Finite Group Databasebeta

← #30639778

Subgroups — #30639778

Order 16256.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16861242C211yesmaximal, characteristic, complemented
2C2×#16861239C211yesmaximal, characteristic, complemented
3#22550454C211yesmaximal, characteristic
4C2×#16861137C211yesmaximal, characteristic, complemented
5#22549782C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861242—22no—
11#16861242—22no—
12#16861239—22no—
13#16861239—22no—
14C2×Dic508—21no—
15C2×#14523238—21no—
16C2×#14523237—21no—
17C2×#14523236—21no—
18C2×#14523232C2211yescharacteristic
19#16861188—21no—
20#16861185—21no—
21#16861183—21no—
22#16861181—22no—
23#16861180—21no—
24C2×C4⋊Dic127C2211yescharacteristic
25#16861137—22no—
26C127⋊#95—22no—
27C2×#41×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29C2×C42×C127C2211yescharacteristic
30#104×C127—22no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254—21no—
39C2×Dic254—21no—
40C2×Dic254D411yescharacteristic
41C2×Dic254D412yescomplemented
42C2×D508D411yescharacteristic
43C2×C22⋊D127—22no—
44C2×C22⋊D127—21no—
45C22×Dic127—21no—
46C22×Dic127D412yes—
47C2×C4×D127—21no—
48Dic508—24no—
49#14523238—24no—
50#14523237—24no—
51#14523236—24no—
52#14523232—21no—
53#14523232D412yescomplemented
54C2×C127⋊C8—21no—
55C4⋊Dic127—21no—
56C4⋊Dic127—21no—
57Dic127⋊C4—22no—
58C22⋊Dic127—22no—
59(C2×D127)⋊C4—22no—
60#41×C127—21no—
61#41×C127—21no—
62#41×C127—21no—
63#41×C127—21no—
64C2×Q8×C127D411yescharacteristic
65C2×D4×C127—21no—
66C2×D4×C127D411yescharacteristic
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70M16×C127—21no—
71M16×C127D412yescomplemented
72C2×C8×C127—21no—
73C42×C127—21no—
74C42×C127—21no—
75C22×D127—21no—
76Dic254—22no—
77Dic254—22no—
78Dic254—21no—
79Dic254—21no—
80Dic254—41no—
81Dic254—41no—
82D508—21no—
83D508—21no—
84C22⋊D127—41no—
85C22⋊D127—41no—
86C22⋊D127—82no—
87C2×Dic127—22no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C2×Dic127—41no—
91C2×Dic127—41no—
92C2×Dic127—42no—
93C4×D127—41no—
94C4×D127—41no—
95C127⋊C8—22no—
96C23×C127—21no—
97C23×C127C2×D411yescharacteristic
98Q8×C127—21no—
99Q8×C127—21no—
100D4×C127—21no—
101D4×C127—21no—
102D4×C127—41no—
103D4×C127—41no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
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—42no—
129C4×C127—41no—
130C4×C127#12011yescharacteristic
131C4×C127#12011yescharacteristic
132C4×C127#12111yescharacteristic
133C4×C127#12111yescharacteristic
134Dic127—41no—
135Dic127—42no—
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×#137—1271no—
148#537—1271no—
149C2×#104—1271no—
150C2×#104D12711yescharacteristic, complemented
151#391—1271no—
152#343—1271no—
153C2×#41—1271no—
154C2×#41C2×D12711yescharacteristic
155C22×Q8—1271no—
156#137—2542no—
157#137—2542no—
158#137—2542no—
159#137—2542no—
160C2×Q16—2541no—
161C2×Q16—2541no—
162C2×SD16—2541no—
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×C42C2×D12711yescharacteristic
173#104—22no—
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—1271no—
185C2×Q8—1272no—
186C2×Q8—2541no—
187C2×Q8—2541no—
188C2×Q8—2541no—
189C2×Q8—2541no—
190C2×Q8—2541no—
191C2×Q8D50811yescharacteristic
192C2×D4—21no—
193C2×D4—1271no—
194C2×D4—2542no—
195C2×D4—2541no—
196C2×D4D50811yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—1272no—
200C22×C4—2541no—
201C22×C4—2541no—
202C22×C4C22×D12711yescharacteristic
203Q16—2544no—
204Q16—2544no—
205SD16—2544no—
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—21no—
219C42—21no—
220C23—21no—
221C23—2541no—
222C23D4×D12711yescharacteristic
223Q8—21no—
224Q8—21no—
225Q8—5081no—
226Q8—5081no—
227Q8—2542no—
228Q8—2541no—
229Q8—2542no—
230Q8—2541no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—10162no—
236D4—5081no—
237D4—5081no—
238D4—2541no—
239D4—2541no—
240C2×C4—22no—
241C2×C4—21no—
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—2541no—
253C2×C4—2542no—
254C2×C4D4×D12711yescharacteristic
255C2×C4D4×D12711yescharacteristic
256C2×C4D4×D12711yescharacteristic
257C2×C4C2×C22⋊D12711yescharacteristic
258C2×C4C2×D50811yescharacteristic
259C2×C4#1452324811yescharacteristic
260C8—22no—
261C8—2542no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—5081no—
268C22—5081no—
269C22—5081no—
270C22#1686117911yescharacteristic
271C22#1686118211yescharacteristic
272C22#1686120511yescharacteristic
273C4—42no—
274C4—41no—
275C4—5081no—
276C4—5082no—
277C4—5082no—
278C4#1686117711yescharacteristic
279C4#1686117911yescharacteristic
280C4#1686118211yescharacteristic
281C4#1686118311yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2—10161no—
286C2#2254978511yescharacteristic
287C2#2255005511yescharacteristic
288C2#2255006011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127#120
prime radical#22550454C2
hypercenterC2×#104D127
hypocenterC127#2007
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C2×#16861239C2semidirect
4. C2×#16861137C2semidirect
5. #22549782C2semidirect
6. #22549644C2semidirect
7. C2×#104×C127C2semidirect
41. C2×Dic254D4semidirect
53. #14523232D4semidirect
71. M16×C127D4semidirect
145. C127#2007semidirect
150. C2×#104D127semidirect
210. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612392
#225504542
C2×#168611372
#225497822
#225496442
C2×#104×C1272
#2007127