Finite Group Databasebeta

← #30641700

Subgroups — #30641700

Order 16256.

560 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic
2#22549776C211yesmaximal, characteristic
3#22549776C211yesmaximal, characteristic
4#22549769C211yesmaximal, characteristic, complemented
5#22549768C211yesmaximal, characteristic, complemented
6#22549766C211yesmaximal, characteristic, complemented
7#22549740C211yesmaximal, characteristic, complemented
8#22549738C211yesmaximal, characteristic, complemented
9#22549737C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic127C211yesmaximal, characteristic, complemented
11C4×Dic127⋊C4C211yesmaximal, characteristic, complemented
12#22549727C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
14C4×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
15#388×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18C22×C4×D127C2211yescharacteristic
19(C22⋊D127)⋊C4C418yescomplemented
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×Dic127⋊C4C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×(C2×D127)⋊C4C2211yescharacteristic
27C2×(C2×D127)⋊C4C2211yescharacteristic
28C2×(C2×D127)⋊C4C2211yescharacteristic
29C2×(C2×D127)⋊C4C2211yescharacteristic
30C2×(C2×D127)⋊C4C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C2×C4×Dic127C2211yescharacteristic
34C2×C4×Dic127C2211yescharacteristic
35C2×C4×Dic127C2211yescharacteristic
36C2×C4×Dic127C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C2×C4⋊C4×C127C2211yescharacteristic
46C2×C22⋊C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C42×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C23×D127C2311yescharacteristic
53C2×C22⋊D127—22no—
54C2×C22⋊D127—22no—
55C2×C22⋊D127C2×C414yes—
56C22×Dic127—41no—
57C22×Dic127—41no—
58C22×Dic127C2×C414yes—
59C22×Dic127C2311yescharacteristic
60C22×Dic127C2311yescharacteristic
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C2×C4×D127—41no—
67C2×C4×D127—41no—
68C2×C4×D127C2×C414yes—
69C4⋊Dic127—22no—
70Dic127⋊C4—22no—
71Dic127⋊C4—22no—
72Dic127⋊C4C2×C414yes—
73C22⋊Dic127—22no—
74C22⋊Dic127—41no—
75(C2×D127)⋊C4—22no—
76(C2×D127)⋊C4—22no—
77(C2×D127)⋊C4—41no—
78(C2×D127)⋊C4—41no—
79(C2×D127)⋊C4C2×C414yes—
80C4×Dic127—22no—
81C4×Dic127—22no—
82C4×Dic127—41no—
83C4×Dic127—41no—
84C4×Dic127—41no—
85C4×Dic127C2×C414yes—
86C24×C127C2311yescharacteristic
87C22×C4×C127C2311yescharacteristic
88C22×C4×C127C2311yescharacteristic
89C22×C4×C127C2311yescharacteristic
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C4⋊C4×C127—22no—
94C22⋊C4×C127—22no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127C2×C414yes—
97C42×C127—41no—
98C22×D127—21no—
99C22×D127—21no—
100C22×D127—21no—
101C22×D127—21no—
102C22×D127—21no—
103C22×D127—21no—
104C22×D127C22×C412yes—
105C22⋊D127—28no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C2×Dic127—21no—
109C2×Dic127—21no—
110C2×Dic127—21no—
111C2×Dic127—21no—
112C2×Dic127—21no—
113C2×Dic127—21no—
114C2×Dic127—21no—
115C2×Dic127—21no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—44no—
122C2×Dic127C22×C412yes—
123C2×Dic127C22×C412yes—
124C2×Dic127C22×C412yes—
125C2×Dic127#4112yes—
126C2×Dic127#4112yes—
127C2×Dic127#4112yes—
128C4×D127—44no—
129C23×C127—21no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127C22×C412yes—
136C23×C127C2411yescharacteristic
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
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×C127C22×C412yes—
153C2×C4×C127C22×C412yes—
154C2×C4×C127#4112yes—
155C22×C127—22no—
156C22×C127—22no—
157C22×C127—22no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—41no—
161C22×C127—41no—
162C22×C127C23×C411yescharacteristic
163C22×C127C2×#4111yescharacteristic
164C22×C127C2×#4111yescharacteristic
165C22×C127C2×#4111yescharacteristic
166C22×C127C2×#4111yescharacteristic
167C22×C127H(5,𝔽2)11yescharacteristic
168C22×C127H(5,𝔽2)11yescharacteristic
169C2×D127—22no—
170C2×D127—22no—
171C2×D127—22no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—41no—
175C2×D127—41no—
176C4×C127—22no—
177C4×C127—22no—
178C4×C127—22no—
179C4×C127—41no—
180C4×C127—41no—
181C4×C127—41no—
182Dic127—22no—
183Dic127—22no—
184Dic127—22no—
185Dic127—41no—
186Dic127—41no—
187Dic127—41no—
188Dic127#11714yes—
189C2×C127—42no—
190C2×C127C2×#11711yescharacteristic
191C2×C127C4×#4111yescharacteristic
192C2×C127#51811yescharacteristic
193C2×C127#55311yescharacteristic
194C2×C127#55311yescharacteristic
195C2×C127#55911yescharacteristic
196C2×C127#55911yescharacteristic
197D127—42no—
198#2357—1271nomaximal, Sylow
199C127#235711yescharacteristic, complemented, Sylow
200C2×C4×D4—1271no—
201#390—1271no—
202#390—1271no—
203#388—1271no—
204#388—1271no—
205#388—1271no—
206#388D12711yescharacteristic, complemented
207#382—1271no—
208#381—1271no—
209#381—1271no—
210C4×C4⋊C4—1271no—
211C4×C4⋊C4—1271no—
212C4×C4⋊C4—1271no—
213C4×C22⋊C4—1271no—
214C4×C22⋊C4—1271no—
215C22×D4—1271no—
216C23×C4—1271no—
217C23×C4—1271no—
218C4×D4—1278no—
219C2×C4⋊C4—1271no—
220C2×C4⋊C4—1271no—
221C2×C4⋊C4—1271no—
222C2×C4⋊C4—1271no—
223C2×C4⋊C4C2×D12711yescharacteristic
224C2×C22⋊C4—1271no—
225C2×C22⋊C4—1271no—
226C2×C22⋊C4—1271no—
227C2×C22⋊C4—1271no—
228C2×C22⋊C4—1271no—
229C2×C22⋊C4—1271no—
230C2×C22⋊C4—1271no—
231C2×C22⋊C4C2×D12711yescharacteristic
232C2×C22⋊C4C2×D12711yescharacteristic
233C2×C22⋊C4C2×D12711yescharacteristic
234C2×C42—1271no—
235C2×C42—1271no—
236C2×C42—1271no—
237C2×C42—1271no—
238C2×C42—1271no—
239C2×C42—1271no—
240C2×C42C2×D12711yescharacteristic
241#95—1271no—
242#95—1271no—
243#95—1271no—
244#95—1271no—
245#95—1271no—
246#95—1271no—
247#95—1271no—
248#95—1271no—
249#95C2×D12711yescharacteristic
250#95C2×D12711yescharacteristic
251C24—1271no—
252C24C22×D12711yescharacteristic
253C2×D4—1274no—
254C2×D4—2542no—
255C2×D4—2542no—
256C22×C4—1271no—
257C22×C4—1274no—
258C22×C4—1271no—
259C22×C4—1271no—
260C22×C4—1271no—
261C22×C4—1274no—
262C22×C4—1271no—
263C22×C4—1271no—
264C22×C4—1271no—
265C22×C4—5081no—
266C22×C4—5081no—
267C22×C4—5081no—
268C22×C4—5081no—
269C22×C4C22×D12711yescharacteristic
270C22×C4C22×D12711yescharacteristic
271C22×C4C22×D12711yescharacteristic
272C22×C4C22×D12711yescharacteristic
273C22×C4C22×D12711yescharacteristic
274C22×C4C22×D12711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—1274no—
277C4⋊C4—2542no—
278C4⋊C4—2542no—
279C4⋊C4—2542no—
280C22⋊C4—22no—
281C22⋊C4—41no—
282C22⋊C4—1274no—
283C22⋊C4—5081no—
284C22⋊C4—5081no—
285C22⋊C4—5081no—
286C22⋊C4—2542no—
287C22⋊C4—2542no—
288C22⋊C4—2542no—
289C22⋊C4C4×D12714yes—
290C42—41no—
291C42—1274no—
292C42—5081no—
293C42—5081no—
294C42—5081no—
295C42—2542no—
296C42—2542no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—1272no—
304C23—2541no—
305C23—2541no—
306C23—2541no—
307C23—2541no—
308C23—2541no—
309C23—2541no—
310C23C2×C4×D12712yes—
311C23C23×D12711yescharacteristic
312D4—2548no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—1272no—
329C2×C4—1272no—
330C2×C4—1272no—
331C2×C4—1272no—
332C2×C4—1272no—
333C2×C4—1272no—
334C2×C4—5084no—
335C2×C4—5084no—
336C2×C4—2541no—
337C2×C4—2541no—
338C2×C4—2541no—
339C2×C4—2541no—
340C2×C4—2541no—
341C2×C4—2541no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4—2541no—
349C2×C4—2541no—
350C2×C4—2541no—
351C2×C4C2×C4×D12712yes—
352C2×C4C2×C4×D12712yes—
353C2×C4#1452324712yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—5081no—
362C22—5081no—
363C22—5081no—
364C22—5081no—
365C22—2542no—
366C22—2542no—
367C22—2542no—
368C22C22×C4×D12711yescharacteristic
369C22#41×D12711yescharacteristic
370C22#41×D12711yescharacteristic
371C22#41×D12711yescharacteristic
372C22C2×#1452324711yescharacteristic
373C22#1686125411yescharacteristic
374C22#1686125511yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1274no—
382C4—5081no—
383C4—5081no—
384C4—5081no—
385C4—2542no—
386C4—2542no—
387C4—2542no—
388C2—42no—
389C2—5082no—
390C2#117×D12711yescharacteristic
391C2C4×#1452324711yescharacteristic
392C2#2255034811yescharacteristic
393C2#2255056911yescharacteristic
394C2#2255057711yescharacteristic
395C2#2255060811yescharacteristic
396C2#2255061011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#388×C127C2
socle, cosocleC23×C127C24
prime radicalC22×C22⋊D127C22
hypercenter#388D127
hypocenterC127#2357
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
4. #22549769C2semidirect
5. #22549768C2semidirect
6. #22549766C2semidirect
7. #22549740C2semidirect
8. #22549738C2semidirect
9. #22549737C2semidirect
10. C4⋊C4×Dic127C2semidirect
11. C4×Dic127⋊C4C2semidirect
12. #22549727C2semidirect
13. C22⋊C4×Dic127C2semidirect
14. C4×(C2×D127)⋊C4C2semidirect
15. #388×C127C2semidirect
19. (C22⋊D127)⋊C4C4semidirect
199. C127#2357semidirect
206. #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
#225497762
#225497692
#225497682
#225497662
#225497402
#225497382
#225497372
C4⋊C4×Dic1272
C4×Dic127⋊C42
#225497272
C22⋊C4×Dic1272
C4×(C2×D127)⋊C42
#388×C1272
#2357127