Finite Group Databasebeta

← #30650148

Subgroups — #30650148

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#22550500C211yesmaximal, characteristic, complemented
2#22550477C211yesmaximal, characteristic, complemented
3#22550182C211yesmaximal, characteristic, complemented
4#22550179C211yesmaximal, characteristic, complemented
5#22550163C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550123C211yesmaximal, characteristic, complemented
8#22550113C211yesmaximal, characteristic, complemented
9#22550104C211yesmaximal, characteristic, complemented
10#22550092C211yesmaximal, characteristic, complemented
11#22550091C211yesmaximal, characteristic, complemented
12#22550075C211yesmaximal, characteristic, complemented
13#22550070C211yesmaximal, characteristic, complemented
14#22549901C211yesmaximal, characteristic, complemented
15#474×C127C211yesmaximal, characteristic, complemented
16C2×#14523237C2211yescharacteristic
17C2×#14523236C2211yescharacteristic
18C2×D1016C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523231C2211yescharacteristic
21C2×C8×D127C2211yescharacteristic
22#16861205C2211yescharacteristic
23#16861204—21no—
24#16861200C2211yescharacteristic
25#16861198—21no—
26#16861197C2211yescharacteristic
27#16861194—21no—
28#16861192—21no—
29#16861191—21no—
30#16861189C2211yescharacteristic
31#16861185—21no—
32#16861184C2211yescharacteristic
33#16861184C2211yescharacteristic
34#16861183—21no—
35#16861182—21no—
36#16861180C2211yescharacteristic
37#16861174C2211yescharacteristic
38C4×D508C2211yescharacteristic
39#16861171C2211yescharacteristic
40#16861166C2211yescharacteristic
41#16861140C2211yescharacteristic
42#16861140C2211yescharacteristic
43C4⋊(C127⋊C8)C2211yescharacteristic
44Dic127⋊C8C2211yescharacteristic
45Dic127⋊C8C2211yescharacteristic
46#16861134C2211yescharacteristic
47#16861132C2211yescharacteristic
48#16861131C2211yescharacteristic
49#16861129C2211yescharacteristic
50#16861129C2211yescharacteristic
51#16861129C2211yescharacteristic
52C4⋊Q8×C127C2211yescharacteristic
53C4×D4×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic
55#107×C127C2211yescharacteristic
56C4⋊C8×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C2×Dic254C2311yescharacteristic
60C2×D508—21no—
61C2×D508—21no—
62C2×D508C2311yescharacteristic
63C2×D508C2311yescharacteristic
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127C2311yescharacteristic
70#14523237—41no—
71#14523236—41no—
72D1016—41no—
73#14523233—41no—
74#14523231—41no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C8×D127—41no—
78C4⋊Dic127—21no—
79C4⋊Dic127—21no—
80C4⋊Dic127C2311yescharacteristic
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84Dic127⋊C4D411yescharacteristic
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)⋊C4D411yescharacteristic
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C2×Q8×C127—21no—
98C2×D4×C127C2311yescharacteristic
99C22×C4×C127—21no—
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127C2×D411yescharacteristic
111Dic254—21no—
112Dic254—21no—
113D508—21no—
114D508—21no—
115D508—21no—
116D508—21no—
117D508—81no—
118D508—81no—
119C22⋊D127—81no—
120C22⋊D127—81no—
121C22⋊D127—81no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127C2×D411yescharacteristic
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C4×D127—21no—
128C4×D127—21no—
129C4×D127—81no—
130C4×D127—81no—
131C127⋊C8—21no—
132C127⋊C8—21no—
133C23×C127—21no—
134Q8×C127—81no—
135D4×C127—21no—
136D4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
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×C127C22×D411yescharacteristic
150C2×D127—21no—
151C2×D127—21no—
152C2×D127—21no—
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—41no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127H(5,𝔽2)11yescharacteristic
165C4×C1272−1+411yescharacteristic
166Dic127—21no—
167Dic127—21no—
168Dic127—21no—
169Dic127—41no—
170Dic127—41no—
171C2×C127—81no—
172C2×C127#53711yescharacteristic
173C2×C127#57611yescharacteristic
174C2×C127#57711yescharacteristic
175D127—41no—
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#474D12711yescharacteristic, complemented
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#424—1271no—
195C2×SD16—1271no—
196C2×SD16—1271no—
197C2×D8—1271no—
198C2×D8—1271no—
199C2×M16—1271no—
200C22×C8—1271no—
201C4⋊Q8C2×D12711yescharacteristic
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×D4C2×D12711yescharacteristic
221#117—1271no—
222#107C2×D12711yescharacteristic
223#107C2×D12711yescharacteristic
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—21no—
239C2×Q8—1271no—
240C2×D4—1271no—
241C2×D4—1271no—
242C2×D4—2541no—
243C2×D4—2541no—
244C2×D4—2541no—
245C2×D4—2541no—
246C2×D4—2541no—
247C2×D4C22×D12711yescharacteristic
248C22×C4—21no—
249C22×C4—1271no—
250C22×C4—2541no—
251C22×C4—2541no—
252SD16—5081no—
253SD16—5081no—
254D8—5081no—
255D8—5081no—
256M16—5081no—
257C2×C8—1271no—
258C2×C8—1271no—
259C2×C8—5081no—
260C2×C8C22×D12711yescharacteristic
261C2×C8C22×D12711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—1271no—
264C4⋊C4—1271no—
265C4⋊C4—1271no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4—2541no—
269C4⋊C4—2541no—
270C4⋊C4C22×D12711yescharacteristic
271C4⋊C4C22×D12711yescharacteristic
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—1271no—
289C23—2541no—
290C23—2541no—
291Q8—81no—
292Q8—2541no—
293Q8—2541no—
294D4—21no—
295D4—21no—
296D4—10161no—
297D4—10161no—
298D4—10161no—
299D4—10161no—
300D4—10161no—
301D4—2541no—
302D4—2541no—
303D4—2541no—
304D4—2541no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—81no—
309C2×C4—1271no—
310C2×C4—1271no—
311C2×C4—1271no—
312C2×C4—10161no—
313C2×C4—10161no—
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—5081no—
329C22—5081no—
330C22—5081no—
331C22—5081no—
332C22—5081no—
333C22—5081no—
334C22—2541no—
335C22—2541no—
336C22—2541no—
337C22C2×D4×D12711yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—5081no—
344C4—5081no—
345C4—2541no—
346C4—2541no—
347C4—2541no—
348C4#1686125511yescharacteristic
349C4#1686125711yescharacteristic
350C2—81no—
351C2—10161no—
352C2—10161no—
353C2—5081no—
354C2#2255045611yescharacteristic
355C2#2255067711yescharacteristic
356C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
hypercenter#474D127
hypocenterC127#3257
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22550500C2semidirect
2. #22550477C2semidirect
3. #22550182C2semidirect
4. #22550179C2semidirect
5. #22550163C2semidirect
6. #22550128C2semidirect
7. #22550123C2semidirect
8. #22550113C2semidirect
9. #22550104C2semidirect
10. #22550092C2semidirect
11. #22550091C2semidirect
12. #22550075C2semidirect
13. #22550070C2semidirect
14. #22549901C2semidirect
15. #474×C127C2semidirect
179. C127#3257semidirect
185. #474D127semidirect

Maximal subgroups

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

subgroupindex
#225505002
#225504772
#225501822
#225501792
#225501632
#225501282
#225501232
#225501132
#225501042
#225500922
#225500912
#225500752
#225500702
#225499012
#474×C1272
#3257127