Finite Group Databasebeta

← #30641556

Subgroups — #30641556

Order 16256.

928 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523247C211yesmaximal, characteristic
2C2×C4×C22⋊D127C211yesmaximal, characteristic, complemented
3C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic, complemented
4#22549780C211yesmaximal, characteristic, complemented
5#22549780C211yesmaximal, characteristic, complemented
6#22549756C211yesmaximal, characteristic, complemented
7#22549752C211yesmaximal, characteristic, complemented
8#22549752C211yesmaximal, characteristic, complemented
9#22549738C211yesmaximal, characteristic, complemented
10#22549737C211yesmaximal, characteristic, complemented
11#22549734C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic127C211yesmaximal, characteristic, complemented
13C2×C127⋊#95C211yesmaximal, characteristic, complemented
14#95×D127C211yesmaximal, characteristic, complemented
15#386×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—24no—
17C2×#14523247—24no—
18C2×#14523247C418yescomplemented
19C22×Dic254C2211yescharacteristic
20C22×C22⋊D127C2211yescharacteristic
21C22×C22⋊D127C2211yescharacteristic
22C23×Dic127C2211yescharacteristic
23C23×Dic127C2211yescharacteristic
24C23×Dic127C2211yescharacteristic
25C22×C4×D127C2211yescharacteristic
26C22×C4×D127C2211yescharacteristic
27C22×C4×D127C2211yescharacteristic
28C4×C22⋊D127—24no—
29(C22⋊D127)⋊C4—24no—
30C2×Dic127⋊C4C2211yescharacteristic
31C2×Dic127⋊C4C2211yescharacteristic
32C2×C22⋊Dic127C2211yescharacteristic
33C2×C22⋊Dic127C2211yescharacteristic
34C2×C22⋊Dic127C2211yescharacteristic
35C2×(C2×D127)⋊C4C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic
37C2×C4×Dic127C2211yescharacteristic
38C2×C4×Dic127C2211yescharacteristic
39C127⋊#95C2211yescharacteristic
40C127⋊#95C2211yescharacteristic
41C127⋊#95C2211yescharacteristic
42C127⋊#95C2211yescharacteristic
43C127⋊#95C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C127⋊#95C2211yescharacteristic
48C127⋊#95C2211yescharacteristic
49C22×D4×C127C2211yescharacteristic
50C23×C4×C127C2211yescharacteristic
51C2×C22⋊C4×C127C2211yescharacteristic
52C2×C22⋊C4×C127C2211yescharacteristic
53C2×C22⋊C4×C127C2211yescharacteristic
54#95×C127C2211yescharacteristic
55#95×C127C2211yescharacteristic
56C23×D127C2311yescharacteristic
57#14523247—232no—
58C2×Dic254—22no—
59C2×Dic254—22no—
60C2×Dic254C2×C414yescomplemented
61C2×C22⋊D127—22no—
62C2×C22⋊D127—22no—
63C2×C22⋊D127C2×C414yes—
64C2×C22⋊D127C2×C414yescomplemented
65C2×C22⋊D127D414yes—
66C2×C22⋊D127D414yes—
67C22×Dic127—22no—
68C22×Dic127—22no—
69C22×Dic127—22no—
70C22×Dic127—22no—
71C22×Dic127—22no—
72C22×Dic127—22no—
73C22×Dic127—41no—
74C22×Dic127C2×C414yescomplemented
75C22×Dic127C2×C414yescomplemented
76C22×Dic127C2311yescharacteristic
77C22×Dic127C2311yescharacteristic
78C22×Dic127C2311yescharacteristic
79C22×Dic127C2311yescharacteristic
80C22×Dic127C2311yescharacteristic
81C22×Dic127C2311yescharacteristic
82C22×Dic127C2311yescharacteristic
83C2×C4×D127—22no—
84C2×C4×D127—22no—
85C2×C4×D127—22no—
86C2×C4×D127—22no—
87C2×C4×D127—22no—
88C2×C4×D127—22no—
89C2×C4×D127—41no—
90C2×C4×D127—41no—
91C2×C4×D127C2×C414yescomplemented
92Dic127⋊C4—22no—
93Dic127⋊C4—22no—
94C22⋊Dic127—22no—
95C22⋊Dic127—22no—
96C22⋊Dic127—41no—
97(C2×D127)⋊C4—22no—
98(C2×D127)⋊C4—22no—
99C4×Dic127—22no—
100C4×Dic127—22no—
101C24×C127C2311yescharacteristic
102C24×C127C2311yescharacteristic
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127C2×C414yescomplemented
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—41no—
109C22×C4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114C22⋊C4×C127—22no—
115C22⋊C4×C127—22no—
116C22⋊C4×C127—41no—
117C22×D127—21no—
118C22×D127—21no—
119C22×D127—21no—
120C22×D127—21no—
121C22×D127C22×C412yes—
122C22×D127C2×D412yes—
123C22×D127C2×D412yes—
124Dic254—28no—
125C22⋊D127—28no—
126C22⋊D127C22⋊C4116yescomplemented
127C2×Dic127—28no—
128C2×Dic127—21no—
129C2×Dic127—21no—
130C2×Dic127—21no—
131C2×Dic127—21no—
132C2×Dic127—21no—
133C2×Dic127—21no—
134C2×Dic127—28no—
135C2×Dic127—21no—
136C2×Dic127—21no—
137C2×Dic127—21no—
138C2×Dic127—21no—
139C2×Dic127—21no—
140C2×Dic127—21no—
141C2×Dic127—21no—
142C2×Dic127—21no—
143C2×Dic127—21no—
144C2×Dic127—44no—
145C2×Dic127C22×C412yes—
146C2×Dic127C22×C412yes—
147C2×Dic127C22×C412yes—
148C2×Dic127C2×D412yes—
149C2×Dic127C2×D412yes—
150C2×Dic127#4112yescomplemented
151C4×D127—28no—
152C4×D127—44no—
153C4×D127—44no—
154C23×C127—21no—
155C23×C127—21no—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127—21no—
162C23×C127—21no—
163C23×C127—21no—
164C23×C127C22×C412yes—
165C23×C127C22×C412yes—
166C23×C127C2×D412yes—
167C23×C127C2×D412yes—
168C23×C127C2411yescharacteristic
169D4×C127—28no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127—21no—
178C2×C4×C127—21no—
179C2×C4×C127—21no—
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—21no—
183C2×C4×C127—44no—
184C2×C4×C127C22×C412yes—
185C2×C4×C127#4112yescomplemented
186C22×C127—22no—
187C22×C127—22no—
188C22×C127—22no—
189C22×C127—22no—
190C22×C127—22no—
191C22×C127—22no—
192C22×C127—22no—
193C22×C127—22no—
194C22×C127—22no—
195C22×C127—41no—
196C22×C127—41no—
197C22×C127—41no—
198C22×C127—41no—
199C22×C127C2×C22⋊C414yes—
200C22×C127C23×C411yescharacteristic
201C22×C127C22×D411yescharacteristic
202C22×C127C22×D411yescharacteristic
203C22×C127C2×#4111yescharacteristic
204C22×C127C2×#4111yescharacteristic
205C22×C127H(5,𝔽2)11yescharacteristic
206C22×C1272−1+411yescharacteristic
207C2×D127—22no—
208C2×D127—22no—
209C2×D127—22no—
210C2×D127—22no—
211C2×D127—22no—
212C2×D127—22no—
213C2×D127C2×C22⋊C414yes—
214C4×C127—22no—
215C4×C127—22no—
216C4×C127—41no—
217C4×C127—41no—
218C4×C127—41no—
219Dic127—22no—
220Dic127—22no—
221Dic127—22no—
222Dic127—41no—
223Dic127—41no—
224Dic127—41no—
225Dic127C2×C22⋊C414yes—
226C2×C127—24no—
227C2×C127—42no—
228C2×C127C22×C22⋊C411yescharacteristic
229C2×C127C4×#4111yescharacteristic
230C2×C127#52011yescharacteristic
231C2×C127#54611yescharacteristic
232C2×C127#54611yescharacteristic
233C2×C127#54711yescharacteristic
234C2×C127#54711yescharacteristic
235D127—24no—
236#2339—1271nomaximal, Sylow
237C127#233911yescharacteristic, complemented, Sylow
238C22×#41—1271no—
239C2×C4×D4—1271no—
240C2×C4×D4—1271no—
241#391—1271no—
242#391—1271no—
243#386—1271no—
244#386—1271no—
245#386—1271no—
246#386D12711yescharacteristic, complemented
247#381—1271no—
248#381—1271no—
249#380—1271no—
250C4×C22⋊C4—1271no—
251C2×#95—1271no—
252C2×#95—1271no—
253C2×#41—1278no—
254C2×#41—2544no—
255C2×#41—2544no—
256C22×Q8—1271no—
257C22×D4—1271no—
258C22×D4—1271no—
259C22×D4C2×D12711yescharacteristic
260C23×C4—1271no—
261C23×C4—1271no—
262C23×C4—1271no—
263C23×C4—1271no—
264C23×C4—1271no—
265C23×C4—1271no—
266C23×C4C2×D12711yescharacteristic
267C4×D4—2544no—
268C4×D4—2544no—
269C2×C4⋊C4—1271no—
270C2×C4⋊C4—1271no—
271C2×C22⋊C4—1271no—
272C2×C22⋊C4—1271no—
273C2×C22⋊C4—1271no—
274C2×C22⋊C4—1271no—
275C2×C22⋊C4—1271no—
276C2×C22⋊C4C2×D12711yescharacteristic
277C2×C22⋊C4C2×D12711yescharacteristic
278C2×C22⋊C4C2×D12711yescharacteristic
279C2×C42—1271no—
280C2×C42—1271no—
281#95—1271no—
282#95—1271no—
283#95—1271no—
284#95—1271no—
285#95—1271no—
286#95—1271no—
287#95—1271no—
288#95—1271no—
289#95—1271no—
290#95—1271no—
291#95C2×D12711yescharacteristic
292#95C2×D12711yescharacteristic
293C24—1271no—
294C24C22×D12711yescharacteristic
295C24C22×D12711yescharacteristic
296#41—25432no—
297C2×Q8—1274no—
298C2×Q8—2542no—
299C2×Q8—2542no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—1274no—
303C2×D4—1274no—
304C2×D4—1274no—
305C2×D4—1274no—
306C2×D4—2542no—
307C2×D4—2542no—
308C2×D4C4×D12714yescomplemented
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—1271no—
313C22×C4—1274no—
314C22×C4—1274no—
315C22×C4—1271no—
316C22×C4—1271no—
317C22×C4—1271no—
318C22×C4—1274no—
319C22×C4—1271no—
320C22×C4—1271no—
321C22×C4—1271no—
322C22×C4—5081no—
323C22×C4—5081no—
324C22×C4—5081no—
325C22×C4—2542no—
326C22×C4—2542no—
327C22×C4—2542no—
328C22×C4—2542no—
329C22×C4—2542no—
330C22×C4—2542no—
331C22×C4—2542no—
332C22×C4—2542no—
333C22×C4—2542no—
334C22×C4—2542no—
335C22×C4—2542no—
336C22×C4—2542no—
337C22×C4C22×D12711yescharacteristic
338C22×C4C22×D12711yescharacteristic
339C22×C4C22×D12711yescharacteristic
340C22×C4C22×D12711yescharacteristic
341C22×C4C22×D12711yescharacteristic
342C4⋊C4—2542no—
343C4⋊C4—2542no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—41no—
347C22⋊C4—5081no—
348C22⋊C4—2542no—
349C22⋊C4—2542no—
350C22⋊C4—2542no—
351C22⋊C4—2542no—
352C42—2542no—
353C42—2542no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—1272no—
365C23—1272no—
366C23—1272no—
367C23—2541no—
368C23—2541no—
369C23—2541no—
370C23—2541no—
371C23C2×C4×D12712yes—
372C23C2×C4×D12712yes—
373C23D4×D12712yes—
374C23D4×D12712yes—
375C23C23×D12711yescharacteristic
376Q8—2548no—
377D4—28no—
378D4—12716no—
379D4—2548no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—44no—
394C2×C4—1272no—
395C2×C4—1272no—
396C2×C4—1272no—
397C2×C4—1272no—
398C2×C4—1272no—
399C2×C4—1272no—
400C2×C4—5084no—
401C2×C4—5084no—
402C2×C4—5084no—
403C2×C4—2548no—
404C2×C4—2548no—
405C2×C4—2541no—
406C2×C4—2541no—
407C2×C4—2541no—
408C2×C4—2541no—
409C2×C4—2541no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4—2541no—
414C2×C4—2541no—
415C2×C4—2541no—
416C2×C4—2541no—
417C2×C4—2541no—
418C2×C4—2541no—
419C2×C4—2541no—
420C2×C4—2548no—
421C2×C4C2×C4×D12712yes—
422C2×C4#1452324712yescomplemented
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—1274no—
437C22—2542no—
438C22—2542no—
439C22—2542no—
440C22—2542no—
441C22—2542no—
442C22—2542no—
443C22C22⋊C4×D12714yes—
444C22C22×C4×D12711yescharacteristic
445C22C2×D4×D12711yescharacteristic
446C22C2×D4×D12711yescharacteristic
447C22#41×D12711yescharacteristic
448C22C2×#1452324711yescharacteristic
449C22#1686125411yescharacteristic
450C22#1686125611yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—1274no—
457C4—5081no—
458C4—5081no—
459C4—5081no—
460C4—2542no—
461C4—2542no—
462C4—2542no—
463C2—24no—
464C2—42no—
465C2—2544no—
466C2C2×C22⋊C4×D12711yescharacteristic
467C2C4×#1452324711yescharacteristic
468C2#2255035511yescharacteristic
469C2#2255051311yescharacteristic
470C2#2255051511yescharacteristic
471C2#2255052411yescharacteristic
472C2#2255052611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#386×C127C2
socle, cosocleC23×C127C24
prime radicalC22×#14523247C2
hypercenter#386D127
hypocenterC127#2339
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D127C2semidirect
3. C2×(C22⋊D127)⋊C4C2semidirect
4. #22549780C2semidirect
5. #22549780C2semidirect
6. #22549756C2semidirect
7. #22549752C2semidirect
8. #22549752C2semidirect
9. #22549738C2semidirect
10. #22549737C2semidirect
11. #22549734C2semidirect
12. C22⋊C4×Dic127C2semidirect
13. C2×C127⋊#95C2semidirect
14. #95×D127C2semidirect
15. #386×C127C2semidirect
18. C2×#14523247C4semidirect
60. C2×Dic254C2×C4semidirect
64. C2×C22⋊D127C2×C4semidirect
74. C22×Dic127C2×C4semidirect
75. C22×Dic127C2×C4semidirect
91. C2×C4×D127C2×C4semidirect
105. C2×D4×C127C2×C4semidirect
126. C22⋊D127C22⋊C4semidirect
150. C2×Dic127#41semidirect
185. C2×C4×C127#41semidirect
237. C127#2339semidirect
246. #386D127semidirect
308. C2×D4C4×D127semidirect
422. 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×C4×C22⋊D1272
C2×(C22⋊D127)⋊C42
#225497802
#225497562
#225497522
#225497382
#225497372
#225497342
C22⋊C4×Dic1272
C2×C127⋊#952
#95×D1272
#386×C1272
#2339127