Finite Group Databasebeta

← #30669334

Subgroups — #30669334

Order 16768.

1246 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D131C211yesmaximal, characteristic, complemented
2C2×#16862220C211yesmaximal, characteristic, complemented
3C2×#16862218C211yesmaximal, characteristic, complemented
4C2×#16862216C211yesmaximal, characteristic, complemented
5C2×#16862216C211yesmaximal, characteristic, complemented
6C22×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
7#22553230C211yesmaximal, characteristic, complemented
8#22553228C211yesmaximal, characteristic, complemented
9#22553222C211yesmaximal, characteristic, complemented
10#22553221C211yesmaximal, characteristic, complemented
11#22553190C211yesmaximal, characteristic, complemented
12#22553189C211yesmaximal, characteristic, complemented
13#22553186C211yesmaximal, characteristic, complemented
14#22553167C211yesmaximal, characteristic, complemented
15#386×C131C211yesmaximal, characteristic, complemented
16C24×D131C2211yescharacteristic
17C24×D131C2211yescharacteristic
18C22×D524C2211yescharacteristic
19C22×D524C2211yescharacteristic
20C22×D524C2211yescharacteristic, complemented
21C22×D524C2211yescharacteristic, complemented
22C22×C22⋊D131C2211yescharacteristic
23C22×C22⋊D131C2211yescharacteristic
24C22×C22⋊D131C2211yescharacteristic, complemented
25C22×C22⋊D131C2211yescharacteristic, complemented
26C2×D4×D131—24no—
27C2×D4×D131—24no—
28C2×D4×D131—24no—
29C23×Dic131C2211yescharacteristic, complemented
30C22×C4×D131C2211yescharacteristic
31#16862220—24no—
32#16862218—24no—
33#16862216—24no—
34#16862216—24no—
35C2×C4⋊Dic131C2211yescharacteristic, complemented
36C2×Dic131⋊C4C2211yescharacteristic, complemented
37C2×C22⋊Dic131C2211yescharacteristic, complemented
38C2×(C2×D131)⋊C4—24no—
39C2×(C2×D131)⋊C4C2211yescharacteristic
40C2×(C2×D131)⋊C4C2211yescharacteristic
41C2×(C2×D131)⋊C4C2211yescharacteristic
42C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
48C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
49C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
50C131⋊#95C2211yescharacteristic, complemented
51C131⋊#95C2211yescharacteristic, complemented
52C22×D4×C131C2211yescharacteristic
53C23×C4×C131C2211yescharacteristic, complemented
54C2×C22⋊C4×C131C2211yescharacteristic, complemented
55C2×C22⋊C4×C131C2211yescharacteristic, complemented
56C2×C22⋊C4×C131C2211yescharacteristic, complemented
57#95×C131C2211yescharacteristic, complemented
58#95×C131C2211yescharacteristic, complemented
59C23×D131—22no—
60C23×D131—22no—
61C23×D131—22no—
62C23×D131—22no—
63C23×D131—22no—
64C23×D131—22no—
65C23×D131—22no—
66C23×D131—22no—
67C23×D131—22no—
68C23×D131—41no—
69C23×D131—41no—
70C23×D131—41no—
71C23×D131—41no—
72C23×D131D414yescomplemented
73C23×D131C2311yescharacteristic
74C23×D131C2311yescharacteristic
75C23×D131C2311yescharacteristic
76C23×D131C2311yescharacteristic
77C2×D524—22no—
78C2×D524—22no—
79C2×D524—22no—
80C2×D524—22no—
81C2×D524—22no—
82C2×D524—22no—
83C2×D524—22no—
84C2×D524—22no—
85C2×D524—22no—
86C2×D524—41no—
87C2×D524—41no—
88C2×D524—41no—
89C2×C22⋊D131—22no—
90C2×C22⋊D131—22no—
91C2×C22⋊D131—22no—
92C2×C22⋊D131—22no—
93C2×C22⋊D131—22no—
94C2×C22⋊D131—22no—
95C2×C22⋊D131—22no—
96C2×C22⋊D131—22no—
97C2×C22⋊D131—22no—
98C2×C22⋊D131—22no—
99C2×C22⋊D131—41no—
100C2×C22⋊D131D414yescomplemented
101D4×D131—232no—
102C22×Dic131—22no—
103C22×Dic131—22no—
104C22×Dic131—41no—
105C22×Dic131C2311yescharacteristic
106C22×Dic131C2311yescharacteristic
107C22×Dic131C2311yescharacteristic
108C22×Dic131C2311yescharacteristic
109C2×C4×D131—22no—
110C2×C4×D131—22no—
111C2×C4×D131—22no—
112C4⋊Dic131—22no—
113Dic131⋊C4—22no—
114C22⋊Dic131—22no—
115(C2×D131)⋊C4—22no—
116(C2×D131)⋊C4—22no—
117(C2×D131)⋊C4—22no—
118(C2×D131)⋊C4—22no—
119(C2×D131)⋊C4—22no—
120(C2×D131)⋊C4—22no—
121(C2×D131)⋊C4—22no—
122(C2×D131)⋊C4—22no—
123(C2×D131)⋊C4—22no—
124(C2×D131)⋊C4—41no—
125(C2×D131)⋊C4D414yescomplemented
126C24×C131C2311yescharacteristic
127C24×C131C2311yescharacteristic
128C2×D4×C131—22no—
129C2×D4×C131—22no—
130C2×D4×C131D414yescomplemented
131C22×C4×C131—22no—
132C22×C4×C131—22no—
133C22×C4×C131—41no—
134C22×C4×C131C2311yescharacteristic
135C22×C4×C131C2311yescharacteristic
136C22×C4×C131C2311yescharacteristic
137C22×C4×C131C2311yescharacteristic
138C22×C4×C131C2311yescharacteristic
139C22⋊C4×C131—22no—
140C22⋊C4×C131—22no—
141C22⋊C4×C131—41no—
142C22×D131—21no—
143C22×D131—21no—
144C22×D131—21no—
145C22×D131—21no—
146C22×D131—28no—
147C22×D131—28no—
148C22×D131—21no—
149C22×D131—21no—
150C22×D131—21no—
151C22×D131—21no—
152C22×D131—21no—
153C22×D131—21no—
154C22×D131—21no—
155C22×D131—21no—
156C22×D131—21no—
157C22×D131—21no—
158C22×D131—21no—
159C22×D131—21no—
160C22×D131—28no—
161C22×D131—21no—
162C22×D131—21no—
163C22×D131—21no—
164C22×D131—21no—
165C22×D131—21no—
166C22×D131—21no—
167C22×D131—21no—
168C22×D131—44no—
169C22×D131—44no—
170C22×D131—44no—
171C22×D131—44no—
172C22×D131—44no—
173C22×D131—44no—
174C22×D131—44no—
175C22×D131—44no—
176C22×D131—44no—
177C22×D131—44no—
178C22×D131—44no—
179C22×D131C2×D412yes—
180C22×D131C2×D412yes—
181C22×D131C2×D412yes—
182C22×D131C2×D412yes—
183C22×D131#4112yes—
184D524—28no—
185D524—44no—
186D524—44no—
187D524—44no—
188C22⋊D131—28no—
189C22⋊D131—28no—
190C22⋊D131—44no—
191C22⋊D131—44no—
192C2×Dic131—21no—
193C2×Dic131—21no—
194C2×Dic131—21no—
195C2×Dic131—21no—
196C2×Dic131—21no—
197C2×Dic131—21no—
198C2×Dic131—21no—
199C2×Dic131—21no—
200C2×Dic131—21no—
201C2×Dic131—21no—
202C2×Dic131—44no—
203C2×Dic131C2×D412yes—
204C2×Dic131C2×D412yescomplemented
205C4×D131—28no—
206C23×C131—21no—
207C23×C131—21no—
208C23×C131—21no—
209C23×C131—21no—
210C23×C131—21no—
211C23×C131—21no—
212C23×C131—21no—
213C23×C131—21no—
214C23×C131—21no—
215C23×C131—21no—
216C23×C131C2×D412yes—
217C23×C131C2×D412yes—
218C23×C131C2×D412yes—
219C23×C131C2×D412yes—
220C23×C131C2411yescharacteristic
221D4×C131—28no—
222C2×C4×C131—21no—
223C2×C4×C131—21no—
224C2×C4×C131—21no—
225C2×C4×C131—21no—
226C2×C4×C131—21no—
227C2×C4×C131—21no—
228C2×C4×C131—21no—
229C2×C4×C131—21no—
230C2×C4×C131—21no—
231C2×C4×C131—21no—
232C2×C4×C131—21no—
233C2×C4×C131—21no—
234C2×C4×C131—21no—
235C2×C4×C131—44no—
236C2×C4×C131C2×D412yes—
237C2×C4×C131C2×D412yescomplemented
238C22×C131—22no—
239C22×C131—22no—
240C22×C131—22no—
241C22×C131—22no—
242C22×C131—22no—
243C22×C131—22no—
244C22×C131—22no—
245C22×C131—22no—
246C22×C131—22no—
247C22×C131—41no—
248C22×C131—41no—
249C22×C131—41no—
250C22×C131—41no—
251C22×C131#12014yescomplemented
252C22×C131C22×D411yescharacteristic
253C22×C131C22×D411yescharacteristic
254C22×C131C22×D411yescharacteristic
255C22×C131C22×D411yescharacteristic
256C22×C131C2×#4111yescharacteristic
257C22×C131H(5,𝔽2)11yescharacteristic
258C22×C131H(5,𝔽2)11yescharacteristic
259C2×D131—22no—
260C2×D131—22no—
261C2×D131—22no—
262C2×D131—22no—
263C2×D131—22no—
264C2×D131—22no—
265C2×D131—22no—
266C2×D131—22no—
267C2×D131—22no—
268C2×D131—22no—
269C2×D131—22no—
270C2×D131—22no—
271C2×D131—416no—
272C2×D131—416no—
273C2×D131—41no—
274C2×D131—41no—
275C2×D131—41no—
276C2×D131—41no—
277C2×D131—41no—
278C2×D131—41no—
279C2×D131—41no—
280C2×D131—41no—
281C2×D131—41no—
282C2×D131—41no—
283C2×D131—41no—
284C2×D131—41no—
285C2×D131—41no—
286C2×D131—41no—
287C2×D131—41no—
288C2×D131#12114yescomplemented
289C4×C131—22no—
290C4×C131—22no—
291C4×C131—41no—
292C4×C131—41no—
293C4×C131—41no—
294Dic131—22no—
295Dic131—22no—
296Dic131—41no—
297Dic131—41no—
298C2×C131—24no—
299C2×C131—42no—
300C2×C131C2×#12011yescharacteristic
301C2×C131C2×#12111yescharacteristic
302C2×C131#53411yescharacteristic
303C2×C131D4211yescharacteristic
304C2×C131D4211yescharacteristic
305C2×C131#54611yescharacteristic
306C2×C131#54611yescharacteristic
307D131—24no—
308D131—42no—
309D131—42no—
310D131—81no—
311C131#241011yescharacteristic, complemented, Sylow
312#2410—1311nomaximal, Sylow
313C23×D4—1311no—
314C2×#121—1311no—
315C2×#121—1311no—
316C2×#120—1311no—
317C2×#120—1311no—
318C22×C22⋊C4—1311no—
319#394—1311no—
320#394—1311no—
321#392—1311no—
322#392—1311no—
323#386—1311no—
324#386—1311no—
325#386D13111yescharacteristic, complemented
326#385—1311no—
327#379—1311no—
328C25—1311no—
329C25—1311no—
330C22×D4—2624no—
331C22×D4—2624no—
332C22×D4—2624no—
333C22×D4—1311no—
334C22×D4—1311no—
335C22×D4—1311no—
336C22×D4—1311no—
337C22×D4—1311no—
338C22×D4—1311no—
339C22×D4—1311no—
340C22×D4—1311no—
341C22×D4C2×D13111yescharacteristic
342C23×C4—1311no—
343C23×C4—1311no—
344C23×C4C2×D13111yescharacteristic, complemented
345#121—2624no—
346#121—2624no—
347#120—2624no—
348#120—2624no—
349C2×C4⋊C4—1311no—
350C2×C4⋊C4—1311no—
351C2×C22⋊C4—2624no—
352C2×C22⋊C4—1311no—
353C2×C22⋊C4—1311no—
354C2×C22⋊C4—1311no—
355C2×C22⋊C4—1311no—
356C2×C22⋊C4—1311no—
357C2×C22⋊C4—1311no—
358C2×C22⋊C4—1311no—
359C2×C22⋊C4—1311no—
360C2×C22⋊C4—1311no—
361C2×C22⋊C4—1311no—
362C2×C22⋊C4—1311no—
363C2×C22⋊C4—1311no—
364C2×C22⋊C4C2×D13111yescharacteristic, complemented
365C2×C22⋊C4C2×D13111yescharacteristic, complemented
366C2×C22⋊C4C2×D13111yescharacteristic, complemented
367#95—1311no—
368#95—1311no—
369#95C2×D13111yescharacteristic, complemented
370#95C2×D13111yescharacteristic, complemented
371C24—2622no—
372C24—2622no—
373C24—2622no—
374C24—2622no—
375C24—2622no—
376C24—2622no—
377C24—2622no—
378C24—2622no—
379C24—2622no—
380C24—5241no—
381C24—5241no—
382C24—5241no—
383C24—5241no—
384C24—1311no—
385C24—1311no—
386C24—1311no—
387C24—1314no—
388C24—1311no—
389C24C22×D13111yescharacteristic
390C24C22×D13111yescharacteristic
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—2622no—
394C2×D4—2622no—
395C2×D4—2622no—
396C2×D4—2622no—
397C2×D4—2622no—
398C2×D4—2622no—
399C2×D4—2622no—
400C2×D4—26232no—
401C2×D4—2622no—
402C2×D4—2622no—
403C2×D4—2622no—
404C2×D4—2622no—
405C2×D4—2622no—
406C2×D4—2622no—
407C2×D4—2622no—
408C2×D4—2622no—
409C2×D4—2622no—
410C2×D4—2622no—
411C2×D4—2622no—
412C2×D4—2622no—
413C2×D4—5241no—
414C2×D4—5241no—
415C2×D4—5241no—
416C2×D4—5241no—
417C2×D4—1314no—
418C2×D4D52414yescomplemented
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—2622no—
423C22×C4—2622no—
424C22×C4—2622no—
425C22×C4—2622no—
426C22×C4—2622no—
427C22×C4—5241no—
428C22×C4—1311no—
429C22×C4—1311no—
430C22×C4—1311no—
431C22×C4—1311no—
432C22×C4C22×D13111yescharacteristic
433C22×C4C22×D13111yescharacteristic
434C22×C4C22×D13111yescharacteristic
435C22×C4C22×D13111yescharacteristic
436C22×C4C22×D13111yescharacteristic
437C4⋊C4—2622no—
438C4⋊C4—2622no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—2622no—
443C22⋊C4—2622no—
444C22⋊C4—2622no—
445C22⋊C4—2622no—
446C22⋊C4—2622no—
447C22⋊C4—2622no—
448C22⋊C4—2622no—
449C22⋊C4—2622no—
450C22⋊C4—2622no—
451C22⋊C4—2622no—
452C22⋊C4—5241no—
453C22⋊C4—1314no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—2628no—
465C23—2628no—
466C23—2621no—
467C23—2621no—
468C23—2621no—
469C23—2621no—
470C23—2628no—
471C23—2621no—
472C23—2621no—
473C23—2621no—
474C23—2621no—
475C23—2621no—
476C23—2621no—
477C23—2621no—
478C23—2621no—
479C23—2621no—
480C23—2621no—
481C23—2621no—
482C23—2621no—
483C23—2621no—
484C23—2621no—
485C23—2621no—
486C23—2621no—
487C23—2621no—
488C23—2621no—
489C23—2621no—
490C23—5244no—
491C23—5244no—
492C23—5244no—
493C23—5244no—
494C23—5244no—
495C23—5244no—
496C23—5244no—
497C23—5244no—
498C23—5244no—
499C23—5244no—
500C23—5244no—
501C23—1312no—
502C23—1312no—
503C23—1312no—
504C23—1312no—
505C23—1312no—
506C23D4×D13112yes—
507C23D4×D13112yes—
508C23C2×D52412yes—
509C23C2×D52412yes—
510C23C23×D13111yescharacteristic
511D4—28no—
512D4—2628no—
513D4—2628no—
514D4—2628no—
515D4—5244no—
516D4—5244no—
517D4—5244no—
518D4—5244no—
519D4—5244no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—44no—
534C2×C4—2628no—
535C2×C4—2621no—
536C2×C4—2621no—
537C2×C4—2621no—
538C2×C4—2621no—
539C2×C4—2621no—
540C2×C4—2621no—
541C2×C4—2621no—
542C2×C4—2621no—
543C2×C4—2621no—
544C2×C4—2621no—
545C2×C4—5244no—
546C2×C4—1312no—
547C2×C4—1312no—
548C2×C4D4×D13112yescomplemented
549C2×C4C2×D52412yes—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—2622no—
564C22—2622no—
565C22—2622no—
566C22—2622no—
567C22—2622no—
568C22—2622no—
569C22—2622no—
570C22—2622no—
571C22—2622no—
572C22—2622no—
573C22—2622no—
574C22—2622no—
575C22—52416no—
576C22—52416no—
577C22—5241no—
578C22—5241no—
579C22—5241no—
580C22—5241no—
581C22—5241no—
582C22—5241no—
583C22—5241no—
584C22—5241no—
585C22—5241no—
586C22—5241no—
587C22—5241no—
588C22—5241no—
589C22—5241no—
590C22—5241no—
591C22—5241no—
592C22—1314no—
593C22#1686221614yescomplemented
594C22C2×D4×D13111yescharacteristic
595C22C2×D4×D13111yescharacteristic
596C22C2×D4×D13111yescharacteristic
597C22C22×D52411yescharacteristic
598C22#41×D13111yescharacteristic
599C22#1686229111yescharacteristic
600C22#1686229211yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—2622no—
607C4—2622no—
608C4—5241no—
609C4—5241no—
610C2—24no—
611C2—42no—
612C2—2624no—
613C2—5242no—
614C2—5242no—
615C2—10481no—
616C2C2×#1686221611yescharacteristic
617C2#121×D13111yescharacteristic
618C2#2255387011yescharacteristic
619C2#2255394411yescharacteristic
620C2D4×D52411yescharacteristic
621C2#2255395411yescharacteristic
622C2#2255395711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#386×C131C2
hypercenter#386D131
hypocenterC131#2410
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C2×#16862220C2semidirect
3. C2×#16862218C2semidirect
4. C2×#16862216C2semidirect
5. C2×#16862216C2semidirect
6. C22×(C2×D131)⋊C4C2semidirect
7. #22553230C2semidirect
8. #22553228C2semidirect
9. #22553222C2semidirect
10. #22553221C2semidirect
11. #22553190C2semidirect
12. #22553189C2semidirect
13. #22553186C2semidirect
14. #22553167C2semidirect
15. #386×C131C2semidirect
20. C22×D524C22semidirect
21. C22×D524C22semidirect
24. C22×C22⋊D131C22semidirect
25. C22×C22⋊D131C22semidirect
29. C23×Dic131C22semidirect
35. C2×C4⋊Dic131C22semidirect
36. C2×Dic131⋊C4C22semidirect
37. C2×C22⋊Dic131C22semidirect
42. C2×(C2×D131)⋊C4C22semidirect
43. C2×(C2×D131)⋊C4C22semidirect
44. C2×(C2×D131)⋊C4C22semidirect
45. C2×(C2×D131)⋊C4C22semidirect
46. C2×(C2×D131)⋊C4C22semidirect
47. C2×(C2×D131)⋊C4C22semidirect
48. C2×(C2×D131)⋊C4C22semidirect
49. C2×(C2×D131)⋊C4C22semidirect
50. C131⋊#95C22semidirect
51. C131⋊#95C22semidirect
53. C23×C4×C131C22semidirect
54. C2×C22⋊C4×C131C22semidirect
55. C2×C22⋊C4×C131C22semidirect
56. C2×C22⋊C4×C131C22semidirect
57. #95×C131C22semidirect
58. #95×C131C22semidirect
72. C23×D131D4semidirect
100. C2×C22⋊D131D4semidirect
125. (C2×D131)⋊C4D4semidirect
130. C2×D4×C131D4semidirect
204. C2×Dic131C2×D4semidirect
237. C2×C4×C131C2×D4semidirect
251. C22×C131#120semidirect
288. C2×D131#121semidirect
311. C131#2410semidirect
325. #386D131semidirect
344. C23×C4C2×D131semidirect
364. C2×C22⋊C4C2×D131semidirect
365. C2×C22⋊C4C2×D131semidirect
366. C2×C22⋊C4C2×D131semidirect
369. #95C2×D131semidirect
370. #95C2×D131semidirect
418. C2×D4D524semidirect
548. C2×C4D4×D131semidirect
593. C22#16862216semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1312
C2×#168622202
C2×#168622182
C2×#168622162
C22×(C2×D131)⋊C42
#225532302
#225532282
#225532222
#225532212
#225531902
#225531892
#225531862
#225531672
#386×C1312
#2410131