Finite Group Databasebeta

← #29586526

Subgroups — #29586526

Order 15616.

1578 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
1C22×#14519701C212yesmaximal, complemented
2C22×C22⋊#92418C211yesmaximal, characteristic
3C2×#16839622C211yesmaximal, characteristic
4C2×#16839524C212yesmaximal, complemented
5C22×C22⋊Q8×C61C211yesmaximal, characteristic
6C23×#92418C2212yes—
7C22×C4⋊Dic61C2212yes—
8C22×C22⋊Dic61—21no—
9C2×#14519701—24no—
10C2×#14519701—24no—
11C2×#14519701—24no—
12C2×C22⋊#92418—24no—
13C2×C22⋊#92418—22no—
14#16839622—24no—
15C2×#14519684—21no—
16#16839524—24no—
17C23×Q8×C61C2211yescharacteristic
18C24×C4×C61C2211yescharacteristic
19C2×C22⋊Q8×C61—24no—
20C2×C22⋊Q8×C61—22no—
21C2×C22⋊Q8×C61—24no—
22C2×C22⋊Q8×C61C414yescomplemented
23C22×C4⋊C4×C61—21no—
24C22×C4⋊C4×C61C2211yescharacteristic
25C22×C22⋊C4×C61—21no—
26C23×Dic61—22no—
27C22×#92418—22no—
28C22×#92418—22no—
29C22×#92418—22no—
30C22×#92418—22no—
31C22×#92418—22no—
32C22×#92418—22no—
33C22×#92418D414yes—
34C2×C4⋊Dic61—22no—
35C2×C4⋊Dic61—22no—
36C2×C4⋊Dic61—22no—
37C2×C4⋊Dic61—22no—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61D414yes—
40C2×C22⋊Dic61—24no—
41C2×C22⋊Dic61—42no—
42C2×C22⋊Dic61—42no—
43#14519701—216no—
44C22⋊#92418—28no—
45#14519684—24no—
46C25×C61D411yescharacteristic
47C22×Q8×C61—21no—
48C22×Q8×C61—22no—
49C22×Q8×C61—22no—
50C22×Q8×C61—22no—
51C22×Q8×C61—22no—
52C22×Q8×C61—22no—
53C22×Q8×C61C2×C412yes—
54C22×Q8×C61D414yes—
55C23×C4×C61—21no—
56C23×C4×C61—21no—
57C23×C4×C61—22no—
58C23×C4×C61—22no—
59C23×C4×C61—22no—
60C23×C4×C61—22no—
61C23×C4×C61—22no—
62C23×C4×C61—22no—
63C23×C4×C61C2×C412yes—
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D412yes—
66C23×C4×C61C2311yescharacteristic
67C22⋊Q8×C61—216no—
68C22⋊Q8×C61—28no—
69C22⋊Q8×C61—28no—
70C2×C4⋊C4×C61—28no—
71C2×C4⋊C4×C61—24no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61C2×C412yes—
75C2×C4⋊C4×C61D414yes—
76C2×C22⋊C4×C61—28no—
77C2×C22⋊C4×C61—24no—
78C24×C61—22no—
79C24×C61—21no—
80C24×C61—22no—
81C24×C61—21no—
82C24×C61—22no—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—22no—
86C24×C61C22⋊C412yes—
87C24×C61SD1612yes—
88C24×C61Q1612yes—
89C24×C61C2×D411yescharacteristic
90C2×Q8×C61—22no—
91C2×Q8×C61—24no—
92C2×Q8×C61—24no—
93C2×Q8×C61—24no—
94C2×Q8×C61—24no—
95C2×Q8×C61—24no—
96C2×Q8×C61—24no—
97C2×Q8×C61—24no—
98C2×Q8×C61—41no—
99C2×Q8×C61—42no—
100C2×Q8×C61—42no—
101C2×Q8×C61—42no—
102C2×Q8×C61—41no—
103C2×Q8×C61—42no—
104C2×Q8×C61—42no—
105C2×Q8×C61C22⋊C414yescomplemented
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—24no—
109C22×C4×C61—24no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—24no—
118C22×C4×C61—24no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—41no—
123C22×C4×C61—42no—
124C22×C4×C61—41no—
125C22×C4×C61—42no—
126C22×C4×C61—41no—
127C22×C4×C61—41no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C22⋊C412yes—
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61#4112yes—
144C22×C4×C61#4112yes—
145C22×Dic61—24no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C4⋊C4×C61—216no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61C22⋊C414yescomplemented
156C22⋊C4×C61—216no—
157C22⋊Dic61—48no—
158C4⋊Dic61—24no—
159C4⋊Dic61—24no—
160C4⋊Dic61—42no—
161C4⋊Dic61—42no—
162C2×#92418—24no—
163C2×#92418—24no—
164C2×#92418—24no—
165C2×#92418—42no—
166C2×#92418—42no—
167C2×#92418—42no—
168C2×#92418—42no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—24no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61#10314yescomplemented
192C23×C61C2×C22⋊C411yescharacteristic
193C23×C61#12012yes—
194C23×C61#12011yescharacteristic
195C23×C61#12112yes—
196C23×C61#12311yescharacteristic
197C23×C61C2×SD1611yescharacteristic
198C23×C61C2×Q1611yescharacteristic
199C23×C61#13512yes—
200C23×C61AGL(1,ℤ/8)11yescharacteristic
201C23×C61#13711yescharacteristic
202C23×C61#13712yes—
203Q8×C61—28no—
204Q8×C61—44no—
205Q8×C61—44no—
206Q8×C61—44no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—44no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—44no—
221C2×C4×C61—44no—
222C2×C4×C61—44no—
223C2×C4×C61—44no—
224C2×C4×C61—44no—
225C2×C4×C61—44no—
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C4×D412yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61#12012yes—
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12012yes—
233C2×C4×C61#12112yes—
234C2×C4×C61#12112yes—
235C2×C4×C61#12112yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#12112yes—
238C2×C4×C61#12312yes—
239C2×C4×C61#12311yescharacteristic
240C2×C4×C61#12412yes—
241C2×C4×C61#12712yes—
242C2×Dic61—44no—
243C2×Dic61—44no—
244C2×Dic61—44no—
245C2×Dic61—82no—
246C2×Dic61—82no—
247C2×Dic61—82no—
248C2×Dic61—82no—
249#92418—44no—
250#10970—611nomaximal, Sylow
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—42no—
262C22×C61#37911yescharacteristic
263C22×C61#38611yescharacteristic
264C22×C61#38611yescharacteristic
265C22×C61#39212yes—
266C22×C61#39412yes—
267C22×C61C2×#10311yescharacteristic
268C22×C61#41611yescharacteristic
269C22×C61#41711yescharacteristic
270C22×C61#41911yescharacteristic
271C22×C61#44812yes—
272C22×C61#44912yes—
273C22×C61#45011yescharacteristic
274C22×C61#45111yescharacteristic
275C22×C61#45112yes—
276C22×C61#45212yes—
277C22×C61#45212yes—
278C22×C61#46512yes—
279C22×C61#46712yes—
280C22×C61#46812yes—
281C22×C61#47012yes—
282C22×C61#48311yescharacteristic
283C22×C61#48411yescharacteristic
284C22×C61#48512yes—
285C4×C61—42no—
286C4×C61—44no—
287C4×C61—44no—
288C4×C61#38612yes—
289C4×C61#39012yes—
290C4×C61#39212yes—
291C4×C61#39412yes—
292Dic61—84no—
293C22×C22⋊Q8D6111yescharacteristic
294C22×#103—612no—
295C22×C22⋊C8—611no—
296C2×#380—611no—
297C2×#340—612no—
298C2×C61—44no—
299C2×C61#143811yescharacteristic
300C2×C61#178711yescharacteristic
301C2×C61#178811yescharacteristic
302C2×C61#187211yescharacteristic
303C2×C61#187311yescharacteristic
304C2×C61#189211yescharacteristic
305C2×C61#189411yescharacteristic
306C2×C61#200012yes—
307C2×C61#200112yes—
308C2×C61#203512yes—
309C2×C61#203612yes—
310C23×Q8C2×D6111yescharacteristic
311C24×C4C2×D6111yescharacteristic
312C23×C8—612no—
313C2×C22⋊Q8—24no—
314C2×C22⋊Q8—22no—
315C2×C22⋊Q8—24no—
316C2×C22⋊Q8Dic6114yescomplemented
317C22×C4⋊C4—21no—
318C22×C4⋊C4—612no—
319C22×C4⋊C4C2×D6111yescharacteristic
320C22×C22⋊C4—21no—
321C22×C22⋊C4—1221no—
322C2×#103—1224no—
323C2×#103—1224no—
324C2×#103—1224no—
325C2×C22⋊C8—1224no—
326C2×C22⋊C8—1222no—
327#380—1224no—
328C2×#95—1221no—
329#340—1224no—
330C61#1097011yescharacteristic, complemented, Sylow
331C25C22⋊D6111yescharacteristic
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8C2×Dic6112yes—
339C22×Q8C22⋊D6114yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—22no—
347C23×C4—22no—
348C23×C4—1222no—
349C23×C4C2×Dic6112yes—
350C23×C4C22⋊D6111yescharacteristic
351C23×C4C22⋊D6112yes—
352C23×C4C22×D6111yescharacteristic
353C22×C8—614no—
354C22×C8—1222no—
355C22×C8—1222no—
356C22×C8—1222no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8—1222no—
360C22⋊Q8—216no—
361C22⋊Q8—28no—
362C22⋊Q8—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—614no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4C2×Dic6112yes—
374C2×C4⋊C4C22⋊D6114yes—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—2442no—
379C2×C22⋊C4—2442no—
380#103—12216no—
381C22⋊C8—1228no—
382#95—1224no—
383C24—22no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—21no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24#1175820112yes—
392C24#1175820212yes—
393C24C22⋊Dic6112yes—
394C24C2×C22⋊D6111yescharacteristic
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—41no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—41no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C22⋊Dic6114yescomplemented
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1224no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4C22⋊Dic6112yes—
449C22×C4D4×D6112yes—
450C22×C4D4×D6112yes—
451C22×C4#1175822412yes—
452C22×C4#1175822412yes—
453C22×C4C22×Dic6111yescharacteristic
454C22×C4C2×C22⋊D6112yes—
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6111yescharacteristic
457C2×C8—1224no—
458C2×C8—1224no—
459C2×C8—1224no—
460C2×C8—2442no—
461C2×C8—2442no—
462C2×C8—2442no—
463C2×C8—2442no—
464C4⋊C4—216no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4C22⋊Dic6114yescomplemented
472C22⋊C4—216no—
473C22⋊C4—2448no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#1451970114yescomplemented
497C23C2×C22⋊Dic6111yescharacteristic
498C23#1451974511yescharacteristic
499C23#1451974612yes—
500C23#1451974912yes—
501C23#1451975811yescharacteristic
502C23C2×#1175820111yescharacteristic
503C23C2×#1175820211yescharacteristic
504C23#1451979712yes—
505C23#1451980211yescharacteristic
506C23#1451980711yescharacteristic
507C23#1451980812yes—
508Q8—28no—
509Q8—44no—
510Q8—44no—
511Q8—44no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—4882no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—2444no—
536C2×C4—2444no—
537C2×C4—2444no—
538C2×C4C2×C22⋊Dic6111yescharacteristic
539C2×C4C2×C22⋊Dic6111yescharacteristic
540C2×C4D4×Dic6112yes—
541C2×C4D4×Dic6112yes—
542C2×C4#1451974512yes—
543C2×C4#1451974511yescharacteristic
544C2×C4#1451974612yes—
545C2×C4#1451974812yes—
546C2×C4#1451974812yes—
547C2×C4#1451974912yes—
548C2×C4#1451974912yes—
549C2×C4#1451974912yes—
550C2×C4#1451975812yes—
551C2×C4#1451975811yescharacteristic
552C2×C4#1451976412yes—
553C2×C4#1451977312yes—
554C8—2444no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22#1683961811yescharacteristic
567C22#1683964211yescharacteristic
568C22#1683964211yescharacteristic
569C22#1683967012yes—
570C22#1683967612yes—
571C22C2×#1451970111yescharacteristic
572C22#1683975811yescharacteristic
573C22#1683976311yescharacteristic
574C22#1683977111yescharacteristic
575C22#1683990112yes—
576C22#1683990712yes—
577C22#1683991211yescharacteristic
578C22#1683991711yescharacteristic
579C22#1683991812yes—
580C22#1683992212yes—
581C22#1683992412yes—
582C22#1683999312yes—
583C22#1684000112yes—
584C22#1684001012yes—
585C22#1684001812yes—
586C22#1684009011yescharacteristic
587C22#1684009511yescharacteristic
588C22#1684010012yes—
589C4—44no—
590C4—42no—
591C4—44no—
592C4—4884no—
593C4#1683964212yes—
594C4#1683966512yes—
595C4#1683967012yes—
596C4#1683967612yes—
597C2—44no—
598C2#2144406011yescharacteristic
599C2#2144575411yescharacteristic
600C2#2144575911yescharacteristic
601C2#2144612711yescharacteristic
602C2#2144613211yescharacteristic
603C2#2144624311yescharacteristic
604C2#2144625311yescharacteristic
605C2#2144682012yes—
606C2#2144682712yes—
607C2#2144701212yes—
608C2#2144701912yes—
subgroupG/H
centerC24C2×C22⋊D61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC22×C22⋊Q8×C61C2
socleC24×C61C2×D4
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenterC22×C22⋊Q8D61
hypocenterC61#10970
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14519701C2semidirect
4. C2×#16839524C2semidirect
22. C2×C22⋊Q8×C61C4semidirect
105. C2×Q8×C61C22⋊C4semidirect
155. C4⋊C4×C61C22⋊C4semidirect
191. C23×C61#103semidirect
316. C2×C22⋊Q8Dic61semidirect
330. C61#10970semidirect
410. C2×Q8C22⋊Dic61semidirect
471. C4⋊C4C22⋊Dic61semidirect
496. C23#14519701semidirect

Maximal subgroups

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

subgroupindex
C22×#145197012
C22×C22⋊#924182
C2×#168396222
C2×#168395242
C22×C22⋊Q8×C612
#1097061