Finite Group Databasebeta

← #30639792

Subgroups — #30639792

Order 16256.

348 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 69 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
2C2×#16861240C211yesmaximal, characteristic, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22549894C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549647C211yesmaximal, characteristic, complemented
7#419×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523248C2211yescharacteristic
10#16861242—21no—
11#16861242—21no—
12#16861240—22no—
13#16861240—22no—
14#16861237—21no—
15#16861237—21no—
16#16861231—22no—
17C2×#14523238—21no—
18C2×#14523235—21no—
19C2×#14523232C2211yescharacteristic
20#16861207—21no—
21#16861198—21no—
22#16861192—22no—
23#16861185—22no—
24#16861168C2211yescharacteristic
25#16861137—22no—
26#16861123—22no—
27C22×Q8×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#103×C127—22no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36#14523248—21no—
37#14523248—21no—
38#14523248—21no—
39C2×Dic254D411yescharacteristic
40C2×Dic254D411yescharacteristic
41C2×D508D411yescharacteristic
42C2×D508D411yescharacteristic
43C2×C22⋊D127—21no—
44C2×C22⋊D127—21no—
45C2×C4×D127—21no—
46C2×C4×D127—21no—
47Dic508—22no—
48#14523238—24no—
49#14523237—22no—
50#14523237—22no—
51#14523235—24no—
52D1016—22no—
53#14523232—21no—
54#14523232D412yes—
55C2×C127⋊C8—21no—
56C4⋊Dic127—21no—
57Dic127⋊C4—24no—
58C22⋊Dic127D412yes—
59(C2×D127)⋊C4—24no—
60C4×Dic127—21no—
61C2×Q8×C127—21no—
62C2×Q8×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127D412yes—
67C22×C4×C127—21no—
68C22×C4×C127C2311yescharacteristic
69M16×C127—21no—
70M16×C127—21no—
71C2×C8×C127D412yes—
72C4⋊C4×C127—21no—
73C22⋊C4×C127—21no—
74C42×C127—21no—
75C22×D127—21no—
76C22×D127—21no—
77Dic254—21no—
78Dic254—21no—
79Dic254—21no—
80Dic254—21no—
81D508—21no—
82D508—21no—
83D508—21no—
84D508—21no—
85C22⋊D127—41no—
86C22⋊D127—41no—
87C22⋊D127—41no—
88C22⋊D127—41no—
89C2×Dic127—22no—
90C2×Dic127—21no—
91C2×Dic127—21no—
92C4×D127—41no—
93C4×D127—41no—
94C4×D127—41no—
95C4×D127—41no—
96C127⋊C8—22no—
97C23×C127C2×D411yescharacteristic
98Q8×C127—22no—
99Q8×C127—22no—
100Q8×C127—41no—
101Q8×C127—41no—
102C2×C4×C127—22no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127#4111yescharacteristic
112C8×C127—22no—
113C22×C127—41no—
114C22×C127#12011yescharacteristic
115C22×C127#12111yescharacteristic
116C22×C127#12711yescharacteristic
117C2×D127—42no—
118C2×D127—41no—
119C2×D127—41no—
120C2×D127—81no—
121C4×C127—42no—
122C4×C127—81no—
123C4×C127#12011yescharacteristic
124C4×C127#12011yescharacteristic
125C4×C127#12111yescharacteristic
126C4×C127#12111yescharacteristic
127Dic127—42no—
128Dic127—41no—
129Dic127—41no—
130C2×C127—21no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127#39211yescharacteristic
134D127—81no—
135D127—81no—
136#2009—1271nomaximal, Sylow
137C127#200911yescharacteristic, complemented, Sylow
138#575—1271no—
139C2×#137—1271no—
140#536—1271no—
141#421—1271no—
142#419D12711yescharacteristic, complemented
143#410—1271no—
144#344—1271no—
145C2×#41—1271no—
146C2×#41—1271no—
147C22×Q8C2×D12711yescharacteristic
148#137—2541no—
149#137—2541no—
150#137—2542no—
151#137—2542no—
152AGL(1,ℤ/8)—2541no—
153AGL(1,ℤ/8)—2541no—
154#135—2542no—
155C2×Q16—2541no—
156C2×SD16—2541no—
157C2×M16—1271no—
158C2×M16C2×D12711yescharacteristic
159C4⋊Q8—2541no—
160#124—2541no—
161#123—2542no—
162C22⋊Q8—2542no—
163#117—1271no—
164#117C2×D12711yescharacteristic
165#104—2542no—
166#103—22no—
167#99—2542no—
168#41—2541no—
169#41—2541no—
170#41—2541no—
171#41—2541no—
172#41—2541no—
173#41—2541no—
174#41—2541no—
175#41—2541no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—1271no—
182C2×Q8—1271no—
183C2×Q8D50812yes—
184C2×D4—1271no—
185C2×D4—1271no—
186C2×D4—2541no—
187C2×D4—2541no—
188C22×C4—21no—
189C22×C4—2541no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192Q16—2542no—
193Q16—2544no—
194SD16—2542no—
195SD16—2542no—
196SD16—2544no—
197D8—2542no—
198M16—21no—
199M16—21no—
200M16—1272no—
201M16—2541no—
202C2×C8—2541no—
203C2×C8C22⋊D12712yes—
204C4⋊C4—21no—
205C4⋊C4—2544no—
206C4⋊C4—2541no—
207C22⋊C4—21no—
208C22⋊C4—1272no—
209C22⋊C4—2544no—
210C42—21no—
211C42—2541no—
212C23—2541no—
213C23—2541no—
214C23D4×D12711yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—41no—
218Q8—41no—
219Q8—2541no—
220Q8—2541no—
221Q8—2541no—
222Q8—2541no—
223D4—5081no—
224D4—5081no—
225D4—5081no—
226D4—5081no—
227D4—2541no—
228D4—2541no—
229D4—2541no—
230D4—2541no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—5081no—
236C2×C4—5081no—
237C2×C4—5081no—
238C2×C4—5081no—
239C2×C4—2542no—
240C2×C4—2541no—
241C2×C4—2541no—
242C2×C4D4×D12711yescharacteristic
243C2×C4D4×D12711yescharacteristic
244C2×C4D4×D12711yescharacteristic
245C2×C4C2×C22⋊D12711yescharacteristic
246C2×C4C2×D50811yescharacteristic
247C2×C4#1452324811yescharacteristic
248C8—22no—
249C8—2542no—
250C22—41no—
251C22—10161no—
252C22—5082no—
253C22—5081no—
254C22—5081no—
255C22#1686117911yescharacteristic
256C22#1686118211yescharacteristic
257C22#1686120511yescharacteristic
258C4—42no—
259C4—81no—
260C4—5082no—
261C4—5081no—
262C4—5081no—
263C4#1686117711yescharacteristic
264C4#1686117911yescharacteristic
265C4#1686118211yescharacteristic
266C4#1686118311yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—10161no—
271C2—10161no—
272C2#2254978511yescharacteristic
subgroupG/H
centerC2#22549785
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC2×C127#392
prime radical#22550676C2
hypercenter#419D127
hypocenterC127#2009
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550450C2semidirect
4. #22549894C2semidirect
5. #22549849C2semidirect
6. #22549647C2semidirect
7. #419×C127C2semidirect
137. C127#2009semidirect
142. #419D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
C2×#168612402
#225504502
#225498942
#225498492
#225496472
#419×C1272
#2009127