Finite Group Databasebeta

← #30642064

Subgroups — #30642064

Order 16256.

864 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523247C211yesmaximal, characteristic
2C2×#16861193C211yesmaximal, characteristic, complemented
3C2×#16861190C211yesmaximal, characteristic, complemented
4C22×C4⋊Dic127C211yesmaximal, characteristic, complemented
5#22549797C212yesmaximal, complemented
6#22549792C212yesmaximal, complemented
7#22549755C211yesmaximal, characteristic, complemented
8#22549751C212yesmaximal, complemented
9#22549750C212yesmaximal, complemented
10#22549737C211yesmaximal, characteristic, complemented
11#386×C127C211yesmaximal, characteristic, complemented
12C2×#14523247—24no—
13C2×#14523247—24no—
14C2×#14523247—24no—
15C22×Dic254C2211yescharacteristic
16C22×C22⋊D127C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C23×Dic127C2211yescharacteristic
19C23×Dic127C2211yescharacteristic
20C23×Dic127C2211yescharacteristic
21C22×C4×D127C2211yescharacteristic
22#16861193—24no—
23#16861190—24no—
24C2×C4⋊Dic127—24no—
25C2×C4⋊Dic127C2212yes—
26C2×C4⋊Dic127C2212yes—
27C2×C4⋊Dic127C2212yes—
28C2×Dic127⋊C4C2212yes—
29C2×C22⋊Dic127C2212yes—
30C2×C22⋊Dic127C2211yescharacteristic
31C2×(C2×D127)⋊C4C2212yes—
32C2×(C2×D127)⋊C4C2212yes—
33C127⋊#95C2212yes—
34C127⋊#95C2212yes—
35C127⋊#95C2212yes—
36C22×D4×C127C2211yescharacteristic
37C23×C4×C127C2211yescharacteristic
38C2×C22⋊C4×C127C2212yes—
39C2×C22⋊C4×C127C2211yescharacteristic
40#95×C127C2212yes—
41C23×D127C2311yescharacteristic
42#14523247—232no—
43C2×Dic254—22no—
44C2×Dic254—22no—
45C2×Dic254—22no—
46C2×C22⋊D127—22no—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127D414yes—
52C22×Dic127—22no—
53C22×Dic127—22no—
54C22×Dic127—22no—
55C22×Dic127—22no—
56C22×Dic127—22no—
57C22×Dic127—22no—
58C22×Dic127—22no—
59C22×Dic127—22no—
60C22×Dic127D414yes—
61C22×Dic127C2312yes—
62C22×Dic127C2312yes—
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C2×C4×D127—22no—
67C2×C4×D127—22no—
68C2×C4×D127—22no—
69C4⋊Dic127—24no—
70C4⋊Dic127—24no—
71C4⋊Dic127—24no—
72Dic127⋊C4—24no—
73C22⋊Dic127—24no—
74C22⋊Dic127—22no—
75(C2×D127)⋊C4—24no—
76(C2×D127)⋊C4—24no—
77C24×C127C2311yescharacteristic
78C24×C127C2311yescharacteristic
79C2×D4×C127—22no—
80C2×D4×C127—22no—
81C2×D4×C127D414yes—
82C22×C4×C127—22no—
83C22×C4×C127—24no—
84C22×C4×C127C2312yes—
85C22×C4×C127C2311yescharacteristic
86C22×C4×C127C2312yes—
87C22⋊C4×C127—22no—
88C22⋊C4×C127—24no—
89C22×D127—21no—
90C22×D127—21no—
91C22×D127—22no—
92C22×D127—22no—
93C22×D127C2×D412yes—
94Dic254—28no—
95C22⋊D127—28no—
96C22⋊D127—28no—
97C2×Dic127—28no—
98C2×Dic127—28no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—22no—
106C2×Dic127—22no—
107C2×Dic127—22no—
108C2×Dic127—22no—
109C2×Dic127—22no—
110C2×Dic127—22no—
111C2×Dic127—44no—
112C2×Dic127C2×D412yes—
113C2×Dic127C2×D412yes—
114C2×Dic127C2×D412yes—
115C4×D127—28no—
116C23×C127—22no—
117C23×C127—22no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—22no—
121C23×C127—22no—
122C23×C127C2×D412yes—
123C23×C127C2×D412yes—
124C23×C127C2×D412yes—
125C23×C127C2×D412yes—
126C23×C127C2411yescharacteristic
127D4×C127—28no—
128C2×C4×C127—28no—
129C2×C4×C127—22no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—22no—
133C2×C4×C127—22no—
134C2×C4×C127—22no—
135C2×C4×C127—22no—
136C2×C4×C127C2×D412yes—
137C2×C4×C127#4114yescomplemented
138C22×C127—22no—
139C22×C127—22no—
140C22×C127—24no—
141C22×C127—24no—
142C22×C127—22no—
143C22×C127—24no—
144C22×C127—41no—
145C22×C127—41no—
146C22×C127—42no—
147C22×C127#12014yes—
148C22×C127C22×D411yescharacteristic
149C22×C127C22×D411yescharacteristic
150C22×C127C22×D411yescharacteristic
151C22×C127C2×#4112yes—
152C22×C1272−1+412yes—
153C2×D127—22no—
154C2×D127—24no—
155C2×D127—41no—
156C2×D127—41no—
157C2×D127—42no—
158C4×C127—22no—
159C4×C127—24no—
160C4×C127—42no—
161Dic127—22no—
162Dic127—22no—
163Dic127—22no—
164Dic127—42no—
165Dic127—42no—
166C2×C127—24no—
167C2×C127—42no—
168C2×C127C2×#12011yescharacteristic
169C2×C127#52511yescharacteristic
170C2×C127#53611yescharacteristic
171C2×C127#54712yes—
172C2×C127#54712yes—
173D127—42no—
174#2406—1271nomaximal, Sylow
175C127#240611yescharacteristic, complemented, Sylow
176C22×#41—1271no—
177C2×#123—1271no—
178C2×#123—1271no—
179C22×C4⋊C4—1271no—
180#395—1272no—
181#394—1272no—
182#386—1271no—
183#386D12711yescharacteristic, complemented
184#385—1272no—
185#385—1272no—
186#381—1271no—
187C2×#41—2544no—
188C2×#41—2544no—
189C2×#41—2544no—
190C22×Q8—1271no—
191C22×D4—1271no—
192C22×D4—1271no—
193C22×D4C2×D12711yescharacteristic
194C23×C4—1271no—
195C23×C4—1271no—
196C23×C4—1271no—
197C23×C4—1271no—
198C23×C4C2×D12711yescharacteristic
199#123—2544no—
200#123—2544no—
201C2×C4⋊C4—1272no—
202C2×C4⋊C4—1272no—
203C2×C4⋊C4—1272no—
204C2×C4⋊C4—1272no—
205C2×C4⋊C4—2544no—
206C2×C22⋊C4—1271no—
207C2×C22⋊C4—1272no—
208C2×C22⋊C4—1272no—
209C2×C22⋊C4—1272no—
210C2×C22⋊C4C2×D12711yescharacteristic
211C2×C22⋊C4C2×D12712yes—
212#95—1272no—
213#95—1272no—
214#95—1272no—
215#95C2×D12712yes—
216C24—1271no—
217C24C22×D12711yescharacteristic
218C24C22×D12711yescharacteristic
219#41—25432no—
220C2×Q8—2542no—
221C2×Q8—2542no—
222C2×Q8—2542no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—1274no—
226C2×D4—2542no—
227C2×D4—2542no—
228C2×D4—2542no—
229C2×D4—2542no—
230C2×D4—2542no—
231C2×D4D50814yes—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—1274no—
235C22×C4—1271no—
236C22×C4—1271no—
237C22×C4—1271no—
238C22×C4—1272no—
239C22×C4—1272no—
240C22×C4—2542no—
241C22×C4—2542no—
242C22×C4—2542no—
243C22×C4—2542no—
244C22×C4—2542no—
245C22×C4—2542no—
246C22×C4—2542no—
247C22×C4—2542no—
248C22×C4—2542no—
249C22×C4—2542no—
250C22×C4—2542no—
251C22×C4C22×D12712yes—
252C22×C4C22×D12711yescharacteristic
253C22×C4C22×D12712yes—
254C4⋊C4—2544no—
255C4⋊C4—2544no—
256C4⋊C4—2544no—
257C4⋊C4—2544no—
258C22⋊C4—22no—
259C22⋊C4—24no—
260C22⋊C4—2544no—
261C22⋊C4—2542no—
262C22⋊C4—2544no—
263C22⋊C4—2544no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—22no—
270C23—1272no—
271C23—2541no—
272C23—2541no—
273C23—2542no—
274C23—2542no—
275C23D4×D12712yes—
276C23D4×D12712yes—
277C23C2×D50812yes—
278C23C2×D50812yes—
279C23C23×D12711yescharacteristic
280Q8—2548no—
281D4—28no—
282D4—2548no—
283D4—2548no—
284C2×C4—28no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—1272no—
293C2×C4—1272no—
294C2×C4—1272no—
295C2×C4—5084no—
296C2×C4—2548no—
297C2×C4—2542no—
298C2×C4—2542no—
299C2×C4—2542no—
300C2×C4—2542no—
301C2×C4—2542no—
302C2×C4—2542no—
303C2×C4—2548no—
304C2×C4—2541no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4—2541no—
309C2×C4—2541no—
310C2×C4—2548no—
311C2×C4C2×D50812yes—
312C2×C4#1452324714yescomplemented
313C22—24no—
314C22—22no—
315C22—22no—
316C22—24no—
317C22—24no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—5082no—
323C22—5081no—
324C22—5081no—
325C22—2542no—
326C22—2544no—
327C22#1686117914yes—
328C22C2×D4×D12711yescharacteristic
329C22C2×D4×D12711yescharacteristic
330C22C22×D50811yescharacteristic
331C22C2×#1452324712yes—
332C22#1686125612yes—
333C4—22no—
334C4—24no—
335C4—42no—
336C4—5082no—
337C4—5082no—
338C4—2542no—
339C4—2542no—
340C4—2542no—
341C2—24no—
342C2—42no—
343C2—5082no—
344C2C2×#1686117911yescharacteristic
345C2#2255038211yescharacteristic
346C2#2255044811yescharacteristic
347C2#2255052612yes—
348C2#2255053012yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#386×C127C2
prime radicalC22×#14523247C2
hypercenter#386D127
hypocenterC127#2406
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#16861193C2semidirect
3. C2×#16861190C2semidirect
4. C22×C4⋊Dic127C2semidirect
5. #22549797C2semidirect
6. #22549792C2semidirect
7. #22549755C2semidirect
8. #22549751C2semidirect
9. #22549750C2semidirect
10. #22549737C2semidirect
11. #386×C127C2semidirect
137. C2×C4×C127#41semidirect
175. C127#2406semidirect
183. #386D127semidirect
312. C2×C4#14523247semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168611932
C2×#168611902
C22×C4⋊Dic1272
#225497972
#225497922
#225497552
#225497512
#225497502
#225497372
#386×C1272
#2406127