Finite Group Databasebeta

← #30641668

Subgroups — #30641668

Order 16256.

588 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic127C211yesmaximal, characteristic
2#22549770C211yesmaximal, characteristic, complemented
3#22549764C211yesmaximal, characteristic, complemented
4#22549761C211yesmaximal, characteristic, complemented
5#22549759C211yesmaximal, characteristic, complemented
6#22549756C211yesmaximal, characteristic
7#22549743C211yesmaximal, characteristic, complemented
8#22549737C211yesmaximal, characteristic, complemented
9#22549734C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
12C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
13#22549721C211yesmaximal, characteristic, complemented
14C2×C127⋊#95C211yesmaximal, characteristic, complemented
15#386×C127C211yesmaximal, characteristic
16C23×Dic127C2211yescharacteristic
17C23×Dic127C2211yescharacteristic
18C23×Dic127C2211yescharacteristic
19D4×Dic127C418yescomplemented
20C2×C4⋊Dic127C2211yescharacteristic
21C2×Dic127⋊C4C2211yescharacteristic
22C2×Dic127⋊C4C2211yescharacteristic
23C2×C22⋊Dic127C2211yescharacteristic
24C2×C22⋊Dic127C2211yescharacteristic
25C2×C22⋊Dic127C2211yescharacteristic
26C2×C22⋊Dic127C2211yescharacteristic
27C2×C22⋊Dic127C2211yescharacteristic
28C2×C4×Dic127C2211yescharacteristic
29C2×C4×Dic127C2211yescharacteristic
30C2×C4×Dic127C2211yescharacteristic
31C2×C4×Dic127C2211yescharacteristic
32C2×C4×Dic127C2211yescharacteristic
33C127⋊#95C2211yescharacteristic
34C127⋊#95C2211yescharacteristic
35C127⋊#95C2211yescharacteristic
36C127⋊#95C2211yescharacteristic
37C127⋊#95C2211yescharacteristic
38C127⋊#95C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C22×D4×C127C2211yescharacteristic
46C23×C4×C127C2211yescharacteristic
47C2×C22⋊C4×C127C2211yescharacteristic
48C2×C22⋊C4×C127C2211yescharacteristic
49C2×C22⋊C4×C127C2211yescharacteristic
50#95×C127C2211yescharacteristic
51#95×C127C2211yescharacteristic
52C22×Dic127—22no—
53C22×Dic127—22no—
54C22×Dic127—41no—
55C22×Dic127—41no—
56C22×Dic127—41no—
57C22×Dic127—41no—
58C22×Dic127—41no—
59C22×Dic127C2×C414yes—
60C22×Dic127C2×C414yescomplemented
61C22×Dic127C2311yescharacteristic
62C22×Dic127C2311yescharacteristic
63C22×Dic127C2311yescharacteristic
64C22×Dic127C2311yescharacteristic
65C22×Dic127C2311yescharacteristic
66C22×Dic127C2311yescharacteristic
67C22×Dic127C2311yescharacteristic
68C22×Dic127C2311yescharacteristic
69C4⋊Dic127C2×C414yescomplemented
70Dic127⋊C4—41no—
71Dic127⋊C4—41no—
72C22⋊Dic127—22no—
73C22⋊Dic127—22no—
74C22⋊Dic127—41no—
75C22⋊Dic127C2×C414yes—
76C22⋊Dic127C2×C414yescomplemented
77C4×Dic127—22no—
78C4×Dic127—22no—
79C4×Dic127—41no—
80C4×Dic127—41no—
81C4×Dic127C2×C414yescomplemented
82C24×C127C2311yescharacteristic
83C24×C127C2311yescharacteristic
84C2×D4×C127—22no—
85C2×D4×C127—22no—
86C2×D4×C127C2×C414yes—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—41no—
90C22×C4×C127C2311yescharacteristic
91C22×C4×C127C2311yescharacteristic
92C22×C4×C127C2311yescharacteristic
93C22×C4×C127C2311yescharacteristic
94C22×C4×C127C2311yescharacteristic
95C22⋊C4×C127—22no—
96C22⋊C4×C127—22no—
97C22⋊C4×C127—41no—
98C2×Dic127—21no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—21no—
105C2×Dic127—21no—
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—44no—
118C2×Dic127—44no—
119C2×Dic127—44no—
120C2×Dic127C22×C412yes—
121C2×Dic127C22×C412yes—
122C2×Dic127C22×C412yes—
123C2×Dic127C22×C412yes—
124C2×Dic127#4112yes—
125C23×C127—21no—
126C23×C127—21no—
127C23×C127—21no—
128C23×C127—21no—
129C23×C127—21no—
130C23×C127—21no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—21no—
135C23×C127C22×C412yes—
136C23×C127C22×C412yes—
137C23×C127#4112yes—
138C23×C127#4112yes—
139C23×C127C2411yescharacteristic
140D4×C127—28no—
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×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—44no—
155C2×C4×C127C22×C412yes—
156C2×C4×C127#4112yes—
157C22×C127—22no—
158C22×C127—22no—
159C22×C127—22no—
160C22×C127—22no—
161C22×C127—22no—
162C22×C127—22no—
163C22×C127—22no—
164C22×C127—22no—
165C22×C127—22no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127#11714yes—
171C22×C127C23×C411yescharacteristic
172C22×C127C2×#4111yescharacteristic
173C22×C127C2×#4111yescharacteristic
174C22×C127C2×#4111yescharacteristic
175C22×C127C2×#4111yescharacteristic
176C22×C127H(5,𝔽2)11yescharacteristic
177C22×C127H(5,𝔽2)11yescharacteristic
178C4×C127—22no—
179C4×C127—22no—
180C4×C127—41no—
181C4×C127—41no—
182C4×C127—41no—
183Dic127—22no—
184Dic127—22no—
185Dic127—22no—
186Dic127—22no—
187Dic127—22no—
188Dic127—41no—
189Dic127—41no—
190Dic127—41no—
191C2×C127—24no—
192C2×C127—42no—
193C2×C127C2×#11711yescharacteristic
194C2×C127C4×#4111yescharacteristic
195C2×C127#51811yescharacteristic
196C2×C127#55111yescharacteristic
197C2×C127#55111yescharacteristic
198C2×C127#55311yescharacteristic
199C2×C127#55311yescharacteristic
200#2353—1271nomaximal, Sylow
201C127#235311yescharacteristic, complemented, Sylow
202C2×C4×D4—1271no—
203#388—1271no—
204#388—1271no—
205#387—1271no—
206#387—1271no—
207#386—1271no—
208#386D12711yescharacteristic
209#383—1271no—
210#381—1271no—
211#380—1271no—
212C4×C22⋊C4—1271no—
213C4×C22⋊C4—1271no—
214C4×C22⋊C4—1271no—
215#376—1271no—
216C2×#95—1271no—
217C22×D4C2×D12711yescharacteristic
218C23×C4—1271no—
219C23×C4—1271no—
220C23×C4—1271no—
221C23×C4C2×D12711yescharacteristic
222C4×D4—1278no—
223C2×C4⋊C4—1271no—
224C2×C4⋊C4—1271no—
225C2×C4⋊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—
239#95—1271no—
240#95—1271no—
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#95—1271no—
250#95—1271no—
251#95C2×D12711yescharacteristic
252#95C2×D12711yescharacteristic
253C24C22×D12711yescharacteristic
254C24C22×D12711yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C4×D12714yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—1274no—
262C22×C4—1274no—
263C22×C4—1271no—
264C22×C4—1271no—
265C22×C4—1271no—
266C22×C4—1271no—
267C22×C4—1271no—
268C22×C4—1271no—
269C22×C4—1271no—
270C22×C4—1271no—
271C22×C4—5081no—
272C22×C4—5081no—
273C22×C4—5081no—
274C22×C4—5081no—
275C22×C4—5081no—
276C22×C4—2542no—
277C22×C4—2542no—
278C22×C4C22×D12711yescharacteristic
279C22×C4C22×D12711yescharacteristic
280C22×C4C22×D12711yescharacteristic
281C22×C4C22×D12711yescharacteristic
282C22×C4C22×D12711yescharacteristic
283C4⋊C4—1274no—
284C4⋊C4—5081no—
285C4⋊C4—5081no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—41no—
289C22⋊C4—1274no—
290C22⋊C4—1274no—
291C22⋊C4—5081no—
292C22⋊C4—2542no—
293C22⋊C4—2542no—
294C42—1274no—
295C42—5081no—
296C42—5081no—
297C42—2542no—
298C42—2542no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C4×D12712yes—
310C23C2×C4×D12712yes—
311C23#1452324712yes—
312C23#1452324712yes—
313C23C23×D12711yescharacteristic
314D4—28no—
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—44no—
329C2×C4—1272no—
330C2×C4—1272no—
331C2×C4—1272no—
332C2×C4—1272no—
333C2×C4—1272no—
334C2×C4—5084no—
335C2×C4—5084no—
336C2×C4—5084no—
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×C4—2541no—
352C2×C4—2541no—
353C2×C4—2541no—
354C2×C4—2541no—
355C2×C4—2541no—
356C2×C4C2×C4×D12712yes—
357C2×C4#1452324712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#1686116514yes—
372C22C22×C4×D12711yescharacteristic
373C22#41×D12711yescharacteristic
374C22C2×#1452324711yescharacteristic
375C22C2×#1452324711yescharacteristic
376C22C2×#1452324711yescharacteristic
377C22#1686125411yescharacteristic
378C22#1686125511yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—5081no—
385C4—5081no—
386C4—5081no—
387C4—2542no—
388C4—2542no—
389C4—2542no—
390C4—2542no—
391C4—2542no—
392C2—24no—
393C2—42no—
394C2C2×#1686116511yescharacteristic
395C2C4×#1452324711yescharacteristic
396C2#2255034811yescharacteristic
397C2#2255054911yescharacteristic
398C2#2255055711yescharacteristic
399C2#2255056911yescharacteristic
400C2#2255057811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#386×C127C2
socle, cosocleC23×C127C24
prime radicalC22×D4×C127C22
hypercenter#386D127
hypocenterC127#2353
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #22549770C2semidirect
3. #22549764C2semidirect
4. #22549761C2semidirect
5. #22549759C2semidirect
7. #22549743C2semidirect
8. #22549737C2semidirect
9. #22549734C2semidirect
10. C22⋊C4×Dic127C2semidirect
11. C22⋊C4×Dic127C2semidirect
12. C4×C22⋊Dic127C2semidirect
13. #22549721C2semidirect
14. C2×C127⋊#95C2semidirect
19. D4×Dic127C4semidirect
60. C22×Dic127C2×C4semidirect
69. C4⋊Dic127C2×C4semidirect
76. C22⋊Dic127C2×C4semidirect
81. C4×Dic127C2×C4semidirect
201. C127#2353semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
#225497702
#225497642
#225497612
#225497592
#225497562
#225497432
#225497372
#225497342
C22⋊C4×Dic1272
C4×C22⋊Dic1272
#225497212
C2×C127⋊#952
#386×C1272
#2353127