Finite Group Databasebeta

← #29575405

Subgroups — #29575405

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839726C211yesmaximal, characteristic, complemented
2#21445729C211yesmaximal, characteristic, complemented
3C2×#16839486C212yesmaximal, complemented
4#21443662C212yesmaximal, complemented
5C2×C61×#386C211yesmaximal, characteristic
6C22×#11758194—21no—
7#16839726—24no—
8C2×C22⋊#92418—21no—
9C2×C22⋊#92418—21no—
10C2×C22⋊#92418—21no—
11C2×C22⋊#92418C2212yes—
12C2×C22⋊#92418C2212yes—
13#16839486—24no—
14C23×D4×C61C411yescharacteristic
15C24×C4×C61C2211yescharacteristic
16C22×C22⋊C4×C61C411yescharacteristic
17C22×C22⋊C4×C61C411yescharacteristic
18C22×C22⋊C4×C61C2211yescharacteristic
19C61×#386—21no—
20C61×#386—21no—
21C61×#386—21no—
22C61×#386—21no—
23C2×#95×C61C411yescharacteristic
24C2×#95×C61C2211yescharacteristic
25C2×#11758194—24no—
26C2×#11758194—42no—
27C2×#11758194—42no—
28C22×#92418—22no—
29C22×#92418—22no—
30C22⋊#92418—24no—
31C22⋊#92418—22no—
32C22⋊#92418—22no—
33C22⋊#92418—22no—
34C22⋊#92418—42no—
35C22⋊#92418—42no—
36C22⋊#92418D414yescomplemented
37C25×C61C2×C411yescharacteristic
38C25×C61C2×C411yescharacteristic
39C22×D4×C61—21no—
40C22×D4×C61—21no—
41C22×D4×C61—22no—
42C22×D4×C61—22no—
43C22×D4×C61—42no—
44C22×D4×C61—42no—
45C23×C4×C61—22no—
46C23×C4×C61—22no—
47C23×C4×C61—42no—
48C23×C4×C61—42no—
49C23×C4×C61C2×C411yescharacteristic
50C23×C4×C61C2×C411yescharacteristic
51C23×C4×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53C23×C4×C61D411yescharacteristic
54C23×C4×C61D411yescharacteristic
55C23×C4×C61D411yescharacteristic
56C23×C4×C61D411yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—22no—
63C2×C22⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—22no—
66C2×C22⋊C4×C61—42no—
67C2×C22⋊C4×C61—42no—
68#95×C61—21no—
69#95×C61—21no—
70#95×C61—21no—
71#95×C61—21no—
72C24×C61—22no—
73C24×C61—22no—
74C24×C61—21no—
75C24×C61—21no—
76C24×C61—21no—
77C24×C61—22no—
78C24×C61—21no—
79C24×C61—22no—
80C24×C61—22no—
81C24×C61—22no—
82C24×C61—41no—
83C24×C61—41no—
84C24×C61—41no—
85C24×C61—41no—
86C24×C61—41no—
87C24×C61—41no—
88C24×C61C22⋊C411yescharacteristic
89C24×C61C22⋊C411yescharacteristic
90C24×C61M1612yescomplemented
91C24×C61C22×C411yescharacteristic
92C2×D4×C61—24no—
93C2×D4×C61—24no—
94C2×D4×C61—48no—
95C2×D4×C61—42no—
96C2×D4×C61—42no—
97C2×D4×C61—48no—
98C2×D4×C61—42no—
99C2×D4×C61—42no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61—41no—
125C22×C4×C61—48no—
126C22×C4×C61—48no—
127C22×C4×C61—41no—
128C22×C4×C61—41no—
129C22×C4×C61—41no—
130C22×C4×C61—41no—
131C22×C4×C61—41no—
132C22×C4×C61—41no—
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61M1612yescomplemented
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61#4112yes—
140C22⋊C4×C61—42no—
141C22⋊C4×C61—42no—
142C22⋊C4×C61—42no—
143C22⋊C4×C61—42no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—42no—
146C22⋊C4×C61—41no—
147#11758194—48no—
148C2×#92418—24no—
149C2×#92418—42no—
150C2×#92418—42no—
151C2×#92418—42no—
152C2×#92418—42no—
153C2×#92418—42no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—22no—
173C23×C61—24no—
174C23×C61—44no—
175C23×C61—44no—
176C23×C61—44no—
177C23×C61—44no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—82no—
194C23×C61—82no—
195C23×C61—82no—
196C23×C61—82no—
197C23×C61#9912yes—
198C23×C61#10012yes—
199C23×C61C2×C22⋊C411yescharacteristic
200C23×C61C4×D412yes—
201C23×C61C2×M1611yescharacteristic
202C23×C61C2×M1611yescharacteristic
203C23×C61#13112yes—
204D4×C61—48no—
205D4×C61—48no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—44no—
221C2×C4×C61—44no—
222C2×C4×C61—44no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—82no—
236C2×C4×C61—82no—
237C2×C4×C61#9912yes—
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12112yes—
240C2×C4×C61#12311yescharacteristic
241#92418—44no—
242#92418—82no—
243#9083—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—48no—
259C22×C61—48no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—84no—
272C22×C61—84no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61#38611yescharacteristic
278C22×C61#40711yescharacteristic
279C22×C61#40811yescharacteristic
280C22×C61C2×#9911yescharacteristic
281C22×C61C2×#9911yescharacteristic
282C22×C61C2×#10011yescharacteristic
283C22×C61#41311yescharacteristic
284C22×C61#43512yes—
285C22×C61#43612yes—
286C4×C61—21no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—21no—
290C4×C61—41no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61—81no—
294C2×#407—611no—
295C2×#386D6111yescharacteristic
296#1779—611no—
297C2×#323—612no—
298#1280—612no—
299C2×C61—22no—
300C2×C61—22no—
301C2×C61—42no—
302C2×C61—42no—
303C2×C61—82no—
304C2×C61#146211yescharacteristic
305C2×C61#146511yescharacteristic
306C2×C61#146811yescharacteristic
307C2×C61#147111yescharacteristic
308C2×C61#187011yescharacteristic
309C2×C61#189811yescharacteristic
310C2×C61#190011yescharacteristic
311C23×D4Dic6111yescharacteristic
312C24×C4C2×D6111yescharacteristic
313C22×M16—1221no—
314C22×C22⋊C4Dic6111yescharacteristic
315C22×C22⋊C4Dic6111yescharacteristic
316C22×C22⋊C4C2×D6111yescharacteristic
317#407—1224no—
318C2×C22⋊C8—612no—
319C2×C22⋊C8—612no—
320C2×C22⋊C8—1221no—
321C2×C22⋊C8—1221no—
322C2×C22⋊C8—1221no—
323#386—21no—
324#386—21no—
325#386—21no—
326#386—21no—
327C2×#95Dic6111yescharacteristic
328C2×#95C2×D6111yescharacteristic
329#323—1224no—
330C61#908311yescharacteristic, complemented, Sylow
331C25C2×Dic6111yescharacteristic
332C25C2×Dic6111yescharacteristic
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—22no—
337C22×D4—42no—
338C22×D4—42no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4C2×Dic6111yescharacteristic
344C23×C4C2×Dic6111yescharacteristic
345C23×C4C2×Dic6111yescharacteristic
346C23×C4C2×Dic6111yescharacteristic
347C23×C4C22⋊D6111yescharacteristic
348C23×C4C22⋊D6111yescharacteristic
349C23×C4C22⋊D6111yescharacteristic
350C23×C4C22⋊D6111yescharacteristic
351C23×C4C22×D6111yescharacteristic
352C2×M16—1224no—
353C2×M16—2442no—
354C2×M16—2442no—
355C22×C8—1222no—
356C22×C8—1222no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C22⋊C8—614no—
368C22⋊C8—1224no—
369C22⋊C8—1222no—
370C22⋊C8—1222no—
371C22⋊C8—1222no—
372C22⋊C8—2442no—
373C22⋊C8—2442no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378C24—22no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24#1175819412yescomplemented
395C24C22⋊Dic6111yescharacteristic
396C24C22⋊Dic6111yescharacteristic
397C24C22×Dic6111yescharacteristic
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—48no—
404C2×D4—42no—
405C2×D4—42no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4#1175819412yescomplemented
440C22×C4C22⋊Dic6111yescharacteristic
441C22×C4C22⋊Dic6111yescharacteristic
442C22×C4D4×D6112yes—
443C22×C4#1175822412yes—
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446M16—2448no—
447C2×C8—1224no—
448C2×C8—2442no—
449C2×C8—2442no—
450C2×C8—2442no—
451C2×C8—2442no—
452C2×C8—2442no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—24no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—24no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23#1451969212yes—
504C23#1451969412yes—
505C23C2×C22⋊Dic6111yescharacteristic
506C23D4×Dic6112yes—
507C23C2×#1175819411yescharacteristic
508C23C2×#1175819411yescharacteristic
509C23#1451978412yes—
510D4—48no—
511D4—48no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#1451969212yes—
544C2×C4#1451974511yescharacteristic
545C2×C4#1451974912yes—
546C2×C4#1451975811yescharacteristic
547C8—4882no—
548C8—2444no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—48no—
564C22—48no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—84no—
577C22—84no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22#1683964211yescharacteristic
583C22#1683972611yescharacteristic
584C22#1683972911yescharacteristic
585C22C2×#1451969211yescharacteristic
586C22C2×#1451969211yescharacteristic
587C22C2×#1451969411yescharacteristic
588C22#1683974611yescharacteristic
589C22#1683983412yes—
590C22#1683984012yes—
591C4—22no—
592C4—21no—
593C4—22no—
594C4—21no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C2—22no—
600C2—22no—
601C2—42no—
602C2—42no—
603C2—82no—
604C2#2144414611yescharacteristic
605C2#2144415711yescharacteristic
606C2#2144416711yescharacteristic
607C2#2144417911yescharacteristic
608C2#2144611711yescharacteristic
609C2#2144627311yescharacteristic
610C2#2144628311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#386C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D4×C61C4
cosocleC23×C4×C61C23
hypercenterC2×#386D61
hypocenterC61#9083
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. C2×#16839726C2semidirect
2. #21445729C2semidirect
3. C2×#16839486C2semidirect
4. #21443662C2semidirect
36. C22⋊#92418D4semidirect
90. C24×C61M16semidirect
135. C22×C4×C61M16semidirect
330. C61#9083semidirect
394. C24#11758194semidirect
439. C22×C4#11758194semidirect

Maximal subgroups

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

subgroupindex
C2×#168397262
#214457292
C2×#168394862
#214436622
C2×C61×#3862
#908361