Finite Group Databasebeta

← #29597503

Subgroups — #29597503

Order 15616.

1088 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519694C211yesmaximal, characteristic, complemented
2C2×#16839549C212yesmaximal, complemented
3#21445734C211yesmaximal, characteristic, complemented
4#21443937C212yesmaximal, complemented
5C61×#2403C211yesmaximal, characteristic
6C22×#11758194—21no—
7C2×#14519694—24no—
8C2×#14519694—21no—
9C2×#14519694C2212yes—
10C2×#14519692—21no—
11C2×#14519692C2212yes—
12#16839618—21no—
13#16839549—24no—
14C23×D4×C61C411yescharacteristic, complemented
15C23×D4×C61C2211yescharacteristic
16C2×#120×C61—21no—
17C2×#120×C61C411yescharacteristic, complemented
18C2×#120×C61C2211yescharacteristic
19C61×#392—21no—
20C61×#392—21no—
21C61×#386—21no—
22C61×#386C411yescharacteristic, complemented
23C61×#386C2211yescharacteristic
24C61×#379C411yescharacteristic, complemented
25C2×#11758194—24no—
26C2×#11758194—22no—
27C2×#11758194—41no—
28C2×#11758194—42no—
29C22×#92418—41no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—41no—
32C2×C22⋊Dic61—41no—
33#14519694—24no—
34#14519694—22no—
35#14519694D414yescomplemented
36#14519692—22no—
37#14519692—22no—
38#14519692—42no—
39C25×C61C2×C411yescharacteristic
40C25×C61C2×C411yescharacteristic
41C25×C61D411yescharacteristic
42C22×D4×C61—21no—
43C22×D4×C61—22no—
44C22×D4×C61—22no—
45C22×D4×C61—21no—
46C22×D4×C61—21no—
47C22×D4×C61—21no—
48C22×D4×C61—42no—
49C22×D4×C61—41no—
50C22×D4×C61—41no—
51C22×D4×C61—44no—
52C22×D4×C61—41no—
53C22×D4×C61—41no—
54C22×D4×C61C2×C411yescharacteristic
55C22×D4×C61D411yescharacteristic
56C22×D4×C61D411yescharacteristic
57C22×D4×C61C2311yescharacteristic
58C23×C4×C61C2×C411yescharacteristic
59C23×C4×C61D411yescharacteristic
60#120×C61—22no—
61#120×C61—22no—
62#120×C61—44no—
63#120×C61—42no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61C2×C411yescharacteristic
71C2×C22⋊C4×C61C2×C411yescharacteristic
72#95×C61—21no—
73#95×C61—21no—
74C24×C61—21no—
75C24×C61—21no—
76C24×C61—21no—
77C24×C61—21no—
78C24×C61—21no—
79C24×C61—22no—
80C24×C61—22no—
81C24×C61—22no—
82C24×C61—41no—
83C24×C61—41no—
84C24×C61—42no—
85C24×C61—41no—
86C24×C61—42no—
87C24×C61—42no—
88C24×C61—41no—
89C24×C61—81no—
90C24×C61—81no—
91C24×C61—81no—
92C24×C61C22⋊C411yescharacteristic
93C24×C61C22⋊C411yescharacteristic
94C24×C61C22⋊C411yescharacteristic
95C24×C61C22×C411yescharacteristic
96C24×C61C2×D411yescharacteristic
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C2×D4×C61—41no—
107C2×D4×C61—41no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C2×D4×C61—44no—
111C2×D4×C61—44no—
112C2×D4×C61—44no—
113C2×D4×C61—44no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—44no—
119C2×D4×C61—44no—
120C2×D4×C61—44no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C2×D4×C61—44no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61—81no—
129C2×D4×C61—81no—
130C2×D4×C61C2×D412yes—
131C2×D4×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—42no—
141C22×C4×C61—41no—
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×Dic61—42no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—42no—
148C22⋊C4×C61—42no—
149C22⋊C4×C61—42no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—81no—
154C22⋊Dic61—42no—
155C22⋊Dic61—42no—
156C22⋊Dic61—82no—
157C22⋊Dic61—82no—
158#11758194—24no—
159#11758194—44no—
160#11758194—42no—
161C2×#92418—44no—
162C2×#92418—42no—
163C23×C61—22no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—22no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—41no—
190C23×C61—42no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—81no—
203C23×C61—81no—
204C23×C61—81no—
205C23×C61—81no—
206C23×C61—82no—
207C23×C61—84no—
208C23×C61—84no—
209C23×C61—84no—
210C23×C61—84no—
211C23×C61—84no—
212C23×C61—84no—
213C23×C61—82no—
214C23×C61#9911yescharacteristic
215C23×C61#9911yescharacteristic
216C23×C61C2×C22⋊C411yescharacteristic
217C23×C61C4×D412yes—
218C23×C61#12112yes—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—44no—
222D4×C61—44no—
223D4×C61—42no—
224D4×C61—42no—
225D4×C61—44no—
226D4×C61—48no—
227D4×C61—44no—
228D4×C61—84no—
229D4×C61—84no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61#9912yes—
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61#12311yescharacteristic
252C2×Dic61—82no—
253C2×Dic61—82no—
254C2×Dic61—82no—
255#92418—44no—
256#13082—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—22no—
264C22×C61—21no—
265C22×C61—22no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—42no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61—84no—
287C22×C61—84no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—84no—
294C22×C61—84no—
295C22×C61—84no—
296C22×C61—84no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61AΣL(1,𝔽16)12yescomplemented
300C22×C61#38611yescharacteristic
301C22×C61C2×#9911yescharacteristic
302C22×C61C2×#9911yescharacteristic
303C4×C61—21no—
304C4×C61—22no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61—81no—
308C4×C61—81no—
309Dic61—162no—
310C22×#100—611no—
311#2403D6111yescharacteristic
312C2×AΣL(1,𝔽16)—612no—
313#1781—611no—
314#1390—612no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—22no—
318C2×C61—42no—
319C2×C61—41no—
320C2×C61—42no—
321C2×C61—81no—
322C2×C61—82no—
323C2×C61—81no—
324C2×C61#189811yescharacteristic
325C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
326C2×C61#212111yescharacteristic
327C23×D4Dic6111yescharacteristic, complemented
328C23×D4C2×D6111yescharacteristic
329C22×M16—1221no—
330C2×#120—21no—
331C2×#120Dic6111yescharacteristic, complemented
332C2×#120C2×D6111yescharacteristic
333C2×#100—612no—
334C2×#100—1224no—
335C2×#100—1221no—
336C2×#99—612no—
337C2×#99—1221no—
338#392—21no—
339#392—21no—
340#386—21no—
341#386Dic6111yescharacteristic, complemented
342#386C2×D6111yescharacteristic
343#379—1221no—
344#379Dic6111yescharacteristic, complemented
345AΣL(1,𝔽16)—1224no—
346C61#1308211yescharacteristic, complemented, Sylow
347C25C2×Dic6111yescharacteristic
348C25C2×Dic6111yescharacteristic
349C25C22⋊D6111yescharacteristic
350C22×D4—21no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4—21no—
356C22×D4—42no—
357C22×D4—41no—
358C22×D4—41no—
359C22×D4—44no—
360C22×D4—41no—
361C22×D4—41no—
362C22×D4C2×Dic6111yescharacteristic
363C22×D4C22⋊D6111yescharacteristic
364C22×D4C22⋊D6111yescharacteristic
365C22×D4C22×D6111yescharacteristic
366C23×C4C2×Dic6111yescharacteristic
367C23×C4C22⋊D6111yescharacteristic
368C2×M16—1224no—
369C2×M16—1222no—
370C2×M16—2441no—
371C2×M16—2442no—
372C22×C8—2441no—
373#120—22no—
374#120—22no—
375#120—44no—
376#120—42no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—2441no—
385C2×C22⋊C4—2441no—
386C2×C22⋊C4C2×Dic6111yescharacteristic
387C2×C22⋊C4C2×Dic6111yescharacteristic
388#100—614no—
389#100—1224no—
390#100—1222no—
391#99—1222no—
392#99—1222no—
393#99—2442no—
394#95—21no—
395#95—21no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—21no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—41no—
411C24—81no—
412C24—81no—
413C24—81no—
414C24C22⋊Dic6111yescharacteristic
415C24C22⋊Dic6111yescharacteristic
416C24C22⋊Dic6111yescharacteristic
417C24C22×Dic6111yescharacteristic
418C24C2×C22⋊D6111yescharacteristic
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—81no—
451C2×D4—81no—
452C2×D4D4×D6112yes—
453C2×D4#1175822412yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—2442no—
465C22×C4C22⋊Dic6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467M16—1224no—
468M16—2444no—
469M16—2442no—
470C2×C8—2444no—
471C2×C8—2442no—
472C22⋊C4—22no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—81no—
481C22⋊C4—4882no—
482C22⋊C4—4882no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—82no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—82no—
536C23#1451969211yescharacteristic
537C23#1451969211yescharacteristic
538C23C2×C22⋊Dic6111yescharacteristic
539C23D4×Dic6112yes—
540C23#1451974912yes—
541D4—42no—
542D4—42no—
543D4—44no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—48no—
549D4—44no—
550D4—84no—
551D4—84no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—4882no—
572C2×C4—4882no—
573C2×C4—4882no—
574C2×C4#1451969212yes—
575C2×C4#1451974511yescharacteristic
576C2×C4#1451975811yescharacteristic
577C8—2444no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—84no—
608C22—84no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—81no—
619C22—81no—
620C22#1683954912yescomplemented
621C22#1683964211yescharacteristic
622C22C2×#1451969211yescharacteristic
623C22C2×#1451969211yescharacteristic
624C4—21no—
625C4—22no—
626C4—21no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—9762no—
631C2—21no—
632C2—21no—
633C2—22no—
634C2—42no—
635C2—41no—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—81no—
640C2#2144627311yescharacteristic
641C2C2×#1683954911yescharacteristic
642C2#2144746511yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
Fitting, prime radicalC61×#2403C2
socleC22×C61C2×#99
cosocleC22×D4×C61C23
hypercenter#2403D61
hypocenterC61#13082
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2403

Complements

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

NH ≅ G/Ntype
1. C22×#14519694C2semidirect
2. C2×#16839549C2semidirect
3. #21445734C2semidirect
4. #21443937C2semidirect
14. C23×D4×C61C4semidirect
17. C2×#120×C61C4semidirect
22. C61×#386C4semidirect
24. C61×#379C4semidirect
35. #14519694D4semidirect
299. C22×C61AΣL(1,𝔽16)semidirect
327. C23×D4Dic61semidirect
331. C2×#120Dic61semidirect
341. #386Dic61semidirect
344. #379Dic61semidirect
346. C61#13082semidirect
620. C22#16839549semidirect

Maximal subgroups

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

subgroupindex
C22×#145196942
C2×#168395492
#214457342
#214439372
C61×#24032
#1308261