Finite Group Databasebeta

← #30650112

Subgroups — #30650112

Order 16256.

386 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550428C211yesmaximal, characteristic, complemented
2#22550421C211yesmaximal, characteristic
3#22550170C211yesmaximal, characteristic, complemented
4#22550169C211yesmaximal, characteristic, complemented
5#22550155C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550124C211yesmaximal, characteristic, complemented
8#22550122C211yesmaximal, characteristic, complemented
9#22550120C211yesmaximal, characteristic, complemented
10#22550080C211yesmaximal, characteristic, complemented
11#22550079C211yesmaximal, characteristic, complemented
12#22550075C211yesmaximal, characteristic, complemented
13#22550074C211yesmaximal, characteristic, complemented
14#22549899C211yesmaximal, characteristic, complemented
15#468×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×#14523237C2211yescharacteristic
18C2×#14523236C2211yescharacteristic
19C2×#14523235C2211yescharacteristic
20C2×#14523232C2211yescharacteristic
21#16861208C2211yescharacteristic
22#16861208C2211yescharacteristic
23#16861207C2211yescharacteristic
24#16861205C2211yescharacteristic
25#16861199—21no—
26#16861195—21no—
27#16861188C2211yescharacteristic
28#16861186—21no—
29#16861183C2211yescharacteristic
30#16861181—21no—
31Q8×Dic127C2211yescharacteristic
32#16861174—21no—
33#16861174C2211yescharacteristic
34D4×Dic127C2211yescharacteristic
35#16861171C2211yescharacteristic
36(C22⋊D127)⋊C4—21no—
37#16861165—21no—
38C2×C4×Dic127C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861142C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43Dic127⋊C8C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45Dic127⋊C8C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861133C2211yescharacteristic
48#16861132C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861130C2211yescharacteristic
51#16861129C2211yescharacteristic
52C2×SD16×C127C2211yescharacteristic
53C2×M16×C127C2211yescharacteristic
54C22⋊Q8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#107×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—44no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254C2311yescharacteristic
63C2×D508C2311yescharacteristic
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C22×Dic127—21no—
68C22×Dic127D411yescharacteristic
69C22×Dic127D411yescharacteristic
70C2×C4×D127—21no—
71#14523237—41no—
72#14523236—41no—
73#14523235—41no—
74#14523232—41no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C4⋊Dic127—21no—
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90C4×Dic127—21no—
91C4×Dic127—21no—
92C4×Dic127D411yescharacteristic
93C4×Dic127D411yescharacteristic
94C4×Dic127C2311yescharacteristic
95C4×Dic127C2311yescharacteristic
96C2×Q8×C127C2311yescharacteristic
97C2×D4×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127C2311yescharacteristic
100SD16×C127—41no—
101M16×C127—41no—
102C2×C8×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C4⋊C4×C127—21no—
105C4⋊C4×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C22×D127—21no—
110Dic254—21no—
111Dic254—21no—
112Dic254—42no—
113Dic254—42no—
114D508—21no—
115D508—21no—
116C22⋊D127—42no—
117C22⋊D127—42no—
118C22⋊D127—42no—
119C2×Dic127—22no—
120C2×Dic127—22no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—42no—
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C4×D127—42no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C23×C127—21no—
133C23×C127C2×D411yescharacteristic
134Q8×C127—21no—
135Q8×C127—21no—
136D4×C127—21no—
137D4×C127—21no—
138D4×C127—42no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2411yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C4×C127—21no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127H(5,𝔽2)11yescharacteristic
163C4×C1272−1+411yescharacteristic
164Dic127—21no—
165Dic127—21no—
166Dic127—41no—
167Dic127—41no—
168Dic127—41no—
169Dic127AGL(1,ℤ/8)12yes—
170Dic127#13712yes—
171C2×C127—41no—
172C2×C127—81no—
173C2×C127#53711yescharacteristic
174C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C127C2×#13711yescharacteristic
176D127—81no—
177#3254—1271nomaximal, Sylow
178C127#325411yescharacteristic, complemented, Sylow
179#533—1271no—
180#532—1271no—
181#477—1271no—
182#477—1271no—
183#474—1271no—
184#474—1271no—
185#468—1271no—
186#468—1271no—
187#468—1271no—
188#468D12711yescharacteristic, complemented
189#461—1271no—
190#461—1271no—
191#460—1271no—
192#460—1271no—
193#423—1271no—
194C2×#41—2541no—
195C2×SD16—1271no—
196C2×SD16—1271no—
197C2×SD16—1271no—
198C2×SD16C2×D12711yescharacteristic
199C2×M16—1271no—
200C2×M16C2×D12711yescharacteristic
201C4⋊Q8—1271no—
202C4⋊Q8—1271no—
203C4⋊Q8—1271no—
204#127—1271no—
205#125—2541no—
206#124—2541no—
207C22⋊Q8—1271no—
208C22⋊Q8—2541no—
209C22⋊Q8C2×D12711yescharacteristic
210#121—1271no—
211#121—2541no—
212#121C2×D12711yescharacteristic
213C4×Q8—1271no—
214C4×Q8—1271no—
215C4×Q8—2541no—
216C4×D4—1271no—
217C4×D4—1271no—
218C4×D4—2541no—
219#117—2541no—
220C2×C42—1271no—
221#107—1271no—
222#107—1271no—
223#107—1271no—
224#107C2×D12711yescharacteristic
225C4⋊C8—1271no—
226C4⋊C8—1271no—
227C4⋊C8—1271no—
228C4⋊C8—1271no—
229#103—1271no—
230#103—1271no—
231#103—1271no—
232#103C2×D12711yescharacteristic
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102C2×D12711yescharacteristic
237#41—5084no—
238C2×Q8—1271no—
239C2×Q8—2541no—
240C2×Q8—2541no—
241C2×Q8C22×D12711yescharacteristic
242C2×D4—21no—
243C2×D4—1271no—
244C2×D4—2541no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4C22×D12711yescharacteristic
248C22×C4—1271no—
249C22×C4—1271no—
250C22×C4—2541no—
251C22×C4—2541no—
252C22×C4C22×D12711yescharacteristic
253SD16—41no—
254SD16—5081no—
255SD16—5081no—
256SD16—5081no—
257M16—41no—
258M16—5081no—
259C2×C8—1271no—
260C2×C8—1271no—
261C2×C8C22×D12711yescharacteristic
262C2×C8C22×D12711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4—2541no—
273C4⋊C4C22×D12711yescharacteristic
274C4⋊C4C22×D12711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C42—1271no—
282C42—1271no—
283C42—1271no—
284C42—1271no—
285C42—2541no—
286C42—2541no—
287C23—21no—
288C23—2541no—
289C23D4×D12711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5082no—
293Q8—5082no—
294Q8—2541no—
295Q8—2541no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—5082no—
300D4—5082no—
301D4—5082no—
302D4—2541no—
303D4—2541no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—1271no—
313C2×C4—5082no—
314C2×C4—5082no—
315C2×C4—2542no—
316C2×C4—2542no—
317C2×C4—2541no—
318C2×C4—2541no—
319C2×C4—2541no—
320C2×C4D4×D12711yescharacteristic
321C2×C4C23×D12711yescharacteristic
322C8—21no—
323C8—21no—
324C8—2541no—
325C8—2541no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—5081no—
333C22—5081no—
334C22—5081no—
335C22C2×D4×D12711yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—1272no—
341C4—1272no—
342C4—5081no—
343C4—5081no—
344C4—5081no—
345C4—2541no—
346C4—2541no—
347C4#1686125411yescharacteristic
348C4#1686125611yescharacteristic
349C2—41no—
350C2—81no—
351C2—10161no—
352C2#2255045311yescharacteristic
353C2AGL(1,ℤ/8)×D12711yescharacteristic
354C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#468×C127C2
socleC22×C127C22×D4
prime radical#22550421C2
hypercenter#468D127
hypocenterC127#3254
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #22550428C2semidirect
3. #22550170C2semidirect
4. #22550169C2semidirect
5. #22550155C2semidirect
6. #22550154C2semidirect
7. #22550124C2semidirect
8. #22550122C2semidirect
9. #22550120C2semidirect
10. #22550080C2semidirect
11. #22550079C2semidirect
12. #22550075C2semidirect
13. #22550074C2semidirect
14. #22549899C2semidirect
15. #468×C127C2semidirect
178. C127#3254semidirect
188. #468D127semidirect

Maximal subgroups

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

subgroupindex
#225504282
#225504212
#225501702
#225501692
#225501552
#225501542
#225501242
#225501222
#225501202
#225500802
#225500792
#225500752
#225500742
#225498992
#468×C1272
#3254127