Finite Group Databasebeta

← #30642074

Subgroups — #30642074

Order 16256.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523248C211yesmaximal, characteristic, complemented
2C2×#16861187C211yesmaximal, characteristic, complemented
3C2×#16861182C211yesmaximal, characteristic, complemented
4C2×#16861171C211yesmaximal, characteristic, complemented
5#22549793C211yesmaximal, characteristic, complemented
6#22549791C211yesmaximal, characteristic, complemented
7#22549786C211yesmaximal, characteristic, complemented
8#22549785C211yesmaximal, characteristic, complemented
9#22549780C211yesmaximal, characteristic, complemented
10#22549776C211yesmaximal, characteristic, complemented
11#22549755C211yesmaximal, characteristic, complemented
12#22549754C211yesmaximal, characteristic, complemented
13#22549747C211yesmaximal, characteristic, complemented
14#22549735C211yesmaximal, characteristic, complemented
15#385×C127C211yesmaximal, characteristic, complemented
16C2×#14523248—24no—
17C2×#14523248—24no—
18C2×#14523248—24no—
19C22×Dic254C2211yescharacteristic
20C22×D508C2211yescharacteristic
21C22×D508C2211yescharacteristic
22C22×D508C2211yescharacteristic
23C22×C22⋊D127C2211yescharacteristic
24C22×C22⋊D127C2211yescharacteristic
25C22×C22⋊D127C2211yescharacteristic
26C22×C4×D127C2211yescharacteristic
27C22×C4×D127C2211yescharacteristic
28C22×C4×D127C2211yescharacteristic
29#16861187—24no—
30#16861182—24no—
31#16861171—24no—
32C2×Dic127⋊C4C2211yescharacteristic
33C2×Dic127⋊C4C2211yescharacteristic
34C2×Dic127⋊C4C2211yescharacteristic
35C2×C22⋊Dic127C2211yescharacteristic
36C2×(C2×D127)⋊C4C2211yescharacteristic
37C2×(C2×D127)⋊C4C2211yescharacteristic
38C2×(C2×D127)⋊C4C2211yescharacteristic
39C2×(C2×D127)⋊C4C2211yescharacteristic
40C2×(C2×D127)⋊C4C2211yescharacteristic
41C2×(C2×D127)⋊C4C2211yescharacteristic
42C2×(C2×D127)⋊C4C2211yescharacteristic
43C2×(C2×D127)⋊C4C2211yescharacteristic
44C2×(C2×D127)⋊C4C2211yescharacteristic
45C2×C4×Dic127C2211yescharacteristic
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
59C23×D127C2311yescharacteristic
60#14523248—232no—
61C2×Dic254—22no—
62C2×Dic254—22no—
63C2×Dic254D414yescomplemented
64C2×D508—22no—
65C2×D508—22no—
66C2×D508—22no—
67C2×D508—22no—
68C2×D508—22no—
69C2×D508—22no—
70C2×D508—22no—
71C2×D508—22no—
72C2×D508—41no—
73C2×C22⋊D127—22no—
74C2×C22⋊D127—22no—
75C2×C22⋊D127—22no—
76C2×C22⋊D127—22no—
77C2×C22⋊D127—22no—
78C2×C22⋊D127—22no—
79C2×C22⋊D127—22no—
80C2×C22⋊D127—41no—
81C2×C22⋊D127D414yescomplemented
82C22×Dic127C2311yescharacteristic
83C22×Dic127C2311yescharacteristic
84C22×Dic127C2311yescharacteristic
85C22×Dic127C2311yescharacteristic
86C22×Dic127C2311yescharacteristic
87C2×C4×D127—22no—
88C2×C4×D127—22no—
89C2×C4×D127—22no—
90C2×C4×D127—22no—
91C2×C4×D127—22no—
92C2×C4×D127—22no—
93C2×C4×D127—22no—
94C2×C4×D127—22no—
95C2×C4×D127D414yescomplemented
96Dic127⋊C4—22no—
97Dic127⋊C4—22no—
98Dic127⋊C4—22no—
99C22⋊Dic127—41no—
100(C2×D127)⋊C4—22no—
101(C2×D127)⋊C4—22no—
102(C2×D127)⋊C4—22no—
103(C2×D127)⋊C4—22no—
104(C2×D127)⋊C4—22no—
105(C2×D127)⋊C4—22no—
106(C2×D127)⋊C4—41no—
107(C2×D127)⋊C4—41no—
108(C2×D127)⋊C4—41no—
109C4×Dic127—22no—
110C24×C127C2311yescharacteristic
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127C2311yescharacteristic
115C22×C4×C127C2311yescharacteristic
116C22×C4×C127C2311yescharacteristic
117C22×C4×C127C2311yescharacteristic
118C22×C4×C127C2311yescharacteristic
119C22×C4×C127C2311yescharacteristic
120C4⋊C4×C127—22no—
121C4⋊C4×C127—22no—
122C22⋊C4×C127—22no—
123C22⋊C4×C127—22no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127—21no—
129C22×D127—21no—
130C22×D127—21no—
131C22×D127—21no—
132C22×D127—21no—
133C22×D127—21no—
134C22×D127—21no—
135C22×D127—21no—
136C22×D127—21no—
137C22×D127—21no—
138C22×D127—21no—
139C22×D127—21no—
140C22×D127—21no—
141C22×D127—21no—
142C22×D127C2×D412yes—
143C22×D127C2×D412yes—
144C22×D127#4112yescomplemented
145Dic254—28no—
146D508—28no—
147D508—28no—
148D508—44no—
149C22⋊D127—28no—
150C22⋊D127—28no—
151C22⋊D127—44no—
152C2×Dic127—21no—
153C2×Dic127—21no—
154C2×Dic127—21no—
155C2×Dic127—21no—
156C2×Dic127—21no—
157C2×Dic127—21no—
158C2×Dic127—21no—
159C2×Dic127—21no—
160C2×Dic127—21no—
161C2×Dic127—21no—
162C2×Dic127—21no—
163C2×Dic127C2×D412yes—
164C2×Dic127C2×D412yes—
165C2×Dic127C2×D412yes—
166C2×Dic127C2×D412yes—
167C4×D127—28no—
168C4×D127—28no—
169C4×D127—44no—
170C23×C127—21no—
171C23×C127—21no—
172C23×C127—21no—
173C23×C127—21no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127C2×D412yes—
177C23×C127C2411yescharacteristic
178C2×C4×C127—28no—
179C2×C4×C127—21no—
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—21no—
183C2×C4×C127—21no—
184C2×C4×C127—21no—
185C2×C4×C127—21no—
186C2×C4×C127—21no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127—21no—
194C2×C4×C127C2×D412yes—
195C2×C4×C127C2×D412yes—
196C2×C4×C127#4112yescomplemented
197C22×C127—22no—
198C22×C127—22no—
199C22×C127—22no—
200C22×C127—41no—
201C22×C127—41no—
202C22×C127—41no—
203C22×C127—41no—
204C22×C127C22×D411yescharacteristic
205C22×C127C22×D411yescharacteristic
206C22×C127C22×D411yescharacteristic
207C22×C127C2×#4111yescharacteristic
208C22×C127C2×#4111yescharacteristic
209C22×C127H(5,𝔽2)11yescharacteristic
210C22×C1272−1+411yescharacteristic
211C2×D127—22no—
212C2×D127—22no—
213C2×D127—22no—
214C2×D127—22no—
215C2×D127—22no—
216C2×D127—22no—
217C2×D127—22no—
218C2×D127—22no—
219C2×D127—22no—
220C2×D127—41no—
221C2×D127—41no—
222C2×D127—41no—
223C2×D127—41no—
224C2×D127—41no—
225C2×D127—41no—
226C2×D127—41no—
227C2×D127—41no—
228C2×D127—41no—
229C2×D127—41no—
230C2×D127—41no—
231C2×D127—41no—
232C4×C127—22no—
233C4×C127—22no—
234C4×C127—22no—
235C4×C127—41no—
236C4×C127—41no—
237C4×C127—41no—
238Dic127—22no—
239Dic127—41no—
240Dic127—41no—
241Dic127—41no—
242Dic127#12014yes—
243C2×C127—42no—
244C2×C127C2×#12011yescharacteristic
245C2×C127#52511yescharacteristic
246C2×C127#53711yescharacteristic
247C2×C127#54611yescharacteristic
248C2×C127#54711yescharacteristic
249C2×C127#54811yescharacteristic
250C2×C127#55011yescharacteristic
251D127—42no—
252D127—42no—
253D127—42no—
254#2407—1271nomaximal, Sylow
255C127#240711yescharacteristic, complemented, Sylow
256C22×#41—1271no—
257C2×C22⋊Q8—1271no—
258C2×#121—1271no—
259C2×C4×D4—1271no—
260#394—1271no—
261#394—1271no—
262#392—1271no—
263#392—1271no—
264#391—1271no—
265#390—1271no—
266#386—1271no—
267#386—1271no—
268#385—1271no—
269#385D12711yescharacteristic, complemented
270#380—1271no—
271C2×#41—2544no—
272C2×#41—2544no—
273C2×#41—2544no—
274C22×Q8—1271no—
275C22×D4—1271no—
276C22×D4—1271no—
277C22×D4—1271no—
278C22×D4—1271no—
279C22×D4—1271no—
280C22×D4—1271no—
281C23×C4—1271no—
282C23×C4—1271no—
283C23×C4—1271no—
284C23×C4C2×D12711yescharacteristic
285C22⋊Q8—2544no—
286#121—2544no—
287C4×D4—2544no—
288C2×C4⋊C4—1271no—
289C2×C4⋊C4—1271no—
290C2×C4⋊C4—1271no—
291C2×C4⋊C4C2×D12711yescharacteristic
292C2×C4⋊C4C2×D12711yescharacteristic
293C2×C22⋊C4—1271no—
294C2×C22⋊C4—1271no—
295C2×C22⋊C4—1271no—
296C2×C22⋊C4—1271no—
297C2×C22⋊C4—1271no—
298C2×C22⋊C4—1271no—
299C2×C22⋊C4—1271no—
300C2×C22⋊C4—1271no—
301C2×C22⋊C4—1271no—
302C2×C22⋊C4—1271no—
303C2×C22⋊C4C2×D12711yescharacteristic
304C2×C22⋊C4C2×D12711yescharacteristic
305C2×C42—1271no—
306#95—1271no—
307#95—1271no—
308#95—1271no—
309#95—1271no—
310#95C2×D12711yescharacteristic
311#95C2×D12711yescharacteristic
312C24—1271no—
313C24—1271no—
314C24—1271no—
315C24C22×D12711yescharacteristic
316#41—25432no—
317C2×Q8—1274no—
318C2×Q8—2542no—
319C2×Q8—2542no—
320C2×D4—1274no—
321C2×D4—5081no—
322C2×D4—5081no—
323C2×D4—2542no—
324C2×D4—2542no—
325C2×D4—2542no—
326C2×D4—2542no—
327C2×D4—2542no—
328C2×D4—2542no—
329C2×D4—2542no—
330C2×D4—2542no—
331C2×D4—2542no—
332C2×D4—2542no—
333C2×D4—2542no—
334C2×D4—2542no—
335C2×D4—2542no—
336C2×D4—2542no—
337C2×D4—2542no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—1271no—
342C22×C4—1271no—
343C22×C4—1274no—
344C22×C4—1271no—
345C22×C4—1271no—
346C22×C4—1271no—
347C22×C4—2542no—
348C22×C4—2542no—
349C22×C4—2542no—
350C22×C4—2542no—
351C22×C4—2542no—
352C22×C4—2542no—
353C22×C4—2542no—
354C22×C4—2542no—
355C22×C4C22×D12711yescharacteristic
356C22×C4C22×D12711yescharacteristic
357C22×C4C22×D12711yescharacteristic
358C22×C4C22×D12711yescharacteristic
359C22×C4C22×D12711yescharacteristic
360C22×C4C22×D12711yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—2542no—
364C4⋊C4—2542no—
365C4⋊C4—2542no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—5081no—
369C22⋊C4—5081no—
370C22⋊C4—5081no—
371C22⋊C4—5081no—
372C22⋊C4—2542no—
373C22⋊C4—2542no—
374C22⋊C4—2542no—
375C22⋊C4—2542no—
376C22⋊C4—2542no—
377C22⋊C4—2542no—
378C42—2542no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—1272no—
386C23—1272no—
387C23—1272no—
388C23—2541no—
389C23—2541no—
390C23—2541no—
391C23—2541no—
392C23—2541no—
393C23—2541no—
394C23—2541no—
395C23—2541no—
396C23—2541no—
397C23—2541no—
398C23—2541no—
399C23—2541no—
400C23—2541no—
401C23—2541no—
402C23—2541no—
403C23—2541no—
404C23—2541no—
405C23—2541no—
406C23D4×D12712yes—
407C23C23×D12711yescharacteristic
408Q8—2548no—
409D4—5084no—
410D4—5084no—
411D4—2548no—
412D4—2548no—
413D4—2548no—
414D4—2548no—
415C2×C4—28no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—1272no—
432C2×C4—1272no—
433C2×C4—1272no—
434C2×C4—1272no—
435C2×C4—5084no—
436C2×C4—2548no—
437C2×C4—2548no—
438C2×C4—2541no—
439C2×C4—2541no—
440C2×C4—2541no—
441C2×C4—2541no—
442C2×C4—2541no—
443C2×C4—2541no—
444C2×C4—2541no—
445C2×C4—2541no—
446C2×C4—2541no—
447C2×C4—2541no—
448C2×C4—2541no—
449C2×C4D4×D12712yes—
450C2×C4D4×D12712yes—
451C2×C4#1452324812yescomplemented
452C22—22no—
453C22—22no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—5081no—
460C22—5081no—
461C22—5081no—
462C22—5081no—
463C22—5081no—
464C22—5081no—
465C22—5081no—
466C22—5081no—
467C22—5081no—
468C22—5081no—
469C22—5081no—
470C22—5081no—
471C22—2542no—
472C22—2542no—
473C22—2542no—
474C22—2542no—
475C22—2542no—
476C22—2542no—
477C22—2542no—
478C22—2542no—
479C22—2542no—
480C22C2×D4×D12711yescharacteristic
481C22C2×D4×D12711yescharacteristic
482C22C2×D4×D12711yescharacteristic
483C22#41×D12711yescharacteristic
484C22C2×#1452324811yescharacteristic
485C22#1686125511yescharacteristic
486C22#1686125711yescharacteristic
487C4—22no—
488C4—22no—
489C4—22no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—1274no—
494C4—5081no—
495C4—5081no—
496C4—5081no—
497C4—2542no—
498C2—42no—
499C2—5082no—
500C2—5082no—
501C2—5082no—
502C2#120×D12711yescharacteristic
503C2#2255038511yescharacteristic
504C2#2255045611yescharacteristic
505C2#2255051811yescharacteristic
506C2#2255052711yescharacteristic
507C2#2255053611yescharacteristic
508C2#2255054711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#385×C127C2
hypercenter#385D127
hypocenterC127#2407
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×#14523248C2semidirect
2. C2×#16861187C2semidirect
3. C2×#16861182C2semidirect
4. C2×#16861171C2semidirect
5. #22549793C2semidirect
6. #22549791C2semidirect
7. #22549786C2semidirect
8. #22549785C2semidirect
9. #22549780C2semidirect
10. #22549776C2semidirect
11. #22549755C2semidirect
12. #22549754C2semidirect
13. #22549747C2semidirect
14. #22549735C2semidirect
15. #385×C127C2semidirect
63. C2×Dic254D4semidirect
81. C2×C22⋊D127D4semidirect
95. C2×C4×D127D4semidirect
144. C22×D127#41semidirect
196. C2×C4×C127#41semidirect
255. C127#2407semidirect
269. #385D127semidirect
451. 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×#168611872
C2×#168611822
C2×#168611712
#225497932
#225497912
#225497862
#225497852
#225497802
#225497762
#225497552
#225497542
#225497472
#225497352
#385×C1272
#2407127