Finite Group Databasebeta

← #30650143

Subgroups — #30650143

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#22550497C211yesmaximal, characteristic, complemented
2#22550481C211yesmaximal, characteristic, complemented
3#22550184C211yesmaximal, characteristic, complemented
4#22550177C211yesmaximal, characteristic, complemented
5#22550167C211yesmaximal, characteristic, complemented
6#22550157C211yesmaximal, characteristic, complemented
7#22550129C211yesmaximal, characteristic, complemented
8#22550122C211yesmaximal, characteristic, complemented
9#22550112C211yesmaximal, characteristic, complemented
10#22550105C211yesmaximal, characteristic, complemented
11#22550092C211yesmaximal, characteristic, complemented
12#22550092C211yesmaximal, characteristic, complemented
13#22550077C211yesmaximal, characteristic, complemented
14#22550072C211yesmaximal, characteristic, complemented
15#424×C127C211yesmaximal, characteristic, complemented
16C2×#14523237C2211yescharacteristic
17C2×#14523237C2211yescharacteristic
18C2×D1016C2211yescharacteristic
19C2×D1016C2211yescharacteristic
20#16861209C2211yescharacteristic
21#16861206C2211yescharacteristic
22#16861204—21no—
23#16861201C2211yescharacteristic
24#16861197C2211yescharacteristic
25#16861195—21no—
26#16861194—21no—
27#16861194—21no—
28#16861191—21no—
29#16861188C2211yescharacteristic
30#16861187—21no—
31#16861183C2211yescharacteristic
32#16861183C2211yescharacteristic
33#16861183C2211yescharacteristic
34#16861182—21no—
35#16861182—21no—
36C4×Dic254C2211yescharacteristic
37C4×D508C2211yescharacteristic
38C4×D508C2211yescharacteristic
39C4×D508C2211yescharacteristic
40#16861143C2211yescharacteristic
41#16861143C2211yescharacteristic
42#16861141C2211yescharacteristic
43#16861141C2211yescharacteristic
44#16861134C2211yescharacteristic
45#16861134C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861131C2211yescharacteristic
50#16861131C2211yescharacteristic
51#16861131C2211yescharacteristic
52C2×M16×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54#117×C127C2211yescharacteristic
55C4⋊C8×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57C4⋊C8×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic
59C2×Dic254—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508—21no—
62C2×D508—21no—
63C2×D508C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72#14523237—41no—
73#14523237—41no—
74D1016—41no—
75D1016—41no—
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85C22⋊Dic127—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94C4×Dic127—21no—
95C22×C4×C127C2311yescharacteristic
96M16×C127—41no—
97C2×C8×C127—41no—
98C2×C8×C127C2311yescharacteristic
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127D411yescharacteristic
103C4⋊C4×C127D411yescharacteristic
104C22⋊C4×C127D411yescharacteristic
105C22⋊C4×C127D411yescharacteristic
106C42×C127C2311yescharacteristic
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—21no—
111Dic254—21no—
112Dic254—21no—
113Dic254—81no—
114D508—21no—
115D508—21no—
116D508—21no—
117D508—21no—
118D508—21no—
119D508—21no—
120D508—81no—
121D508—81no—
122C22⋊D127—81no—
123C22⋊D127—81no—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C4×D127—81no—
131C4×D127—81no—
132C4×D127—81no—
133C23×C127C2×D411yescharacteristic
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127C2×D411yescharacteristic
137C2×C4×C127C2×D411yescharacteristic
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2×D411yescharacteristic
140C2×C4×C127C2×D411yescharacteristic
141C2×C4×C127C2411yescharacteristic
142C8×C127—21no—
143C8×C127—21no—
144C8×C127—21no—
145C8×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127C22×D411yescharacteristic
150C2×D127—41no—
151C2×D127—41no—
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127H(5,𝔽2)11yescharacteristic
165C4×C1272−1+411yescharacteristic
166Dic127—41no—
167Dic127—41no—
168Dic127—41no—
169Dic127—41no—
170Dic127—41no—
171C2×C127—41no—
172C2×C127#53711yescharacteristic
173C2×C127#57611yescharacteristic
174C2×C127#57711yescharacteristic
175D127—81no—
176D127—81no—
177D127—81no—
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#459—1271no—
194#424D12711yescharacteristic, complemented
195C2×SD16—1271no—
196C2×SD16—1271no—
197C2×D8—1271no—
198C2×D8—1271no—
199C2×M16C2×D12711yescharacteristic
200C22×C8C2×D12711yescharacteristic
201C4⋊Q8—1271no—
202#127—1271no—
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×D4—1271no—
221#117C2×D12711yescharacteristic
222#107—1271no—
223#107—1271no—
224#106—1271no—
225#106—1271no—
226C4⋊C8C2×D12711yescharacteristic
227C4⋊C8C2×D12711yescharacteristic
228C4⋊C8C2×D12711yescharacteristic
229C4⋊C8C2×D12711yescharacteristic
230#103—1271no—
231#103—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102—1271no—
238C2×Q8—1271no—
239C2×Q8—2541no—
240C2×D4—1271no—
241C2×D4—1271no—
242C2×D4—1271no—
243C2×D4—2541no—
244C2×D4—2541no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4—2541no—
248C22×C4—2541no—
249C22×C4—2541no—
250C22×C4—2541no—
251C22×C4C22×D12711yescharacteristic
252SD16—5081no—
253SD16—5081no—
254D8—5081no—
255D8—5081no—
256M16—41no—
257C2×C8—41no—
258C2×C8C22×D12711yescharacteristic
259C2×C8C22×D12711yescharacteristic
260C2×C8C22×D12711yescharacteristic
261C2×C8C22×D12711yescharacteristic
262C4⋊C4—1271no—
263C4⋊C4—1271no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4—2541no—
271C4⋊C4D50811yescharacteristic
272C4⋊C4D50811yescharacteristic
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C22⋊C4—2541no—
277C22⋊C4—2541no—
278C22⋊C4—2541no—
279C22⋊C4—2541no—
280C22⋊C4—2541no—
281C22⋊C4—2541no—
282C22⋊C4D50811yescharacteristic
283C22⋊C4D50811yescharacteristic
284C42—2541no—
285C42C22×D12711yescharacteristic
286C42C22×D12711yescharacteristic
287C23—2541no—
288C23—2541no—
289C23—2541no—
290C23C2×D50811yescharacteristic
291Q8—10161no—
292Q8—2541no—
293Q8—2541no—
294D4—10161no—
295D4—10161no—
296D4—10161no—
297D4—10161no—
298D4—10161no—
299D4—2541no—
300D4—2541no—
301D4—2541no—
302D4—2541no—
303D4—2541no—
304D4—2541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—10161no—
308C2×C4—10161no—
309C2×C4—10161no—
310C2×C4—2541no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4—2541no—
315C2×C4C2×D50811yescharacteristic
316C2×C4C2×D50811yescharacteristic
317C2×C4C2×D50811yescharacteristic
318C2×C4C2×D50811yescharacteristic
319C2×C4C2×D50811yescharacteristic
320C2×C4C23×D12711yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—5081no—
329C22—5081no—
330C22—5081no—
331C22—5081no—
332C22—5081no—
333C22—5081no—
334C22—5081no—
335C22—5081no—
336C22—5081no—
337C22C22×D50811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—5081no—
344C4—5081no—
345C4—5081no—
346C4—5081no—
347C4—5081no—
348C4#1686125411yescharacteristic
349C4#1686125711yescharacteristic
350C2—41no—
351C2—10161no—
352C2—10161no—
353C2—10161no—
354C2#2255045711yescharacteristic
355C2#2255068111yescharacteristic
356C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#424×C127C2
socleC22×C127C22×D4
hypercenter#424D127
hypocenterC127#3257
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #22550497C2semidirect
2. #22550481C2semidirect
3. #22550184C2semidirect
4. #22550177C2semidirect
5. #22550167C2semidirect
6. #22550157C2semidirect
7. #22550129C2semidirect
8. #22550122C2semidirect
9. #22550112C2semidirect
10. #22550105C2semidirect
11. #22550092C2semidirect
12. #22550092C2semidirect
13. #22550077C2semidirect
14. #22550072C2semidirect
15. #424×C127C2semidirect
179. C127#3257semidirect
194. #424D127semidirect

Maximal subgroups

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

subgroupindex
#225504972
#225504812
#225501842
#225501772
#225501672
#225501572
#225501292
#225501222
#225501122
#225501052
#225500922
#225500772
#225500722
#424×C1272
#3257127