Finite Group Databasebeta

← #46839731

Subgroups — #46839731

Order 34688.

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×D271C211yesmaximal, characteristic, complemented
2C2×#22576133C211yesmaximal, characteristic, complemented
3C2×#22576131C211yesmaximal, characteristic, complemented
4C2×#22576128C211yesmaximal, characteristic, complemented
5C2×#22576128C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D271C211yesmaximal, characteristic, complemented
7#31773813C211yesmaximal, characteristic, complemented
8#31773811C211yesmaximal, characteristic, complemented
9#31773805C211yesmaximal, characteristic, complemented
10#31773775C211yesmaximal, characteristic, complemented
11#31773773C211yesmaximal, characteristic, complemented
12#31773773C211yesmaximal, characteristic, complemented
13#31773771C211yesmaximal, characteristic, complemented
14#31773752C211yesmaximal, characteristic, complemented
15#392×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C24×D271C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C22⋊D271C2211yescharacteristic
21C22×C22⋊D271C2211yescharacteristic
22C22×C22⋊D271C2211yescharacteristic, complemented
23C22×C22⋊D271C2211yescharacteristic, complemented
24C2×D4×D271—24no—
25C2×D4×D271—24no—
26C2×D4×D271—24no—
27C22×C4×D271C2211yescharacteristic
28C22×C4×D271C2211yescharacteristic, complemented
29C22×C4×D271C2211yescharacteristic, complemented
30#22576133—24no—
31#22576131—24no—
32#22576128—24no—
33#22576128—24no—
34C2×C4⋊Dic271C2211yescharacteristic, complemented
35C2×Dic271⋊C4C2211yescharacteristic, complemented
36C2×C22⋊Dic271C2211yescharacteristic, complemented
37C2×C22⋊Dic271C2211yescharacteristic, complemented
38C2×C22⋊Dic271C2211yescharacteristic, complemented
39C2×C22⋊Dic271C2211yescharacteristic, complemented
40C2×C22⋊Dic271C2211yescharacteristic, complemented
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic
43C2×(C2×D271)⋊C4C2211yescharacteristic
44C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
47C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
48C22⋊C4×D271—24no—
49C271⋊#95C2211yescharacteristic, complemented
50C271⋊#95C2211yescharacteristic, complemented
51C271⋊#95C2211yescharacteristic, complemented
52C22×D4×C271C2211yescharacteristic
53C22×D4×C271C2211yescharacteristic, complemented
54C22×D4×C271C2211yescharacteristic, complemented
55C2×C22⋊C4×C271C2211yescharacteristic, complemented
56C2×C22⋊C4×C271C2211yescharacteristic, complemented
57C2×C22⋊C4×C271C2211yescharacteristic, complemented
58#95×C271C2211yescharacteristic, complemented
59C23×D271—22no—
60C23×D271—22no—
61C23×D271—22no—
62C23×D271—22no—
63C23×D271—22no—
64C23×D271—22no—
65C23×D271—22no—
66C23×D271—22no—
67C23×D271—22no—
68C23×D271—41no—
69C23×D271—41no—
70C23×D271—41no—
71C23×D271—41no—
72C23×D271D414yescomplemented
73C23×D271C2311yescharacteristic
74C23×D271C2311yescharacteristic
75C23×D271C2311yescharacteristic
76C2×D1084—22no—
77C2×D1084—22no—
78C2×D1084D414yescomplemented
79C2×C22⋊D271—22no—
80C2×C22⋊D271—22no—
81C2×C22⋊D271—22no—
82C2×C22⋊D271—22no—
83C2×C22⋊D271—22no—
84C2×C22⋊D271—22no—
85C2×C22⋊D271—22no—
86C2×C22⋊D271—22no—
87C2×C22⋊D271—22no—
88C2×C22⋊D271—22no—
89C2×C22⋊D271—22no—
90C2×C22⋊D271—22no—
91C2×C22⋊D271—22no—
92C2×C22⋊D271—41no—
93C2×C22⋊D271D414yescomplemented
94D4×D271—232no—
95C22×Dic271C2311yescharacteristic
96C22×Dic271C2311yescharacteristic
97C22×Dic271C2311yescharacteristic
98C22×Dic271C2311yescharacteristic
99C22×Dic271C2311yescharacteristic
100C2×C4×D271—22no—
101C2×C4×D271—22no—
102C2×C4×D271—22no—
103C2×C4×D271—22no—
104C2×C4×D271—22no—
105C2×C4×D271—22no—
106C2×C4×D271—22no—
107C2×C4×D271—41no—
108C2×C4×D271—41no—
109C4⋊Dic271—22no—
110Dic271⋊C4—22no—
111C22⋊Dic271—22no—
112C22⋊Dic271—22no—
113C22⋊Dic271—22no—
114C22⋊Dic271—22no—
115C22⋊Dic271—22no—
116(C2×D271)⋊C4—22no—
117(C2×D271)⋊C4—22no—
118(C2×D271)⋊C4—22no—
119(C2×D271)⋊C4—22no—
120(C2×D271)⋊C4—22no—
121(C2×D271)⋊C4—41no—
122(C2×D271)⋊C4D414yescomplemented
123C24×C271C2311yescharacteristic
124C24×C271C2311yescharacteristic
125C24×C271C2311yescharacteristic
126C2×D4×C271—22no—
127C2×D4×C271—22no—
128C2×D4×C271—22no—
129C2×D4×C271—22no—
130C2×D4×C271—22no—
131C2×D4×C271—22no—
132C2×D4×C271—41no—
133C2×D4×C271—41no—
134C2×D4×C271—41no—
135C22×C4×C271C2311yescharacteristic
136C22×C4×C271C2311yescharacteristic
137C22×C4×C271C2311yescharacteristic
138C22×C4×C271C2311yescharacteristic
139C22⋊C4×C271—22no—
140C22⋊C4×C271—22no—
141C22⋊C4×C271—41no—
142C22×D271—21no—
143C22×D271—21no—
144C22×D271—21no—
145C22×D271—21no—
146C22×D271—21no—
147C22×D271—21no—
148C22×D271—21no—
149C22×D271—21no—
150C22×D271—28no—
151C22×D271—21no—
152C22×D271—21no—
153C22×D271—21no—
154C22×D271—21no—
155C22×D271—21no—
156C22×D271—21no—
157C22×D271—28no—
158C22×D271—28no—
159C22×D271—44no—
160C22×D271—44no—
161C22×D271—44no—
162C22×D271—44no—
163C22×D271—44no—
164C22×D271—44no—
165C22×D271—44no—
166C22×D271—44no—
167C22×D271—44no—
168C22×D271—44no—
169C22×D271—44no—
170C22×D271C2×D412yes—
171C22×D271C2×D412yes—
172C22×D271C2×D412yes—
173C22×D271C2×D412yes—
174C22×D271C2×D412yes—
175C22×D271C2×D412yes—
176C22×D271#4112yes—
177D1084—28no—
178C22⋊D271—28no—
179C22⋊D271—28no—
180C22⋊D271—44no—
181C22⋊D271—44no—
182C22⋊D271—44no—
183C2×Dic271—21no—
184C2×Dic271—21no—
185C2×Dic271—21no—
186C2×Dic271—21no—
187C2×Dic271—21no—
188C2×Dic271—21no—
189C2×Dic271—21no—
190C2×Dic271—21no—
191C2×Dic271—21no—
192C2×Dic271—21no—
193C2×Dic271—21no—
194C2×Dic271—21no—
195C2×Dic271—21no—
196C2×Dic271C2×D412yes—
197C2×Dic271C2×D412yescomplemented
198C4×D271—28no—
199C4×D271—44no—
200C4×D271—44no—
201C23×C271—21no—
202C23×C271—21no—
203C23×C271—21no—
204C23×C271—21no—
205C23×C271—21no—
206C23×C271—21no—
207C23×C271—21no—
208C23×C271—21no—
209C23×C271—21no—
210C23×C271—21no—
211C23×C271—21no—
212C23×C271—21no—
213C23×C271—21no—
214C23×C271—21no—
215C23×C271—21no—
216C23×C271—21no—
217C23×C271—21no—
218C23×C271—21no—
219C23×C271—21no—
220C23×C271C2×D412yes—
221C23×C271C2×D412yes—
222C23×C271C2411yescharacteristic
223D4×C271—28no—
224D4×C271—44no—
225D4×C271—44no—
226C2×C4×C271—21no—
227C2×C4×C271—21no—
228C2×C4×C271—21no—
229C2×C4×C271—21no—
230C2×C4×C271—21no—
231C2×C4×C271—21no—
232C2×C4×C271—21no—
233C2×C4×C271—21no—
234C2×C4×C271—21no—
235C2×C4×C271—21no—
236C2×C4×C271C2×D412yes—
237C2×C4×C271C2×D412yescomplemented
238C22×C271—22no—
239C22×C271—22no—
240C22×C271—22no—
241C22×C271—22no—
242C22×C271—22no—
243C22×C271—22no—
244C22×C271—41no—
245C22×C271—41no—
246C22×C271—41no—
247C22×C271—41no—
248C22×C271—41no—
249C22×C271—41no—
250C22×C271—41no—
251C22×C271—41no—
252C22×C271—41no—
253C22×C271—41no—
254C22×C271—41no—
255C22×C271—41no—
256C22×C271—41no—
257C22×C271—41no—
258C22×C271—41no—
259C22×C271C22×D411yescharacteristic
260C22×C271C22×D411yescharacteristic
261C22×C271C22×D411yescharacteristic
262C22×C271C22×D411yescharacteristic
263C22×C271C2×#4111yescharacteristic
264C22×C271H(5,𝔽2)11yescharacteristic
265C22×C271H(5,𝔽2)11yescharacteristic
266C2×D271—22no—
267C2×D271—22no—
268C2×D271—22no—
269C2×D271—22no—
270C2×D271—22no—
271C2×D271—22no—
272C2×D271—22no—
273C2×D271—22no—
274C2×D271—22no—
275C2×D271—22no—
276C2×D271—22no—
277C2×D271—22no—
278C2×D271—22no—
279C2×D271—22no—
280C2×D271—22no—
281C2×D271—416no—
282C2×D271—416no—
283C2×D271—41no—
284C2×D271—41no—
285C2×D271—41no—
286C2×D271—41no—
287C2×D271#12014yescomplemented
288C2×D271#12114yescomplemented
289C4×C271—22no—
290C4×C271—22no—
291C4×C271—41no—
292C4×C271—41no—
293Dic271—22no—
294Dic271—22no—
295Dic271—41no—
296Dic271—41no—
297Dic271—41no—
298C2×C271—42no—
299C2×C271—42no—
300C2×C271—81no—
301C2×C271C2×#12011yescharacteristic
302C2×C271C2×#12111yescharacteristic
303C2×C271#53411yescharacteristic
304C2×C271D4211yescharacteristic
305C2×C271D4211yescharacteristic
306C2×C271#54611yescharacteristic
307C2×C271#54611yescharacteristic
308D271—24no—
309D271—24no—
310D271—42no—
311C271#241011yescharacteristic, complemented, Sylow
312#2410—2711nomaximal, Sylow
313C23×D4—2711no—
314C2×#121—2711no—
315C2×#121—2711no—
316C2×#120—2711no—
317C2×#120—2711no—
318C22×C22⋊C4—2711no—
319#394—2711no—
320#394—2711no—
321#392—2711no—
322#392D27111yescharacteristic, complemented
323#386—2711no—
324#386—2711no—
325#386—2711no—
326#385—2711no—
327#379—2711no—
328C25—2711no—
329C25—2711no—
330C22×D4—2711no—
331C22×D4—2711no—
332C22×D4—2711no—
333C22×D4—2711no—
334C22×D4—2711no—
335C22×D4—2711no—
336C22×D4—5424no—
337C22×D4—5424no—
338C22×D4—5424no—
339C22×D4C2×D27111yescharacteristic
340C22×D4C2×D27111yescharacteristic, complemented
341C22×D4C2×D27111yescharacteristic, complemented
342C23×C4—2711no—
343C23×C4—2711no—
344C23×C4—2711no—
345#121—5424no—
346#121—5424no—
347#120—5424no—
348#120—5424no—
349C2×C4⋊C4—2711no—
350C2×C4⋊C4—2711no—
351C2×C22⋊C4—2711no—
352C2×C22⋊C4—2711no—
353C2×C22⋊C4—2711no—
354C2×C22⋊C4—2711no—
355C2×C22⋊C4—2711no—
356C2×C22⋊C4—2711no—
357C2×C22⋊C4—2711no—
358C2×C22⋊C4—2711no—
359C2×C22⋊C4—2711no—
360C2×C22⋊C4—2711no—
361C2×C22⋊C4—2711no—
362C2×C22⋊C4—2711no—
363C2×C22⋊C4—5424no—
364C2×C22⋊C4C2×D27111yescharacteristic, complemented
365C2×C22⋊C4C2×D27111yescharacteristic, complemented
366C2×C22⋊C4C2×D27111yescharacteristic, complemented
367#95—2711no—
368#95—2711no—
369#95—2711no—
370#95C2×D27111yescharacteristic, complemented
371C24—2711no—
372C24—2711no—
373C24—2711no—
374C24—2714no—
375C24—5422no—
376C24—5422no—
377C24—5422no—
378C24—5422no—
379C24—5422no—
380C24—5422no—
381C24—5422no—
382C24—5422no—
383C24—5422no—
384C24—10841no—
385C24—10841no—
386C24—10841no—
387C24—10841no—
388C24C22×D27111yescharacteristic
389C24C22×D27111yescharacteristic
390C24C22×D27111yescharacteristic
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—2714no—
401C2×D4—2714no—
402C2×D4—5422no—
403C2×D4—5422no—
404C2×D4—5422no—
405C2×D4—5422no—
406C2×D4—54232no—
407C2×D4—5422no—
408C2×D4—5422no—
409C2×D4—5422no—
410C2×D4—5422no—
411C2×D4—5422no—
412C2×D4—5422no—
413C2×D4—5422no—
414C2×D4—5422no—
415C2×D4—5422no—
416C2×D4—5422no—
417C2×D4—5422no—
418C2×D4—10841no—
419C22×C4—2711no—
420C22×C4—2711no—
421C22×C4—2711no—
422C22×C4—2711no—
423C22×C4—2711no—
424C22×C4—5422no—
425C22×C4—5422no—
426C22×C4—5422no—
427C22×C4—5422no—
428C22×C4—5422no—
429C22×C4—5422no—
430C22×C4—5422no—
431C22×C4—10841no—
432C22×C4—10841no—
433C22×C4C22×D27111yescharacteristic
434C22×C4C22×D27111yescharacteristic
435C22×C4C22×D27111yescharacteristic
436C22×C4C22×D27111yescharacteristic
437C4⋊C4—5422no—
438C4⋊C4—5422no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—2714no—
443C22⋊C4—5422no—
444C22⋊C4—5422no—
445C22⋊C4—5422no—
446C22⋊C4—5422no—
447C22⋊C4—5422no—
448C22⋊C4—5422no—
449C22⋊C4—5422no—
450C22⋊C4—5422no—
451C22⋊C4—5422no—
452C22⋊C4—5422no—
453C22⋊C4—10841no—
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—2712no—
474C23—2712no—
475C23—2712no—
476C23—2712no—
477C23—2712no—
478C23—2712no—
479C23—2712no—
480C23—5428no—
481C23—5428no—
482C23—5428no—
483C23—5421no—
484C23—5421no—
485C23—5421no—
486C23—5421no—
487C23—5421no—
488C23—5421no—
489C23—5421no—
490C23—5421no—
491C23—5421no—
492C23—5421no—
493C23—5421no—
494C23—5421no—
495C23—5421no—
496C23—5421no—
497C23—10844no—
498C23—10844no—
499C23—10844no—
500C23—10844no—
501C23—10844no—
502C23—10844no—
503C23—10844no—
504C23—10844no—
505C23—10844no—
506C23—10844no—
507C23—10844no—
508C23D4×D27112yes—
509C23D4×D27112yes—
510C23C23×D27111yescharacteristic
511D4—28no—
512D4—44no—
513D4—44no—
514D4—5428no—
515D4—5428no—
516D4—5428no—
517D4—10844no—
518D4—10844no—
519D4—10844no—
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—2712no—
531C2×C4—2712no—
532C2×C4—5428no—
533C2×C4—5421no—
534C2×C4—5421no—
535C2×C4—5421no—
536C2×C4—5421no—
537C2×C4—5421no—
538C2×C4—5421no—
539C2×C4—5421no—
540C2×C4—5421no—
541C2×C4—5421no—
542C2×C4—5421no—
543C2×C4—5421no—
544C2×C4—5421no—
545C2×C4—5421no—
546C2×C4—10844no—
547C2×C4—10844no—
548C2×C4D4×D27112yes—
549C2×C4D4×D27112yescomplemented
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—2714no—
572C22—2714no—
573C22—5422no—
574C22—5422no—
575C22—5422no—
576C22—5422no—
577C22—5422no—
578C22—5422no—
579C22—5422no—
580C22—5422no—
581C22—5422no—
582C22—5422no—
583C22—5422no—
584C22—5422no—
585C22—5422no—
586C22—5422no—
587C22—5422no—
588C22—108416no—
589C22—108416no—
590C22—10841no—
591C22—10841no—
592C22—10841no—
593C22—10841no—
594C22C2×D4×D27111yescharacteristic
595C22C2×D4×D27111yescharacteristic
596C22C2×D4×D27111yescharacteristic
597C22C2×D4×D27111yescharacteristic
598C22#41×D27111yescharacteristic
599C22#2257620511yescharacteristic
600C22#2257620511yescharacteristic
601C4—22no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—5422no—
606C4—5422no—
607C4—10841no—
608C4—10841no—
609C4—10841no—
610C2—42no—
611C2—42no—
612C2—81no—
613C2—5424no—
614C2—5424no—
615C2—10842no—
616C2#120×D27111yescharacteristic
617C2#121×D27111yescharacteristic
618C2#3177445711yescharacteristic
619C2#3177452911yescharacteristic
620C2#3177453211yescharacteristic
621C2#3177453611yescharacteristic
622C2#3177454411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#392×C271C2
hypercenter#392D271
hypocenterC271#2410
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D271C2semidirect
2. C2×#22576133C2semidirect
3. C2×#22576131C2semidirect
4. C2×#22576128C2semidirect
5. C2×#22576128C2semidirect
6. C2×C22⋊C4×D271C2semidirect
7. #31773813C2semidirect
8. #31773811C2semidirect
9. #31773805C2semidirect
10. #31773775C2semidirect
11. #31773773C2semidirect
12. #31773773C2semidirect
13. #31773771C2semidirect
14. #31773752C2semidirect
15. #392×C271C2semidirect
22. C22×C22⋊D271C22semidirect
23. C22×C22⋊D271C22semidirect
28. C22×C4×D271C22semidirect
29. C22×C4×D271C22semidirect
34. C2×C4⋊Dic271C22semidirect
35. C2×Dic271⋊C4C22semidirect
36. C2×C22⋊Dic271C22semidirect
37. C2×C22⋊Dic271C22semidirect
38. C2×C22⋊Dic271C22semidirect
39. C2×C22⋊Dic271C22semidirect
40. C2×C22⋊Dic271C22semidirect
44. C2×(C2×D271)⋊C4C22semidirect
45. C2×(C2×D271)⋊C4C22semidirect
46. C2×(C2×D271)⋊C4C22semidirect
47. C2×(C2×D271)⋊C4C22semidirect
49. C271⋊#95C22semidirect
50. C271⋊#95C22semidirect
51. C271⋊#95C22semidirect
53. C22×D4×C271C22semidirect
54. C22×D4×C271C22semidirect
55. C2×C22⋊C4×C271C22semidirect
56. C2×C22⋊C4×C271C22semidirect
57. C2×C22⋊C4×C271C22semidirect
58. #95×C271C22semidirect
72. C23×D271D4semidirect
78. C2×D1084D4semidirect
93. C2×C22⋊D271D4semidirect
122. (C2×D271)⋊C4D4semidirect
197. C2×Dic271C2×D4semidirect
237. C2×C4×C271C2×D4semidirect
287. C2×D271#120semidirect
288. C2×D271#121semidirect
311. C271#2410semidirect
322. #392D271semidirect
340. C22×D4C2×D271semidirect
341. C22×D4C2×D271semidirect
364. C2×C22⋊C4C2×D271semidirect
365. C2×C22⋊C4C2×D271semidirect
366. C2×C22⋊C4C2×D271semidirect
370. #95C2×D271semidirect
549. C2×C4D4×D271semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2712
C2×#225761332
C2×#225761312
C2×#225761282
C2×C22⋊C4×D2712
#317738132
#317738112
#317738052
#317737752
#317737732
#317737712
#317737522
#392×C2712
#2410271