Finite Group Databasebeta

← #30647099

Subgroups — #30647099

Order 16256.

524 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861199C211yesmaximal, characteristic, complemented
2C2×#16861195C211yesmaximal, characteristic
3#22549780C211yesmaximal, characteristic, complemented
4#22549780C211yesmaximal, characteristic, complemented
5#22549776C211yesmaximal, characteristic
6#22549776C211yesmaximal, characteristic
7#22549770C211yesmaximal, characteristic, complemented
8#22549769C211yesmaximal, characteristic, complemented
9#22549768C211yesmaximal, characteristic, complemented
10#22549767C211yesmaximal, characteristic, complemented
11#22549766C211yesmaximal, characteristic, complemented
12#22549765C211yesmaximal, characteristic, complemented
13#22549764C211yesmaximal, characteristic, complemented
14C42×Dic127C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C22×Dic254C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18#16861199—24no—
19#16861195—24no—
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×(C2×D127)⋊C4C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C2×C4×Dic127C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C2×C4×Dic127C2211yescharacteristic
39C2×C4×Dic127C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C2×C4⋊C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C42×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53C23×D127C2311yescharacteristic
54C2×Dic254—22no—
55C2×Dic254—22no—
56C2×Dic254—22no—
57C2×C22⋊D127—22no—
58C2×C22⋊D127—22no—
59C2×C22⋊D127—22no—
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C4⋊Dic127—22no—
68Dic127⋊C4—22no—
69Dic127⋊C4—22no—
70Dic127⋊C4—22no—
71Dic127⋊C4—22no—
72C22⋊Dic127—22no—
73C22⋊Dic127—22no—
74C22⋊Dic127—22no—
75(C2×D127)⋊C4—22no—
76(C2×D127)⋊C4—22no—
77(C2×D127)⋊C4—22no—
78(C2×D127)⋊C4—22no—
79(C2×D127)⋊C4—22no—
80(C2×D127)⋊C4—22no—
81C4×Dic127—22no—
82C4×Dic127—22no—
83C4×Dic127—22no—
84C4×Dic127—22no—
85C4×Dic127—22no—
86C4×Dic127D414yes—
87C24×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C4⋊C4×C127—22no—
95C22⋊C4×C127—22no—
96C22⋊C4×C127—22no—
97C22⋊C4×C127—22no—
98C42×C127—22no—
99C22×D127—21no—
100C22×D127—21no—
101C22×D127—21no—
102C22×D127—21no—
103C22×D127—21no—
104C22×D127—21no—
105C22×D127—21no—
106Dic254—44no—
107C22⋊D127—44no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127C2×D412yes—
124C2×Dic127C2×D412yes—
125C2×Dic127#4112yes—
126C2×Dic127#4112yes—
127C2×Dic127#4112yes—
128C2×Dic127#4112yes—
129C23×C127—21no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127—21no—
136C23×C127C2411yescharacteristic
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127C2×D412yes—
153C2×C4×C127#4112yes—
154C2×C4×C127#4112yes—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127C22×D411yescharacteristic
163C22×C127C2×#4111yescharacteristic
164C22×C127C2×#4111yescharacteristic
165C22×C127C2×#4111yescharacteristic
166C22×C127C2×#4111yescharacteristic
167C22×C127C2×#4111yescharacteristic
168C22×C127C2×#4111yescharacteristic
169C2×D127—41no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C4×C127—22no—
177C4×C127—22no—
178C4×C127—22no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182Dic127—22no—
183Dic127—22no—
184Dic127—22no—
185Dic127—41no—
186Dic127—41no—
187Dic127—41no—
188Dic127#12414yes—
189C2×C127—81no—
190C2×C127C2×#12411yescharacteristic
191C2×C127#52911yescharacteristic
192C2×C127#52911yescharacteristic
193C2×C127#52911yescharacteristic
194C2×C127#52911yescharacteristic
195C2×C127#53211yescharacteristic
196C2×C127#53211yescharacteristic
197D127—81no—
198#2861—1271nomaximal, Sylow
199C127#286111yescharacteristic, complemented, Sylow
200C2×#125—1271no—
201C2×#124—1271no—
202#391—1271no—
203#391—1271no—
204#390—1271no—
205#390—1271no—
206#388—1271no—
207#388—1271no—
208#388—1271no—
209#388—1271no—
210#388—1271no—
211#388—1271no—
212#388—1271no—
213#388D12711yescharacteristic, complemented
214C43—1271no—
215C22×Q8—1271no—
216C22×D4—1271no—
217#125—2544no—
218#124—2544no—
219C2×C4⋊C4—1271no—
220C2×C4⋊C4—1271no—
221C2×C4⋊C4—1271no—
222C2×C4⋊C4—1271no—
223C2×C4⋊C4—1271no—
224C2×C4⋊C4C2×D12711yescharacteristic
225C2×C22⋊C4—1271no—
226C2×C22⋊C4—1271no—
227C2×C22⋊C4—1271no—
228C2×C22⋊C4—1271no—
229C2×C22⋊C4—1271no—
230C2×C22⋊C4—1271no—
231C2×C22⋊C4—1271no—
232C2×C22⋊C4—1271no—
233C2×C22⋊C4—1271no—
234C2×C22⋊C4C2×D12711yescharacteristic
235C2×C22⋊C4C2×D12711yescharacteristic
236C2×C22⋊C4C2×D12711yescharacteristic
237C2×C42—1271no—
238C2×C42—1271no—
239C2×C42—1271no—
240C2×C42—1271no—
241C2×C42—1271no—
242C2×C42—1271no—
243C2×C42C2×D12711yescharacteristic
244#95—1271no—
245#95—1271no—
246#95—1271no—
247#95—1271no—
248#95—1271no—
249#95—1271no—
250#95C2×D12711yescharacteristic
251#95C2×D12711yescharacteristic
252C24—1271no—
253C24C22×D12711yescharacteristic
254C2×Q8—2542no—
255C2×Q8—2542no—
256C2×Q8—2542no—
257C2×D4—2542no—
258C2×D4—2542no—
259C2×D4—2542no—
260C22×C4—1271no—
261C22×C4—1271no—
262C22×C4—1271no—
263C22×C4—1271no—
264C22×C4—1271no—
265C22×C4—1271no—
266C22×C4—1271no—
267C22×C4C22×D12711yescharacteristic
268C22×C4C22×D12711yescharacteristic
269C22×C4C22×D12711yescharacteristic
270C22×C4C22×D12711yescharacteristic
271C22×C4C22×D12711yescharacteristic
272C22×C4C22×D12711yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—2542no—
275C4⋊C4—2542no—
276C4⋊C4—2542no—
277C4⋊C4—2542no—
278C4⋊C4—2542no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—2542no—
283C22⋊C4—2542no—
284C22⋊C4—2542no—
285C22⋊C4—2542no—
286C22⋊C4—2542no—
287C22⋊C4—2542no—
288C22⋊C4—2542no—
289C22⋊C4—2542no—
290C22⋊C4—2542no—
291C42—22no—
292C42—1274no—
293C42—2542no—
294C42—2542no—
295C42—2542no—
296C42—2542no—
297C42—2542no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—2541no—
306C23—2541no—
307C23—2541no—
308C23—2541no—
309C23—2541no—
310C23—2541no—
311C23—2541no—
312C23C23×D12711yescharacteristic
313Q8—5084no—
314D4—5084no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—1272no—
331C2×C4—1272no—
332C2×C4—1272no—
333C2×C4—1272no—
334C2×C4—1272no—
335C2×C4—1272no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4D4×D12712yes—
352C2×C4#1452324712yes—
353C2×C4#1452324812yes—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—5081no—
365C22—5081no—
366C22—5081no—
367C22—5081no—
368C22C2×D4×D12711yescharacteristic
369C22#41×D12711yescharacteristic
370C22#41×D12711yescharacteristic
371C22#41×D12711yescharacteristic
372C22#41×D12711yescharacteristic
373C22C2×#1452324711yescharacteristic
374C22C2×#1452324811yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1274no—
382C4—5081no—
383C4—5081no—
384C4—5081no—
385C4—2542no—
386C4—2542no—
387C4—2542no—
388C2—81no—
389C2—10161no—
390C2#124×D12711yescharacteristic
391C2#2255040311yescharacteristic
392C2#2255040411yescharacteristic
393C2#2255040611yescharacteristic
394C2#2255040911yescharacteristic
395C2#2255042111yescharacteristic
396C2#2255042311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#388×C127C2
prime radicalC22×C22⋊D127C22
hypercenter#388D127
hypocenterC127#2861
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#16861199C2semidirect
3. #22549780C2semidirect
4. #22549780C2semidirect
7. #22549770C2semidirect
8. #22549769C2semidirect
9. #22549768C2semidirect
10. #22549767C2semidirect
11. #22549766C2semidirect
12. #22549765C2semidirect
13. #22549764C2semidirect
14. C42×Dic127C2semidirect
15. #388×C127C2semidirect
199. C127#2861semidirect
213. #388D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611992
C2×#168611952
#225497802
#225497762
#225497702
#225497692
#225497682
#225497672
#225497662
#225497652
#225497642
C42×Dic1272
#388×C1272
#2861127