Finite Group Databasebeta

← #29707896

Subgroups — #29707896

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840560C211yesmaximal, characteristic
2C22×#14519713C211yesmaximal, characteristic, complemented
3#21454338C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic, complemented
5C2×#16839744C211yesmaximal, characteristic, complemented
6#21447117C214yesmaximal, complemented
7#21447110C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C2×C61×#408C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840560—24no—
12#16840560—24no—
13C2×#14519808—21no—
14C2×#14519807C2212yes—
15C2×#14519803—21no—
16C2×#14519802C2212yes—
17C2×#14519797—22no—
18C22×#11758194C2211yescharacteristic
19C22×#11758194C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C2×#14519713—24no—
22C2×#14519713C2214yescomplemented
23#16839771—21no—
24#16839767—21no—
25#16839763C2214yes—
26#16839759—22no—
27#16839759—22no—
28#16839758—22no—
29C2×#14519702C2212yescomplemented
30C2×#14519699C2212yescomplemented
31#16839744—24no—
32C2×#14519695C2212yescomplemented
33C2×#14519693C2212yescomplemented
34#16839530C2214yescomplemented
35C22×#41×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C23×C8×C61C2211yescharacteristic
38C61×#408—22no—
39C61×#408C2214yes—
40C2×C22⋊C8×C61—22no—
41C2×#11758222—22no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D414yes—
49C22×Dic122C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C4×D61—21no—
53#14519808—28no—
54#14519807—24no—
55#14519807—24no—
56#14519803—28no—
57#14519802—24no—
58#14519802—24no—
59#14519797—216no—
60C2×#11758202—22no—
61C2×#11758200—22no—
62C2×#11758201—22no—
63C2×#11758199—22no—
64C2×#11758194—21no—
65C2×#11758194—22no—
66C2×#11758194—21no—
67C2×#11758194—22no—
68C2×#11758194—22no—
69C2×#11758194D414yes—
70C2×#11758194C2312yes—
71C2×#11758194C2312yes—
72C22×#92418—21no—
73C22×#92418—21no—
74#14519736—21no—
75#14519736—21no—
76#14519736—21no—
77#14519736—21no—
78C2×C4⋊Dic61C2312yes—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81#14519713—28no—
82#14519702—24no—
83#14519701—24no—
84#14519699—24no—
85#14519698—24no—
86#14519695—24no—
87#14519693—24no—
88C2×#41×C61—21no—
89C2×#41×C61—21no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61D412yes—
95C2×#41×C61C2312yes—
96C22×Q8×C61D411yescharacteristic
97C22×D4×C61—21no—
98C22×D4×C61D411yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—21no—
102C2×M16×C61—22no—
103C2×M16×C61—22no—
104C2×M16×C61C2312yes—
105C22×C8×C61—22no—
106C22×C8×C61—22no—
107C22×C8×C61—21no—
108C22×C8×C61D412yes—
109C22×C8×C61D412yes—
110C22×C8×C61C2312yes—
111C22⋊C8×C61—28no—
112C24×C61—21no—
113C24×C61#4111yescharacteristic
114C23×D61—21no—
115#41×C61—24no—
116#41×C61—24no—
117#41×C61—22no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—42no—
124#41×C61—42no—
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61C2×D412yes—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4112yes—
149C22×C4×C61#4111yescharacteristic
150C22×C4×C61#4112yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—24no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—42no—
156C22×Dic61—21no—
157C22×Dic61—22no—
158#11758222—24no—
159#11758222—24no—
160#11758222—24no—
161#11758222—24no—
162#11758222—42no—
163#11758222—42no—
164#11758222—42no—
165#11758222—42no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244C2×D412yes—
172C2×C4×D61—24no—
173C2×C4×D61—41no—
174C2×C4×D61—41no—
175C2×C4×D61—42no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122C2×D412yes—
182M16×C61—22no—
183M16×C61—22no—
184M16×C61—42no—
185C2×C8×C61—24no—
186C2×C8×C61—24no—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C2×C8×C61—42no—
191C2×C8×C61C2×D414yes—
192C22⋊Dic61—41no—
193C22⋊Dic61—41no—
194#11758202—28no—
195#11758201—28no—
196#11758200—28no—
197#11758199—28no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4×Dic61—41no—
201C4×Dic61—41no—
202#11758194—24no—
203#11758194—22no—
204#11758194—22no—
205#11758194—22no—
206#11758194—42no—
207#11758194C2×D414yes—
208C2×#92418—24no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—42no—
212C23×C61—22no—
213C23×C61—42no—
214C23×C61—44no—
215C23×C61—42no—
216C23×C61#12111yescharacteristic
217C23×C61#12111yescharacteristic
218C23×C61#12312yes—
219C23×C61#12412yes—
220C23×C61C2×#4111yescharacteristic
221C22×D61—22no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—82no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—42no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—44no—
233D4×C61—42no—
234C2×C4×C61—22no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—44no—
239C2×C4×C61#12012yes—
240C2×C4×C61#12012yes—
241C2×C4×C61#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61#12312yes—
248C2×C4×C61#12312yes—
249C2×C4×C61#12312yes—
250C2×C4×C61#12412yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C22⋊D61—44no—
258C22⋊D61—44no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—42no—
263C2×Dic61—82no—
264D244—22no—
265D244—22no—
266D244—41no—
267D244—41no—
268C4×D61—44no—
269C4×D61—44no—
270Dic122—22no—
271Dic122—22no—
272Dic122—41no—
273Dic122—41no—
274C8×C61—24no—
275C8×C61—42no—
276#92418—24no—
277#92418—42no—
278#23530—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—44no—
286C22×C61—84no—
287C22×C61#39412yes—
288C22×C61#39412yes—
289C22×C61C2×#12111yescharacteristic
290C22×C61C2×#12311yescharacteristic
291C22×C61C2×#12411yescharacteristic
292C2×D61—42no—
293C2×D61—82no—
294C2×D61—82no—
295C2×D61—82no—
296C4×C61—42no—
297C4×C61#39412yes—
298C4×C61#39412yes—
299C4×C61C2×#12011yescharacteristic
300C4×C61C2×#12111yescharacteristic
301C4×C61C2×#12111yescharacteristic
302C4×C61C2×#12311yescharacteristic
303Dic61—42no—
304Dic61—82no—
305C2×#575—611no—
306C22×#108—611no—
307#2896—611no—
308C2×#416—611no—
309C2×#413—611no—
310C2×#408D6111yescharacteristic, complemented
311#2052—614no—
312#2051—614no—
313#1742—611no—
314C2×C61—21no—
315C2×C61—21no—
316C2×C61—22no—
317C2×C61—22no—
318C2×C61—82no—
319C2×C61C2×#39411yescharacteristic
320C2×C61#306712yes—
321D61—82no—
322C22×#41—611no—
323C22×#41C2×D6111yescharacteristic
324#575—1224no—
325#575—1224no—
326C2×#137—612no—
327C2×#137—1221no—
328C2×AGL(1,ℤ/8)—612no—
329C2×AGL(1,ℤ/8)—1221no—
330C2×#135—1222no—
331C22×M16—611no—
332C22×M16—611no—
333C22×M16C2×D6111yescharacteristic
334C23×C8C2×D6111yescharacteristic
335C2×#117—611no—
336C2×#108—614no—
337C2×#108—1224no—
338#419—1221no—
339#418—1221no—
340#417—614no—
341#416—1222no—
342#416—1222no—
343#416—1222no—
344C2×#103—612no—
345C2×#102—612no—
346#413—1224no—
347C2×#101—612no—
348C2×#100—612no—
349#408—22no—
350#408C2×D6114yes—
351C2×C22⋊C8—22no—
352#343—614no—
353C61#2353011yescharacteristic, complemented, Sylow
354C2×#41—21no—
355C2×#41—21no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—614no—
361C2×#41—1222no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1222no—
365C2×#41—1222no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41D24412yes—
369C2×#41C22×D6112yes—
370C22×Q8—611no—
371C22×Q8D24411yescharacteristic
372C22×D4—21no—
373C22×D4—611no—
374C22×D4—1221no—
375C22×D4D24411yescharacteristic
376C23×C4—21no—
377C23×C4—1221no—
378C23×C4C22×D6111yescharacteristic
379#137—1224no—
380#137—1228no—
381#137—1224no—
382AGL(1,ℤ/8)—1224no—
383AGL(1,ℤ/8)—1228no—
384AGL(1,ℤ/8)—1224no—
385#135—12216no—
386C2×Q16—1222no—
387C2×SD16—1222no—
388C2×SD16—1222no—
389C2×D8—1222no—
390C2×M16—21no—
391C2×M16—22no—
392C2×M16—22no—
393C2×M16—614no—
394C2×M16—612no—
395C2×M16—612no—
396C2×M16—1221no—
397C2×M16—1222no—
398C2×M16—1221no—
399C2×M16—1222no—
400C2×M16—1222no—
401C2×M16C22×D6112yes—
402C22×C8—22no—
403C22×C8—22no—
404C22×C8—21no—
405C22×C8—1221no—
406C22×C8—1221no—
407C22×C8C22⋊D6112yes—
408C22×C8C22⋊D6112yes—
409C22×C8C22×D6112yes—
410#117—1221no—
411#117—1221no—
412#117—1221no—
413#117—1221no—
414C2×C4⋊C4—612no—
415C2×C22⋊C4—1221no—
416C2×C42—1221no—
417#108—1228no—
418#103—1224no—
419#103—1224no—
420#102—1224no—
421#102—1224no—
422#101—1224no—
423#100—1224no—
424C22⋊C8—28no—
425C24—21no—
426C24—1221no—
427C24#1175822211yescharacteristic
428#41—22no—
429#41—22no—
430#41—22no—
431#41—22no—
432#41—24no—
433#41—24no—
434#41—42no—
435#41—42no—
436#41—42no—
437#41—42no—
438#41—1224no—
439#41—1224no—
440#41—1224no—
441#41—1224no—
442#41—2442no—
443#41—2442no—
444#41—2442no—
445#41—2442no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—612no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8C2×D24412yes—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—612no—
464C2×D4—1224no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—2441no—
471C2×D4—2441no—
472C2×D4—2442no—
473C2×D4C2×D24412yes—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—1221no—
480C22×C4—1224no—
481C22×C4—1222no—
482C22×C4—2441no—
483C22×C4—2441no—
484C22×C4—2442no—
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4#1175822211yescharacteristic
488C22×C4D4×D6112yes—
489C22×C4D4×D6112yes—
490C22×C4#1175822412yes—
491C22×C4#1175822412yes—
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C23×D6111yescharacteristic
495Q16—1228no—
496SD16—1228no—
497SD16—1228no—
498D8—1228no—
499M16—22no—
500M16—22no—
501M16—42no—
502M16—614no—
503M16—1222no—
504M16—1224no—
505M16—1222no—
506M16—1222no—
507M16—2442no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—42no—
514C2×C8—1222no—
515C2×C8—1224no—
516C2×C8—1222no—
517C2×C8—2442no—
518C2×C8C2×C22⋊D6114yes—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C22⋊C4—2441no—
522C22⋊C4—2441no—
523C42—2441no—
524C42—2441no—
525C23—22no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—1222no—
530C23—4882no—
531C23—2441no—
532C23—2441no—
533C23#1451975111yescharacteristic
534C23#1451975111yescharacteristic
535C23#1451975912yes—
536C23#1451976312yes—
537C23C2×#1175822211yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—1222no—
542Q8—1222no—
543Q8—2441no—
544Q8—2441no—
545D4—22no—
546D4—22no—
547D4—42no—
548D4—42no—
549D4—44no—
550D4—42no—
551D4—1222no—
552D4—1222no—
553D4—2444no—
554D4—2444no—
555D4—2441no—
556D4—2441no—
557C2×C4—22no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—1222no—
563C2×C4—4882no—
564C2×C4—2442no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2444no—
568C2×C4—2444no—
569C2×C4#1451974712yes—
570C2×C4#1451974712yes—
571C2×C4#1451974812yes—
572C2×C4#1451974812yes—
573C2×C4#1451974912yes—
574C2×C4#1451974912yes—
575C2×C4#1451975111yescharacteristic
576C2×C4#1451975111yescharacteristic
577C2×C4#1451975912yes—
578C2×C4#1451976112yes—
579C2×C4#1451976112yes—
580C2×C4#1451976312yes—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C22×D24411yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4C2×#1175822411yescharacteristic
587C8—24no—
588C8—42no—
589C8—1224no—
590C8—2442no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—84no—
599C22—4882no—
600C22—4882no—
601C22—4882no—
602C22—2442no—
603C22#1683967812yes—
604C22#1683967812yes—
605C22C2×#1451975111yescharacteristic
606C22C2×#1451975911yescharacteristic
607C22C2×#1451976311yescharacteristic
608C4—42no—
609C4—4882no—
610C4—2442no—
611C4#1683967812yes—
612C4#1683967812yes—
613C4C2×#1451974711yescharacteristic
614C4C2×#1451974811yescharacteristic
615C4C2×#1451974911yescharacteristic
616C4C2×#1451976111yescharacteristic
617C2—22no—
618C2—21no—
619C2—21no—
620C2—22no—
621C2—82no—
622C2—4882no—
623C2C2×#1683967811yescharacteristic
624C2#2145553912yes—
subgroupG/H
centerC2×C4#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#408C2
socleC22×C61C2×#121
prime radicalC2×#16840560C2
hypercenterC2×#408D61
hypocenterC61#23530
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C22×#14519713C2semidirect
3. #21454338C2semidirect
4. C2×#16839759C2semidirect
5. C2×#16839744C2semidirect
6. #21447117C2semidirect
7. #21447110C2semidirect
8. #21445557C2semidirect
9. C2×C61×#408C2semidirect
22. C2×#14519713C22semidirect
29. C2×#14519702C22semidirect
30. C2×#14519699C22semidirect
32. C2×#14519695C22semidirect
33. C2×#14519693C22semidirect
34. #16839530C22semidirect
310. C2×#408D61semidirect
353. C61#23530semidirect

Maximal subgroups

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

subgroupindex
C2×#168405602
C22×#145197132
#214543382
C2×#168397592
C2×#168397442
#214471172
#214471102
#214455572
C2×C61×#4082
#2353061