Finite Group Databasebeta

← #30650291

Subgroups — #30650291

Order 16256.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861238C211yesmaximal, characteristic, complemented
2C2×#16861228C211yesmaximal, characteristic, complemented
3#22550534C211yesmaximal, characteristic, complemented
4#22550530C211yesmaximal, characteristic, complemented
5#22550271C211yesmaximal, characteristic, complemented
6#22550121C211yesmaximal, characteristic, complemented
7#22550116C211yesmaximal, characteristic, complemented
8#22550087C211yesmaximal, characteristic, complemented
9#22550039C211yesmaximal, characteristic, complemented
10#22550035C211yesmaximal, characteristic, complemented
11#22550033C211yesmaximal, characteristic, complemented
12#22550018C211yesmaximal, characteristic, complemented
13#22549976C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×#14523247C2211yescharacteristic
18C2×#14523247C2211yescharacteristic
19C2×Q8×D127C2211yescharacteristic
20#16861238—24no—
21#16861228—24no—
22C2×Dic508C2211yescharacteristic
23C2×Dic508C2211yescharacteristic
24C2×#14523238C2211yescharacteristic
25C2×#14523238C2211yescharacteristic
26C2×#14523236C2211yescharacteristic
27C2×#14523236C2211yescharacteristic
28C2×#14523236C2211yescharacteristic
29C2×#14523231C2211yescharacteristic
30C2×C8×D127C2211yescharacteristic
31#16861209C2211yescharacteristic
32#16861195—21no—
33#16861193—21no—
34#16861193—21no—
35#16861189C2211yescharacteristic
36#16861189C2211yescharacteristic
37#16861186—21no—
38#16861185C2211yescharacteristic
39#16861183—21no—
40#16861182—21no—
41#16861180C2211yescharacteristic
42C4×Dic254C2211yescharacteristic
43C4×D508C2211yescharacteristic
44(C22⋊D127)⋊C4—21no—
45C2×C4⋊Dic127—21no—
46#16861142C2211yescharacteristic
47#16861134C2211yescharacteristic
48#16861134C2211yescharacteristic
49#16861132C2211yescharacteristic
50#16861130C2211yescharacteristic
51(C2×D127)⋊C8C2211yescharacteristic
52(C2×D127)⋊C8C2211yescharacteristic
53#16861120C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic
57C4×D4×C127C2211yescharacteristic
58C4⋊C8×C127C2211yescharacteristic
59#103×C127C2211yescharacteristic
60#102×C127C2211yescharacteristic
61#14523247—22no—
62#14523247—22no—
63#14523247—22no—
64#14523247—41no—
65#14523247—41no—
66#14523247—41no—
67C2×Dic254—21no—
68C2×Dic254C2311yescharacteristic
69C2×Dic254C2311yescharacteristic
70C2×Dic254C2311yescharacteristic
71Q8×D127—22no—
72Q8×D127—41no—
73C2×D508D411yescharacteristic
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77C2×C22⋊D127—21no—
78C22×Dic127—21no—
79C22×Dic127—21no—
80C22×Dic127—21no—
81C2×C4×D127—21no—
82C2×C4×D127C2311yescharacteristic
83C2×C4×D127C2311yescharacteristic
84Dic508—22no—
85Dic508—22no—
86#14523238—22no—
87#14523238—41no—
88#14523236—22no—
89#14523236—41no—
90#14523236D414yescomplemented
91#14523231—22no—
92C2×C127⋊C8C2311yescharacteristic
93C2×C127⋊C8C2311yescharacteristic
94C8×D127—22no—
95C4⋊Dic127—21no—
96C4⋊Dic127—21no—
97C4⋊Dic127—21no—
98C4⋊Dic127—21no—
99C4⋊Dic127D411yescharacteristic
100C4⋊Dic127C2311yescharacteristic
101Dic127⋊C4—21no—
102Dic127⋊C4—21no—
103C22⋊Dic127—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108(C2×D127)⋊C4D411yescharacteristic
109C4×Dic127—21no—
110C2×Q8×C127C2311yescharacteristic
111C2×D4×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C22×C4×C127—21no—
114SD16×C127—22no—
115D8×C127—22no—
116C2×C8×C127C2311yescharacteristic
117C2×C8×C127C2311yescharacteristic
118C4⋊C4×C127C2311yescharacteristic
119C22⋊C4×C127—21no—
120C22⋊C4×C127—21no—
121C22⋊C4×C127—21no—
122C42×C127C2311yescharacteristic
123C22×D127C2×D411yescharacteristic
124C22×D127C2×D411yescharacteristic
125Dic254—21no—
126Dic254—21no—
127Dic254—21no—
128Dic254—21no—
129Dic254—21no—
130Dic254—42no—
131Dic254C2×D412yes—
132D508—41no—
133C22⋊D127—42no—
134C22⋊D127—42no—
135C22⋊D127—42no—
136C22⋊D127—42no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—42no—
142C2×Dic127—42no—
143C2×Dic127—42no—
144C2×Dic127C2×D411yescharacteristic
145C2×Dic127C2×D411yescharacteristic
146C4×D127—21no—
147C4×D127—21no—
148C4×D127—21no—
149C4×D127—21no—
150C4×D127—42no—
151C127⋊C8—21no—
152C127⋊C8C2×D412yes—
153C23×C127—21no—
154C23×C127—21no—
155Q8×C127—21no—
156Q8×C127—21no—
157D4×C127—21no—
158D4×C127—21no—
159D4×C127—21no—
160D4×C127C2×D412yes—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—42no—
164C2×C4×C127C2×D411yescharacteristic
165C2×C4×C127C2×D411yescharacteristic
166C2×C4×C127C2411yescharacteristic
167C8×C127—21no—
168C8×C127—21no—
169C22×C127—22no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127—41no—
174C22×C127—41no—
175C22×C127C22×D411yescharacteristic
176C2×D127—21no—
177C2×D127—21no—
178C2×D127—21no—
179C2×D127—21no—
180C2×D127—21no—
181C2×D127—21no—
182C4×C127—21no—
183C4×C127—21no—
184C4×C127—41no—
185C4×C127—41no—
186C4×C127C22×D411yescharacteristic
187C4×C127H(5,𝔽2)11yescharacteristic
188Dic127—21no—
189Dic127—21no—
190Dic127—22no—
191Dic127—41no—
192Dic127—41no—
193Dic127—41no—
194C2×C127—42no—
195C2×C127—81no—
196C2×C127D4211yescharacteristic
197C2×C127#57511yescharacteristic
198C2×C127#57811yescharacteristic
199D127—41no—
200D127—41no—
201#3269—1271nomaximal, Sylow
202C127#326911yescharacteristic, complemented, Sylow
203C2×#137—1271no—
204C2×#135—1271no—
205#548—1271no—
206#547—1271no—
207#497—1271no—
208#468—1271no—
209#467—1271no—
210#463D12711yescharacteristic, complemented
211#462—1271no—
212#452—1271no—
213#452—1271no—
214#451—1271no—
215#449—1271no—
216#440—1271no—
217#435—1271no—
218C2×#41—1271no—
219C2×#41—1271no—
220C2×#41—1271no—
221C22×Q8—1271no—
222#137—2544no—
223#135—2544no—
224C2×Q16—1271no—
225C2×Q16—1271no—
226C2×Q16—1271no—
227C2×Q16—1271no—
228C2×SD16—1271no—
229C2×SD16—1271no—
230C2×SD16—1271no—
231C2×SD16C2×D12711yescharacteristic
232C2×D8C2×D12711yescharacteristic
233C2×M16—1271no—
234C22×C8—1271no—
235C4⋊Q8—1271no—
236#124—2541no—
237#124C2×D12711yescharacteristic
238#123—2541no—
239#123—2541no—
240C22⋊Q8—1271no—
241C22⋊Q8—1271no—
242C22⋊Q8—1271no—
243C22⋊Q8—2541no—
244#121—1271no—
245#121—2541no—
246#121—2541no—
247C4×Q8—1271no—
248C4×D4—1271no—
249C4×D4—2541no—
250C4×D4C2×D12711yescharacteristic
251C2×C4⋊C4—2541no—
252#107—1271no—
253C4⋊C8C2×D12711yescharacteristic
254#103—1271no—
255#103—1271no—
256#103—1271no—
257#103C2×D12711yescharacteristic
258#102—1271no—
259#102C2×D12711yescharacteristic
260C22⋊C8—1271no—
261C22⋊C8—1271no—
262#97—1271no—
263#41—5081no—
264#41—5081no—
265#41—5081no—
266#41—2542no—
267#41—2542no—
268#41—2542no—
269C2×Q8—1271no—
270C2×Q8—1271no—
271C2×Q8—1271no—
272C2×Q8—5081no—
273C2×Q8—2541no—
274C2×Q8—2542no—
275C2×Q8C22×D12711yescharacteristic
276C2×D4—1271no—
277C2×D4—2541no—
278C2×D4—2541no—
279C2×D4—2541no—
280C2×D4—2541no—
281C2×D4C22×D12711yescharacteristic
282C2×D4C22×D12711yescharacteristic
283C22×C4—21no—
284C22×C4—1271no—
285C22×C4—1271no—
286C22×C4—2541no—
287C22×C4—2541no—
288C22×C4—2541no—
289C22×C4—2541no—
290Q16—5081no—
291Q16—2542no—
292Q16—2542no—
293Q16—2542no—
294SD16—22no—
295SD16—1274no—
296SD16—5081no—
297SD16—2542no—
298D8—22no—
299M16—2542no—
300C2×C8—1271no—
301C2×C8—1271no—
302C2×C8—2542no—
303C2×C8C22×D12711yescharacteristic
304C2×C8C22×D12711yescharacteristic
305C4⋊C4—1271no—
306C4⋊C4—1271no—
307C4⋊C4—2541no—
308C4⋊C4—2541no—
309C4⋊C4—2541no—
310C4⋊C4—2541no—
311C4⋊C4—2541no—
312C4⋊C4—2541no—
313C4⋊C4C22×D12711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—1271no—
318C22⋊C4—1271no—
319C22⋊C4—2541no—
320C22⋊C4—2541no—
321C22⋊C4—2541no—
322C22⋊C4—2541no—
323C42—2541no—
324C42C22×D12711yescharacteristic
325C23—21no—
326C23—21no—
327C23—1271no—
328C23—1271no—
329Q8—21no—
330Q8—21no—
331Q8—1272no—
332Q8—5082no—
333Q8—2541no—
334Q8—2541no—
335Q8—2541no—
336Q8—2541no—
337Q8—2541no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—5081no—
342D4—5082no—
343D4—5082no—
344D4—5082no—
345D4—5082no—
346D4D4×D12712yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—1271no—
351C2×C4—1271no—
352C2×C4—5082no—
353C2×C4—5082no—
354C2×C4—5082no—
355C2×C4—5082no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4—2541no—
364C2×C4D4×D12711yescharacteristic
365C2×C4D4×D12711yescharacteristic
366C2×C4C23×D12711yescharacteristic
367C8—21no—
368C8—21no—
369C8—1272no—
370C8—2541no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—2541no—
378C22—2541no—
379C22—2541no—
380C22—2541no—
381C22—2541no—
382C22—2541no—
383C22C2×D4×D12711yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—5081no—
389C4—5081no—
390C4—5081no—
391C4—2541no—
392C4—2541no—
393C4—2542no—
394C4C2×D4×D12711yescharacteristic
395C4#1686125511yescharacteristic
396C2—42no—
397C2—81no—
398C2—5081no—
399C2—5081no—
400C2#2255051011yescharacteristic
401C2#2255067211yescharacteristic
402C2#2255069011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3269
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861238C2semidirect
2. C2×#16861228C2semidirect
3. #22550534C2semidirect
4. #22550530C2semidirect
5. #22550271C2semidirect
6. #22550121C2semidirect
7. #22550116C2semidirect
8. #22550087C2semidirect
9. #22550039C2semidirect
10. #22550035C2semidirect
11. #22550033C2semidirect
12. #22550018C2semidirect
13. #22549976C2semidirect
14. #22549946C2semidirect
15. #463×C127C2semidirect
90. #14523236D4semidirect
202. C127#3269semidirect
210. #463D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612382
C2×#168612282
#225505342
#225505302
#225502712
#225501212
#225501162
#225500872
#225500392
#225500352
#225500332
#225500182
#225499762
#225499462
#463×C1272
#3269127