Finite Group Databasebeta

← #13389466

Subgroups — #13389466

Order 1792.

1606 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1142C211yesmaximal, characteristic, complemented
2C7×#2408C211yesmaximal, characteristic
3C2×#89813C212yesmaximal, complemented
4#11739794C211yesmaximal, characteristic, complemented
5C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
6#11729882C211yesmaximal, characteristic, complemented
7#11729439C212yesmaximal, complemented
8#11727405C212yesmaximal, complemented
9#11727400C212yesmaximal, complemented
10#11727279C211yesmaximal, characteristic, complemented
11#11727265C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C23×Dic14C2211yescharacteristic
14C2×C7×#121C2212yes—
15C2×C7×#120—21no—
16C2×C4×C7×D4C2211yescharacteristic
17C22×#1142—28no—
18C22×#1142—24no—
19C22×#1142C2212yes—
20C22×#1142C2212yes—
21C2×#6487C2212yescomplemented
22C23×C7⋊C8C2211yescharacteristic
23C23×C7⋊C8C2211yescharacteristic, complemented
24C7×#394—21no—
25C7×#392—21no—
26C7×#390C2212yes—
27C7×#380C2211yescharacteristic
28C7×#379—21no—
29#89992—21no—
30C2×#6406C2212yes—
31C2×#6406C2212yes—
32#89984C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic, complemented
35#89813—28no—
36D4×C7⋊C8—24no—
37#89751—21no—
38C2×#6384C2212yescomplemented
39C2×#6384C2212yescomplemented
40C2×#6382C2212yescomplemented
41#89719—21no—
42#89718—21no—
43#89701C2212yescomplemented
44C2×C4⋊(C7⋊C8)C2211yescharacteristic, complemented
45C2×C4×C7⋊C8C2211yescharacteristic, complemented
46#89324C2212yescomplemented
47#24264—71nomaximal, Sylow
48C25×C7—21no—
49C22×C7×D4—24no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—21no—
53C22×C7×D4—22no—
54C22×C7×D4—42no—
55C22×C7×D4D411yescharacteristic
56C22×C7×D4C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C23×Dic7—21no—
60C22×Dic14—22no—
61C22×Dic14—24no—
62C22×Dic14—22no—
63C22×Dic14—21no—
64C22×Dic14—42no—
65C22×Dic14C2312yes—
66C7×#121—24no—
67C7×#121—42no—
68C7×#120—44no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4D411yescharacteristic
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—22no—
81C2×C22⋊Dic7—21no—
82C2×#1142—232no—
83C2×#1142—24no—
84C2×#1142—24no—
85C2×#1142—24no—
86C2×#1142—24no—
87C2×#1142—24no—
88C2×#1142D418yescomplemented
89C2×C4⋊Dic7C2312yes—
90#6487—24no—
91#6487—42no—
92C2×Dic7⋊C4—22no—
93C22×C7⋊C8—24no—
94C22×C7⋊C8—22no—
95C22×C7⋊C8—24no—
96C22×C7⋊C8D414yescomplemented
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C7×#95—21no—
102#6406—24no—
103#6406—42no—
104C7⋊#95—22no—
105C7⋊#95—22no—
106C7⋊#95—21no—
107C22⋊(C7⋊C8)—22no—
108C22⋊(C7⋊C8)—22no—
109#6384—24no—
110#6384—42no—
111#6382—24no—
112C4⋊(C7⋊C8)—22no—
113C4×C7⋊C8—22no—
114C23×SD16—71no—
115C2×#465—72no—
116C2×C8×D4—71no—
117#2409—71no—
118#2408D711yescharacteristic
119#2055—72no—
120#2000—72no—
121#1968—72no—
122#1807—71no—
123#1787—71no—
124#1786—71no—
125C24×C7—22no—
126C24×C7—44no—
127C24×C7—42no—
128C24×C7—42no—
129C24×C7—42no—
130C24×C7—82no—
131C24×C7C2×D411yescharacteristic
132C24×C7C2×D411yescharacteristic
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—28no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—48no—
142C2×C7×D4—44no—
143C2×C7×D4—44no—
144C2×C7×D4—44no—
145C2×C7×D4—44no—
146C2×C7×D4—42no—
147C2×C7×D4—82no—
148C2×C7×D4SD1614yescomplemented
149C2×C7×D4C2×D414yes—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—21no—
153C22×C4×C7—22no—
154C22×C4×C7—21no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7#4112yes—
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—44no—
167C22×Dic7—42no—
168C2×Dic14—28no—
169C2×Dic14—22no—
170C2×Dic14—22no—
171C2×Dic14—22no—
172C2×Dic14—22no—
173C2×Dic14—22no—
174C2×Dic14—48no—
175C2×Dic14—44no—
176C2×Dic14—44no—
177C2×Dic14—42no—
178C2×Dic14C2×D414yes—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—44no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—82no—
188C42×C7—21no—
189C42×C7—21no—
190C22⋊Dic7—44no—
191C22⋊Dic7—42no—
192#1142—216no—
193#1142—216no—
194C4⋊Dic7—22no—
195C4⋊Dic7—22no—
196Dic7⋊C4—44no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—28no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—44no—
204C2×C7⋊C8C2×D412yes—
205C2×C7⋊C8C2×D414yes—
206C2×C7⋊C8C2×D412yes—
207C23×Q8—71no—
208C23×D4C2×D711yescharacteristic
209C22×SD16—72no—
210C22×SD16—72no—
211C22×SD16—144no—
212C22×SD16—148no—
213C23×C8—71no—
214C23×C8—71no—
215C2×C22⋊Q8—72no—
216C2×#121C2×D712yes—
217C2×#120—21no—
218C2×C4×D4C2×D711yescharacteristic
219#465—148no—
220C8×D4—144no—
221C2×#106—72no—
222C2×C4⋊C8—71no—
223C2×#103—72no—
224C2×#103—72no—
225C2×#102—72no—
226C2×#102—72no—
227C2×C22⋊C8—71no—
228C2×C22⋊C8—71no—
229C2×C4×C8—71no—
230#395—141no—
231#394—21no—
232#393—141no—
233#392—21no—
234#391—72no—
235#390C2×D712yes—
236#386—141no—
237#385—141no—
238#380—71no—
239#380C2×D711yescharacteristic
240#379—21no—
241#340—72no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—21no—
245C23×C7—22no—
246C23×C7—21no—
247C23×C7—22no—
248C23×C7—24no—
249C23×C7—48no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—88no—
257C23×C7—88no—
258C23×C7—84no—
259C23×C7—84no—
260C23×C7#12112yes—
261C23×C7C2×SD1612yes—
262C23×C7C2×SD1612yescomplemented
263C23×C7#13512yes—
264C23×C7C22×D411yescharacteristic
265C7×D4—28no—
266C7×D4—24no—
267C7×D4—24no—
268C7×D4—48no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—88no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—24no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7#12112yes—
289C2×C4×C7C2×SD1612yescomplemented
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D412yes—
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7H(5,𝔽2)12yes—
294C2×Dic7—24no—
295C2×Dic7—48no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301Dic14—24no—
302Dic14—24no—
303Dic14—48no—
304Dic14—42no—
305Dic14—42no—
306C7⋊C8—22no—
307C7⋊C8—41no—
308C7⋊C8—41no—
309C7⋊C8#12014yescomplemented
310C25—21no—
311C22×Q8—72no—
312C22×Q8—144no—
313C22×Q8—141no—
314C22×Q8—142no—
315C22×Q8—142no—
316C22×Q8—282no—
317C22×D4—24no—
318C22×D4—22no—
319C22×D4—22no—
320C22×D4—22no—
321C22×D4—21no—
322C22×D4—42no—
323C22×D4C22⋊D711yescharacteristic
324C22×D4C22×D712yes—
325C23×C4—141no—
326C23×C4C22×D711yescharacteristic
327C23×C4C22×D711yescharacteristic
328C2×SD16—78no—
329C2×SD16—144no—
330C2×SD16—144no—
331C2×SD16—144no—
332C2×SD16—144no—
333C2×SD16—1432no—
334C2×SD16—144no—
335C22×C8—71no—
336C22×C8—74no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—144no—
341C22×C8—142no—
342C22×C8—144no—
343C22⋊Q8—144no—
344C22⋊Q8—282no—
345#121—24no—
346#121—42no—
347#120—44no—
348C4×D4—22no—
349C4×D4—22no—
350C2×C4⋊C4—72no—
351C2×C4⋊C4—142no—
352C2×C4⋊C4C22⋊D711yescharacteristic
353C2×C4⋊C4C22×D712yes—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4C22⋊D711yescharacteristic
361C2×C22⋊C4C22⋊D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363#106—144no—
364C4⋊C8—142no—
365#103—144no—
366#103—282no—
367#102—144no—
368#102—282no—
369C22⋊C8—142no—
370C22⋊C8—142no—
371C4×C8—142no—
372#95—21no—
373#95—141no—
374#95—142no—
375#95—142no—
376C22×C7—22no—
377C22×C7—24no—
378C22×C7—24no—
379C22×C7—22no—
380C22×C7—24no—
381C22×C7—22no—
382C22×C7—41no—
383C22×C7—42no—
384C22×C7—41no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7—44no—
388C22×C7—816no—
389C22×C7—82no—
390C22×C7—82no—
391C22×C7—82no—
392C22×C7—82no—
393C22×C7#46514yescomplemented
394C22×C7C2×#12111yescharacteristic
395C22×C7D4212yes—
396C22×C7C22×SD1611yescharacteristic
397C22×C7C2×#13511yescharacteristic
398C22×C7#57712yes—
399C4×C7—22no—
400C4×C7—22no—
401C4×C7—41no—
402C4×C7—41no—
403C4×C7—82no—
404C4×C7C2×#12011yescharacteristic
405C4×C7#53511yescharacteristic
406C4×C7#54612yes—
407Dic7—44no—
408Dic7—82no—
409C24—22no—
410C24—44no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—82no—
415C24C2×C22⋊D711yescharacteristic
416C24C2×C22⋊D711yescharacteristic
417C2×Q8—74no—
418C2×Q8—148no—
419C2×Q8—142no—
420C2×Q8—142no—
421C2×Q8—142no—
422C2×Q8—142no—
423C2×Q8—142no—
424C2×Q8—284no—
425C2×Q8—288no—
426C2×Q8—284no—
427C2×Q8—282no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—28no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—82no—
443C2×D4#114214yescomplemented
444C2×D4D4×D714yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—284no—
455C22×C4—282no—
456C22×C4D4×D712yes—
457C22×C4#115912yes—
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C23×D711yescharacteristic
463SD16—1416no—
464SD16—1416no—
465C2×C8—72no—
466C2×C8—74no—
467C2×C8—72no—
468C2×C8—148no—
469C2×C8—142no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—142no—
473C2×C8—142no—
474C2×C8—284no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—142no—
479C4⋊C4—142no—
480C4⋊C4—284no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—82no—
487C22⋊C4—284no—
488C22⋊C4—282no—
489C42—21no—
490C42—21no—
491C2×C7—24no—
492C2×C7—42no—
493C2×C7—84no—
494C2×C7#240811yescharacteristic
495C2×C7C2×#46511yescharacteristic
496C2×C7#305711yescharacteristic
497C2×C7D4×SD1612yes—
498C2×C7#328212yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—24no—
506C23—48no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—88no—
514C23—88no—
515C23—84no—
516C23—84no—
517C23C2×#114212yes—
518C23C2×#114212yescomplemented
519C23#650212yes—
520C23#651312yes—
521C23C22×C22⋊D711yescharacteristic
522Q8—144no—
523Q8—144no—
524Q8—288no—
525Q8—282no—
526Q8—282no—
527D4—28no—
528D4—24no—
529D4—24no—
530D4—48no—
531D4—42no—
532D4—42no—
533D4—88no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—144no—
551C2×C4—288no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4C2×#114212yescomplemented
558C2×C4#650212yes—
559C2×C4C2×D4×D712yes—
560C2×C4C2×D4×D711yescharacteristic
561C2×C4C2×#115911yescharacteristic
562C2×C4#654812yes—
563C8—74no—
564C8—142no—
565C8—281no—
566C8—281no—
567C7#2426411yescharacteristic, complemented, Sylow
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—24no—
573C22—22no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—816no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22#8981314yescomplemented
586C22C22×#114211yescharacteristic
587C22D4×C22⋊D712yes—
588C22C2×#650211yescharacteristic
589C22C2×#651311yescharacteristic
590C22#9052712yes—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—82no—
596C4—284no—
597C4—562no—
598C4D7×#12011yescharacteristic
599C4#9028212yes—
600C4#9028311yescharacteristic
601C2—24no—
602C2—42no—
603C2—84no—
604C2#1174002711yescharacteristic
605C2C2×#8981311yescharacteristic
606C2#1174082211yescharacteristic
607C2D4×#114212yes—
608C2#1174095412yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2408C2
socleC23×C7C22×D4
hypercenter#2408D7
hypocenterC7#24264
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C23×#1142C2semidirect
3. C2×#89813C2semidirect
4. #11739794C2semidirect
5. C2×D4×C7⋊C8C2semidirect
6. #11729882C2semidirect
7. #11729439C2semidirect
8. #11727405C2semidirect
9. #11727400C2semidirect
10. #11727279C2semidirect
11. #11727265C2semidirect
21. C2×#6487C22semidirect
23. C23×C7⋊C8C22semidirect
34. C2×C22⋊(C7⋊C8)C22semidirect
38. C2×#6384C22semidirect
39. C2×#6384C22semidirect
40. C2×#6382C22semidirect
43. #89701C22semidirect
44. C2×C4⋊(C7⋊C8)C22semidirect
45. C2×C4×C7⋊C8C22semidirect
46. #89324C22semidirect
88. C2×#1142D4semidirect
96. C22×C7⋊C8D4semidirect
148. C2×C7×D4SD16semidirect
262. C23×C7C2×SD16semidirect
289. C2×C4×C7C2×SD16semidirect
309. C7⋊C8#120semidirect
393. C22×C7#465semidirect
443. C2×D4#1142semidirect
518. C23C2×#1142semidirect
557. C2×C4C2×#1142semidirect
567. C7#24264semidirect
585. C22#89813semidirect

Maximal subgroups

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

subgroupindex
C23×#11422
C7×#24082
C2×#898132
#117397942
C2×D4×C7⋊C82
#117298822
#117294392
#117274052
#117274002
#117272792
#117272652
#242647