Finite Group Databasebeta

← #30642063

Subgroups — #30642063

Order 16256.

864 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523248C211yesmaximal, characteristic
2C2×#16861194C211yesmaximal, characteristic, complemented
3C2×#16861191C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
5#22549797C211yesmaximal, characteristic, complemented
6#22549796C211yesmaximal, characteristic, complemented
7#22549792C211yesmaximal, characteristic, complemented
8#22549791C211yesmaximal, characteristic, complemented
9#22549754C211yesmaximal, characteristic, complemented
10#22549752C211yesmaximal, characteristic, complemented
11#22549750C211yesmaximal, characteristic, complemented
12#22549749C211yesmaximal, characteristic, complemented
13#22549747C211yesmaximal, characteristic, complemented
14#22549738C211yesmaximal, characteristic, complemented
15#385×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—24no—
17C2×#14523248—24no—
18C2×#14523248—24no—
19C22×Dic254C2211yescharacteristic
20C22×D508C2211yescharacteristic
21C22×C22⋊D127C2211yescharacteristic
22C22×C22⋊D127C2211yescharacteristic
23C22×C4×D127C2211yescharacteristic
24C22×C4×D127C2211yescharacteristic
25C22×C4×D127C2211yescharacteristic
26C22×C4×D127C2211yescharacteristic
27#16861194—24no—
28#16861191—24no—
29C2×C4⋊Dic127C2211yescharacteristic
30C2×C4⋊Dic127C2211yescharacteristic
31C2×Dic127⋊C4C2211yescharacteristic
32C2×Dic127⋊C4C2211yescharacteristic
33C2×Dic127⋊C4C2211yescharacteristic
34C2×Dic127⋊C4C2211yescharacteristic
35C4⋊C4×D127—24no—
36C2×C22⋊Dic127C2211yescharacteristic
37C2×C22⋊Dic127C2211yescharacteristic
38C2×(C2×D127)⋊C4C2211yescharacteristic
39C2×(C2×D127)⋊C4C2211yescharacteristic
40C2×(C2×D127)⋊C4C2211yescharacteristic
41C2×(C2×D127)⋊C4C2211yescharacteristic
42C2×(C2×D127)⋊C4C2211yescharacteristic
43C2×(C2×D127)⋊C4C2211yescharacteristic
44C127⋊#95C2211yescharacteristic
45C127⋊#95C2211yescharacteristic
46C127⋊#95C2211yescharacteristic
47C127⋊#95C2211yescharacteristic
48C127⋊#95C2211yescharacteristic
49C127⋊#95C2211yescharacteristic
50C23×C4×C127C2211yescharacteristic
51C2×C4⋊C4×C127C2211yescharacteristic
52C2×C4⋊C4×C127C2211yescharacteristic
53C2×C22⋊C4×C127C2211yescharacteristic
54C2×C22⋊C4×C127C2211yescharacteristic
55#95×C127C2211yescharacteristic
56#95×C127C2211yescharacteristic
57C23×D127C2311yescharacteristic
58C23×D127C2311yescharacteristic
59#14523248—232no—
60C2×Dic254—22no—
61C2×Dic254—22no—
62C2×Dic254—22no—
63C2×D508—22no—
64C2×D508—22no—
65C2×D508D414yes—
66C2×C22⋊D127—22no—
67C2×C22⋊D127—22no—
68C2×C22⋊D127—22no—
69C2×C22⋊D127—22no—
70C2×C22⋊D127—22no—
71C2×C22⋊D127D414yes—
72C22×Dic127C2311yescharacteristic
73C22×Dic127C2311yescharacteristic
74C22×Dic127C2311yescharacteristic
75C22×Dic127C2311yescharacteristic
76C22×Dic127C2311yescharacteristic
77C22×Dic127C2311yescharacteristic
78C2×C4×D127—22no—
79C2×C4×D127—22no—
80C2×C4×D127—22no—
81C2×C4×D127—22no—
82C2×C4×D127—22no—
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×D127D414yes—
90C4⋊Dic127—22no—
91C4⋊Dic127—22no—
92Dic127⋊C4—22no—
93Dic127⋊C4—22no—
94Dic127⋊C4—22no—
95Dic127⋊C4—22no—
96C22⋊Dic127—22no—
97C22⋊Dic127—22no—
98(C2×D127)⋊C4—22no—
99(C2×D127)⋊C4—22no—
100(C2×D127)⋊C4—22no—
101(C2×D127)⋊C4—22no—
102(C2×D127)⋊C4—22no—
103(C2×D127)⋊C4—22no—
104C24×C127C2311yescharacteristic
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127C2311yescharacteristic
109C22×C4×C127C2311yescharacteristic
110C22×C4×C127C2311yescharacteristic
111C22×C4×C127C2311yescharacteristic
112C22×C4×C127C2311yescharacteristic
113C22×C4×C127C2311yescharacteristic
114C4⋊C4×C127—22no—
115C4⋊C4×C127—22no—
116C22⋊C4×C127—22no—
117C22⋊C4×C127—22no—
118C22×D127—21no—
119C22×D127—21no—
120C22×D127—21no—
121C22×D127—21no—
122C22×D127—21no—
123C22×D127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127C2×D412yes—
129C22×D127C2×D412yes—
130C22×D127C2×D412yes—
131C22×D127C2×D412yes—
132Dic254—28no—
133D508—28no—
134C22⋊D127—28no—
135C22⋊D127—28no—
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—21no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—21no—
148C2×Dic127—21no—
149C2×Dic127—21no—
150C2×Dic127—21no—
151C2×Dic127C2×D412yes—
152C2×Dic127C2×D412yes—
153C2×Dic127#4112yescomplemented
154C4×D127—28no—
155C4×D127—28no—
156C4×D127—28no—
157C4×D127—44no—
158C23×C127—21no—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127—21no—
162C23×C127—21no—
163C23×C127—21no—
164C23×C127C2×D412yes—
165C23×C127C2411yescharacteristic
166C2×C4×C127—28no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
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×C127C2×D412yes—
183C2×C4×C127C2×D412yes—
184C2×C4×C127#4112yescomplemented
185C22×C127—22no—
186C22×C127—22no—
187C22×C127—22no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127—41no—
191C22×C127—41no—
192C22×C127C22×D411yescharacteristic
193C22×C127C22×D411yescharacteristic
194C22×C127C22×D411yescharacteristic
195C22×C127C2×#4111yescharacteristic
196C22×C127C2×#4111yescharacteristic
197C22×C1272−1+411yescharacteristic
198C22×C1272−1+411yescharacteristic
199C2×D127—22no—
200C2×D127—22no—
201C2×D127—22no—
202C2×D127—22no—
203C2×D127—22no—
204C2×D127—22no—
205C2×D127—22no—
206C2×D127—22no—
207C2×D127—22no—
208C2×D127—41no—
209C2×D127—41no—
210C2×D127—41no—
211C2×D127—41no—
212C2×D127#12014yes—
213C4×C127—22no—
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—
225C2×C127—42no—
226C2×C127C2×#12011yescharacteristic
227C2×C127#52511yescharacteristic
228C2×C127#53611yescharacteristic
229C2×C127#54711yescharacteristic
230C2×C127#54711yescharacteristic
231C2×C127#54711yescharacteristic
232C2×C127#54711yescharacteristic
233D127—24no—
234D127—42no—
235#2406—1271nomaximal, Sylow
236C127#240611yescharacteristic, complemented, Sylow
237C22×#41—1271no—
238C2×#123—1271no—
239C2×#123—1271no—
240C22×C4⋊C4—1271no—
241#395—1271no—
242#395—1271no—
243#394—1271no—
244#394—1271no—
245#386—1271no—
246#386—1271no—
247#385—1271no—
248#385—1271no—
249#385—1271no—
250#385D12711yescharacteristic, complemented
251#381—1271no—
252C2×#41—2544no—
253C2×#41—2544no—
254C2×#41—2544no—
255C22×Q8—1271no—
256C22×D4—1271no—
257C22×D4—1271no—
258C22×D4—1271no—
259C23×C4—1271no—
260C23×C4—1271no—
261C23×C4—1271no—
262C23×C4—1271no—
263C23×C4C2×D12711yescharacteristic
264#123—2544no—
265#123—2544no—
266C2×C4⋊C4—1271no—
267C2×C4⋊C4—1271no—
268C2×C4⋊C4—1271no—
269C2×C4⋊C4—1271no—
270C2×C4⋊C4—1271no—
271C2×C4⋊C4—1271no—
272C2×C4⋊C4—2544no—
273C2×C4⋊C4C2×D12711yescharacteristic
274C2×C4⋊C4C2×D12711yescharacteristic
275C2×C22⋊C4—1271no—
276C2×C22⋊C4—1271no—
277C2×C22⋊C4—1271no—
278C2×C22⋊C4—1271no—
279C2×C22⋊C4—1271no—
280C2×C22⋊C4—1271no—
281C2×C22⋊C4—1271no—
282C2×C22⋊C4—1271no—
283C2×C22⋊C4C2×D12711yescharacteristic
284C2×C22⋊C4C2×D12711yescharacteristic
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—
294C24—1271no—
295C24C22×D12711yescharacteristic
296#41—25432no—
297C2×Q8—2542no—
298C2×Q8—2542no—
299C2×Q8—2542no—
300C2×D4—1274no—
301C2×D4—1274no—
302C2×D4—2542no—
303C2×D4—2542no—
304C2×D4—2542no—
305C2×D4—2542no—
306C2×D4—2542no—
307C2×D4—2542no—
308C2×D4—2542no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—1271no—
313C22×C4—1271no—
314C22×C4—1271no—
315C22×C4—1274no—
316C22×C4—1271no—
317C22×C4—1271no—
318C22×C4—1271no—
319C22×C4—2542no—
320C22×C4—2542no—
321C22×C4—2542no—
322C22×C4—2542no—
323C22×C4—2542no—
324C22×C4—2542no—
325C22×C4—2542no—
326C22×C4—2542no—
327C22×C4—2542no—
328C22×C4—2542no—
329C22×C4—2542no—
330C22×C4C22×D12711yescharacteristic
331C22×C4C22×D12711yescharacteristic
332C22×C4C22×D12711yescharacteristic
333C22×C4C22×D12711yescharacteristic
334C22×C4C22×D12711yescharacteristic
335C22×C4C22×D12711yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—2542no—
339C4⋊C4—2542no—
340C4⋊C4—2542no—
341C4⋊C4—2542no—
342C4⋊C4—2542no—
343C4⋊C4—2542no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—2542no—
347C22⋊C4—2542no—
348C22⋊C4—2542no—
349C22⋊C4—2542no—
350C22⋊C4—2542no—
351C22⋊C4—2542no—
352C22⋊C4—2542no—
353C22⋊C4—2542no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—1272no—
361C23—1272no—
362C23—1272no—
363C23—1272no—
364C23—2541no—
365C23—2541no—
366C23—2541no—
367C23—2541no—
368C23—2541no—
369C23—2541no—
370C23—2541no—
371C23—2541no—
372C23—2541no—
373C23—2541no—
374C23D4×D12712yes—
375C23C23×D12711yescharacteristic
376Q8—2548no—
377D4—2548no—
378D4—2548no—
379D4—2548no—
380C2×C4—28no—
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—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—1272no—
397C2×C4—1272no—
398C2×C4—1272no—
399C2×C4—5084no—
400C2×C4—2548no—
401C2×C4—2548no—
402C2×C4—2548no—
403C2×C4—2541no—
404C2×C4—2541no—
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×C4D4×D12712yes—
419C2×C4D4×D12712yes—
420C2×C4#1452324812yescomplemented
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—1274no—
429C22—5081no—
430C22—5081no—
431C22—5081no—
432C22—5081no—
433C22—2542no—
434C22—2542no—
435C22—2542no—
436C22—2542no—
437C22—2542no—
438C22—2542no—
439C22—2542no—
440C22—2542no—
441C22—2542no—
442C22C2×D4×D12711yescharacteristic
443C22C2×D4×D12711yescharacteristic
444C22C2×D4×D12711yescharacteristic
445C22#41×D12711yescharacteristic
446C22C2×#1452324811yescharacteristic
447C22#1686125611yescharacteristic
448C22#1686125711yescharacteristic
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—5081no—
456C4—5081no—
457C4—5081no—
458C4—2542no—
459C4—2542no—
460C4—2542no—
461C2—42no—
462C2—5082no—
463C2—2544no—
464C2#120×D12711yescharacteristic
465C2#2255038511yescharacteristic
466C2#2255044911yescharacteristic
467C2#2255052411yescharacteristic
468C2#2255052711yescharacteristic
469C2#2255052811yescharacteristic
470C2#2255053111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#385×C127C2
prime radicalC22×#14523248C2
hypercenter#385D127
hypocenterC127#2406
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#16861194C2semidirect
3. C2×#16861191C2semidirect
4. C2×C4⋊C4×D127C2semidirect
5. #22549797C2semidirect
6. #22549796C2semidirect
7. #22549792C2semidirect
8. #22549791C2semidirect
9. #22549754C2semidirect
10. #22549752C2semidirect
11. #22549750C2semidirect
12. #22549749C2semidirect
13. #22549747C2semidirect
14. #22549738C2semidirect
15. #385×C127C2semidirect
153. C2×Dic127#41semidirect
184. C2×C4×C127#41semidirect
236. C127#2406semidirect
250. #385D127semidirect
420. C2×C4#14523248semidirect

Maximal subgroups

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

subgroupindex
C22×#145232482
C2×#168611942
C2×#168611912
C2×C4⋊C4×D1272
#225497972
#225497962
#225497922
#225497912
#225497542
#225497522
#225497502
#225497492
#225497472
#225497382
#385×C1272
#2406127