Finite Group Databasebeta

← #30650137

Subgroups — #30650137

Order 16256.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550491C211yesmaximal, characteristic, complemented
2#22550480C211yesmaximal, characteristic, complemented
3#22550184C211yesmaximal, characteristic, complemented
4#22550176C211yesmaximal, characteristic, complemented
5#22550164C211yesmaximal, characteristic, complemented
6#22550154C211yesmaximal, characteristic, complemented
7#22550126C211yesmaximal, characteristic, complemented
8#22550121C211yesmaximal, characteristic, complemented
9#22550109C211yesmaximal, characteristic, complemented
10#22550104C211yesmaximal, characteristic, complemented
11#22550089C211yesmaximal, characteristic, complemented
12#22550089C211yesmaximal, characteristic, complemented
13#22550073C211yesmaximal, characteristic, complemented
14#22549901C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×#14523236C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861208C2211yescharacteristic
21#16861202—21no—
22#16861201C2211yescharacteristic
23#16861196C2211yescharacteristic
24#16861195—21no—
25#16861192—21no—
26#16861190—21no—
27#16861190—21no—
28#16861189C2211yescharacteristic
29#16861188—21no—
30#16861181—21no—
31#16861181—21no—
32#16861180C2211yescharacteristic
33#16861180C2211yescharacteristic
34#16861180C2211yescharacteristic
35C4×Dic254C2211yescharacteristic
36D4×Dic127C2211yescharacteristic
37D4×Dic127C2211yescharacteristic
38#16861166C2211yescharacteristic
39#16861143C2211yescharacteristic
40#16861142C2211yescharacteristic
41#16861141C2211yescharacteristic
42#16861140C2211yescharacteristic
43C4⋊(C127⋊C8)C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45Dic127⋊C8C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861134C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50#16861130C2211yescharacteristic
51#16861130C2211yescharacteristic
52C2×D8×C127C2211yescharacteristic
53C2×D8×C127C2211yescharacteristic
54#127×C127C2211yescharacteristic
55C4×D4×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C2×C22⋊D127—21no—
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C22×Dic127—21no—
65C22×Dic127—21no—
66C2×C4×D127C2311yescharacteristic
67#14523236—41no—
68#14523236—41no—
69#14523231—41no—
70C2×C127⋊C8C2311yescharacteristic
71C2×C127⋊C8C2311yescharacteristic
72C8×D127—41no—
73C4⋊Dic127—21no—
74C4⋊Dic127C2311yescharacteristic
75C4⋊Dic127C2311yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77Dic127⋊C4—21no—
78Dic127⋊C4—21no—
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4D411yescharacteristic
82Dic127⋊C4D411yescharacteristic
83C22⋊Dic127—21no—
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4D411yescharacteristic
92(C2×D127)⋊C4D411yescharacteristic
93C4×Dic127—21no—
94C4×Dic127C2311yescharacteristic
95C2×D4×C127—21no—
96C2×D4×C127—21no—
97C2×D4×C127C2311yescharacteristic
98C2×D4×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C22×C4×C127—21no—
101D8×C127—41no—
102D8×C127—41no—
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127C2×D411yescharacteristic
109Dic254—21no—
110Dic254—21no—
111Dic254—81no—
112C22⋊D127—81no—
113C22⋊D127—81no—
114C22⋊D127—81no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C2×Dic127—81no—
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C2×Dic127C2×D411yescharacteristic
124C4×D127—21no—
125C4×D127—21no—
126C127⋊C8—21no—
127C127⋊C8—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127—21no—
131D4×C127—21no—
132D4×C127—21no—
133D4×C127—21no—
134D4×C127—21no—
135D4×C127—21no—
136D4×C127—21no—
137D4×C127—81no—
138D4×C127—81no—
139C2×C4×C127—21no—
140C2×C4×C127—81no—
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2411yescharacteristic
144C8×C127—21no—
145C8×C127—21no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127C22×D411yescharacteristic
156C2×D127—21no—
157C2×D127—21no—
158C2×D127—21no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—41no—
162C4×C127H(5,𝔽2)11yescharacteristic
163C4×C1272−1+411yescharacteristic
164Dic127—21no—
165Dic127—21no—
166Dic127—21no—
167Dic127—41no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171C2×C127—81no—
172C2×C127—81no—
173C2×C127—81no—
174C2×C127#53711yescharacteristic
175C2×C127#57611yescharacteristic
176C2×C127#57711yescharacteristic
177D127—41no—
178#3257—1271nomaximal, Sylow
179C127#325711yescharacteristic, complemented, Sylow
180#542—1271no—
181#540—1271no—
182#479—1271no—
183#478—1271no—
184#476—1271no—
185#474—1271no—
186#469—1271no—
187#468—1271no—
188#466—1271no—
189#465—1271no—
190#463—1271no—
191#463—1271no—
192#460—1271no—
193#459D12711yescharacteristic, complemented
194#424—1271no—
195C2×SD16—1271no—
196C2×SD16—1271no—
197C2×D8C2×D12711yescharacteristic
198C2×D8C2×D12711yescharacteristic
199C2×M16—1271no—
200C22×C8—1271no—
201C4⋊Q8—1271no—
202#127C2×D12711yescharacteristic
203#126—2541no—
204#125—1271no—
205#124—1271no—
206#124—2541no—
207#123—2541no—
208#123—2541no—
209#123—2541no—
210C22⋊Q8—1271no—
211C22⋊Q8—2541no—
212#121—1271no—
213#121—1271no—
214#121—1271no—
215#121—2541no—
216#121—2541no—
217C4×Q8—1271no—
218C4×D4—1271no—
219C4×D4—1271no—
220C4×D4C2×D12711yescharacteristic
221#117—1271no—
222#107—1271no—
223#107—1271no—
224#106—1271no—
225#106—1271no—
226C4⋊C8—1271no—
227C4⋊C8—1271no—
228C4⋊C8—1271no—
229C4⋊C8C2×D12711yescharacteristic
230#103—1271no—
231#103—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102C2×D12711yescharacteristic
237#102C2×D12711yescharacteristic
238C2×Q8—1271no—
239C2×Q8—2541no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—2541no—
243C2×D4—2541no—
244C2×D4—2541no—
245C2×D4C22×D12711yescharacteristic
246C2×D4C22×D12711yescharacteristic
247C2×D4C22×D12711yescharacteristic
248C22×C4—21no—
249C22×C4—1271no—
250C22×C4—2541no—
251C22×C4—2541no—
252SD16—5081no—
253SD16—5081no—
254D8—41no—
255D8—41no—
256M16—5081no—
257C2×C8—1271no—
258C2×C8—1271no—
259C2×C8—5081no—
260C2×C8C22×D12711yescharacteristic
261C2×C8C22×D12711yescharacteristic
262C4⋊C4—1271no—
263C4⋊C4—1271no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—1271no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4—2541no—
272C4⋊C4C22×D12711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—1271no—
275C22⋊C4—1271no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4—2541no—
283C22⋊C4—2541no—
284C42—1271no—
285C42—2541no—
286C42C22×D12711yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23—1271no—
291Q8—10161no—
292Q8—2541no—
293Q8—2541no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—10161no—
303D4—10161no—
304D4—10161no—
305C2×C4—21no—
306C2×C4—81no—
307C2×C4—1271no—
308C2×C4—1271no—
309C2×C4—1271no—
310C2×C4—10161no—
311C2×C4—10161no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4—2541no—
316C2×C4—2541no—
317C2×C4—2541no—
318C2×C4D4×D12711yescharacteristic
319C2×C4D4×D12711yescharacteristic
320C2×C4C23×D12711yescharacteristic
321C8—21no—
322C8—21no—
323C8—2541no—
324C8—2541no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—2541no—
335C22—2541no—
336C22—2541no—
337C22C2×D4×D12711yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—5081no—
342C4—5081no—
343C4—5081no—
344C4—5081no—
345C4—2541no—
346C4—2541no—
347C4—2541no—
348C4#1686125411yescharacteristic
349C4#1686125611yescharacteristic
350C2—81no—
351C2—81no—
352C2—81no—
353C2—5081no—
354C2#2255045511yescharacteristic
355C2#2255067711yescharacteristic
356C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3257
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550491C2semidirect
2. #22550480C2semidirect
3. #22550184C2semidirect
4. #22550176C2semidirect
5. #22550164C2semidirect
6. #22550154C2semidirect
7. #22550126C2semidirect
8. #22550121C2semidirect
9. #22550109C2semidirect
10. #22550104C2semidirect
11. #22550089C2semidirect
12. #22550089C2semidirect
13. #22550073C2semidirect
14. #22549901C2semidirect
15. #459×C127C2semidirect
179. C127#3257semidirect
193. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225504912
#225504802
#225501842
#225501762
#225501642
#225501542
#225501262
#225501212
#225501092
#225501042
#225500892
#225500732
#225499012
#459×C1272
#3257127