Finite Group Databasebeta

← #30649015

Subgroups — #30649015

Order 16256.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550479C211yesmaximal, characteristic
2#22550354C211yesmaximal, characteristic, complemented
3#22550201C211yesmaximal, characteristic, complemented
4#22550197C211yesmaximal, characteristic, complemented
5#22550185C211yesmaximal, characteristic, complemented
6#22550181C211yesmaximal, characteristic, complemented
7#22550074C211yesmaximal, characteristic, complemented
8#22550074C211yesmaximal, characteristic, complemented
9#22550015C211yesmaximal, characteristic, complemented
10#22549979C211yesmaximal, characteristic, complemented
11#22549977C211yesmaximal, characteristic, complemented
12SD16×Dic127C211yesmaximal, characteristic, complemented
13#22549961C211yesmaximal, characteristic, complemented
14#22549934C211yesmaximal, characteristic, complemented
15#448×C127C211yesmaximal, characteristic, complemented
16C2×#14523235C2211yescharacteristic
17C2×#14523235C2211yescharacteristic
18#16861205C2211yescharacteristic
19#16861200C2211yescharacteristic
20#16861193—21no—
21#16861191—21no—
22#16861183C2211yescharacteristic
23#16861181—21no—
24#16861180—21no—
25Q8×Dic127—21no—
26Q8×Dic127—21no—
27Q8×Dic127C2211yescharacteristic
28Q8×Dic127C2211yescharacteristic
29Q8×Dic127C2211yescharacteristic
30Q8×Dic127C2211yescharacteristic
31D4×Dic127C2211yescharacteristic
32#16861171C2211yescharacteristic
33#16861168—21no—
34#16861168—21no—
35#16861168C2211yescharacteristic
36#16861168C2211yescharacteristic
37#16861143C2211yescharacteristic
38#16861142C2211yescharacteristic
39#16861141C2211yescharacteristic
40#16861140C2211yescharacteristic
41Dic127⋊C8C2211yescharacteristic
42Dic127⋊C8C2211yescharacteristic
43#16861133C2211yescharacteristic
44#16861133C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50#16861119C2211yescharacteristic
51C8×Dic127C2211yescharacteristic
52C22×Q8×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic
56#103×C127C2211yescharacteristic
57#103×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic
59C2×D508C2311yescharacteristic
60C2×C22⋊D127—21no—
61C2×C22⋊D127—21no—
62C2×C22⋊D127—21no—
63C22×Dic127—21no—
64C2×C4×D127—21no—
65#14523235—22no—
66#14523235—22no—
67C2×C127⋊C8C2311yescharacteristic
68C2×C127⋊C8C2311yescharacteristic
69C4⋊Dic127—21no—
70C4⋊Dic127—21no—
71C4⋊Dic127—21no—
72C4⋊Dic127—21no—
73C4⋊Dic127—21no—
74C4⋊Dic127D411yescharacteristic
75C4⋊Dic127D411yescharacteristic
76C4⋊Dic127C2311yescharacteristic
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80C22⋊Dic127—21no—
81C22⋊Dic127—21no—
82C22⋊Dic127D411yescharacteristic
83C22⋊Dic127D411yescharacteristic
84(C2×D127)⋊C4—21no—
85(C2×D127)⋊C4—21no—
86C4×Dic127—21no—
87C4×Dic127—21no—
88C4×Dic127—21no—
89C4×Dic127—21no—
90C4×Dic127—21no—
91C4×Dic127C2311yescharacteristic
92C4×Dic127C2311yescharacteristic
93C2×Q8×C127—22no—
94C2×Q8×C127—22no—
95C2×Q8×C127—21no—
96C2×Q8×C127C2311yescharacteristic
97C2×Q8×C127C2311yescharacteristic
98C2×D4×C127—21no—
99C2×D4×C127C2311yescharacteristic
100C22×C4×C127—21no—
101C22×C4×C127C2311yescharacteristic
102SD16×C127—22no—
103SD16×C127—22no—
104C2×C8×C127C2311yescharacteristic
105C2×C8×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C22⋊C4×C127—21no—
108C22×D127—21no—
109D508—21no—
110D508—21no—
111C22⋊D127—81no—
112C22⋊D127—81no—
113C22⋊D127—81no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—81no—
118C2×Dic127C2×D411yescharacteristic
119C2×Dic127C2×D411yescharacteristic
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C4×D127—81no—
123C127⋊C8—21no—
124C127⋊C8—21no—
125C23×C127—21no—
126C23×C127C2×D411yescharacteristic
127Q8×C127—24no—
128Q8×C127—21no—
129Q8×C127—21no—
130Q8×C127#4112yes—
131Q8×C127#4112yes—
132D4×C127—21no—
133D4×C127—21no—
134D4×C127—81no—
135C2×C4×C127—24no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127C2×D411yescharacteristic
142C2×C4×C127C2411yescharacteristic
143C8×C127—21no—
144C8×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127C22×D411yescharacteristic
152C2×D127—41no—
153C2×D127—41no—
154C2×D127—41no—
155C4×C127—22no—
156C4×C127—22no—
157C4×C127—21no—
158C4×C127—41no—
159C4×C127C2×#4111yescharacteristic
160C4×C127C2×#4111yescharacteristic
161Dic127—21no—
162Dic127—21no—
163Dic127—21no—
164Dic127—21no—
165Dic127—41no—
166Dic127—41no—
167Dic127—41no—
168C2×C127—41no—
169C2×C127—81no—
170C2×C127#52511yescharacteristic
171C2×C127#57711yescharacteristic
172C2×C127#57711yescharacteristic
173D127—81no—
174#3126—1271nomaximal, Sylow
175C127#312611yescharacteristic, complemented, Sylow
176#540—1271no—
177#519—1271no—
178#482—1271no—
179#482—1271no—
180#479—1271no—
181#479—1271no—
182#460—1271no—
183#460—1271no—
184#448—1271no—
185#448D12711yescharacteristic, complemented
186#440—1271no—
187#440—1271no—
188C4×SD16—1271no—
189C4×SD16—1271no—
190#433—1271no—
191C22×Q8C2×D12711yescharacteristic
192C2×SD16—1271no—
193C2×SD16—1271no—
194C2×SD16C2×D12711yescharacteristic
195C2×SD16C2×D12711yescharacteristic
196#127—1271no—
197#125—1271no—
198#123—2541no—
199#123—2541no—
200#121—1271no—
201#121—2541no—
202#121—2541no—
203#121C2×D12711yescharacteristic
204C4×Q8—1271no—
205C4×Q8—1271no—
206C4×Q8—1271no—
207C4×Q8—1271no—
208C4×Q8—2541no—
209C4×Q8—2541no—
210C4×D4—1271no—
211C4×D4—1271no—
212#117—1271no—
213#117—1271no—
214#117—2541no—
215#117—2541no—
216#107—1271no—
217#107—1271no—
218#106—1271no—
219#106—1271no—
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222#103—1271no—
223#103—1271no—
224#103C2×D12711yescharacteristic
225#103C2×D12711yescharacteristic
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230C22⋊C8—1271no—
231C22⋊C8C2×D12711yescharacteristic
232#97—1271no—
233C4×C8—1271no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D12711yescharacteristic
238C2×Q8C22×D12711yescharacteristic
239C2×D4—21no—
240C2×D4—1271no—
241C2×D4—2541no—
242C2×D4—2541no—
243C2×D4—2541no—
244C2×D4C22×D12711yescharacteristic
245C22×C4—21no—
246C22×C4—2541no—
247C22×C4—2541no—
248C22×C4C22×D12711yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—2542no—
252SD16—2542no—
253C2×C8—1271no—
254C2×C8—1271no—
255C2×C8C22×D12711yescharacteristic
256C2×C8C22×D12711yescharacteristic
257C4⋊C4—1271no—
258C4⋊C4—1271no—
259C4⋊C4—1271no—
260C4⋊C4—1271no—
261C4⋊C4—1271no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4—2541no—
265C4⋊C4—2541no—
266C4⋊C4—2541no—
267C4⋊C4—2541no—
268C4⋊C4C22×D12711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—1271no—
271C22⋊C4—1271no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C22⋊C4—2541no—
276C42—1271no—
277C42—1271no—
278C42—2541no—
279C42—2541no—
280C42—2541no—
281C42—2541no—
282C42—2541no—
283C23—21no—
284C23—2541no—
285C23D4×D12711yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1452324712yes—
290Q8#1452324712yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—10161no—
295D4—10161no—
296D4—10161no—
297D4—2541no—
298D4—2541no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—1271no—
306C2×C4—1271no—
307C2×C4—1271no—
308C2×C4—1271no—
309C2×C4—10161no—
310C2×C4—10161no—
311C2×C4—2541no—
312C2×C4—2541no—
313C2×C4—2541no—
314C2×C4D4×D12711yescharacteristic
315C2×C4C23×D12711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2541no—
319C8—2541no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—5081no—
327C22—5081no—
328C22—5081no—
329C22C2×D4×D12711yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—5081no—
335C4—5081no—
336C4—5081no—
337C4—2541no—
338C4—2541no—
339C4—2541no—
340C4—2541no—
341C4C2×#1452324711yescharacteristic
342C4C2×#1452324711yescharacteristic
343C2—41no—
344C2—81no—
345C2—10161no—
346C2#2255038211yescharacteristic
347C2#2255068211yescharacteristic
348C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#448×C127C2
socleC22×C127C22×D4
prime radical#22550479C2
hypercenter#448D127
hypocenterC127#3126
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #22550354C2semidirect
3. #22550201C2semidirect
4. #22550197C2semidirect
5. #22550185C2semidirect
6. #22550181C2semidirect
7. #22550074C2semidirect
8. #22550074C2semidirect
9. #22550015C2semidirect
10. #22549979C2semidirect
11. #22549977C2semidirect
12. SD16×Dic127C2semidirect
13. #22549961C2semidirect
14. #22549934C2semidirect
15. #448×C127C2semidirect
175. C127#3126semidirect
185. #448D127semidirect

Maximal subgroups

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

subgroupindex
#225504792
#225503542
#225502012
#225501972
#225501852
#225501812
#225500742
#225500152
#225499792
#225499772
SD16×Dic1272
#225499612
#225499342
#448×C1272
#3126127