Finite Group Databasebeta

← #30641319

Subgroups — #30641319

Order 16256.

880 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523248C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
4#22549780C212yesmaximal, complemented
5#22549776C212yesmaximal, complemented
6#22549754C212yesmaximal, complemented
7#22549752C212yesmaximal, complemented
8#22549736C211yesmaximal, characteristic, complemented
9#22549733C211yesmaximal, characteristic, complemented
10#22549733C211yesmaximal, characteristic, complemented
11#380×C127C211yesmaximal, characteristic, complemented
12C2×#14523248—24no—
13C2×#14523248—24no—
14C2×#14523248C418yescomplemented
15C22×Dic254C2211yescharacteristic
16C22×D508C2211yescharacteristic
17C22×C22⋊D127C2211yescharacteristic
18C22×C22⋊D127C2211yescharacteristic
19C22×C4×D127C2211yescharacteristic
20C22×C4×D127C2212yes—
21C22×C4×D127C2211yescharacteristic
22#16861167—24no—
23C2×C4⋊Dic127C2211yescharacteristic
24C2×C4⋊Dic127C2211yescharacteristic
25C2×Dic127⋊C4C2212yes—
26C4⋊C4×D127—24no—
27C2×C22⋊Dic127C2212yes—
28C2×(C2×D127)⋊C4C2212yes—
29C2×(C2×D127)⋊C4C2212yes—
30C2×(C2×D127)⋊C4C2212yes—
31C2×C4×Dic127C2212yes—
32C127⋊#95C2212yes—
33C127⋊#95C2212yes—
34C127⋊#95C2212yes—
35C23×C4×C127C2211yescharacteristic
36C2×C4⋊C4×C127C2211yescharacteristic
37C2×C4⋊C4×C127C2211yescharacteristic
38C2×C22⋊C4×C127C2212yes—
39#95×C127C2212yes—
40C23×D127C2311yescharacteristic
41C23×D127C2311yescharacteristic
42#14523248—232no—
43C2×Dic254—22no—
44C2×Dic254—22no—
45C2×Dic254C2×C414yescomplemented
46C2×D508—22no—
47C2×D508—22no—
48C2×D508C2×C414yes—
49C2×C22⋊D127—22no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—22no—
52C2×C22⋊D127—22no—
53C2×C22⋊D127C2×C414yes—
54C2×C22⋊D127C2×C414yescomplemented
55C22×Dic127C2312yes—
56C22×Dic127C2312yes—
57C22×Dic127C2311yescharacteristic
58C22×Dic127C2311yescharacteristic
59C2×C4×D127—24no—
60C2×C4×D127—22no—
61C2×C4×D127—22no—
62C2×C4×D127—42no—
63C2×C4×D127—42no—
64C2×C4×D127C2×C414yes—
65C2×C4×D127C2×C414yescomplemented
66C2×C4×D127D414yes—
67C2×C4×D127D414yes—
68C4⋊Dic127—22no—
69C4⋊Dic127—22no—
70Dic127⋊C4—24no—
71C22⋊Dic127—42no—
72(C2×D127)⋊C4—24no—
73(C2×D127)⋊C4—42no—
74(C2×D127)⋊C4—42no—
75C4×Dic127—24no—
76C24×C127C2311yescharacteristic
77C22×C4×C127—22no—
78C22×C4×C127—22no—
79C22×C4×C127C2×C414yescomplemented
80C22×C4×C127C2312yes—
81C22×C4×C127C2312yes—
82C22×C4×C127C2311yescharacteristic
83C22×C4×C127C2311yescharacteristic
84C4⋊C4×C127—22no—
85C4⋊C4×C127—22no—
86C22⋊C4×C127—42no—
87C22×D127—21no—
88C22×D127—21no—
89C22×D127—21no—
90C22×D127—21no—
91C22×D127—21no—
92C22×D127—21no—
93C22×D127—21no—
94C22×D127—21no—
95C22×D127—21no—
96C22×D127—21no—
97C22×D127C22×C412yes—
98C22×D127C22×C412yes—
99C22×D127C2×D412yes—
100C22×D127C2×D412yes—
101Dic254—28no—
102D508—28no—
103C22⋊D127—28no—
104C22⋊D127—28no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—22no—
108C2×Dic127—22no—
109C2×Dic127—22no—
110C2×Dic127—22no—
111C2×Dic127—22no—
112C2×Dic127—22no—
113C2×Dic127C22×C412yes—
114C2×Dic127C22×C412yes—
115C2×Dic127C2×D412yes—
116C2×Dic127C2×D412yes—
117C4×D127—28no—
118C4×D127—48no—
119C4×D127C22⋊C4116yescomplemented
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127—21no—
125C23×C127—21no—
126C23×C127C22×C412yes—
127C23×C127C2411yescharacteristic
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—28no—
131C2×C4×C127—22no—
132C2×C4×C127—22no—
133C2×C4×C127—22no—
134C2×C4×C127—22no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127C22×C412yes—
138C2×C4×C127C22×C412yes—
139C2×C4×C127C2×D412yes—
140C2×C4×C127C2×D412yes—
141C22×C127—22no—
142C22×C127—22no—
143C22×C127—22no—
144C22×C127—41no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127C23×C411yescharacteristic
149C22×C127C22×D411yescharacteristic
150C22×C127C22×D411yescharacteristic
151C22×C127H(5,𝔽2)11yescharacteristic
152C22×C127H(5,𝔽2)11yescharacteristic
153C22×C1272−1+411yescharacteristic
154C22×C1272−1+411yescharacteristic
155C2×D127—22no—
156C2×D127—22no—
157C2×D127—22no—
158C2×D127—22no—
159C2×D127—22no—
160C2×D127—22no—
161C2×D127—22no—
162C2×D127—22no—
163C2×D127—22no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—41no—
168C2×D127C2×C22⋊C414yes—
169C4×C127—22no—
170C4×C127—42no—
171C4×C127—42no—
172C4×C127C2×C22⋊C414yes—
173Dic127—22no—
174Dic127—42no—
175Dic127—42no—
176Dic127C2×C22⋊C414yes—
177C2×C127—42no—
178C2×C127C22×C22⋊C411yescharacteristic
179C2×C127#52011yescharacteristic
180C2×C127#52011yescharacteristic
181C2×C127#53511yescharacteristic
182C2×C127#53611yescharacteristic
183C2×C127#53711yescharacteristic
184C2×C127#53711yescharacteristic
185D127—24no—
186D127—42no—
187#2311—1271nomaximal, Sylow
188C127#231111yescharacteristic, complemented, Sylow
189C22×#41—1271no—
190C2×#117—1271no—
191C22×C4⋊C4—1271no—
192#391—1272no—
193#390—1272no—
194#386—1272no—
195#386—1272no—
196#380—1271no—
197#380—1271no—
198#380—1271no—
199#380D12711yescharacteristic, complemented
200C2×#41—1278no—
201C2×#41—2544no—
202C2×#41—2544no—
203C22×Q8—1271no—
204C22×D4—1271no—
205C22×D4—1271no—
206C22×D4—1271no—
207C23×C4—1271no—
208C23×C4—1271no—
209C23×C4—1272no—
210C23×C4C2×D12711yescharacteristic
211#117—2544no—
212C2×C4⋊C4—1271no—
213C2×C4⋊C4—1272no—
214C2×C4⋊C4—1271no—
215C2×C4⋊C4—2544no—
216C2×C4⋊C4C2×D12711yescharacteristic
217C2×C4⋊C4C2×D12711yescharacteristic
218C2×C22⋊C4—1272no—
219C2×C22⋊C4—1272no—
220C2×C22⋊C4—1272no—
221C2×C22⋊C4—1272no—
222C2×C22⋊C4C2×D12712yes—
223C2×C42—1272no—
224#95—1272no—
225#95—1272no—
226#95—1272no—
227#95C2×D12712yes—
228C24—1271no—
229C24—1271no—
230C24C22×D12711yescharacteristic
231#41—25432no—
232C2×Q8—1274no—
233C2×Q8—2542no—
234C2×Q8—2542no—
235C2×D4—1274no—
236C2×D4—1274no—
237C2×D4—1274no—
238C2×D4—2542no—
239C2×D4—2542no—
240C2×D4—2542no—
241C2×D4—2542no—
242C2×D4—2542no—
243C2×D4—2542no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—1271no—
247C22×C4—1274no—
248C22×C4—1274no—
249C22×C4—1274no—
250C22×C4—1271no—
251C22×C4—1274no—
252C22×C4—1272no—
253C22×C4—1272no—
254C22×C4—5082no—
255C22×C4—5082no—
256C22×C4—2544no—
257C22×C4—2542no—
258C22×C4—2542no—
259C22×C4C4×D12714yescomplemented
260C22×C4C22×D12712yes—
261C22×C4C22×D12712yes—
262C22×C4C22×D12711yescharacteristic
263C22×C4C22×D12711yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—2544no—
267C4⋊C4—2542no—
268C4⋊C4—2542no—
269C22⋊C4—42no—
270C22⋊C4—5082no—
271C22⋊C4—5082no—
272C22⋊C4—5082no—
273C22⋊C4—2544no—
274C42—2544no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—1272no—
282C23—1272no—
283C23—1272no—
284C23—1272no—
285C23—2541no—
286C23—2541no—
287C23—2541no—
288C23—2541no—
289C23—2541no—
290C23—2541no—
291C23—2541no—
292C23—2541no—
293C23—2541no—
294C23—2541no—
295C23C2×C4×D12712yes—
296C23C23×D12711yescharacteristic
297Q8—2548no—
298D4—2548no—
299D4—2548no—
300D4—2548no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—1272no—
311C2×C4—1272no—
312C2×C4—1272no—
313C2×C4—12716no—
314C2×C4—1272no—
315C2×C4—5088no—
316C2×C4—2542no—
317C2×C4—2542no—
318C2×C4—2542no—
319C2×C4—2542no—
320C2×C4—2542no—
321C2×C4—2542no—
322C2×C4—2541no—
323C2×C4—2541no—
324C2×C4—2548no—
325C2×C4C2×C4×D12712yes—
326C2×C4C2×C4×D12712yes—
327C2×C4D4×D12712yes—
328C2×C4D4×D12712yes—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—1274no—
337C22—5081no—
338C22—5081no—
339C22—5081no—
340C22—5081no—
341C22—2542no—
342C22—2542no—
343C22—2542no—
344C22—2542no—
345C22—2542no—
346C22—2542no—
347C22—2542no—
348C22—2542no—
349C22—2542no—
350C22C22×C4×D12711yescharacteristic
351C22C2×D4×D12711yescharacteristic
352C22C2×D4×D12711yescharacteristic
353C22#1686125411yescharacteristic
354C22#1686125511yescharacteristic
355C22#1686125611yescharacteristic
356C22#1686125711yescharacteristic
357C4—22no—
358C4—42no—
359C4—42no—
360C4—1274no—
361C4—5082no—
362C4—5082no—
363C4—2542no—
364C4C22⋊C4×D12714yes—
365C2—42no—
366C2—5082no—
367C2—2544no—
368C2C2×C22⋊C4×D12711yescharacteristic
369C2#2255035611yescharacteristic
370C2#2255035711yescharacteristic
371C2#2255044211yescharacteristic
372C2#2255044911yescharacteristic
373C2#2255045511yescharacteristic
374C2#2255045611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#380×C127C2
socle, cosocleC23×C127C24
prime radicalC22×#14523248C2
hypercenter#380D127
hypocenterC127#2311
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#16861167C2semidirect
3. C2×C4⋊C4×D127C2semidirect
4. #22549780C2semidirect
5. #22549776C2semidirect
6. #22549754C2semidirect
7. #22549752C2semidirect
8. #22549736C2semidirect
9. #22549733C2semidirect
10. #22549733C2semidirect
11. #380×C127C2semidirect
14. C2×#14523248C4semidirect
45. C2×Dic254C2×C4semidirect
54. C2×C22⋊D127C2×C4semidirect
65. C2×C4×D127C2×C4semidirect
79. C22×C4×C127C2×C4semidirect
119. C4×D127C22⋊C4semidirect
188. C127#2311semidirect
199. #380D127semidirect
259. C22×C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168611672
C2×C4⋊C4×D1272
#225497802
#225497762
#225497542
#225497522
#225497362
#225497332
#380×C1272
#2311127