Finite Group Databasebeta

← #30639775

Subgroups — #30639775

Order 16256.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861240C211yesmaximal, characteristic, complemented
2#22550454C211yesmaximal, characteristic
3C2×#16861136C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549783C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240—22no—
11C2×#14523238—21no—
12C2×#14523235—21no—
13C2×#14523232C2211yescharacteristic
14#16861188—21no—
15#16861185—21no—
16#16861183—21no—
17#16861181—22no—
18#16861180—21no—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×C4×Dic127C2211yescharacteristic
21#16861136—22no—
22#16861135—22no—
23C127⋊#95—22no—
24C2×#41×C127C2211yescharacteristic
25C22×Q8×C127C2211yescharacteristic
26#137×C127—22no—
27#137×C127—22no—
28C2×Q16×C127—21no—
29C2×SD16×C127—21no—
30C2×M16×C127C2211yescharacteristic
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254D411yescharacteristic
36C2×D508D411yescharacteristic
37C2×C22⋊D127—22no—
38C2×C22⋊D127—21no—
39C22×Dic127—21no—
40C22×Dic127D412yes—
41C2×C4×D127—21no—
42#14523238—24no—
43#14523235—24no—
44#14523232—21no—
45#14523232D412yescomplemented
46C2×C127⋊C8—21no—
47C4⋊Dic127—21no—
48C4⋊Dic127—21no—
49Dic127⋊C4—22no—
50C22⋊Dic127—22no—
51(C2×D127)⋊C4—22no—
52C4×Dic127—21no—
53C4×Dic127—21no—
54#41×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127—21no—
62C2×Q8×C127—21no—
63C2×Q8×C127D411yescharacteristic
64C2×Q8×C127D412yescomplemented
65C2×D4×C127—21no—
66C2×D4×C127D411yescharacteristic
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70Q16×C127—24no—
71SD16×C127—24no—
72M16×C127—21no—
73M16×C127D412yescomplemented
74C2×C8×C127—21no—
75C22×D127—21no—
76Dic254—21no—
77Dic254—21no—
78D508—21no—
79D508—21no—
80C22⋊D127—41no—
81C22⋊D127—41no—
82C22⋊D127—82no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—21no—
86C2×Dic127—41no—
87C2×Dic127—41no—
88C2×Dic127—42no—
89C4×D127—41no—
90C4×D127—41no—
91C127⋊C8—22no—
92C23×C127—21no—
93C23×C127C2×D411yescharacteristic
94Q8×C127—22no—
95Q8×C127—21no—
96Q8×C127—22no—
97Q8×C127—21no—
98Q8×C127—41no—
99Q8×C127—41no—
100D4×C127—21no—
101D4×C127—21no—
102D4×C127—41no—
103D4×C127—41no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127#4111yescharacteristic
116C8×C127—22no—
117C22×C127—21no—
118C22×C127—21no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127#12011yescharacteristic
123C22×C127#12111yescharacteristic
124C22×C127#12711yescharacteristic
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—41no—
128C4×C127—41no—
129C4×C127—42no—
130C4×C127#12011yescharacteristic
131C4×C127#12011yescharacteristic
132C4×C127#12111yescharacteristic
133C4×C127#12111yescharacteristic
134Dic127—42no—
135Dic127—41no—
136Dic127—42no—
137C2×C127—21no—
138C2×C127—21no—
139C2×C127—81no—
140C2×C127#39211yescharacteristic
141C2×C127#45511yescharacteristic
142C2×C127#45611yescharacteristic
143D127—81no—
144#2007—1271nomaximal, Sylow
145C127#200711yescharacteristic, complemented, Sylow
146C2×#137—1271no—
147C2×#137D12711yescharacteristic, complemented
148#537—1271no—
149C2×#104—1271no—
150C2×#104—1271no—
151#391—1271no—
152#343—1271no—
153C2×#41—1271no—
154C2×#41C2×D12711yescharacteristic
155C22×Q8C2×D12711yescharacteristic
156#137—22no—
157#137—22no—
158#137—2542no—
159#137—2542no—
160C2×Q16—21no—
161C2×Q16—2541no—
162C2×SD16—21no—
163C2×SD16—2541no—
164C2×M16—1271no—
165C2×M16C2×D12711yescharacteristic
166C22⋊Q8—2541no—
167C22⋊Q8—2541no—
168#121—2541no—
169#121—2542no—
170#121—2541no—
171C2×C4⋊C4—1271no—
172C2×C42—1271no—
173#104—2542no—
174#104—2542no—
175#95—2542no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—2541no—
181#41—2541no—
182#41—2541no—
183#41—2541no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—1271no—
190C2×Q8C22⋊D12711yescharacteristic
191C2×Q8C22⋊D12712yescomplemented
192C2×D4—21no—
193C2×D4—1271no—
194C2×D4—2542no—
195C2×D4—2541no—
196C2×D4C22⋊D12711yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—1272no—
200C22×C4—2541no—
201C22×C4—2541no—
202C22×C4C22×D12711yescharacteristic
203Q16—24no—
204Q16—2544no—
205SD16—24no—
206SD16—2544no—
207M16—21no—
208M16—1272no—
209M16—2541no—
210M16C22⋊D12712yescomplemented
211C2×C8—21no—
212C2×C8—2541no—
213C4⋊C4—2542no—
214C4⋊C4—2541no—
215C4⋊C4—2541no—
216C22⋊C4—2542no—
217C22⋊C4—2542no—
218C42—2541no—
219C42—2541no—
220C23—21no—
221C23—2541no—
222C23D4×D12711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—21no—
226Q8—21no—
227Q8—41no—
228Q8—41no—
229Q8—2541no—
230Q8—2541no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—10162no—
236D4—5081no—
237D4—5081no—
238D4—2541no—
239D4—2541no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—5081no—
247C2×C4—5081no—
248C2×C4—5081no—
249C2×C4—5081no—
250C2×C4—5082no—
251C2×C4—2542no—
252C2×C4—2542no—
253C2×C4—2541no—
254C2×C4D4×D12711yescharacteristic
255C2×C4D4×D12711yescharacteristic
256C2×C4C2×C22⋊D12711yescharacteristic
257C2×C4C2×C22⋊D12711yescharacteristic
258C2×C4C2×C22⋊D12711yescharacteristic
259C2×C4#1452324711yescharacteristic
260C8—22no—
261C8—2542no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—5081no—
268C22—5081no—
269C22—5081no—
270C22#1686117711yescharacteristic
271C22#1686118111yescharacteristic
272C22#1686120511yescharacteristic
273C4—41no—
274C4—42no—
275C4—5082no—
276C4—5081no—
277C4—5082no—
278C4#1686117711yescharacteristic
279C4#1686117811yescharacteristic
280C4#1686118011yescharacteristic
281C4#1686118111yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2—10161no—
286C2#2254978411yescharacteristic
287C2#2255005211yescharacteristic
288C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC22×C127#120
prime radical#22550454C2
hypercenterC2×#137D127
hypocenterC127#2007
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C2×#16861136C2semidirect
4. C2×#16861135C2semidirect
5. #22549783C2semidirect
6. #22549644C2semidirect
7. C2×#137×C127C2semidirect
45. #14523232D4semidirect
64. C2×Q8×C127D4semidirect
73. M16×C127D4semidirect
145. C127#2007semidirect
147. C2×#137D127semidirect
191. C2×Q8C22⋊D127semidirect
210. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225504542
C2×#168611362
C2×#168611352
#225497832
#225496442
C2×#137×C1272
#2007127