Finite Group Databasebeta

← #30642077

Subgroups — #30642077

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×#14523249C211yesmaximal, characteristic, complemented
2C2×#16861183C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
4#22549793C212yesmaximal, complemented
5#22549785C212yesmaximal, complemented
6#22549783C211yesmaximal, characteristic, complemented
7#22549777C211yesmaximal, characteristic, complemented
8#22549752C212yesmaximal, complemented
9#22549747C212yesmaximal, complemented
10#22549733C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C127C211yesmaximal, characteristic, complemented
12C2×#14523249—24no—
13C2×#14523249—24no—
14C2×#14523249—24no—
15C22×D508C2212yes—
16C22×D508C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C22⋊D127C2212yes—
19C22×C4×D127C2212yes—
20C22×C4×D127C2211yescharacteristic
21#16861183—24no—
22C4×C22⋊D127—24no—
23C2×C4⋊Dic127C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×(C2×D127)⋊C4C2212yes—
27C2×(C2×D127)⋊C4C2211yescharacteristic
28C2×(C2×D127)⋊C4C2212yes—
29C2×(C2×D127)⋊C4C2212yes—
30C2×(C2×D127)⋊C4C2212yes—
31C2×C4×Dic127C2211yescharacteristic
32C127⋊#95C2212yes—
33C127⋊#95C2212yes—
34C127⋊#95C2212yes—
35C22×Q8×C127C2211yescharacteristic
36C23×C4×C127C2211yescharacteristic
37C22⋊Q8×C127—24no—
38C2×C4⋊C4×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2212yes—
40C2×C22⋊C4×C127C2212yes—
41C23×D127C2312yes—
42C23×D127C2311yescharacteristic
43#14523249—232no—
44C2×D508—24no—
45C2×D508—24no—
46C2×D508—22no—
47C2×D508—24no—
48C2×D508—22no—
49C2×D508D414yescomplemented
50C2×C22⋊D127—22no—
51C2×C22⋊D127—24no—
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54C2×C22⋊D127—24no—
55C2×C22⋊D127—42no—
56C22×Dic127C2312yes—
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C22×Dic127C2311yescharacteristic
60C2×C4×D127—24no—
61C2×C4×D127—24no—
62C2×C4×D127—22no—
63C2×C4×D127—24no—
64C2×C4×D127—22no—
65C2×C4×D127D414yescomplemented
66C4⋊Dic127—22no—
67Dic127⋊C4—22no—
68C22⋊Dic127—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—24no—
71(C2×D127)⋊C4—24no—
72(C2×D127)⋊C4—42no—
73(C2×D127)⋊C4—42no—
74C4×Dic127—22no—
75C24×C127C2311yescharacteristic
76C2×Q8×C127—22no—
77C2×Q8×C127—22no—
78C2×Q8×C127D414yescomplemented
79C22×C4×C127—22no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
82C22×C4×C127C2312yes—
83C22×C4×C127C2311yescharacteristic
84C22×C4×C127C2312yes—
85C22×C4×C127C2311yescharacteristic
86C4⋊C4×C127—22no—
87C4⋊C4×C127—24no—
88C22⋊C4×C127—24no—
89C22×D127—22no—
90C22×D127—21no—
91C22×D127—22no—
92C22×D127—21no—
93C22×D127—22no—
94C22×D127—21no—
95C22×D127—21no—
96C22×D127—22no—
97C22×D127—21no—
98C22×D127—22no—
99C22×D127—21no—
100C22×D127—22no—
101C22×D127C2×D414yes—
102C22×D127C2×D412yes—
103D508—216no—
104D508—28no—
105C22⋊D127—28no—
106C22⋊D127—48no—
107C2×Dic127—21no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—22no—
113C2×Dic127—22no—
114C2×Dic127—21no—
115C2×Dic127—22no—
116C2×Dic127—21no—
117C2×Dic127C2×D412yes—
118C2×Dic127#4112yescomplemented
119C4×D127—216no—
120C4×D127—28no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127#4112yescomplemented
128C23×C127C2411yescharacteristic
129Q8×C127—28no—
130C2×C4×C127—22no—
131C2×C4×C127—22no—
132C2×C4×C127—22no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—44no—
139C2×C4×C127C2×D412yes—
140C2×C4×C127C2×D412yes—
141C2×C4×C127C2×D414yes—
142C2×C4×C127C2×D412yes—
143C22×C127—22no—
144C22×C127—22no—
145C22×C127—22no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127C22×D411yescharacteristic
151C22×C127C22×D411yescharacteristic
152C22×C127C22×D411yescharacteristic
153C22×C127C2×#4111yescharacteristic
154C22×C127C2×#4111yescharacteristic
155C22×C127H(5,𝔽2)11yescharacteristic
156C22×C1272−1+411yescharacteristic
157C2×D127—24no—
158C2×D127—22no—
159C2×D127—22no—
160C2×D127—22no—
161C2×D127—24no—
162C2×D127—24no—
163C2×D127—42no—
164C2×D127—42no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127—42no—
169C2×D127—42no—
170C2×D127—41no—
171C4×C127—24no—
172C4×C127—41no—
173C4×C127—42no—
174C4×C127#12014yes—
175Dic127—22no—
176Dic127—22no—
177Dic127—41no—
178Dic127—42no—
179C2×C127—42no—
180C2×C127C2×#12011yescharacteristic
181C2×C127#52511yescharacteristic
182C2×C127#53711yescharacteristic
183C2×C127#54611yescharacteristic
184C2×C127#54711yescharacteristic
185C2×C127#54811yescharacteristic
186C2×C127#55011yescharacteristic
187D127—42no—
188D127—44no—
189#2407—1271nomaximal, Sylow
190C127#240711yescharacteristic, complemented, Sylow
191C22×#41—1271no—
192C2×C22⋊Q8D12711yescharacteristic, complemented
193C2×#121—1271no—
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×Q8C2×D12711yescharacteristic
206C22×D4—1272no—
207C22×D4—1271no—
208C22×D4—1271no—
209C22×D4—1272no—
210C23×C4—1271no—
211C23×C4—1272no—
212C23×C4C2×D12711yescharacteristic
213C22⋊Q8—24no—
214#121—2544no—
215C4×D4—2544no—
216C2×C4⋊C4—1271no—
217C2×C4⋊C4—1271no—
218C2×C4⋊C4C2×D12711yescharacteristic
219C2×C4⋊C4C2×D12712yes—
220C2×C22⋊C4—1272no—
221C2×C22⋊C4—1272no—
222C2×C22⋊C4—1271no—
223C2×C22⋊C4—1271no—
224C2×C22⋊C4—1272no—
225C2×C22⋊C4—1272no—
226C2×C22⋊C4C2×D12712yes—
227C2×C42—1271no—
228#95—1272no—
229#95—1272no—
230#95—1272no—
231C24—1271no—
232C24—1272no—
233C24C22×D12711yescharacteristic
234#41—25432no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8C22⋊D12714yescomplemented
238C2×D4—1274no—
239C2×D4—5082no—
240C2×D4—2544no—
241C2×D4—2544no—
242C2×D4—2542no—
243C2×D4—2544no—
244C2×D4—2544no—
245C2×D4—2542no—
246C2×D4—2542no—
247C2×D4—2542no—
248C2×D4—2544no—
249C2×D4—2542no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—1271no—
254C22×C4—1271no—
255C22×C4—1274no—
256C22×C4—1272no—
257C22×C4—1271no—
258C22×C4—2544no—
259C22×C4—2544no—
260C22×C4—2542no—
261C22×C4—2544no—
262C22×C4—2542no—
263C22×C4C22×D12712yes—
264C22×C4C22×D12711yescharacteristic
265C22×C4C22×D12712yes—
266C22×C4C22×D12711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—2542no—
270C4⋊C4—2542no—
271C22⋊C4—24no—
272C22⋊C4—5082no—
273C22⋊C4—5082no—
274C22⋊C4—2544no—
275C22⋊C4—2542no—
276C22⋊C4—2544no—
277C22⋊C4—2542no—
278C42—2542no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—1272no—
286C23—1274no—
287C23—2541no—
288C23—2542no—
289C23—2542no—
290C23—2541no—
291C23—2542no—
292C23—2541no—
293C23—2542no—
294C23—2541no—
295C23—2541no—
296C23—2542no—
297C23—2541no—
298C23—2542no—
299C23#1452324912yescomplemented
300C23C23×D12711yescharacteristic
301Q8—28no—
302D4—5088no—
303D4—25416no—
304D4—2548no—
305D4—2548no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—44no—
315C2×C4—1272no—
316C2×C4—1272no—
317C2×C4—25416no—
318C2×C4—2541no—
319C2×C4—2542no—
320C2×C4—2542no—
321C2×C4—2541no—
322C2×C4—2542no—
323C2×C4—2541no—
324C2×C4—2541no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4—2541no—
328C2×C4—2548no—
329C2×C4D4×D12712yes—
330C2×C4D4×D12712yes—
331C2×C4C2×C22⋊D12714yes—
332C2×C4C2×C22⋊D12712yes—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—5081no—
341C22—5081no—
342C22—5082no—
343C22—5082no—
344C22—5081no—
345C22—5082no—
346C22—5082no—
347C22—5081no—
348C22—2544no—
349C22—2544no—
350C22—2542no—
351C22—2542no—
352C22—2542no—
353C22—2544no—
354C22C2×D4×D12711yescharacteristic
355C22C2×D4×D12711yescharacteristic
356C22C22×C22⋊D12711yescharacteristic
357C22#41×D12711yescharacteristic
358C22C2×#1452324911yescharacteristic
359C22#1686125511yescharacteristic
360C22#1686125711yescharacteristic
361C4—24no—
362C4—42no—
363C4—41no—
364C4—5081no—
365C4—5082no—
366C4—2542no—
367C4—2542no—
368C4#1686117714yes—
369C2—42no—
370C2—5082no—
371C2—5084no—
372C2C2×#1686117711yescharacteristic
373C2#2255038911yescharacteristic
374C2#2255045611yescharacteristic
375C2#2255051711yescharacteristic
376C2#2255052811yescharacteristic
377C2#2255053711yescharacteristic
378C2#2255054511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×C22⋊Q8×C127C2
hypercenterC2×C22⋊Q8D127
hypocenterC127#2407
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C2×#16861183C2semidirect
3. C2×C4×C22⋊D127C2semidirect
4. #22549793C2semidirect
5. #22549785C2semidirect
6. #22549783C2semidirect
7. #22549777C2semidirect
8. #22549752C2semidirect
9. #22549747C2semidirect
10. #22549733C2semidirect
11. C2×C22⋊Q8×C127C2semidirect
49. C2×D508D4semidirect
65. C2×C4×D127D4semidirect
78. C2×Q8×C127D4semidirect
118. C2×Dic127#41semidirect
127. C23×C127#41semidirect
190. C127#2407semidirect
192. C2×C22⋊Q8D127semidirect
237. C2×Q8C22⋊D127semidirect
299. C23#14523249semidirect

Maximal subgroups

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

subgroupindex
C22×#145232492
C2×#168611832
C2×C4×C22⋊D1272
#225497932
#225497852
#225497832
#225497772
#225497522
#225497472
#225497332
C2×C22⋊Q8×C1272
#2407127