Finite Group Databasebeta

← #30669328

Subgroups — #30669328

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×#16862219C211yesmaximal, characteristic, complemented
3C2×#16862217C211yesmaximal, characteristic, complemented
4C2×#16862214C211yesmaximal, characteristic, complemented
5C2×#16862214C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
7#22553228C211yesmaximal, characteristic, complemented
8#22553226C211yesmaximal, characteristic, complemented
9#22553220C211yesmaximal, characteristic, complemented
10#22553190C211yesmaximal, characteristic, complemented
11#22553188C211yesmaximal, characteristic, complemented
12#22553188C211yesmaximal, characteristic, complemented
13#22553186C211yesmaximal, characteristic, complemented
14#22553167C211yesmaximal, characteristic, complemented
15#392×C131C211yesmaximal, characteristic, complemented
16C24×D131C2211yescharacteristic
17C24×D131C2211yescharacteristic
18C22×D524C2211yescharacteristic
19C22×C22⋊D131C2211yescharacteristic
20C22×C22⋊D131C2211yescharacteristic
21C22×C22⋊D131C2211yescharacteristic
22C22×C22⋊D131C2211yescharacteristic, complemented
23C22×C22⋊D131C2211yescharacteristic, complemented
24C2×D4×D131—24no—
25C2×D4×D131—24no—
26C2×D4×D131—24no—
27C22×C4×D131C2211yescharacteristic
28C22×C4×D131C2211yescharacteristic, complemented
29C22×C4×D131C2211yescharacteristic, complemented
30#16862219—24no—
31#16862217—24no—
32#16862214—24no—
33#16862214—24no—
34C2×C4⋊Dic131C2211yescharacteristic, complemented
35C2×Dic131⋊C4C2211yescharacteristic, complemented
36C2×C22⋊Dic131C2211yescharacteristic, complemented
37C2×C22⋊Dic131C2211yescharacteristic, complemented
38C2×C22⋊Dic131C2211yescharacteristic, complemented
39C2×C22⋊Dic131C2211yescharacteristic, complemented
40C2×C22⋊Dic131C2211yescharacteristic, complemented
41C2×(C2×D131)⋊C4C2211yescharacteristic
42C2×(C2×D131)⋊C4C2211yescharacteristic
43C2×(C2×D131)⋊C4C2211yescharacteristic
44C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
48C22⋊C4×D131—24no—
49C131⋊#95C2211yescharacteristic, complemented
50C131⋊#95C2211yescharacteristic, complemented
51C131⋊#95C2211yescharacteristic, complemented
52C22×D4×C131C2211yescharacteristic
53C22×D4×C131C2211yescharacteristic, complemented
54C22×D4×C131C2211yescharacteristic, complemented
55C2×C22⋊C4×C131C2211yescharacteristic, complemented
56C2×C22⋊C4×C131C2211yescharacteristic, complemented
57C2×C22⋊C4×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
76C2×D524—22no—
77C2×D524—22no—
78C2×D524D414yescomplemented
79C2×C22⋊D131—22no—
80C2×C22⋊D131—22no—
81C2×C22⋊D131—22no—
82C2×C22⋊D131—22no—
83C2×C22⋊D131—22no—
84C2×C22⋊D131—22no—
85C2×C22⋊D131—22no—
86C2×C22⋊D131—22no—
87C2×C22⋊D131—22no—
88C2×C22⋊D131—22no—
89C2×C22⋊D131—22no—
90C2×C22⋊D131—22no—
91C2×C22⋊D131—22no—
92C2×C22⋊D131—41no—
93C2×C22⋊D131D414yescomplemented
94D4×D131—232no—
95C22×Dic131C2311yescharacteristic
96C22×Dic131C2311yescharacteristic
97C22×Dic131C2311yescharacteristic
98C22×Dic131C2311yescharacteristic
99C22×Dic131C2311yescharacteristic
100C2×C4×D131—22no—
101C2×C4×D131—22no—
102C2×C4×D131—22no—
103C2×C4×D131—22no—
104C2×C4×D131—22no—
105C2×C4×D131—22no—
106C2×C4×D131—22no—
107C2×C4×D131—41no—
108C2×C4×D131—41no—
109C4⋊Dic131—22no—
110Dic131⋊C4—22no—
111C22⋊Dic131—22no—
112C22⋊Dic131—22no—
113C22⋊Dic131—22no—
114C22⋊Dic131—22no—
115C22⋊Dic131—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—41no—
122(C2×D131)⋊C4D414yescomplemented
123C24×C131C2311yescharacteristic
124C24×C131C2311yescharacteristic
125C24×C131C2311yescharacteristic
126C2×D4×C131—22no—
127C2×D4×C131—22no—
128C2×D4×C131—22no—
129C2×D4×C131—22no—
130C2×D4×C131—22no—
131C2×D4×C131—22no—
132C2×D4×C131—41no—
133C2×D4×C131—41no—
134C2×D4×C131—41no—
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—21no—
147C22×D131—21no—
148C22×D131—21no—
149C22×D131—21no—
150C22×D131—28no—
151C22×D131—21no—
152C22×D131—21no—
153C22×D131—21no—
154C22×D131—21no—
155C22×D131—21no—
156C22×D131—21no—
157C22×D131—28no—
158C22×D131—28no—
159C22×D131—44no—
160C22×D131—44no—
161C22×D131—44no—
162C22×D131—44no—
163C22×D131—44no—
164C22×D131—44no—
165C22×D131—44no—
166C22×D131—44no—
167C22×D131—44no—
168C22×D131—44no—
169C22×D131—44no—
170C22×D131C2×D412yes—
171C22×D131C2×D412yes—
172C22×D131C2×D412yes—
173C22×D131C2×D412yes—
174C22×D131C2×D412yes—
175C22×D131C2×D412yes—
176C22×D131#4112yes—
177D524—28no—
178C22⋊D131—28no—
179C22⋊D131—28no—
180C22⋊D131—44no—
181C22⋊D131—44no—
182C22⋊D131—44no—
183C2×Dic131—21no—
184C2×Dic131—21no—
185C2×Dic131—21no—
186C2×Dic131—21no—
187C2×Dic131—21no—
188C2×Dic131—21no—
189C2×Dic131—21no—
190C2×Dic131—21no—
191C2×Dic131—21no—
192C2×Dic131—21no—
193C2×Dic131—21no—
194C2×Dic131—21no—
195C2×Dic131—21no—
196C2×Dic131C2×D412yes—
197C2×Dic131C2×D412yescomplemented
198C4×D131—28no—
199C4×D131—44no—
200C4×D131—44no—
201C23×C131—21no—
202C23×C131—21no—
203C23×C131—21no—
204C23×C131—21no—
205C23×C131—21no—
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×C131—21no—
217C23×C131—21no—
218C23×C131—21no—
219C23×C131—21no—
220C23×C131C2×D412yes—
221C23×C131C2×D412yes—
222C23×C131C2411yescharacteristic
223D4×C131—28no—
224D4×C131—44no—
225D4×C131—44no—
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—21no—
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—41no—
245C22×C131—41no—
246C22×C131—41no—
247C22×C131—41no—
248C22×C131—41no—
249C22×C131—41no—
250C22×C131—41no—
251C22×C131—41no—
252C22×C131—41no—
253C22×C131—41no—
254C22×C131—41no—
255C22×C131—41no—
256C22×C131—41no—
257C22×C131—41no—
258C22×C131—41no—
259C22×C131C22×D411yescharacteristic
260C22×C131C22×D411yescharacteristic
261C22×C131C22×D411yescharacteristic
262C22×C131C22×D411yescharacteristic
263C22×C131C2×#4111yescharacteristic
264C22×C131H(5,𝔽2)11yescharacteristic
265C22×C131H(5,𝔽2)11yescharacteristic
266C2×D131—22no—
267C2×D131—22no—
268C2×D131—22no—
269C2×D131—22no—
270C2×D131—22no—
271C2×D131—22no—
272C2×D131—22no—
273C2×D131—22no—
274C2×D131—22no—
275C2×D131—22no—
276C2×D131—22no—
277C2×D131—22no—
278C2×D131—22no—
279C2×D131—22no—
280C2×D131—22no—
281C2×D131—416no—
282C2×D131—416no—
283C2×D131—41no—
284C2×D131—41no—
285C2×D131—41no—
286C2×D131—41no—
287C2×D131#12014yescomplemented
288C2×D131#12114yescomplemented
289C4×C131—22no—
290C4×C131—22no—
291C4×C131—41no—
292C4×C131—41no—
293Dic131—22no—
294Dic131—22no—
295Dic131—41no—
296Dic131—41no—
297Dic131—41no—
298C2×C131—42no—
299C2×C131—42no—
300C2×C131—81no—
301C2×C131C2×#12011yescharacteristic
302C2×C131C2×#12111yescharacteristic
303C2×C131#53411yescharacteristic
304C2×C131D4211yescharacteristic
305C2×C131D4211yescharacteristic
306C2×C131#54611yescharacteristic
307C2×C131#54611yescharacteristic
308D131—24no—
309D131—24no—
310D131—42no—
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#392D13111yescharacteristic, complemented
323#386—1311no—
324#386—1311no—
325#386—1311no—
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×D4C2×D13111yescharacteristic
340C22×D4C2×D13111yescharacteristic, complemented
341C22×D4C2×D13111yescharacteristic, complemented
342C23×C4—1311no—
343C23×C4—1311no—
344C23×C4—1311no—
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#95—1311no—
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—1314no—
386C24—1311no—
387C24—1311no—
388C24C22×D13111yescharacteristic
389C24C22×D13111yescharacteristic
390C24C22×D13111yescharacteristic
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—2622no—
401C2×D4—2622no—
402C2×D4—26232no—
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—2622no—
414C2×D4—2622no—
415C2×D4—2622no—
416C2×D4—5241no—
417C2×D4—1314no—
418C2×D4—1314no—
419C22×C4—2622no—
420C22×C4—2622no—
421C22×C4—2622no—
422C22×C4—2622no—
423C22×C4—2622no—
424C22×C4—2622no—
425C22×C4—2622no—
426C22×C4—5241no—
427C22×C4—5241no—
428C22×C4—1311no—
429C22×C4—1311no—
430C22×C4—1311no—
431C22×C4—1311no—
432C22×C4—1311no—
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—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—2628no—
474C23—2628no—
475C23—2628no—
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—
506C23—1312no—
507C23—1312no—
508C23D4×D13112yes—
509C23D4×D13112yes—
510C23C23×D13111yescharacteristic
511D4—28no—
512D4—44no—
513D4—44no—
514D4—2628no—
515D4—2628no—
516D4—2628no—
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—2628no—
531C2×C4—2621no—
532C2×C4—2621no—
533C2×C4—2621no—
534C2×C4—2621no—
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—5244no—
545C2×C4—5244no—
546C2×C4—1312no—
547C2×C4—1312no—
548C2×C4D4×D13112yes—
549C2×C4D4×D13112yescomplemented
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—2622no—
572C22—2622no—
573C22—2622no—
574C22—2622no—
575C22—2622no—
576C22—2622no—
577C22—2622no—
578C22—2622no—
579C22—2622no—
580C22—2622no—
581C22—2622no—
582C22—2622no—
583C22—2622no—
584C22—2622no—
585C22—2622no—
586C22—52416no—
587C22—52416no—
588C22—5241no—
589C22—5241no—
590C22—5241no—
591C22—5241no—
592C22—1314no—
593C22—1314no—
594C22C2×D4×D13111yescharacteristic
595C22C2×D4×D13111yescharacteristic
596C22C2×D4×D13111yescharacteristic
597C22C2×D4×D13111yescharacteristic
598C22#41×D13111yescharacteristic
599C22#1686229111yescharacteristic
600C22#1686229111yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—2622no—
606C4—2622no—
607C4—5241no—
608C4—5241no—
609C4—5241no—
610C2—42no—
611C2—42no—
612C2—2624no—
613C2—2624no—
614C2—81no—
615C2—5242no—
616C2#120×D13111yescharacteristic
617C2#121×D13111yescharacteristic
618C2#2255387211yescharacteristic
619C2#2255394411yescharacteristic
620C2#2255394711yescharacteristic
621C2#2255395111yescharacteristic
622C2#2255395911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#392×C131C2
hypercenter#392D131
hypocenterC131#2410
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C2×#16862219C2semidirect
3. C2×#16862217C2semidirect
4. C2×#16862214C2semidirect
5. C2×#16862214C2semidirect
6. C2×C22⋊C4×D131C2semidirect
7. #22553228C2semidirect
8. #22553226C2semidirect
9. #22553220C2semidirect
10. #22553190C2semidirect
11. #22553188C2semidirect
12. #22553188C2semidirect
13. #22553186C2semidirect
14. #22553167C2semidirect
15. #392×C131C2semidirect
22. C22×C22⋊D131C22semidirect
23. C22×C22⋊D131C22semidirect
28. C22×C4×D131C22semidirect
29. C22×C4×D131C22semidirect
34. C2×C4⋊Dic131C22semidirect
35. C2×Dic131⋊C4C22semidirect
36. C2×C22⋊Dic131C22semidirect
37. C2×C22⋊Dic131C22semidirect
38. C2×C22⋊Dic131C22semidirect
39. C2×C22⋊Dic131C22semidirect
40. C2×C22⋊Dic131C22semidirect
44. C2×(C2×D131)⋊C4C22semidirect
45. C2×(C2×D131)⋊C4C22semidirect
46. C2×(C2×D131)⋊C4C22semidirect
47. C2×(C2×D131)⋊C4C22semidirect
49. C131⋊#95C22semidirect
50. C131⋊#95C22semidirect
51. C131⋊#95C22semidirect
53. C22×D4×C131C22semidirect
54. C22×D4×C131C22semidirect
55. C2×C22⋊C4×C131C22semidirect
56. C2×C22⋊C4×C131C22semidirect
57. C2×C22⋊C4×C131C22semidirect
58. #95×C131C22semidirect
72. C23×D131D4semidirect
78. C2×D524D4semidirect
93. C2×C22⋊D131D4semidirect
122. (C2×D131)⋊C4D4semidirect
197. C2×Dic131C2×D4semidirect
237. C2×C4×C131C2×D4semidirect
287. C2×D131#120semidirect
288. C2×D131#121semidirect
311. C131#2410semidirect
322. #392D131semidirect
340. C22×D4C2×D131semidirect
341. C22×D4C2×D131semidirect
364. C2×C22⋊C4C2×D131semidirect
365. C2×C22⋊C4C2×D131semidirect
366. C2×C22⋊C4C2×D131semidirect
370. #95C2×D131semidirect
549. C2×C4D4×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×#168622192
C2×#168622172
C2×#168622142
C2×C22⋊C4×D1312
#225532282
#225532262
#225532202
#225531902
#225531882
#225531862
#225531672
#392×C1312
#2410131