteristic
123#428—1271no—
124C2×#104—1271no—
125#417—1271no—
126C2×#101—1271no—
127#391—1271no—
128#339—1271no—
129C2×#41C2×D12711yescharacteristic
130C22×Q8C2×D12711yescharacteristic
131C2×M16—1271no—
132C2×M16—1271no—
133C4⋊Q8—21no—
134#124—21no—
135#123—22no—
136C22⋊Q8—21no—
137C22⋊Q8—21no—
138#117C2×D12711yescharacteristic
139C2×C4⋊C4—1271no—
140C2×C42—1271no—
141#107—2541no—
142#106—2541no—
143#104—2542no—
144#103—2541no—
145#102—2541no—
146#101—2542no—
147#95—2542no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8C22⋊D12711yescharacteristic
159C2×D4—21no—
160C2×D4C22⋊D12711yescharacteristic
161C22×C4—21no—
162C22×C4—2541no—
163C22×C4—2541no—
164C22×C4C22⋊D12712yes—
165C22×C4C22×D12711yescharacteristic
166M16—1272no—
167M16—2541no—
168M16—2541no—
169M16—2541no—
170C2×C8—2541no—
171C2×C8—2541no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—2541no—
175C4⋊C4—2541no—
176C4⋊C4C22⋊D12711yescharacteristic
177C4⋊C4C22⋊D12711yescharacteristic
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—21no—
181C42—21no—
182C42—2541no—
183C42—2541no—
184C23—21no—
185C23C2×C22⋊D12711yescharacteristic
186Q8—21no—
187Q8—21no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—41no—
192D4—21no—
193D4—21no—
194D4—41no—
195D4—41no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—5082no—
205C2×C4—5081no—
206C2×C4—5081no—
207C2×C4—5081no—
208C2×C4—2542no—
209C2×C4D4×D12711yescharacteristic
210C2×C4C2×C22⋊D12711yescharacteristic
211C2×C4C2×C22⋊D12711yescharacteristic
212C2×C4#1452324711yescharacteristic
213C2×C4#1452324711yescharacteristic
214C2×C4#1452324711yescharacteristic
215C8—5081no—
216C8—2542no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22#1686117811yescharacteristic
223C22#1686118111yescharacteristic
224C22#1686119011yescharacteristic
225C4—42no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—10161no—
230C4—5082no—
231C4#1686118011yescharacteristic
232C4#1686118111yescharacteristic
233C4#1686119011yescharacteristic
234C4#1686119611yescharacteristic
235C2—21no—
236C2—21no—
237C2—81no—
238C2#2254979011yescharacteristic
239C2#2255004911yescharacteristic
240C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#536×C127C2
socleC22×C127#120
prime radicalC2×#41×C127C22
hypercenter#536D127
hypocenterC127#2038
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #22549919C2semidirect
4. C2×#16861128C2semidirect
5. #22549783C2semidirect
6. #22549635C2semidirect
121. C127#2038semidirect

Maximal subgroups

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

subgroupindex
#225499192
C2×#168611362
#225498772
C2×#168611282
#225497832
#225496352
#536×C1272
#2038127