Finite Group Databasebeta

← #30639093

Subgroups — #30639093

Order 16256.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#22550676C211yesmaximal, characteristic
2#22550642C211yesmaximal, characteristic, complemented
3#22549892C211yesmaximal, characteristic, complemented
4#22549873C211yesmaximal, characteristic, complemented
5#22549842C211yesmaximal, characteristic, complemented
6#22549648C211yesmaximal, characteristic, complemented
7#341×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523248C2211yescharacteristic
10#16861242—21no—
11#16861242C412yescomplemented
12#16861237—21no—
13#16861237C412yescomplemented
14#16861231—22no—
15#16861217—22no—
16#16861215—22no—
17C2×#14523232C2211yescharacteristic
18M16×D127—22no—
19#16861168C2211yescharacteristic
20#16861135—22no—
21#16861134—21no—
22#16861131—21no—
23(C2×D127)⋊C8—22no—
24C2×M16×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33#14523248C2×C412yes—
34#14523248D412yes—
35C2×Dic254C2×C411yescharacteristic
36C2×Dic254D411yescharacteristic
37C2×D508C2×C411yescharacteristic
38C2×D508D411yescharacteristic
39C2×C22⋊D127—21no—
40C2×C22⋊D127—21no—
41C2×C4×D127—21no—
42C2×C4×D127—21no—
43Dic508—22no—
44#14523237—22no—
45#14523237—22no—
46D1016—22no—
47#14523232—21no—
48#14523232—21no—
49#14523231—24no—
50C2×C127⋊C8D412yes—
51C8×D127—24no—
52C4⋊Dic127—21no—
53C22⋊Dic127—21no—
54C4×Dic127—21no—
55C22×C4×C127C2311yescharacteristic
56M16×C127—21no—
57M16×C127—21no—
58M16×C127C2×C412yes—
59M16×C127D412yes—
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127—21no—
65C22×D127—21no—
66C22×D127—21no—
67Dic254—21no—
68Dic254—21no—
69Dic254—21no—
70Dic254C22⋊C412yes—
71D508—21no—
72D508—21no—
73D508—21no—
74D508C22⋊C412yes—
75C22⋊D127—22no—
76C22⋊D127—22no—
77C22⋊D127—41no—
78C22⋊D127—41no—
79C2×Dic127—21no—
80C2×Dic127—21no—
81C2×Dic127—41no—
82C4×D127—22no—
83C4×D127—22no—
84C4×D127—41no—
85C4×D127—41no—
86C127⋊C8—22no—
87C23×C127#4111yescharacteristic
88C2×C4×C127C22×C411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127#4111yescharacteristic
94C8×C127—22no—
95C8×C127—22no—
96C8×C127—41no—
97C22×C127—41no—
98C22×C127C4×D411yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127#12311yescharacteristic
101C2×D127—22no—
102C2×D127—42no—
103C2×D127—41no—
104C2×D127—81no—
105C4×C127C2×C22⋊C411yescharacteristic
106C4×C127C4×D411yescharacteristic
107C4×C127#12011yescharacteristic
108C4×C127#12111yescharacteristic
109Dic127—22no—
110Dic127—41no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127#38611yescharacteristic
116D127—42no—
117D127—81no—
118#1881—1271nomaximal, Sylow
119C127#188111yescharacteristic, complemented, Sylow
120#575—1271no—
121#568—1271no—
122#421—1271no—
123#416—1271no—
124#408—1271no—
125#344—1271no—
126#341D12711yescharacteristic, complemented
127C2×#41—1271no—
128C2×#41—1271no—
129#137—1272no—
130#137—2541no—
131AGL(1,ℤ/8)—1272no—
132AGL(1,ℤ/8)—2541no—
133#135—2542no—
134#131—2542no—
135#131—2542no—
136C2×M16—1271no—
137C2×M16—2542no—
138C2×M16C2×D12711yescharacteristic
139C2×M16C2×D12711yescharacteristic
140C22×C8C2×D12711yescharacteristic
141#117—1271no—
142#104—2542no—
143#103—2541no—
144#102—2541no—
145C22⋊C8—2542no—
146#41—1272no—
147#41—1272no—
148#41—2541no—
149#41—2541no—
150#41—2541no—
151#41—2541no—
152#41—2541no—
153#41—2541no—
154C2×Q8—1271no—
155C2×Q8—1271no—
156C2×D4—1271no—
157C2×D4—1271no—
158C2×D4—2541no—
159C2×D4—2541no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163Q16—2542no—
164SD16—2542no—
165SD16—2542no—
166D8—2542no—
167M16—21no—
168M16—21no—
169M16—2544no—
170M16—2541no—
171M16—2541no—
172M16C4×D12712yes—
173M16D50812yes—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—1272no—
180C2×C8—2544no—
181C4⋊C4—2541no—
182C22⋊C4—2541no—
183C42—2541no—
184C23—2541no—
185C23—2541no—
186C23#1452324911yescharacteristic
187Q8—1272no—
188Q8—2541no—
189Q8—2541no—
190Q8—2541no—
191D4—1272no—
192D4—5081no—
193D4—5081no—
194D4—2542no—
195D4—2542no—
196D4—2541no—
197D4—2541no—
198D4—2541no—
199C2×C4—5081no—
200C2×C4—5081no—
201C2×C4—5081no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4C2×C4×D12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4D4×D12711yescharacteristic
209C2×C4D4×D12711yescharacteristic
210C2×C4C2×D50811yescharacteristic
211C2×C4#1452324811yescharacteristic
212C8—22no—
213C8—22no—
214C8—41no—
215C8—2542no—
216C22—41no—
217C22—10161no—
218C22—5082no—
219C22—5081no—
220C22—2542no—
221C22#1686117111yescharacteristic
222C22#1686118411yescharacteristic
223C22#1686119411yescharacteristic
224C4—10161no—
225C4—5081no—
226C4—2542no—
227C4C22⋊C4×D12711yescharacteristic
228C4C4×D50811yescharacteristic
229C4#1686117911yescharacteristic
230C4#1686118211yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—10161no—
235C2—5082no—
236C2#2254975411yescharacteristic
subgroupG/H
centerC4#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#341×C127C2
socleC2×C127#386
prime radical#22550676C2
cosocleC22×C4×C127C23
hypercenter#341D127
hypocenterC127#1881
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #341

Complements

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

NH ≅ G/Ntype
2. #22550642C2semidirect
3. #22549892C2semidirect
4. #22549873C2semidirect
5. #22549842C2semidirect
6. #22549648C2semidirect
7. #341×C127C2semidirect
11. #16861242C4semidirect
13. #16861237C4semidirect
119. C127#1881semidirect
126. #341D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225506422
#225498922
#225498732
#225498422
#225496482
#341×C1272
#1881127