Finite Group Databasebeta

← #30641676

Subgroups — #30641676

Order 16256.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic
2C2×#16861166C211yesmaximal, characteristic, complemented
3C2×#16861165C211yesmaximal, characteristic, complemented
4#22549776C211yesmaximal, characteristic
5#22549776C211yesmaximal, characteristic
6#22549769C211yesmaximal, characteristic, complemented
7#22549768C211yesmaximal, characteristic, complemented
8#22549766C211yesmaximal, characteristic, complemented
9#22549745C211yesmaximal, characteristic, complemented
10#22549735C211yesmaximal, characteristic, complemented
11#22549734C211yesmaximal, characteristic, complemented
12C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
13#22549727C211yesmaximal, characteristic, complemented
14#22549727C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19(C22⋊D127)⋊C4C418yescomplemented
20#16861166—24no—
21#16861165—24no—
22C2×C4⋊Dic127C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×Dic127⋊C4C2211yescharacteristic
25C2×Dic127⋊C4C2211yescharacteristic
26C2×Dic127⋊C4C2211yescharacteristic
27C2×Dic127⋊C4C2211yescharacteristic
28C2×Dic127⋊C4C2211yescharacteristic
29C2×Dic127⋊C4C2211yescharacteristic
30C2×C22⋊Dic127C2211yescharacteristic
31C2×C22⋊Dic127C2211yescharacteristic
32C2×(C2×D127)⋊C4C2211yescharacteristic
33C2×(C2×D127)⋊C4C2211yescharacteristic
34C2×(C2×D127)⋊C4C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C2×C4×Dic127C2211yescharacteristic
39C2×C4×Dic127C2211yescharacteristic
40C2×C4×Dic127C2211yescharacteristic
41C2×C4×Dic127C2211yescharacteristic
42C2×C4×Dic127C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C2×C4⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50C2×C22⋊C4×C127C2211yescharacteristic
51C2×C42×C127C2211yescharacteristic
52#95×C127C2211yescharacteristic
53#95×C127C2211yescharacteristic
54C23×D127C2311yescharacteristic
55C2×C22⋊D127—22no—
56C2×C22⋊D127—22no—
57C2×C22⋊D127C2×C414yes—
58C22×Dic127—22no—
59C22×Dic127—22no—
60C22×Dic127C2×C414yes—
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C2×C4×D127—22no—
69C2×C4×D127—22no—
70C2×C4×D127C2×C414yes—
71C4⋊Dic127—41no—
72Dic127⋊C4—22no—
73Dic127⋊C4—22no—
74Dic127⋊C4—22no—
75Dic127⋊C4—22no—
76Dic127⋊C4—22no—
77Dic127⋊C4C2×C414yes—
78Dic127⋊C4D414yescomplemented
79C22⋊Dic127—22no—
80C22⋊Dic127—41no—
81(C2×D127)⋊C4—22no—
82(C2×D127)⋊C4—22no—
83(C2×D127)⋊C4—41no—
84(C2×D127)⋊C4—41no—
85(C2×D127)⋊C4C2×C414yes—
86C4×Dic127—22no—
87C4×Dic127—22no—
88C4×Dic127—22no—
89C4×Dic127—22no—
90C4×Dic127—22no—
91C4×Dic127C2×C414yes—
92C24×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C22×C4×C127C2311yescharacteristic
95C22×C4×C127C2311yescharacteristic
96C22×C4×C127C2311yescharacteristic
97C22×C4×C127C2311yescharacteristic
98C22×C4×C127C2311yescharacteristic
99C4⋊C4×C127—41no—
100C22⋊C4×C127—22no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127C2×C414yes—
103C42×C127—22no—
104C22×D127—21no—
105C22×D127—21no—
106C22×D127—21no—
107C22×D127—21no—
108C22×D127—21no—
109C22×D127—21no—
110C22×D127C22×C412yes—
111C22⋊D127—28no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—28no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127C22×C412yes—
127C2×Dic127C22×C412yes—
128C2×Dic127C22×C412yes—
129C2×Dic127C2×D412yes—
130C2×Dic127C2×D412yes—
131C2×Dic127#4112yes—
132C2×Dic127#4112yes—
133C2×Dic127#4112yes—
134C4×D127—28no—
135C23×C127—21no—
136C23×C127—21no—
137C23×C127—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127C22×C412yes—
142C23×C127C2411yescharacteristic
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×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127C22×C412yes—
159C2×C4×C127C22×C412yes—
160C2×C4×C127C2×D412yes—
161C22×C127—22no—
162C22×C127—22no—
163C22×C127—22no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127C23×C411yescharacteristic
169C22×C127C22×D411yescharacteristic
170C22×C127C2×#4111yescharacteristic
171C22×C127C2×#4111yescharacteristic
172C22×C127C2×#4111yescharacteristic
173C22×C127H(5,𝔽2)11yescharacteristic
174C22×C1272−1+411yescharacteristic
175C2×D127—22no—
176C2×D127—22no—
177C2×D127—22no—
178C2×D127—41no—
179C2×D127—41no—
180C2×D127—41no—
181C2×D127—41no—
182C4×C127—22no—
183C4×C127—22no—
184C4×C127—22no—
185C4×C127—41no—
186C4×C127—41no—
187C4×C127—41no—
188Dic127—22no—
189Dic127—22no—
190Dic127—41no—
191Dic127—41no—
192Dic127—41no—
193Dic127#11714yescomplemented
194Dic127C4×D414yes—
195C2×C127—42no—
196C2×C127C2×#11711yescharacteristic
197C2×C127C2×C4×D411yescharacteristic
198C2×C127#52011yescharacteristic
199C2×C127#54711yescharacteristic
200C2×C127#55011yescharacteristic
201C2×C127#55511yescharacteristic
202C2×C127#55611yescharacteristic
203D127—42no—
204#2354—1271nomaximal, Sylow
205C127#235411yescharacteristic, complemented, Sylow
206C2×C4×D4—1271no—
207C2×#117—1271no—
208C2×#117—1271no—
209#390—1271no—
210#390—1271no—
211#388—1271no—
212#388—1271no—
213#388—1271no—
214#388D12711yescharacteristic, complemented
215#384—1271no—
216#380—1271no—
217#380—1271no—
218C4×C4⋊C4—1271no—
219C4×C4⋊C4—1271no—
220C4×C4⋊C4—1271no—
221C22×D4—1271no—
222C23×C4—1271no—
223C23×C4—1271no—
224C4×D4—1278no—
225#117—2544no—
226#117—2544no—
227C2×C4⋊C4—1271no—
228C2×C4⋊C4—1271no—
229C2×C4⋊C4—1271no—
230C2×C4⋊C4—1271no—
231C2×C4⋊C4—1271no—
232C2×C4⋊C4—1271no—
233C2×C4⋊C4—1271no—
234C2×C4⋊C4—1271no—
235C2×C4⋊C4C2×D12711yescharacteristic
236C2×C22⋊C4—1271no—
237C2×C22⋊C4—1271no—
238C2×C22⋊C4—1271no—
239C2×C22⋊C4—1271no—
240C2×C22⋊C4—1271no—
241C2×C22⋊C4—1271no—
242C2×C22⋊C4—1271no—
243C2×C22⋊C4C2×D12711yescharacteristic
244C2×C22⋊C4C2×D12711yescharacteristic
245C2×C22⋊C4C2×D12711yescharacteristic
246C2×C42—1271no—
247C2×C42—1271no—
248C2×C42—1271no—
249C2×C42—1271no—
250C2×C42—1271no—
251C2×C42—1271no—
252C2×C42C2×D12711yescharacteristic
253#95—1271no—
254#95—1271no—
255#95—1271no—
256#95—1271no—
257#95C2×D12711yescharacteristic
258#95C2×D12711yescharacteristic
259C24—1271no—
260C24C22×D12711yescharacteristic
261C2×D4—1274no—
262C2×D4—2542no—
263C2×D4—2542no—
264C22×C4—1271no—
265C22×C4—1271no—
266C22×C4—1274no—
267C22×C4—1271no—
268C22×C4—1271no—
269C22×C4—1274no—
270C22×C4—1271no—
271C22×C4—1271no—
272C22×C4—1271no—
273C22×C4—2542no—
274C22×C4—2542no—
275C22×C4—2542no—
276C22×C4—2542no—
277C22×C4C22×D12711yescharacteristic
278C22×C4C22×D12711yescharacteristic
279C22×C4C22×D12711yescharacteristic
280C22×C4C22×D12711yescharacteristic
281C22×C4C22×D12711yescharacteristic
282C22×C4C22×D12711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—1274no—
285C4⋊C4—1274no—
286C4⋊C4—5081no—
287C4⋊C4—2542no—
288C4⋊C4—2542no—
289C4⋊C4—2542no—
290C4⋊C4—2542no—
291C4⋊C4—2542no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—1274no—
295C22⋊C4—5081no—
296C22⋊C4—5081no—
297C22⋊C4—5081no—
298C22⋊C4—2542no—
299C22⋊C4—2542no—
300C22⋊C4—2542no—
301C22⋊C4C4×D12714yes—
302C42—22no—
303C42—1274no—
304C42—2542no—
305C42—2542no—
306C42—2542no—
307C42—2542no—
308C42—2542no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—1272no—
316C23—2541no—
317C23—2541no—
318C23—2541no—
319C23—2541no—
320C23—2541no—
321C23—2541no—
322C23C2×C4×D12712yes—
323C23C23×D12711yescharacteristic
324D4—2548no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—1272no—
341C2×C4—1272no—
342C2×C4—1272no—
343C2×C4—1272no—
344C2×C4—1272no—
345C2×C4—1272no—
346C2×C4—1272no—
347C2×C4—1272no—
348C2×C4—2548no—
349C2×C4—2548no—
350C2×C4—2541no—
351C2×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4—2541no—
357C2×C4—2541no—
358C2×C4—2541no—
359C2×C4—2541no—
360C2×C4—2541no—
361C2×C4—2541no—
362C2×C4—2541no—
363C2×C4C2×C4×D12712yes—
364C2×C4C2×C4×D12712yes—
365C2×C4D4×D12712yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—5081no—
374C22—5081no—
375C22—5081no—
376C22—5081no—
377C22—2542no—
378C22—2542no—
379C22—2542no—
380C22C22×C4×D12711yescharacteristic
381C22C2×D4×D12711yescharacteristic
382C22#41×D12711yescharacteristic
383C22#41×D12711yescharacteristic
384C22#41×D12711yescharacteristic
385C22#1686125411yescharacteristic
386C22#1686125611yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—1274no—
394C4—1274no—
395C4—5081no—
396C4—5081no—
397C4—5081no—
398C4—2542no—
399C4—2542no—
400C2—42no—
401C2—5082no—
402C2#117×D12711yescharacteristic
403C2C4×D4×D12711yescharacteristic
404C2#2255035511yescharacteristic
405C2#2255052411yescharacteristic
406C2#2255054411yescharacteristic
407C2#2255058811yescharacteristic
408C2#2255059311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC22×C22⋊D127C22
hypercenter#388D127
hypocenterC127#2354
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#16861166C2semidirect
3. C2×#16861165C2semidirect
6. #22549769C2semidirect
7. #22549768C2semidirect
8. #22549766C2semidirect
9. #22549745C2semidirect
10. #22549735C2semidirect
11. #22549734C2semidirect
12. C4×Dic127⋊C4C2semidirect
13. #22549727C2semidirect
14. #22549727C2semidirect
15. #388×C127C2semidirect
19. (C22⋊D127)⋊C4C4semidirect
78. Dic127⋊C4D4semidirect
193. Dic127#117semidirect
205. C127#2354semidirect
214. #388D127semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D127)⋊C42
C2×#168611662
C2×#168611652
#225497762
#225497692
#225497682
#225497662
#225497452
#225497352
#225497342
C4×Dic127⋊C42
#225497272
#388×C1272
#2354127