Finite Group Databasebeta

← #30669283

Subgroups — #30669283

Order 16768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D131C211yesmaximal, characteristic
2C2×#16862231C211yesmaximal, characteristic, complemented
3C2×#16862228C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
5#22553228C211yesmaximal, characteristic, complemented
6#22553227C211yesmaximal, characteristic, complemented
7#22553226C211yesmaximal, characteristic, complemented
8#22553190C211yesmaximal, characteristic, complemented
9#22553188C211yesmaximal, characteristic, complemented
10#22553186C211yesmaximal, characteristic, complemented
11#22553183C211yesmaximal, characteristic, complemented
12#22553167C211yesmaximal, characteristic, complemented
13#22553167C211yesmaximal, characteristic, complemented
14#22553167C211yesmaximal, characteristic, complemented
15#394×C131C211yesmaximal, characteristic, complemented
16C24×D131C2211yescharacteristic
17C24×D131C2211yescharacteristic
18C22×D524C2211yescharacteristic
19C22×C22⋊D131C2211yescharacteristic
20C22×C22⋊D131C2211yescharacteristic
21C2×D4×D131—24no—
22C2×D4×D131—24no—
23C2×D4×D131—24no—
24C22×C4×D131C2211yescharacteristic
25C22×C4×D131C2211yescharacteristic, complemented
26C22×C4×D131C2211yescharacteristic, complemented
27#16862231—24no—
28#16862228—24no—
29C2×C4⋊Dic131C2211yescharacteristic, complemented
30C2×Dic131⋊C4C2211yescharacteristic, complemented
31C2×Dic131⋊C4C2211yescharacteristic, complemented
32C2×C22⋊Dic131C2211yescharacteristic, complemented
33C2×C22⋊Dic131C2211yescharacteristic, complemented
34C2×C22⋊Dic131C2211yescharacteristic, complemented
35C2×C22⋊Dic131C2211yescharacteristic, complemented
36C2×(C2×D131)⋊C4C2211yescharacteristic
37C2×(C2×D131)⋊C4C2211yescharacteristic
38C2×(C2×D131)⋊C4C2211yescharacteristic
39C2×(C2×D131)⋊C4C2211yescharacteristic
40C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
46C22⋊C4×D131—24no—
47C131⋊#95C2211yescharacteristic, complemented
48C131⋊#95C2211yescharacteristic, complemented
49C131⋊#95C2211yescharacteristic, complemented
50C22×D4×C131C2211yescharacteristic
51C2×C4⋊C4×C131C2211yescharacteristic, complemented
52C2×C22⋊C4×C131C2211yescharacteristic, complemented
53C2×C22⋊C4×C131C2211yescharacteristic, complemented
54C2×C22⋊C4×C131C2211yescharacteristic, complemented
55C2×C22⋊C4×C131C2211yescharacteristic, complemented
56#95×C131C2211yescharacteristic, complemented
57C23×D131—22no—
58C23×D131—22no—
59C23×D131—22no—
60C23×D131—22no—
61C23×D131—22no—
62C23×D131—22no—
63C23×D131—22no—
64C23×D131—22no—
65C23×D131—22no—
66C23×D131—41no—
67C23×D131—41no—
68C23×D131—41no—
69C23×D131—41no—
70C23×D131D414yes—
71C23×D131C2311yescharacteristic
72C23×D131C2311yescharacteristic
73C23×D131C2311yescharacteristic
74C2×D524—22no—
75C2×D524—22no—
76C2×D524D414yes—
77C2×C22⋊D131—22no—
78C2×C22⋊D131—22no—
79C2×C22⋊D131—22no—
80C2×C22⋊D131—22no—
81C2×C22⋊D131—22no—
82C2×C22⋊D131D414yes—
83D4×D131—232no—
84C22×Dic131C2311yescharacteristic
85C22×Dic131C2311yescharacteristic
86C22×Dic131C2311yescharacteristic
87C22×Dic131C2311yescharacteristic
88C22×Dic131C2311yescharacteristic
89C2×C4×D131—22no—
90C2×C4×D131—22no—
91C2×C4×D131—22no—
92C2×C4×D131—22no—
93C2×C4×D131—22no—
94C2×C4×D131—22no—
95C2×C4×D131—22no—
96C2×C4×D131—41no—
97C2×C4×D131—41no—
98C4⋊Dic131—22no—
99Dic131⋊C4—22no—
100Dic131⋊C4—22no—
101C22⋊Dic131—22no—
102C22⋊Dic131—22no—
103C22⋊Dic131—41no—
104C22⋊Dic131—41no—
105(C2×D131)⋊C4—22no—
106(C2×D131)⋊C4—22no—
107(C2×D131)⋊C4—22no—
108(C2×D131)⋊C4—22no—
109(C2×D131)⋊C4—22no—
110(C2×D131)⋊C4—22no—
111(C2×D131)⋊C4—22no—
112(C2×D131)⋊C4—22no—
113(C2×D131)⋊C4—41no—
114(C2×D131)⋊C4—41no—
115C24×C131C2311yescharacteristic
116C24×C131C2311yescharacteristic
117C2×D4×C131—22no—
118C2×D4×C131—22no—
119C2×D4×C131—22no—
120C22×C4×C131C2311yescharacteristic
121C22×C4×C131C2311yescharacteristic
122C22×C4×C131C2311yescharacteristic
123C22×C4×C131C2311yescharacteristic
124C22×C4×C131C2311yescharacteristic
125C4⋊C4×C131—22no—
126C22⋊C4×C131—22no—
127C22⋊C4×C131—22no—
128C22⋊C4×C131—41no—
129C22⋊C4×C131—41no—
130C22×D131—21no—
131C22×D131—21no—
132C22×D131—21no—
133C22×D131—21no—
134C22×D131—21no—
135C22×D131—21no—
136C22×D131—21no—
137C22×D131—21no—
138C22×D131—28no—
139C22×D131—21no—
140C22×D131—21no—
141C22×D131—21no—
142C22×D131—21no—
143C22×D131—21no—
144C22×D131—21no—
145C22×D131—28no—
146C22×D131—28no—
147C22×D131—44no—
148C22×D131—44no—
149C22×D131—44no—
150C22×D131—44no—
151C22×D131—44no—
152C22×D131—44no—
153C22×D131—44no—
154C22×D131—44no—
155C22×D131—44no—
156C22×D131—44no—
157C22×D131—44no—
158C22×D131C2×D412yes—
159C22×D131C2×D412yes—
160C22×D131C2×D412yes—
161C22×D131C2×D412yes—
162C22×D131C2×D412yes—
163C22×D131#4112yes—
164C22×D131#4112yes—
165D524—28no—
166C22⋊D131—28no—
167C22⋊D131—28no—
168C2×Dic131—21no—
169C2×Dic131—21no—
170C2×Dic131—21no—
171C2×Dic131—21no—
172C2×Dic131—21no—
173C2×Dic131—21no—
174C2×Dic131—21no—
175C2×Dic131—21no—
176C2×Dic131—21no—
177C2×Dic131—21no—
178C2×Dic131—21no—
179C2×Dic131—21no—
180C2×Dic131—21no—
181C2×Dic131—21no—
182C2×Dic131C2×D412yes—
183C4×D131—28no—
184C4×D131—44no—
185C4×D131—44no—
186C23×C131—21no—
187C23×C131—21no—
188C23×C131—21no—
189C23×C131—21no—
190C23×C131—21no—
191C23×C131—21no—
192C23×C131—21no—
193C23×C131—21no—
194C23×C131—21no—
195C23×C131—21no—
196C23×C131—21no—
197C23×C131—21no—
198C23×C131C2×D412yes—
199C23×C131C2×D412yes—
200C23×C131C2411yescharacteristic
201D4×C131—28no—
202C2×C4×C131—21no—
203C2×C4×C131—21no—
204C2×C4×C131—21no—
205C2×C4×C131—21no—
206C2×C4×C131—21no—
207C2×C4×C131—21no—
208C2×C4×C131—21no—
209C2×C4×C131—21no—
210C2×C4×C131—21no—
211C2×C4×C131—21no—
212C2×C4×C131—21no—
213C2×C4×C131—21no—
214C2×C4×C131—21no—
215C2×C4×C131—21no—
216C2×C4×C131C2×D412yes—
217C22×C131—22no—
218C22×C131—22no—
219C22×C131—22no—
220C22×C131—22no—
221C22×C131—22no—
222C22×C131—22no—
223C22×C131—41no—
224C22×C131—41no—
225C22×C131—41no—
226C22×C131—41no—
227C22×C131—41no—
228C22×C131—41no—
229C22×C131—41no—
230C22×C131—41no—
231C22×C131C22×D411yescharacteristic
232C22×C131C22×D411yescharacteristic
233C22×C131C22×D411yescharacteristic
234C22×C131C2×#4111yescharacteristic
235C22×C131C2×#4111yescharacteristic
236C22×C131H(5,𝔽2)11yescharacteristic
237C22×C131H(5,𝔽2)11yescharacteristic
238C2×D131—22no—
239C2×D131—22no—
240C2×D131—22no—
241C2×D131—22no—
242C2×D131—22no—
243C2×D131—22no—
244C2×D131—22no—
245C2×D131—22no—
246C2×D131—22no—
247C2×D131—22no—
248C2×D131—22no—
249C2×D131—22no—
250C2×D131—22no—
251C2×D131—22no—
252C2×D131—22no—
253C2×D131—416no—
254C2×D131—416no—
255C2×D131—41no—
256C2×D131—41no—
257C2×D131—41no—
258C2×D131—41no—
259C2×D131#12014yes—
260C2×D131#12314yescomplemented
261C4×C131—22no—
262C4×C131—41no—
263C4×C131—41no—
264C4×C131—41no—
265C4×C131—41no—
266Dic131—22no—
267Dic131—41no—
268Dic131—41no—
269Dic131—41no—
270Dic131—41no—
271C2×C131—42no—
272C2×C131—42no—
273C2×C131C2×#12011yescharacteristic
274C2×C131C2×#12311yescharacteristic
275C2×C131#53411yescharacteristic
276C2×C131#54611yescharacteristic
277C2×C131#54611yescharacteristic
278C2×C131#54611yescharacteristic
279C2×C131#54611yescharacteristic
280D131—24no—
281D131—24no—
282D131—42no—
283C131#240511yescharacteristic, complemented, Sylow
284#2405—1311nomaximal, Sylow
285C23×D4—1311no—
286C2×#123—1311no—
287C2×#123—1311no—
288C22×C22⋊C4—1311no—
289#394—1311no—
290#394—1311no—
291#394—1311no—
292#394D13111yescharacteristic, complemented
293#386—1311no—
294#386—1311no—
295#385—1311no—
296#385—1311no—
297#379—1311no—
298#379—1311no—
299#379—1311no—
300C25—1311no—
301C25—1311no—
302C22×D4—2624no—
303C22×D4—2624no—
304C22×D4—2624no—
305C22×D4—1311no—
306C22×D4—1311no—
307C22×D4—1311no—
308C22×D4C2×D13111yescharacteristic
309C23×C4—1311no—
310C23×C4—1311no—
311C23×C4—1311no—
312#123—2624no—
313#123—2624no—
314C2×C4⋊C4—1311no—
315C2×C4⋊C4—1311no—
316C2×C4⋊C4—1311no—
317C2×C4⋊C4C2×D13111yescharacteristic, complemented
318C2×C22⋊C4—2624no—
319C2×C22⋊C4—1311no—
320C2×C22⋊C4—1311no—
321C2×C22⋊C4—1311no—
322C2×C22⋊C4—1311no—
323C2×C22⋊C4—1311no—
324C2×C22⋊C4—1311no—
325C2×C22⋊C4—1311no—
326C2×C22⋊C4—1311no—
327C2×C22⋊C4—1311no—
328C2×C22⋊C4—1311no—
329C2×C22⋊C4—1311no—
330C2×C22⋊C4—1311no—
331C2×C22⋊C4—1311no—
332C2×C22⋊C4—1311no—
333C2×C22⋊C4C2×D13111yescharacteristic, complemented
334C2×C22⋊C4C2×D13111yescharacteristic, complemented
335C2×C22⋊C4C2×D13111yescharacteristic, complemented
336C2×C22⋊C4C2×D13111yescharacteristic, complemented
337#95—1311no—
338#95—1311no—
339#95—1311no—
340#95C2×D13111yescharacteristic, complemented
341C24—2622no—
342C24—2622no—
343C24—2622no—
344C24—2622no—
345C24—2622no—
346C24—2622no—
347C24—2622no—
348C24—2622no—
349C24—2622no—
350C24—5241no—
351C24—5241no—
352C24—5241no—
353C24—5241no—
354C24—1311no—
355C24—1314no—
356C24—1311no—
357C24—1311no—
358C24C22×D13111yescharacteristic
359C24C22×D13111yescharacteristic
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—2622no—
364C2×D4—2622no—
365C2×D4—26232no—
366C2×D4—2622no—
367C2×D4—2622no—
368C2×D4—2622no—
369C2×D4—2622no—
370C2×D4—2622no—
371C2×D4—1314no—
372C2×D4—1314no—
373C22×C4—2622no—
374C22×C4—2622no—
375C22×C4—2622no—
376C22×C4—2622no—
377C22×C4—2622no—
378C22×C4—2622no—
379C22×C4—2622no—
380C22×C4—5241no—
381C22×C4—5241no—
382C22×C4—1311no—
383C22×C4—1311no—
384C22×C4—1311no—
385C22×C4—1311no—
386C22×C4—1311no—
387C22×C4C22×D13111yescharacteristic
388C22×C4C22×D13111yescharacteristic
389C22×C4C22×D13111yescharacteristic
390C22×C4C22×D13111yescharacteristic
391C22×C4C22×D13111yescharacteristic
392C4⋊C4—22no—
393C4⋊C4—2622no—
394C4⋊C4—2622no—
395C4⋊C4—2622no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—2622no—
401C22⋊C4—2622no—
402C22⋊C4—2622no—
403C22⋊C4—2622no—
404C22⋊C4—2622no—
405C22⋊C4—2622no—
406C22⋊C4—2622no—
407C22⋊C4—2622no—
408C22⋊C4—2622no—
409C22⋊C4—2622no—
410C22⋊C4—5241no—
411C22⋊C4—5241no—
412C22⋊C4—5241no—
413C22⋊C4—5241no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—2628no—
427C23—2628no—
428C23—2628no—
429C23—2621no—
430C23—2621no—
431C23—2621no—
432C23—2621no—
433C23—2621no—
434C23—2621no—
435C23—2621no—
436C23—2621no—
437C23—2621no—
438C23—2621no—
439C23—2621no—
440C23—2621no—
441C23—2621no—
442C23—2621no—
443C23—5244no—
444C23—5244no—
445C23—5244no—
446C23—5244no—
447C23—5244no—
448C23—5244no—
449C23—5244no—
450C23—5244no—
451C23—5244no—
452C23—5244no—
453C23—5244no—
454C23—1312no—
455C23—1312no—
456C23—1312no—
457C23—1312no—
458C23—1312no—
459C23—1312no—
460C23—1312no—
461C23D4×D13112yes—
462C23D4×D13112yes—
463C23C23×D13111yescharacteristic
464D4—28no—
465D4—2628no—
466D4—2628no—
467D4—2628no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—2628no—
483C2×C4—2621no—
484C2×C4—2621no—
485C2×C4—2621no—
486C2×C4—2621no—
487C2×C4—2621no—
488C2×C4—2621no—
489C2×C4—2621no—
490C2×C4—2621no—
491C2×C4—2621no—
492C2×C4—2621no—
493C2×C4—2621no—
494C2×C4—2621no—
495C2×C4—2621no—
496C2×C4—2621no—
497C2×C4—5244no—
498C2×C4—5244no—
499C2×C4—1312no—
500C2×C4D4×D13112yes—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—2622no—
516C22—2622no—
517C22—2622no—
518C22—2622no—
519C22—2622no—
520C22—2622no—
521C22—2622no—
522C22—2622no—
523C22—2622no—
524C22—2622no—
525C22—2622no—
526C22—2622no—
527C22—2622no—
528C22—2622no—
529C22—2622no—
530C22—52416no—
531C22—52416no—
532C22—5241no—
533C22—5241no—
534C22—5241no—
535C22—5241no—
536C22—1314no—
537C22—1314no—
538C22C2×D4×D13111yescharacteristic
539C22C2×D4×D13111yescharacteristic
540C22C2×D4×D13111yescharacteristic
541C22#41×D13111yescharacteristic
542C22#41×D13111yescharacteristic
543C22#1686229111yescharacteristic
544C22#1686229211yescharacteristic
545C4—22no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—2622no—
551C4—5241no—
552C4—5241no—
553C4—5241no—
554C4—5241no—
555C2—42no—
556C2—42no—
557C2—2624no—
558C2—2624no—
559C2—5242no—
560C2#120×D13111yescharacteristic
561C2#123×D13111yescharacteristic
562C2#2255387011yescharacteristic
563C2#2255395111yescharacteristic
564C2#2255395411yescharacteristic
565C2#2255395511yescharacteristic
566C2#2255395911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#394×C131C2
prime radicalC22×D4×D131C2
hypercenter#394D131
hypocenterC131#2405
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#16862231C2semidirect
3. C2×#16862228C2semidirect
4. C2×C22⋊C4×D131C2semidirect
5. #22553228C2semidirect
6. #22553227C2semidirect
7. #22553226C2semidirect
8. #22553190C2semidirect
9. #22553188C2semidirect
10. #22553186C2semidirect
11. #22553183C2semidirect
12. #22553167C2semidirect
13. #22553167C2semidirect
14. #22553167C2semidirect
15. #394×C131C2semidirect
25. C22×C4×D131C22semidirect
26. C22×C4×D131C22semidirect
29. C2×C4⋊Dic131C22semidirect
30. C2×Dic131⋊C4C22semidirect
31. C2×Dic131⋊C4C22semidirect
32. C2×C22⋊Dic131C22semidirect
33. C2×C22⋊Dic131C22semidirect
34. C2×C22⋊Dic131C22semidirect
35. C2×C22⋊Dic131C22semidirect
40. C2×(C2×D131)⋊C4C22semidirect
41. C2×(C2×D131)⋊C4C22semidirect
42. C2×(C2×D131)⋊C4C22semidirect
43. C2×(C2×D131)⋊C4C22semidirect
44. C2×(C2×D131)⋊C4C22semidirect
45. C2×(C2×D131)⋊C4C22semidirect
47. C131⋊#95C22semidirect
48. C131⋊#95C22semidirect
49. C131⋊#95C22semidirect
51. C2×C4⋊C4×C131C22semidirect
52. C2×C22⋊C4×C131C22semidirect
53. C2×C22⋊C4×C131C22semidirect
54. C2×C22⋊C4×C131C22semidirect
55. C2×C22⋊C4×C131C22semidirect
56. #95×C131C22semidirect
260. C2×D131#123semidirect
283. C131#2405semidirect
292. #394D131semidirect
317. C2×C4⋊C4C2×D131semidirect
333. C2×C22⋊C4C2×D131semidirect
334. C2×C22⋊C4C2×D131semidirect
335. C2×C22⋊C4C2×D131semidirect
336. C2×C22⋊C4C2×D131semidirect
340. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1312
C2×#168622312
C2×#168622282
C2×C22⋊C4×D1312
#225532282
#225532272
#225532262
#225531902
#225531882
#225531862
#225531832
#225531672
#394×C1312
#2405131