Finite Group Databasebeta

← #29601767

Subgroups — #29601767

Order 15616.

1790 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
1C22×C4×Dic122C211yesmaximal, characteristic
2C2×C4×#11758196C211yesmaximal, characteristic
3C2×#16839608C211yesmaximal, characteristic
4#21444089C212yesmaximal
5#21444085C212yesmaximal
6#21444071C212yesmaximal
7#21444070C212yesmaximal
8C4×#14519684C211yesmaximal, characteristic
9C4×#14519684C211yesmaximal, characteristic
10C4×#14519684C211yesmaximal, characteristic
11C4×#95×C61C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×C4×Dic122—24no—
14C2×C4×Dic122—24no—
15C2×C4×Dic122C418yescomplemented
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758196C2212yes—
18C22×#11758196C2211yescharacteristic
19C22×#11758196C2211yescharacteristic
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24C4×#11758196—24no—
25#16839608—24no—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C42×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C22×C42×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2212yes—
40C22×Dic122—24no—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—24no—
44C22×Dic122C2×C414yes—
45C23×Dic61C2311yescharacteristic
46C23×Dic61C2312yes—
47C23×Dic61C2312yes—
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2311yescharacteristic
51C4×Dic122—232no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61C2×C414yes—
55C2×#11758196—24no—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×#11758196—22no—
60C2×#11758196—24no—
61C2×#11758196—24no—
62C2×#11758196C2×C414yes—
63C2×#11758196C2×C414yes—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61C2×C414yes—
72C2×C4×Dic61C2×C414yes—
73C2×C4×Dic61D414yes—
74C2×C4×Dic61D414yes—
75C2×C4×Dic61Q814yescomplemented
76#14519684—24no—
77#14519684—24no—
78#14519684—24no—
79#14519684—24no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684—24no—
84#14519684—24no—
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2312yes—
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C2×C42×C61—22no—
91C2×C42×C61—22no—
92C2×C42×C61—22no—
93C2×C42×C61—22no—
94C2×C42×C61—22no—
95C2×C42×C61—22no—
96C2×C42×C61—22no—
97C2×C42×C61C2×C414yes—
98C2×C42×C61Q814yescomplemented
99#95×C61—24no—
100#95×C61—24no—
101C24×C61C2411yescharacteristic
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×Q814yes—
127C22×C4×C61C2×Q812yes—
128C22×C4×C61C2×Q812yes—
129C22×C4×C61#4114yes—
130C22×C4×C61#4114yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61C22×C412yes—
157C22×Dic61C22×C412yes—
158C22×Dic61C22×C412yes—
159C22×Dic61C22×C412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D412yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×Q814yes—
165C22×Dic61#4114yes—
166C22×Dic61#4114yes—
167C2×Dic122—216no—
168C2×Dic122—216no—
169C2×Dic122—28no—
170C2×Dic122—28no—
171C2×Dic122—28no—
172C42×C61—28no—
173C42×C61—28no—
174C42×C61—44no—
175C4⋊Dic61—28no—
176#11758196—216no—
177#11758196—28no—
178#11758196—28no—
179C4×Dic61—28no—
180C4×Dic61—28no—
181C4×Dic61—44no—
182C4×Dic61C22⋊C4116yes—
183C23×C61C23×C411yescharacteristic
184C23×C61C22×D411yescharacteristic
185C23×C61C22×D411yescharacteristic
186C23×C61C22×Q811yescharacteristic
187C23×C61C22×Q811yescharacteristic
188C23×C61C2×#4112yes—
189C23×C61C2×#4112yes—
190C23×C61C2×#4111yescharacteristic
191C23×C61C2×#4111yescharacteristic
192C23×C61C2×#4112yes—
193C23×C61C2×#4112yes—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61C2×C22⋊C414yes—
218C2×C4×C61#11714yes—
219C2×C4×C61C4×Q814yes—
220C2×C4×C61C4×Q814yes—
221C2×C4×C61C22⋊Q814yes—
222C2×C4×C61C22⋊Q814yes—
223C2×C4×C61#12414yes—
224C2×C4×C61C4⋊Q814yes—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61C2×C22⋊C414yes—
257C2×Dic61C2×C22⋊C414yes—
258C2×Dic61#11714yes—
259Dic122—232no—
260#13850—611nomaximal, Sylow
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61C2×#11711yescharacteristic
263C22×C61C2×#11711yescharacteristic
264C22×C61C2×C4×Q811yescharacteristic
265C22×C61C2×C4×Q811yescharacteristic
266C22×C61C4×#4111yescharacteristic
267C22×C61C4×#4111yescharacteristic
268C22×C61C2×C22⋊Q811yescharacteristic
269C22×C61C2×C22⋊Q811yescharacteristic
270C22×C61#52512yes—
271C22×C61#52512yes—
272C22×C61#52512yes—
273C22×C61C2×#12411yescharacteristic
274C22×C61#52912yes—
275C22×C61#52912yes—
276C22×C61#52912yes—
277C22×C61#52912yes—
278C22×C61C2×C4⋊Q811yescharacteristic
279C22×C61#53212yes—
280C22×C61#53312yes—
281C22×C61#53312yes—
282C22×C61#53312yes—
283C22×C61#53312yes—
284C4×C61—24no—
285C4×C61—44no—
286C4×C61—44no—
287C4×C61—42no—
288C4×C61#39118yes—
289Dic61—24no—
290Dic61—24no—
291Dic61—24no—
292Dic61—44no—
293Dic61—44no—
294Dic61—42no—
295C22×C4×Q8—611no—
296C2×C4×C4⋊C4—611no—
297C2×#376—611no—
298#1446—612no—
299#1445—612no—
300#1441—612no—
301#1441—612no—
302C4×#95—611no—
303C4×#95—611no—
304C4×#95—611no—
305C4×#95D6111yescharacteristic
306C2×C61#228011yescharacteristic
307C2×C61C2×#39111yescharacteristic
308C2×C61#229511yescharacteristic
309C2×C61#229511yescharacteristic
310C2×C61#229511yescharacteristic
311C2×C61#229511yescharacteristic
312C2×C61#229711yescharacteristic
313C2×C61#239612yes—
314C2×C61#239612yes—
315C2×C61#239812yes—
316C2×C61#239912yes—
317C23×Q8—611no—
318C2×C4×Q8—618no—
319C2×C4×Q8—1224no—
320C2×C4×Q8—1224no—
321C22×C4⋊C4—611no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—611no—
324C22×C4⋊C4—612no—
325C22×C42—611no—
326C22×C42—611no—
327C22×C42—611no—
328C22×C42—611no—
329C22×C42C2×D6111yescharacteristic
330C22×C42C2×D6111yescharacteristic
331C22×C42C2×D6111yescharacteristic
332C4×C4⋊C4—1224no—
333#376—1224no—
334C2×#95—612no—
335C2×#95—612no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—612no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95C2×D6112yes—
344C2×#95C2×D6112yes—
345C61#1385011yescharacteristic, complemented, Sylow
346C22×Q8—614no—
347C22×Q8—1222no—
348C22×Q8—1222no—
349C22×Q8—1224no—
350C22×Q8—1224no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—612no—
356C23×C4—612no—
357C23×C4C22×D6112yes—
358C23×C4C22×D6111yescharacteristic
359C23×C4C22×D6112yes—
360C23×C4C22×D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C4×Q8—12232no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1224no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1224no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—614no—
383C2×C42—614no—
384C2×C42—614no—
385C2×C42—614no—
386C2×C42—614no—
387C2×C42—1222no—
388C2×C42—1222no—
389C2×C42—1222no—
390C2×C42—1222no—
391C2×C42—1222no—
392C2×C42—1222no—
393C2×C42—1222no—
394C2×C42Dic12214yescomplemented
395C2×C42C4×D6114yes—
396#95—24no—
397#95—24no—
398#95—1224no—
399#95—1224no—
400#95—1224no—
401#95—1224no—
402#95—1224no—
403#95—1224no—
404#95—1224no—
405#95—1224no—
406#95—1224no—
407C24C23×D6111yescharacteristic
408C2×Q8—1228no—
409C2×Q8—1228no—
410C2×Q8—12216no—
411C2×Q8—12216no—
412C2×Q8—1228no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—614no—
440C22×C4—612no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4C2×Dic12214yes—
467C22×C4C2×Dic12212yes—
468C22×C4C2×C4×D6112yes—
469C22×C4C2×C4×D6112yes—
470C22×C4C2×C4×D6112yes—
471C22×C4#1175822214yes—
472C22×C4#1175822214yes—
473C22×C4#1175822212yes—
474C22×C4D4×D6112yes—
475C22×C4D4×D6112yes—
476C22×C4#1175822412yes—
477C22×C4Q8×D6112yes—
478C4⋊C4—1228no—
479C4⋊C4—1228no—
480C4⋊C4—1228no—
481C4⋊C4—12216no—
482C42—28no—
483C42—28no—
484C42—44no—
485C42—6116no—
486C42—1228no—
487C42—1228no—
488C42—2444no—
489C23C22×C4×D6111yescharacteristic
490C23C2×D4×D6111yescharacteristic
491C23C2×D4×D6111yescharacteristic
492C23C2×Q8×D6111yescharacteristic
493C23C22×Dic12211yescharacteristic
494C23#41×D6112yes—
495C23#41×D6112yes—
496C23C2×#1175822411yescharacteristic
497C23C2×#1175822212yes—
498C23C2×#1175822211yescharacteristic
499C23C2×#1175822212yes—
500Q8—12232no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—614no—
525C2×C4—614no—
526C2×C4—614no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4C22⋊C4×D6114yes—
559C2×C4#1451973414yes—
560C2×C4#1451974214yes—
561C2×C4C4×Dic12214yes—
562C2×C4#1451975414yes—
563C2×C4#1451975514yes—
564C2×C4#1451976314yes—
565C2×C4#1451977614yes—
566C22C2×C22⋊C4×D6111yescharacteristic
567C22#117×D6111yescharacteristic
568C22C2×#1451973411yescharacteristic
569C22C2×#1451974211yescharacteristic
570C22C2×C4×Dic12211yescharacteristic
571C22#1684023011yescharacteristic
572C22C4×#1175822211yescharacteristic
573C22C2×#1451975411yescharacteristic
574C22C2×#1451975511yescharacteristic
575C22#1684027112yes—
576C22#1684027112yes—
577C22#1684027112yes—
578C22C2×#1451976311yescharacteristic
579C22#1684029012yes—
580C22#1684029212yes—
581C22#1684029412yes—
582C22#1684029612yes—
583C22C2×#1451977611yescharacteristic
584C22#1684030912yes—
585C22#1684031512yes—
586C22#1684031612yes—
587C22#1684031612yes—
588C22#1684031912yes—
589C4—24no—
590C4—42no—
591C4—44no—
592C4—44no—
593C4—1224no—
594C4—1224no—
595C4—1224no—
596C4—2442no—
597C4—2444no—
598C4—2444no—
599C4#1683966618yes—
600C2#2144820011yescharacteristic
601C2C2×#1683966611yescharacteristic
602C2#2144829111yescharacteristic
603C2#2144829311yescharacteristic
604C2#2144829511yescharacteristic
605C2#2144829611yescharacteristic
606C2#2144830511yescharacteristic
607C2#2144909312yes—
608C2#2144909512yes—
609C2#2144910612yes—
610C2#2144911312yes—
subgroupG/H
centerC23×C4C22×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC4×#95×C61C2
socle, prime radical, cosocleC24×C61C24
hypercenterC4×#95D61
hypocenterC61#13850
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23×C4 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic122C4semidirect
75. C2×C4×Dic61Q8semidirect
98. C2×C42×C61Q8semidirect
345. C61#13850semidirect
394. C2×C42Dic122semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1222
C2×C4×#117581962
C2×#168396082
#214440892
#214440852
#214440712
#214440702
C4×#145196842
C4×#95×C612
#1385061