Finite Group Databasebeta

← #30648096

Subgroups — #30648096

Order 16256.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861256C211yesmaximal, characteristic, complemented
2#22550676C211yesmaximal, characteristic, complemented
3#22550674C211yesmaximal, characteristic, complemented
4#22550452C211yesmaximal, characteristic, complemented
5#22550060C214yesmaximal, complemented
6#22550050C214yesmaximal, complemented
7#22549894C211yesmaximal, characteristic, complemented
8C2×#16861127C211yesmaximal, characteristic, complemented
9#421×C127C211yesmaximal, characteristic, complemented
10#16861256—22no—
11#16861256—22no—
12#16861256—22no—
13#16861256C2214yes—
14C2×#14523248—21no—
15C2×#14523248C2211yescharacteristic
16C2×#14523248C2211yescharacteristic
17C2×#14523247—22no—
18C2×#14523247—21no—
19C22×Dic254—21no—
20C22×Dic254C2211yescharacteristic
21C2×Q8×D127—21no—
22#16861242—21no—
23#16861242C2212yescomplemented
24#16861240—21no—
25#16861239C2212yescomplemented
26#16861237—21no—
27#16861237C2212yescomplemented
28#16861235—21no—
29#16861234C2212yescomplemented
30#16861231—22no—
31#16861229—22no—
32C2×#14523232C2211yescharacteristic
33#16861209C2212yescomplemented
34#16861197C2212yescomplemented
35#16861193—22no—
36#16861189—22no—
37#16861137C2214yescomplemented
38#16861127C2214yescomplemented
39C2×#41×C127C2211yescharacteristic
40C2×M16×C127C2211yescharacteristic
41#117×C127C2211yescharacteristic
42#104×C127C2214yescomplemented
43#14523248—24no—
44#14523248—24no—
45#14523248—21no—
46#14523248—21no—
47#14523248—21no—
48#14523248—21no—
49#14523248—21no—
50#14523248—21no—
51#14523248D412yes—
52#14523248C2312yes—
53#14523247—28no—
54#14523247—24no—
55#14523247—28no—
56#14523247—24no—
57C2×Dic254—22no—
58C2×Dic254—24no—
59C2×Dic254—24no—
60C2×Dic254—21no—
61C2×Dic254—21no—
62C2×Dic254—21no—
63C2×Dic254—21no—
64C2×Dic254D411yescharacteristic
65C2×Dic254D412yes—
66C2×Dic254C2311yescharacteristic
67C2×Dic254C2312yes—
68Q8×D127—24no—
69Q8×D127—24no—
70C2×D508D411yescharacteristic
71C2×D508C2311yescharacteristic
72C2×C22⋊D127—22no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C22×Dic127—22no—
76C22×Dic127D412yes—
77C2×C4×D127—22no—
78C2×C4×D127—21no—
79C2×C4×D127—21no—
80Dic508—22no—
81#14523238—22no—
82#14523237—22no—
83#14523237—22no—
84#14523236—22no—
85#14523235—22no—
86D1016—22no—
87#14523233—22no—
88#14523232—21no—
89#14523232C2312yes—
90C2×C127⋊C8—21no—
91C4⋊Dic127—24no—
92(C2×D127)⋊C4—24no—
93#41×C127—21no—
94#41×C127—21no—
95#41×C127D412yes—
96#41×C127C2312yes—
97C2×Q8×C127D411yescharacteristic
98C2×D4×C127—21no—
99C2×D4×C127D411yescharacteristic
100C22×C4×C127—21no—
101C22×C4×C127C2311yescharacteristic
102M16×C127—21no—
103M16×C127C2312yes—
104C2×C8×C127—21no—
105C4⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127C2312yes—
108C22×D127—21no—
109C22×D127—21no—
110Dic254—28no—
111Dic254—22no—
112Dic254—21no—
113Dic254—22no—
114Dic254—22no—
115Dic254—22no—
116Dic254—21no—
117Dic254—21no—
118Dic254C2×D412yes—
119D508—21no—
120D508—21no—
121D508—21no—
122D508C2×D412yes—
123C22⋊D127—28no—
124C22⋊D127—22no—
125C22⋊D127—22no—
126C22⋊D127—41no—
127C22⋊D127—41no—
128C2×Dic127—28no—
129C2×Dic127—21no—
130C2×Dic127—22no—
131C2×Dic127—22no—
132C2×Dic127—21no—
133C2×Dic127C2×D414yes—
134C4×D127—28no—
135C4×D127—22no—
136C4×D127—22no—
137C4×D127—41no—
138C4×D127—41no—
139C127⋊C8—22no—
140C23×C127—21no—
141C23×C127C2×D411yescharacteristic
142Q8×C127—21no—
143Q8×C127C2×D412yes—
144D4×C127—22no—
145D4×C127—22no—
146D4×C127—21no—
147D4×C127C2×D412yes—
148C2×C4×C127—21no—
149C2×C4×C127—22no—
150C2×C4×C127—22no—
151C2×C4×C127—22no—
152C2×C4×C127C2×D411yescharacteristic
153C2×C4×C127C2×D411yescharacteristic
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2×D411yescharacteristic
156C2×C4×C127C2×D411yescharacteristic
157C2×C4×C127C2411yescharacteristic
158C8×C127—22no—
159C22×C127—22no—
160C22×C127—41no—
161C22×C127—42no—
162C22×C127#12011yescharacteristic
163C22×C127#12011yescharacteristic
164C22×C127C22×D411yescharacteristic
165C2×D127—22no—
166C2×D127—42no—
167C2×D127—42no—
168C2×D127—41no—
169C4×C127—22no—
170C4×C127—42no—
171C4×C127#12011yescharacteristic
172C4×C127#12011yescharacteristic
173C4×C127C22×D411yescharacteristic
174C4×C127C22×D411yescharacteristic
175Dic127—22no—
176Dic127—24no—
177Dic127—41no—
178C2×C127—21no—
179C2×C127—21no—
180C2×C127—21no—
181C2×C127—42no—
182C2×C127C2×#12011yescharacteristic
183D127—42no—
184D127—81no—
185#3020—1271nomaximal, Sylow
186C127#302011yescharacteristic, complemented, Sylow
187C2×2−1+4—1271no—
188#575—1271no—
189#575—1271no—
190#536—1271no—
191#456—1274no—
192#454—1274no—
193#421—1271no—
194#421D12711yescharacteristic, complemented
195C2×#101—1271no—
1962−1+4—1274no—
1972−1+4—2542no—
1982−1+4—2542no—
1992−1+4—2542no—
200C2×#41—1271no—
201C2×#41—1271no—
202C2×#41—2541no—
203C2×#41—2542no—
204C2×#41—2541no—
205C2×#41C2×D12711yescharacteristic
206C22×Q8—1271no—
207C22×Q8—2541no—
208C22×Q8—2541no—
209#137—1272no—
210#137—1272no—
211#137—2541no—
212#137—2541no—
213AGL(1,ℤ/8)—1272no—
214AGL(1,ℤ/8)—1272no—
215AGL(1,ℤ/8)—2541no—
216AGL(1,ℤ/8)—2541no—
217#135—2542no—
218#135—2542no—
219C2×M16—1271no—
220C2×M16C2×D12711yescharacteristic
221C4⋊Q8—1272no—
222#124—1272no—
223#123—2542no—
224C22⋊Q8—2542no—
225#117C2×D12711yescharacteristic
226#104—1274no—
227#104C2×D12714yescomplemented
228#101—1274no—
229#41—21no—
230#41—21no—
231#41—1272no—
232#41—1272no—
233#41—2544no—
234#41—2544no—
235#41—2548no—
236#41—2541no—
237#41—2541no—
238#41—2541no—
239#41—2541no—
240#41—2548no—
241#41—2544no—
242#41—2544no—
243#41—2541no—
244#41—2541no—
245#41D50812yes—
246#41C22×D12712yes—
247C2×Q8—1272no—
248C2×Q8—1272no—
249C2×Q8—1271no—
250C2×Q8—1271no—
251C2×Q8—2544no—
252C2×Q8—2542no—
253C2×Q8—2544no—
254C2×Q8—2544no—
255C2×Q8—2544no—
256C2×Q8—2541no—
257C2×Q8—2541no—
258C2×Q8—2541no—
259C2×Q8—2541no—
260C2×Q8D50811yescharacteristic
261C2×D4—21no—
262C2×D4—1271no—
263C2×D4—1271no—
264C2×D4—2541no—
265C2×D4—2542no—
266C2×D4—2541no—
267C2×D4D50811yescharacteristic
268C22×C4—21no—
269C22×C4—1272no—
270C22×C4—2542no—
271C22×C4—2541no—
272C22×C4—2542no—
273C22×C4—2541no—
274C22×C4C22×D12711yescharacteristic
275Q16—2542no—
276Q16—2542no—
277SD16—2542no—
278SD16—2542no—
279SD16—2542no—
280SD16—2542no—
281D8—2542no—
282D8—2542no—
283M16—21no—
284M16—1272no—
285M16—2541no—
286M16C22×D12712yes—
287C2×C8—21no—
288C2×C8—2541no—
289C4⋊C4—21no—
290C4⋊C4—2544no—
291C22⋊C4—21no—
292C22⋊C4—2544no—
293C42C22×D12712yes—
294C23—21no—
295C23—2541no—
296C23—2541no—
297C23D4×D12711yescharacteristic
298Q8—21no—
299Q8—1272no—
300Q8—2548no—
301Q8—2541no—
302Q8—2541no—
303Q8—2542no—
304Q8—2542no—
305Q8—2542no—
306Q8—2542no—
307Q8—2541no—
308Q8C2×D50812yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—1272no—
313D4—5081no—
314D4—5081no—
315D4—2548no—
316D4—2542no—
317D4—2542no—
318D4—2541no—
319D4—2541no—
320D4—2541no—
321D4C2×D50812yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—1274no—
327C2×C4—5081no—
328C2×C4—5081no—
329C2×C4—2548no—
330C2×C4—2541no—
331C2×C4—2548no—
332C2×C4—2542no—
333C2×C4—2542no—
334C2×C4—2541no—
335C2×C4—2542no—
336C2×C4—2542no—
337C2×C4D4×D12711yescharacteristic
338C2×C4D4×D12711yescharacteristic
339C2×C4D4×D12711yescharacteristic
340C2×C4C2×D50811yescharacteristic
341C2×C4C2×D50811yescharacteristic
342C2×C4C23×D12711yescharacteristic
343C8—22no—
344C8—2542no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—5082no—
349C22—5082no—
350C22—5081no—
351C22—2542no—
352C22#1686117911yescharacteristic
353C22#1686117911yescharacteristic
354C22C2×D4×D12711yescharacteristic
355C4—22no—
356C4—42no—
357C4—5081no—
358C4—2544no—
359C4—2542no—
360C4#1686117911yescharacteristic
361C4#1686117911yescharacteristic
362C4C2×D4×D12711yescharacteristic
363C4C22×D50811yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—10161no—
369C2—5082no—
370C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#421×C127C2
socleC2×C127C2×#120
hypercenter#421D127
hypocenterC127#3020
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#16861256C2semidirect
2. #22550676C2semidirect
3. #22550674C2semidirect
4. #22550452C2semidirect
5. #22550060C2semidirect
6. #22550050C2semidirect
7. #22549894C2semidirect
8. C2×#16861127C2semidirect
9. #421×C127C2semidirect
23. #16861242C22semidirect
25. #16861239C22semidirect
27. #16861237C22semidirect
29. #16861234C22semidirect
33. #16861209C22semidirect
34. #16861197C22semidirect
37. #16861137C22semidirect
38. #16861127C22semidirect
42. #104×C127C22semidirect
186. C127#3020semidirect
194. #421D127semidirect
227. #104C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
#225506762
#225506742
#225504522
#225500602
#225500502
#225498942
C2×#168611272
#421×C1272
#3020127