Finite Group Databasebeta

← #30669330

Subgroups — #30669330

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×#16862221C211yesmaximal, characteristic, complemented
3C2×#16862219C211yesmaximal, characteristic, complemented
4C2×#16862216C211yesmaximal, characteristic, complemented
5C2×#16862216C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
7#22553227C211yesmaximal, characteristic, complemented
8#22553221C211yesmaximal, characteristic, complemented
9#22553220C211yesmaximal, characteristic, complemented
10#22553190C211yesmaximal, characteristic, complemented
11#22553190C211yesmaximal, characteristic, complemented
12#22553188C211yesmaximal, characteristic, complemented
13#22553183C211yesmaximal, characteristic, complemented
14#22553167C211yesmaximal, characteristic, complemented
15#394×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—
29C22×C4×D131C2211yescharacteristic
30C22×C4×D131C2211yescharacteristic, complemented
31C22×C4×D131C2211yescharacteristic, complemented
32#16862221—24no—
33#16862219—24no—
34#16862216—24no—
35#16862216—24no—
36C2×Dic131⋊C4C2211yescharacteristic, complemented
37C2×C22⋊Dic131C2211yescharacteristic, complemented
38C2×C22⋊Dic131C2211yescharacteristic, complemented
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
48C22⋊C4×D131—24no—
49C131⋊#95C2211yescharacteristic, complemented
50C131⋊#95C2211yescharacteristic, complemented
51C131⋊#95C2211yescharacteristic, complemented
52C22×D4×C131C2211yescharacteristic
53C2×C4⋊C4×C131C2211yescharacteristic, complemented
54C2×C22⋊C4×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
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—22no—
87C2×D524—41no—
88C2×D524D414yescomplemented
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—41no—
98C2×C22⋊D131—41no—
99C2×C22⋊D131—41no—
100C2×C22⋊D131D414yescomplemented
101D4×D131—232no—
102C22×Dic131C2311yescharacteristic
103C22×Dic131C2311yescharacteristic
104C22×Dic131C2311yescharacteristic
105C22×Dic131C2311yescharacteristic
106C2×C4×D131—22no—
107C2×C4×D131—22no—
108C2×C4×D131—22no—
109C2×C4×D131—22no—
110C2×C4×D131—22no—
111C2×C4×D131—22no—
112C2×C4×D131—22no—
113C2×C4×D131—41no—
114C2×C4×D131—41no—
115Dic131⋊C4—22no—
116C22⋊Dic131—22no—
117C22⋊Dic131—41no—
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—22no—
125(C2×D131)⋊C4—41no—
126(C2×D131)⋊C4D414yescomplemented
127C24×C131C2311yescharacteristic
128C24×C131C2311yescharacteristic
129C2×D4×C131—22no—
130C2×D4×C131—22no—
131C2×D4×C131—22no—
132C22×C4×C131C2311yescharacteristic
133C22×C4×C131C2311yescharacteristic
134C22×C4×C131C2311yescharacteristic
135C22×C4×C131C2311yescharacteristic
136C22×C4×C131C2311yescharacteristic
137C4⋊C4×C131—22no—
138C22⋊C4×C131—22no—
139C22⋊C4×C131—22no—
140C22⋊C4×C131—22no—
141C22⋊C4×C131—22no—
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—21no—
160C22×D131—21no—
161C22×D131—21no—
162C22×D131—21no—
163C22×D131—21no—
164C22×D131—21no—
165C22×D131—21no—
166C22×D131—44no—
167C22×D131—44no—
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×D131C2×D412yes—
178C22×D131C2×D412yes—
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×Dic131C2×D412yes—
203C2×Dic131C2×D412yescomplemented
204C4×D131—28no—
205C4×D131—44no—
206C4×D131—44no—
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×C131C2×D412yes—
220C23×C131C2×D412yes—
221C23×C131C2411yescharacteristic
222D4×C131—28no—
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—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×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—22no—
272C2×D131—22no—
273C2×D131—22no—
274C2×D131—416no—
275C2×D131—416no—
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#12014yescomplemented
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—42no—
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—24no—
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#394D13111yescharacteristic, complemented
321#392—1311no—
322#392—1311no—
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×D4—1311no—
340C22×D4—1311no—
341C22×D4C2×D13111yescharacteristic
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⋊C4C2×D13111yescharacteristic, complemented
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⋊C4C2×D13111yescharacteristic, complemented
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—1311no—
386C24—1314no—
387C24—1311no—
388C24—1311no—
389C24C22×D13111yescharacteristic
390C24C22×D13111yescharacteristic
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—2622no—
395C2×D4—2622no—
396C2×D4—2622no—
397C2×D4—2622no—
398C2×D4—2622no—
399C2×D4—2622no—
400C2×D4—2622no—
401C2×D4—2622no—
402C2×D4—2622no—
403C2×D4—26232no—
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×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×C4C22×D13111yescharacteristic
433C22×C4C22×D13111yescharacteristic
434C22×C4C22×D13111yescharacteristic
435C22×C4C22×D13111yescharacteristic
436C22×C4C22×D13111yescharacteristic
437C4⋊C4—22no—
438C4⋊C4—2622no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
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—5241no—
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—2628no—
467C23—2628no—
468C23—2628no—
469C23—2621no—
470C23—2621no—
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—
506C23—1312no—
507C23—1312no—
508C23D4×D13112yes—
509C23D4×D13112yes—
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—2628no—
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—2622no—
565C22—2622no—
566C22—2622no—
567C22—2622no—
568C22—2622no—
569C22—2622no—
570C22—2622no—
571C22—2622no—
572C22—2622no—
573C22—2622no—
574C22—2622no—
575C22—2622no—
576C22—2622no—
577C22—2622no—
578C22—2622no—
579C22—52416no—
580C22—52416no—
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—1314no—
594C22C2×D4×D13111yescharacteristic
595C22C2×D4×D13111yescharacteristic
596C22C2×D4×D13111yescharacteristic
597C22C2×D4×D13111yescharacteristic
598C22#41×D13111yescharacteristic
599C22#1686229211yescharacteristic
600C22#1686229211yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—2622no—
607C4—2622no—
608C4—5241no—
609C4—5241no—
610C2—42no—
611C2—42no—
612C2—2624no—
613C2—2624no—
614C2—5242no—
615C2—10481no—
616C2#120×D13111yescharacteristic
617C2#121×D13111yescharacteristic
618C2#2255387311yescharacteristic
619C2#2255394611yescharacteristic
620C2#2255394611yescharacteristic
621C2#2255395411yescharacteristic
622C2#2255395511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#394×C131C2
hypercenter#394D131
hypocenterC131#2410
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C2×#16862221C2semidirect
3. C2×#16862219C2semidirect
4. C2×#16862216C2semidirect
5. C2×#16862216C2semidirect
6. C2×C22⋊C4×D131C2semidirect
7. #22553227C2semidirect
8. #22553221C2semidirect
9. #22553220C2semidirect
10. #22553190C2semidirect
11. #22553190C2semidirect
12. #22553188C2semidirect
13. #22553183C2semidirect
14. #22553167C2semidirect
15. #394×C131C2semidirect
20. C22×D524C22semidirect
21. C22×D524C22semidirect
24. C22×C22⋊D131C22semidirect
25. C22×C22⋊D131C22semidirect
30. C22×C4×D131C22semidirect
31. C22×C4×D131C22semidirect
36. C2×Dic131⋊C4C22semidirect
37. C2×C22⋊Dic131C22semidirect
38. 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
49. C131⋊#95C22semidirect
50. C131⋊#95C22semidirect
51. C131⋊#95C22semidirect
53. C2×C4⋊C4×C131C22semidirect
54. C2×C22⋊C4×C131C22semidirect
55. C2×C22⋊C4×C131C22semidirect
56. C2×C22⋊C4×C131C22semidirect
57. C2×C22⋊C4×C131C22semidirect
58. #95×C131C22semidirect
72. C23×D131D4semidirect
88. C2×D524D4semidirect
100. C2×C22⋊D131D4semidirect
126. (C2×D131)⋊C4D4semidirect
203. C2×Dic131C2×D4semidirect
237. C2×C4×C131C2×D4semidirect
287. C2×D131#120semidirect
288. C2×D131#121semidirect
311. C131#2410semidirect
320. #394D131semidirect
350. C2×C4⋊C4C2×D131semidirect
363. C2×C22⋊C4C2×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×#168622212
C2×#168622192
C2×#168622162
C2×C22⋊C4×D1312
#225532272
#225532212
#225532202
#225531902
#225531882
#225531832
#225531672
#394×C1312
#2410131