Finite Group Databasebeta

← #30642071

Subgroups — #30642071

Order 16256.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523247C211yesmaximal, characteristic, complemented
2C2×#16861188C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
4#22549791C212yesmaximal, complemented
5#22549784C212yesmaximal, complemented
6#22549781C211yesmaximal, characteristic, complemented
7#22549779C211yesmaximal, characteristic, complemented
8#22549756C212yesmaximal, complemented
9#22549748C212yesmaximal, complemented
10#22549733C211yesmaximal, characteristic, complemented
11C2×#121×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—24no—
13C2×#14523247—24no—
14C2×#14523247—24no—
15C22×Dic254C2211yescharacteristic
16C22×C22⋊D127C2211yescharacteristic
17C22×C22⋊D127C2212yes—
18C23×Dic127C2212yes—
19C22×C4×D127C2211yescharacteristic
20#16861188—24no—
21C4×C22⋊D127—24no—
22C2×C4⋊Dic127C2211yescharacteristic
23C2×Dic127⋊C4C2212yes—
24C2×Dic127⋊C4C2211yescharacteristic
25C2×C22⋊Dic127C2212yes—
26C2×C22⋊Dic127C2212yes—
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C22⋊Dic127C2212yes—
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2212yes—
31C2×C4×Dic127C2211yescharacteristic
32C127⋊#95C2212yes—
33C127⋊#95C2212yes—
34C127⋊#95C2212yes—
35C22×D4×C127C2212yes—
36C22×D4×C127C2211yescharacteristic
37C23×C4×C127C2211yescharacteristic
38#121×C127—24no—
39C2×C4⋊C4×C127C2211yescharacteristic
40C2×C22⋊C4×C127C2212yes—
41C23×D127C2311yescharacteristic
42#14523247—232no—
43C2×Dic254—22no—
44C2×Dic254—22no—
45C2×Dic254D414yescomplemented
46C2×C22⋊D127—24no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—24no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127—24no—
51C2×C22⋊D127—22no—
52C22×Dic127—24no—
53C22×Dic127—24no—
54C22×Dic127—24no—
55C22×Dic127C2311yescharacteristic
56C22×Dic127C2312yes—
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2312yes—
59C22×Dic127C2311yescharacteristic
60C2×C4×D127—22no—
61C2×C4×D127—22no—
62C2×C4×D127D414yescomplemented
63C4⋊Dic127—22no—
64Dic127⋊C4—24no—
65Dic127⋊C4—22no—
66C22⋊Dic127—24no—
67C22⋊Dic127—22no—
68C22⋊Dic127—24no—
69C22⋊Dic127—42no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—42no—
72C4×Dic127—22no—
73C24×C127C2312yes—
74C24×C127C2311yescharacteristic
75C2×D4×C127—24no—
76C2×D4×C127—24no—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127—42no—
80C2×D4×C127D414yescomplemented
81C22×C4×C127—22no—
82C22×C4×C127—22no—
83C22×C4×C127—22no—
84C22×C4×C127C2311yescharacteristic
85C22×C4×C127C2312yes—
86C22×C4×C127C2311yescharacteristic
87C4⋊C4×C127—22no—
88C22⋊C4×C127—24no—
89C22×D127—21no—
90C22×D127—21no—
91C22×D127—21no—
92C22×D127—21no—
93C22×D127—21no—
94C22×D127—21no—
95C22×D127C2×D412yes—
96Dic254—28no—
97C22⋊D127—28no—
98C22⋊D127—216no—
99C2×Dic127—216no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—22no—
104C2×Dic127—22no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—22no—
108C2×Dic127—22no—
109C2×Dic127—22no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127C2×D412yes—
113C2×Dic127C2×D414yes—
114C2×Dic127#4112yescomplemented
115C4×D127—28no—
116C23×C127—21no—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—22no—
120C23×C127—22no—
121C23×C127—22no—
122C23×C127—22no—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—22no—
126C23×C127—22no—
127C23×C127—21no—
128C23×C127C2×D414yes—
129C23×C127#4112yescomplemented
130C23×C127C2411yescharacteristic
131D4×C127—28no—
132D4×C127—48no—
133C2×C4×C127—21no—
134C2×C4×C127—22no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—44no—
140C2×C4×C127C2×D412yes—
141C2×C4×C127C2×D412yes—
142C2×C4×C127C2×D412yes—
143C22×C127—24no—
144C22×C127—22no—
145C22×C127—22no—
146C22×C127—24no—
147C22×C127—24no—
148C22×C127—22no—
149C22×C127—41no—
150C22×C127—42no—
151C22×C127—41no—
152C22×C127—42no—
153C22×C127—41no—
154C22×C127—42no—
155C22×C127—41no—
156C22×C127—42no—
157C22×C127C22×D411yescharacteristic
158C22×C127C22×D411yescharacteristic
159C22×C127C22×D411yescharacteristic
160C22×C127C2×#4111yescharacteristic
161C22×C127C2×#4111yescharacteristic
162C22×C127H(5,𝔽2)11yescharacteristic
163C22×C1272−1+411yescharacteristic
164C2×D127—22no—
165C2×D127—22no—
166C2×D127—22no—
167C2×D127—41no—
168C2×D127—41no—
169C2×D127—41no—
170C2×D127—41no—
171C4×C127—41no—
172C4×C127—42no—
173C4×C127#12014yes—
174Dic127—22no—
175Dic127—24no—
176Dic127—22no—
177Dic127—41no—
178Dic127—42no—
179C2×C127—44no—
180C2×C127—42no—
181C2×C127C2×#12011yescharacteristic
182C2×C127#52511yescharacteristic
183C2×C127#53711yescharacteristic
184C2×C127#54611yescharacteristic
185C2×C127#54711yescharacteristic
186C2×C127#54811yescharacteristic
187C2×C127#55011yescharacteristic
188D127—42no—
189#2407—1271nomaximal, Sylow
190C127#240711yescharacteristic, complemented, Sylow
191C22×#41—1271no—
192C2×C22⋊Q8—1271no—
193C2×#121D12711yescharacteristic, complemented
194C2×C4×D4—1271no—
195#394—1272no—
196#392—1272no—
197#391—1271no—
198#390—1271no—
199#386—1272no—
200#385—1272no—
201#380—1271no—
202C2×#41—2544no—
203C2×#41—2544no—
204C2×#41—2544no—
205C22×Q8—1271no—
206C22×D4—1271no—
207C22×D4—1272no—
208C22×D4C2×D12712yes—
209C22×D4C2×D12711yescharacteristic
210C23×C4—1271no—
211C23×C4—1272no—
212C23×C4C2×D12711yescharacteristic
213C22⋊Q8—2544no—
214#121—24no—
215C4×D4—2544no—
216C2×C4⋊C4—1271no—
217C2×C4⋊C4—1271no—
218C2×C4⋊C4—1272no—
219C2×C4⋊C4C2×D12711yescharacteristic
220C2×C22⋊C4—1272no—
221C2×C22⋊C4—1272no—
222C2×C22⋊C4—1271no—
223C2×C22⋊C4—1272no—
224C2×C22⋊C4—1272no—
225C2×C22⋊C4—1271no—
226C2×C22⋊C4C2×D12712yes—
227C2×C42—1271no—
228#95—1272no—
229#95—1272no—
230#95—1272no—
231C24—1271no—
232C24C22×D12712yes—
233C24C22×D12711yescharacteristic
234#41—25432no—
235C2×Q8—1274no—
236C2×Q8—2542no—
237C2×Q8—2542no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—2542no—
244C2×D4—2544no—
245C2×D4—2542no—
246C2×D4—2544no—
247C2×D4—2544no—
248C2×D4—2542no—
249C2×D4C22⋊D12714yescomplemented
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—1271no—
254C22×C4—1271no—
255C22×C4—1272no—
256C22×C4—1274no—
257C22×C4—1271no—
258C22×C4—1272no—
259C22×C4—2544no—
260C22×C4—2544no—
261C22×C4—2544no—
262C22×C4—2542no—
263C22×C4—2542no—
264C22×C4C22×D12712yes—
265C22×C4C22×D12711yescharacteristic
266C22×C4C22×D12711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—2542no—
269C4⋊C4—2544no—
270C4⋊C4—2542no—
271C22⋊C4—24no—
272C22⋊C4—5082no—
273C22⋊C4—5082no—
274C22⋊C4—2542no—
275C22⋊C4—2544no—
276C22⋊C4—2542no—
277C22⋊C4—2544no—
278C42—2542no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—1272no—
292C23—2541no—
293C23—2541no—
294C23—2541no—
295C23—2541no—
296C23—2541no—
297C23—2541no—
298C23C2×C22⋊D12714yes—
299C23#1452324712yescomplemented
300C23C23×D12711yescharacteristic
301Q8—2548no—
302D4—28no—
303D4—48no—
304D4—25416no—
305D4—2548no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—44no—
313C2×C4—1272no—
314C2×C4—1274no—
315C2×C4—1272no—
316C2×C4—2541no—
317C2×C4—2542no—
318C2×C4—2542no—
319C2×C4—2542no—
320C2×C4—2541no—
321C2×C4—2541no—
322C2×C4—25416no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2542no—
326C2×C4—2541no—
327C2×C4—2542no—
328C2×C4—2541no—
329C2×C4—2548no—
330C2×C4D4×D12712yes—
331C2×C4D4×D12712yes—
332C2×C4C2×C22⋊D12712yes—
333C22—22no—
334C22—24no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—24no—
339C22—42no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—5081no—
348C22—5081no—
349C22—5081no—
350C22—5081no—
351C22—2542no—
352C22—2542no—
353C22—2542no—
354C22C2×D4×D12711yescharacteristic
355C22C2×D4×D12711yescharacteristic
356C22C22×C22⋊D12711yescharacteristic
357C22#41×D12711yescharacteristic
358C22C2×#1452324711yescharacteristic
359C22#1686125411yescharacteristic
360C22#1686125611yescharacteristic
361C4—41no—
362C4—42no—
363C4—5082no—
364C4—5081no—
365C4—2542no—
366C4—2544no—
367C4—2542no—
368C4#1686117714yes—
369C2—42no—
370C2—44no—
371C2—5082no—
372C2C2×#1686117711yescharacteristic
373C2#2255038611yescharacteristic
374C2#2255045511yescharacteristic
375C2#2255052011yescharacteristic
376C2#2255052511yescharacteristic
377C2#2255053211yescharacteristic
378C2#2255054411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×#121×C127C2
hypercenterC2×#121D127
hypocenterC127#2407
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14523247C2semidirect
2. C2×#16861188C2semidirect
3. C2×C4×C22⋊D127C2semidirect
4. #22549791C2semidirect
5. #22549784C2semidirect
6. #22549781C2semidirect
7. #22549779C2semidirect
8. #22549756C2semidirect
9. #22549748C2semidirect
10. #22549733C2semidirect
11. C2×#121×C127C2semidirect
45. C2×Dic254D4semidirect
62. C2×C4×D127D4semidirect
80. C2×D4×C127D4semidirect
114. C2×Dic127#41semidirect
129. C23×C127#41semidirect
190. C127#2407semidirect
193. C2×#121D127semidirect
249. C2×D4C22⋊D127semidirect
299. C23#14523247semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168611882
C2×C4×C22⋊D1272
#225497912
#225497842
#225497812
#225497792
#225497562
#225497482
#225497332
C2×#121×C1272
#2407127