Finite Group Databasebeta

← #30648447

Subgroups — #30648447

Order 16256.

452 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#22550674C211yesmaximal, characteristic, complemented
2C2×#16861229C211yesmaximal, characteristic, complemented
3C2×#16861216C211yesmaximal, characteristic, complemented
4#22550149C212yesmaximal, complemented
5#22550144C212yesmaximal, complemented
6#22550141C212yesmaximal, complemented
7#22550137C212yesmaximal, complemented
8C2×#16861144C211yesmaximal, characteristic, complemented
9#22549860C211yesmaximal, characteristic, complemented
10#22549858C211yesmaximal, characteristic, complemented
11#575×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861240—21no—
14#16861239C2212yes—
15#16861235—21no—
16#16861234C2212yes—
17#16861229—21no—
18#16861229—21no—
19#16861229—21no—
20#16861229—21no—
21#16861229—22no—
22C2×#14523238C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523235C2211yescharacteristic, complemented
25C2×#14523233C2211yescharacteristic, complemented
26#16861216—21no—
27#16861216—21no—
28#16861216C2212yes—
29#16861216C2212yes—
30C2×#14523232—21no—
31C2×#14523232C2211yescharacteristic
32C2×#14523232C2211yescharacteristic
33C22×C127⋊C8—21no—
34C22×C127⋊C8C2211yescharacteristic
35#16861144C2212yescomplemented
36#16861144C2212yescomplemented
37#16861128C2212yescomplemented
38#16861127C2212yescomplemented
39#16861126C2212yescomplemented
40#16861125C2212yescomplemented
41C2×#41×C127C2211yescharacteristic
42C2×#41×C127C2211yescharacteristic
43#137×C127—21no—
44#137×C127C2212yes—
45AGL(1,ℤ/8)×C127—21no—
46AGL(1,ℤ/8)×C127C2212yes—
47#135×C127—22no—
48C2×M16×C127C2211yescharacteristic
49#14523248—21no—
50#14523248—21no—
51#14523248—21no—
52#14523248—21no—
53C2×Dic254C2311yescharacteristic
54C2×D508C2311yescharacteristic
55C2×C22⋊D127—21no—
56C2×C4×D127—21no—
57#14523238—21no—
58#14523238—21no—
59#14523238—22no—
60#14523236—21no—
61#14523236—21no—
62#14523236—22no—
63#14523235—21no—
64#14523235—21no—
65#14523235—22no—
66#14523233—21no—
67#14523233—21no—
68#14523233—22no—
69#14523232—22no—
70#14523232—22no—
71#14523232—21no—
72#14523232—21no—
73#14523232C2312yes—
74#14523232C2312yes—
75C2×C127⋊C8—22no—
76C2×C127⋊C8—22no—
77C2×C127⋊C8—21no—
78C2×C127⋊C8—21no—
79C2×C127⋊C8D411yescharacteristic
80C2×C127⋊C8D411yescharacteristic
81C2×C127⋊C8D411yescharacteristic
82C2×C127⋊C8D411yescharacteristic
83C2×C127⋊C8C2311yescharacteristic
84C2×C127⋊C8C2311yescharacteristic
85#41×C127—21no—
86#41×C127—21no—
87#41×C127—21no—
88#41×C127—21no—
89#41×C127—21no—
90#41×C127—21no—
91#41×C127D412yes—
92#41×C127C2312yes—
93C2×Q8×C127D411yescharacteristic
94C2×Q8×C127C2311yescharacteristic
95C2×D4×C127—21no—
96C2×D4×C127—21no—
97C2×D4×C127D411yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100C22×C4×C127—21no—
101C22×C4×C127C2311yescharacteristic
102Q16×C127—22no—
103SD16×C127—22no—
104SD16×C127—22no—
105D8×C127—22no—
106M16×C127—21no—
107M16×C127C2312yes—
108C2×C8×C127—21no—
109C22×D127—21no—
110Dic254—21no—
111Dic254—21no—
112D508—21no—
113D508—21no—
114C22⋊D127—41no—
115C22⋊D127—41no—
116C2×Dic127—21no—
117C4×D127—41no—
118C4×D127—41no—
119C127⋊C8—22no—
120C127⋊C8—22no—
121C127⋊C8C2×D411yescharacteristic
122C127⋊C8C2×D411yescharacteristic
123C127⋊C8C2×D411yescharacteristic
124C127⋊C8C2×D411yescharacteristic
125C23×C127—21no—
126C23×C127—21no—
127C23×C127#4111yescharacteristic
128Q8×C127—21no—
129Q8×C127—21no—
130Q8×C127—21no—
131Q8×C127C2×D412yes—
132D4×C127—22no—
133D4×C127—22no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—41no—
138D4×C127—41no—
139D4×C127C2×D412yes—
140C2×C4×C127—21no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—21no—
144C2×C4×C127—41no—
145C2×C4×C127—41no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127#4111yescharacteristic
151C2×C4×C127C2411yescharacteristic
152C8×C127—22no—
153C22×C127—22no—
154C22×C127—41no—
155C22×C127—42no—
156C22×C127—41no—
157C22×C127—81no—
158C22×C127#12111yescharacteristic
159C22×C127#12111yescharacteristic
160C22×C127C2×#4111yescharacteristic
161C2×D127—41no—
162C2×D127—81no—
163C4×C127—22no—
164C4×C127—41no—
165C4×C127#12111yescharacteristic
166C4×C127#12111yescharacteristic
167C4×C127C22×D411yescharacteristic
168C4×C127C22×D411yescharacteristic
169Dic127—41no—
170C2×C127—21no—
171C2×C127—21no—
172C2×C127—21no—
173C2×C127—42no—
174C2×C127—81no—
175C2×C127C2×#12111yescharacteristic
176D127—81no—
177#3067—1271nomaximal, Sylow
178C127#306711yescharacteristic, complemented, Sylow
179#575—1271no—
180#575D12711yescharacteristic, complemented
181C2×#135—1271no—
182C2×#131—1271no—
183#473—1272no—
184#472—1272no—
185#471—1272no—
186#471—1272no—
187C2×#108—1271no—
188#413—1271no—
189#413—1271no—
190C2×#41—1271no—
191C2×#41C2×D12711yescharacteristic
192C2×#41C2×D12711yescharacteristic
193#137—21no—
194#137—1272no—
195#137—2541no—
196#137C2×D12712yes—
197AGL(1,ℤ/8)—21no—
198AGL(1,ℤ/8)—1272no—
199AGL(1,ℤ/8)—2541no—
200AGL(1,ℤ/8)C2×D12712yes—
201#135—22no—
202#135—2541no—
203#135—2541no—
204#135—2541no—
205#135—2541no—
206#135—2542no—
207C2×Q16—1271no—
208C2×SD16—1271no—
209C2×SD16—1271no—
210C2×D8—1271no—
211#131—1272no—
212#131—1272no—
213#131—2541no—
214#131—2541no—
215C2×M16—1271no—
216C2×M16—1271no—
217C2×M16—2541no—
218C2×M16C2×D12711yescharacteristic
219C22×C8—1271no—
220C22×C8—2541no—
221#108—1272no—
222#108—1272no—
223#101—1272no—
224#101—1272no—
225#100—1272no—
226#100—1272no—
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—2541no—
234#41—2541no—
235#41—2541no—
236#41—2541no—
237#41C22⋊D12712yes—
238#41C22×D12712yes—
239C2×Q8—1271no—
240C2×Q8C22⋊D12711yescharacteristic
241C2×Q8C22×D12711yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—1271no—
245C2×D4—2541no—
246C2×D4C22⋊D12711yescharacteristic
247C2×D4C22×D12711yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—2541no—
251C22×C4C22×D12711yescharacteristic
252Q16—22no—
253Q16—2541no—
254Q16—2541no—
255Q16—2542no—
256SD16—22no—
257SD16—22no—
258SD16—2541no—
259SD16—2541no—
260SD16—2541no—
261SD16—2541no—
262SD16—2542no—
263SD16—2542no—
264D8—22no—
265D8—2541no—
266D8—2541no—
267D8—2542no—
268M16—21no—
269M16—1272no—
270M16—1272no—
271M16—2542no—
272M16—2542no—
273M16—2541no—
274M16—2541no—
275M16C22×D12712yes—
276C2×C8—21no—
277C2×C8—1271no—
278C2×C8—1271no—
279C2×C8—1271no—
280C2×C8—1271no—
281C2×C8—1271no—
282C2×C8—1271no—
283C2×C8—2542no—
284C2×C8—2542no—
285C2×C8—2541no—
286C2×C8—2541no—
287C23—21no—
288C23—21no—
289C23—2541no—
290C23#1452324711yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—2541no—
295Q8—2541no—
296Q8C2×C22⋊D12712yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—5081no—
305D4—5081no—
306D4—2541no—
307D4—2541no—
308D4C2×C22⋊D12712yes—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—5081no—
316C2×C4—5081no—
317C2×C4—2541no—
318C2×C4D4×D12711yescharacteristic
319C2×C4D4×D12711yescharacteristic
320C2×C4C2×C22⋊D12711yescharacteristic
321C2×C4C2×C22⋊D12711yescharacteristic
322C2×C4#1452324711yescharacteristic
323C2×C4C23×D12711yescharacteristic
324C8—22no—
325C8—1271no—
326C8—1271no—
327C8—1271no—
328C8—1271no—
329C8—2542no—
330C8—2542no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—81no—
336C22—10161no—
337C22—5081no—
338C22#1686118111yescharacteristic
339C22#1686118111yescharacteristic
340C22C2×#1452324711yescharacteristic
341C4—22no—
342C4—41no—
343C4—5081no—
344C4#1686118111yescharacteristic
345C4#1686118111yescharacteristic
346C4C2×D4×D12711yescharacteristic
347C4C22×C22⋊D12711yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2—81no—
353C2—10161no—
354C2C2×#1686118111yescharacteristic
subgroupG/H
centerC4#16861181
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#575×C127C2
socleC2×C127C2×#121
hypercenter#575D127
hypocenterC127#3067
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. C2×#16861229C2semidirect
3. C2×#16861216C2semidirect
4. #22550149C2semidirect
5. #22550144C2semidirect
6. #22550141C2semidirect
7. #22550137C2semidirect
8. C2×#16861144C2semidirect
9. #22549860C2semidirect
10. #22549858C2semidirect
11. #575×C127C2semidirect
22. C2×#14523238C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523235C22semidirect
25. C2×#14523233C22semidirect
35. #16861144C22semidirect
36. #16861144C22semidirect
37. #16861128C22semidirect
38. #16861127C22semidirect
39. #16861126C22semidirect
40. #16861125C22semidirect
178. C127#3067semidirect
180. #575D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612292
C2×#168612162
#225501492
#225501442
#225501412
#225501372
C2×#168611442
#225498602
#225498582
#575×C1272
#3067127