Finite Group Databasebeta

← #30638685

Subgroups — #30638685

Order 16256.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861255C211yesmaximal, characteristic
2C2×#16861137C211yesmaximal, characteristic, complemented
3#22549878C211yesmaximal, characteristic, complemented
4#22549877C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549778C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8#16861255—22no—
9#16861255—22no—
10#16861255—22no—
11#16861255C414yescomplemented
12C2×#14523249—21no—
13C2×#14523248—21no—
14C2×#14523248C2211yescharacteristic
15C22×D508—21no—
16C22×D508C2211yescharacteristic
17C2×D4×D127—22no—
18C2×D4×D127—21no—
19C2×#14523232C2211yescharacteristic, complemented
20C2×C4⋊Dic127C2211yescharacteristic, complemented
21C2×(C2×D127)⋊C4—22no—
22#16861137—22no—
23#16861134—21no—
24#16861133—21no—
25#16861131—21no—
26#16861130—21no—
27#16861125—22no—
28C2×#41×C127C2211yescharacteristic
29C2×M16×C127C2211yescharacteristic, complemented
30C2×C42×C127C2211yescharacteristic, complemented
31#104×C127—22no—
32C23×D127—22no—
33C23×D127D412yes—
34#14523249—24no—
35#14523249—24no—
36#14523248—24no—
37#14523248—24no—
38#14523248—21no—
39#14523248—21no—
40#14523248C2×C412yes—
41#14523248D412yes—
42C2×Dic254D411yescharacteristic
43C2×D508—22no—
44C2×D508—24no—
45C2×D508—24no—
46C2×D508—21no—
47C2×D508—21no—
48C2×D508—21no—
49C2×D508—21no—
50C2×D508C2×C412yes—
51C2×D508D412yes—
52C2×D508D411yescharacteristic
53C2×C22⋊D127—22no—
54C2×C22⋊D127—21no—
55D4×D127—28no—
56D4×D127—24no—
57D4×D127—28no—
58D4×D127—24no—
59C22×Dic127—21no—
60C2×C4×D127—22no—
61C2×C4×D127—21no—
62#14523232—21no—
63#14523232—21no—
64C2×C127⋊C8—21no—
65C4⋊Dic127—21no—
66C4⋊Dic127—21no—
67(C2×D127)⋊C4—44no—
68#41×C127—21no—
69#41×C127—21no—
70#41×C127C2×C412yes—
71#41×C127D412yes—
72C2×Q8×C127D411yescharacteristic
73C2×D4×C127—21no—
74C2×D4×C127D411yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78M16×C127—21no—
79M16×C127—21no—
80C2×C8×C127—21no—
81C42×C127—21no—
82C42×C127—21no—
83C22×D127—22no—
84C22×D127—21no—
85C22×D127—28no—
86C22×D127—22no—
87C22×D127—42no—
88C22×D127—42no—
89C22×D127—82no—
90C22×D127—82no—
91C22×D127C22⋊C414yes—
92Dic254—21no—
93Dic254C22⋊C412yescomplemented
94D508—28no—
95D508—22no—
96D508—22no—
97D508—22no—
98D508—22no—
99D508—21no—
100D508C22⋊C412yescomplemented
101C22⋊D127—28no—
102C22⋊D127—22no—
103C22⋊D127—22no—
104C2×Dic127—21no—
105C2×Dic127—42no—
106C2×Dic127—41no—
107C4×D127—28no—
108C4×D127—22no—
109C4×D127—22no—
110C127⋊C8—41no—
111C23×C127—21no—
112C23×C127C2×D411yescharacteristic
113Q8×C127—21no—
114Q8×C127C22⋊C412yescomplemented
115D4×C127—22no—
116D4×C127—22no—
117D4×C127—21no—
118D4×C127C22⋊C412yescomplemented
119C2×C4×C127—21no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127C22×C411yescharacteristic
126C2×C4×C127C2×D411yescharacteristic
127C2×C4×C127C2×D411yescharacteristic
128C2×C4×C127C2×D411yescharacteristic
129C2×C4×C127C2×D411yescharacteristic
130C2×C4×C127C2×D411yescharacteristic
131C8×C127—41no—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—22no—
135C22×C127—41no—
136C22×C127—41no—
137C22×C127C2×C22⋊C411yescharacteristic
138C22×C127#12011yescharacteristic
139C22×C127#12011yescharacteristic
140C2×D127—24no—
141C2×D127—22no—
142C2×D127—42no—
143C2×D127—41no—
144C2×D127—42no—
145C2×D127—41no—
146C2×D127—44no—
147C2×D127—82no—
148C2×D127—88no—
149C4×C127—22no—
150C4×C127—42no—
151C4×C127C2×C22⋊C411yescharacteristic
152C4×C127C2×C22⋊C411yescharacteristic
153C4×C127#12011yescharacteristic
154C4×C127#12011yescharacteristic
155Dic127—22no—
156Dic127—81no—
157C2×C127—21no—
158C2×C127—21no—
159C2×C127—42no—
160C2×C127#37911yescharacteristic
161C2×C127#45311yescharacteristic
162C2×C127#45511yescharacteristic
163D127—44no—
164D127—42no—
165#1792—1271nomaximal, Sylow
166C127#179211yescharacteristic, complemented, Sylow
167C2×H(5,𝔽2)—1271no—
168C2×#104—1271no—
169C2×#104D12711yescharacteristic, complemented
170#417—1271no—
171#417—1271no—
172C2×#100—1271no—
173#390—1271no—
174H(5,𝔽2)—1274no—
175H(5,𝔽2)—2542no—
176H(5,𝔽2)—2542no—
177H(5,𝔽2)—2542no—
178C2×#41—1271no—
179C2×#41—2541no—
180C2×#41—2541no—
181C2×#41C2×D12711yescharacteristic
182C22×D4—1271no—
183C22×D4—2541no—
184C22×D4—2542no—
185C22×D4—2541no—
186C2×M16—1271no—
187C2×M16C2×D12711yescharacteristic, complemented
188C2×C4⋊C4—1271no—
189C2×C22⋊C4—2542no—
190C2×C42C2×D12711yescharacteristic, complemented
191#104—22no—
192#104—2542no—
193#103—2541no—
194#103—2541no—
195#102—2541no—
196#102—2541no—
197#100—2542no—
198C24—1272no—
199C24—2542no—
200#41—21no—
201#41—21no—
202#41—1272no—
203#41—1272no—
204#41—2544no—
205#41—2544no—
206#41—2544no—
207#41—2544no—
208#41—2541no—
209#41—2541no—
210#41C4×D12712yes—
211#41C22⋊D12712yes—
212C2×Q8—1271no—
213C2×Q8D50811yescharacteristic
214C2×D4—21no—
215C2×D4—1272no—
216C2×D4—1271no—
217C2×D4—1272no—
218C2×D4—2544no—
219C2×D4—2542no—
220C2×D4—2544no—
221C2×D4—2548no—
222C2×D4—2548no—
223C2×D4—2544no—
224C2×D4—2542no—
225C2×D4—2541no—
226C2×D4—2544no—
227C2×D4—2541no—
228C2×D4—2541no—
229C2×D4—2541no—
230C2×D4—2541no—
231C2×D4D50811yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—2542no—
235C22×C4—2541no—
236C22×C4—2541no—
237C22×C4C22×D12711yescharacteristic
238M16—21no—
239M16—21no—
240M16—2541no—
241M16—2541no—
242C2×C8—21no—
243C2×C8—2541no—
244C4⋊C4—2541no—
245C4⋊C4—2541no—
246C22⋊C4—5084no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—1274no—
251C23—10162no—
252C23—10162no—
253C23—5082no—
254C23—5082no—
255C23—2548no—
256C23—2542no—
257C23—2541no—
258C23—2542no—
259C23D4×D12711yescharacteristic
260Q8—21no—
261Q8—1272no—
262Q8—2541no—
263Q8(C2×D127)⋊C412yescomplemented
264D4—22no—
265D4—22no—
266D4—21no—
267D4—1272no—
268D4—2548no—
269D4—2548no—
270D4—2542no—
271D4—2542no—
272D4—2542no—
273D4—2542no—
274D4—2541no—
275D4—2542no—
276D4—2542no—
277D4(C2×D127)⋊C412yescomplemented
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—5082no—
285C2×C4—5081no—
286C2×C4—2548no—
287C2×C4—2541no—
288C2×C4—2542no—
289C2×C4—2542no—
290C2×C4C2×C4×D12711yescharacteristic
291C2×C4D4×D12711yescharacteristic
292C2×C4D4×D12711yescharacteristic
293C2×C4D4×D12711yescharacteristic
294C2×C4C2×C22⋊D12711yescharacteristic
295C2×C4C2×D50811yescharacteristic
296C8—41no—
297C8—5081no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—10162no—
304C22—10168no—
305C22—5084no—
306C22—5081no—
307C22—5082no—
308C22—5081no—
309C22—5082no—
310C22—2542no—
311C22—2544no—
312C22C22⋊C4×D12711yescharacteristic
313C22#1686117711yescharacteristic
314C22#1686117911yescharacteristic
315C4—22no—
316C4—42no—
317C4—10161no—
318C4—2542no—
319C4C22⋊C4×D12711yescharacteristic
320C4C2×(C2×D127)⋊C411yescharacteristic
321C4#1686117711yescharacteristic
322C4#1686117911yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2—5082no—
327C2—5084no—
328C2#2254973111yescharacteristic
329C2#2255004511yescharacteristic
330C2#2255005511yescharacteristic
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127#120
prime radicalC2×#16861255C2
cosocleC22×C4×C127C23
hypercenterC2×#104D127
hypocenterC127#1792
LCSG ▷ C2×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
2. C2×#16861137C2semidirect
3. #22549878C2semidirect
4. #22549877C2semidirect
5. C2×#16861125C2semidirect
6. #22549778C2semidirect
7. C2×#104×C127C2semidirect
11. #16861255C4semidirect
19. C2×#14523232C22semidirect
20. C2×C4⋊Dic127C22semidirect
29. C2×M16×C127C22semidirect
30. C2×C42×C127C22semidirect
93. Dic254C22⋊C4semidirect
100. D508C22⋊C4semidirect
114. Q8×C127C22⋊C4semidirect
118. D4×C127C22⋊C4semidirect
166. C127#1792semidirect
169. C2×#104D127semidirect
187. C2×M16C2×D127semidirect
190. C2×C42C2×D127semidirect
263. Q8(C2×D127)⋊C4semidirect
277. D4(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168611372
#225498782
#225498772
C2×#168611252
#225497782
C2×#104×C1272
#1792127