Finite Group Databasebeta

← #30648284

Subgroups — #30648284

Order 16256.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861229C211yesmaximal, characteristic
2#22550450C211yesmaximal, characteristic, complemented
3#22550359C211yesmaximal, characteristic, complemented
4#22550100C212yesmaximal, complemented
5#22550093C212yesmaximal, complemented
6#22550084C212yesmaximal, complemented
7#22550079C212yesmaximal, complemented
8#22549904C211yesmaximal, characteristic, complemented
9#22549878C211yesmaximal, characteristic, complemented
10C2×#16861133C211yesmaximal, characteristic, complemented
11C2×#137×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861229—24no—
14C2×#14523238C2211yescharacteristic
15C2×#14523236C2211yescharacteristic
16C2×#14523235C2211yescharacteristic
17C2×#14523233C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861207C2212yes—
20#16861198C2212yes—
21#16861192—22no—
22#16861185—21no—
23#16861185—21no—
24Q8×Dic127C2212yes—
25D4×Dic127—22no—
26D4×Dic127C2212yes—
27#16861168—21no—
28#16861168C2211yescharacteristic
29C2×C4⋊Dic127—21no—
30C2×C4⋊Dic127C2211yescharacteristic
31Dic127⋊C8C2212yes—
32Dic127⋊C8C2212yes—
33#16861134C2212yes—
34#16861133C2212yes—
35#16861133C2212yes—
36#16861131C2212yes—
37C2×#41×C127C2211yescharacteristic
38C22×Q8×C127C2211yescharacteristic
39#137×C127—24no—
40C2×Q16×C127C2212yes—
41C2×SD16×C127C2212yes—
42C2×M16×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254C2311yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C22×Dic127—22no—
49C22×Dic127—21no—
50C2×C4×D127—21no—
51#14523238—22no—
52#14523236—22no—
53#14523235—22no—
54#14523233—22no—
55C2×C127⋊C8—22no—
56C2×C127⋊C8C2311yescharacteristic
57C2×C127⋊C8C2311yescharacteristic
58C4⋊Dic127—22no—
59C4⋊Dic127—21no—
60C4⋊Dic127D411yescharacteristic
61C4⋊Dic127D411yescharacteristic
62C4⋊Dic127C2312yes—
63Dic127⋊C4—22no—
64Dic127⋊C4—22no—
65C22⋊Dic127—22no—
66C22⋊Dic127D412yes—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69C4×Dic127—22no—
70C4×Dic127C2312yes—
71#41×C127—22no—
72#41×C127D414yes—
73C2×Q8×C127—21no—
74C2×Q8×C127—22no—
75C2×Q8×C127—22no—
76C2×Q8×C127C2312yes—
77C2×Q8×C127C2311yescharacteristic
78C2×D4×C127—21no—
79C2×D4×C127C2311yescharacteristic
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127C2311yescharacteristic
83Q16×C127—24no—
84SD16×C127—24no—
85M16×C127—22no—
86C2×C8×C127C2312yes—
87C22×D127—21no—
88Dic254—21no—
89Dic254—21no—
90D508—21no—
91D508—21no—
92C22⋊D127—42no—
93C2×Dic127—22no—
94C2×Dic127—21no—
95C2×Dic127—82no—
96C2×Dic127—81no—
97C2×Dic127C2×D412yes—
98C2×Dic127C2×D412yes—
99C4×D127—42no—
100C127⋊C8—21no—
101C127⋊C8—21no—
102C23×C127—21no—
103C23×C127C2×D411yescharacteristic
104Q8×C127—21no—
105Q8×C127—22no—
106Q8×C127—22no—
107Q8×C127—42no—
108Q8×C127C2×D412yes—
109D4×C127—24no—
110D4×C127—21no—
111D4×C127C2×D412yes—
112C2×C4×C127—21no—
113C2×C4×C127—24no—
114C2×C4×C127—22no—
115C2×C4×C127—42no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D412yes—
118C2×C4×C127#4112yes—
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C22×C127—21no—
122C22×C127—21no—
123C22×C127—22no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127#12112yes—
127C22×C127C22×D411yescharacteristic
128C2×D127—41no—
129C2×D127—41no—
130C2×D127—41no—
131C4×C127—22no—
132C4×C127—42no—
133C4×C127#12112yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C2×#4111yescharacteristic
136Dic127—22no—
137Dic127—22no—
138Dic127—41no—
139Dic127—42no—
140C2×C127—22no—
141C2×C127—42no—
142C2×C127C2×#12111yescharacteristic
143C2×C127#57711yescharacteristic
144C2×C127#57811yescharacteristic
145D127—81no—
146#3046—1271nomaximal, Sylow
147C127#304611yescharacteristic, complemented, Sylow
148C2×#137D12711yescharacteristic, complemented
149C2×#135—1271no—
150#536—1271no—
151#520—1271no—
152#464—1272no—
153#463—1272no—
154#462—1272no—
155#461—1272no—
156#424—1271no—
157#417—1271no—
158C2×#103—1271no—
159C2×#41—1271no—
160C2×#41C2×D12711yescharacteristic
161C22×Q8C2×D12711yescharacteristic
162#137—24no—
163#135—2544no—
164C2×Q16—1271no—
165C2×Q16C2×D12712yes—
166C2×SD16—1271no—
167C2×SD16—1271no—
168C2×SD16C2×D12712yes—
169C2×D8—1271no—
170C2×M16C2×D12711yescharacteristic
171C22×C8—1271no—
172C4⋊Q8—1272no—
173#124—1272no—
174#123—2542no—
175C22⋊Q8—2541no—
176C22⋊Q8—2541no—
177C4×Q8—1272no—
178C4×D4—1272no—
179C4×D4—2542no—
180#117—1271no—
181#117—2541no—
182C2×C4⋊C4—1271no—
183C2×C4⋊C4—2541no—
184C4⋊C8—1272no—
185C4⋊C8—1272no—
186#103—1272no—
187#103—1272no—
188#103—1272no—
189#102—1272no—
190#41—22no—
191#41—2542no—
192#41—2542no—
193#41C22⋊D12714yes—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—1271no—
198C2×Q8C22×D12712yes—
199C2×Q8C22×D12711yescharacteristic
200C2×D4—21no—
201C2×D4—1271no—
202C2×D4—2541no—
203C2×D4C22×D12711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—2542no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4C22×D12711yescharacteristic
210Q16—24no—
211Q16—2542no—
212SD16—24no—
213SD16—2542no—
214SD16—2542no—
215D8—2542no—
216M16—22no—
217C2×C8—1271no—
218C2×C8—1271no—
219C2×C8—2542no—
220C2×C8C22×D12712yes—
221C4⋊C4—1271no—
222C4⋊C4—1271no—
223C4⋊C4—1272no—
224C4⋊C4—2542no—
225C4⋊C4—2542no—
226C4⋊C4—2542no—
227C4⋊C4—2541no—
228C22⋊C4—1272no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C22⋊C4—2542no—
232C42—1272no—
233C42—2542no—
234C23—21no—
235C23—2541no—
236C23D4×D12711yescharacteristic
237Q8—21no—
238Q8—22no—
239Q8—22no—
240Q8—42no—
241Q8—2541no—
242Q8—2541no—
243Q8C2×C22⋊D12712yes—
244D4—24no—
245D4—21no—
246D4—5082no—
247D4—2541no—
248D4—2541no—
249D4C2×C22⋊D12712yes—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—1272no—
255C2×C4—1272no—
256C2×C4—10161no—
257C2×C4—10162no—
258C2×C4—5082no—
259C2×C4—2542no—
260C2×C4—2541no—
261C2×C4D4×D12711yescharacteristic
262C2×C4C2×C22⋊D12712yes—
263C2×C4#1452324712yes—
264C2×C4C23×D12711yescharacteristic
265C8—22no—
266C8—2541no—
267C8—2541no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—5081no—
274C22—5081no—
275C22—5081no—
276C22#1686118112yes—
277C22C2×D4×D12711yescharacteristic
278C4—22no—
279C4—42no—
280C4—5081no—
281C4—5082no—
282C4—2542no—
283C4—2542no—
284C4#1686118112yes—
285C4C22×C22⋊D12711yescharacteristic
286C4C2×#1452324711yescharacteristic
287C2—22no—
288C2—42no—
289C2—10161no—
290C2C2×#1686118111yescharacteristic
291C2#2255068511yescharacteristic
292C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861229C2
hypercenterC2×#137D127
hypocenterC127#3046
LCSG ▷ C2×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
2. #22550450C2semidirect
3. #22550359C2semidirect
4. #22550100C2semidirect
5. #22550093C2semidirect
6. #22550084C2semidirect
7. #22550079C2semidirect
8. #22549904C2semidirect
9. #22549878C2semidirect
10. C2×#16861133C2semidirect
11. C2×#137×C127C2semidirect
147. C127#3046semidirect
148. C2×#137D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
#225504502
#225503592
#225501002
#225500932
#225500842
#225500792
#225499042
#225498782
C2×#168611332
C2×#137×C1272
#3046127