Finite Group Databasebeta

← #30608714

Subgroups — #30608714

Order 15616.

2182 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519856C211yesmaximal, characteristic
2C2×C4⋊C4×#6923C212yesmaximal, complemented
3C2×C4×C22⋊#6923C211yesmaximal, characteristic
4#21461000C214yesmaximal
5#21460996C212yesmaximal, complemented
6#21460994C212yesmaximal, complemented
7#95×#6923C212yesmaximal, complemented
8#380×D61C211yesmaximal, characteristic
9C2×#14519856—28no—
10C2×#14519856C418yescomplemented
11C22×C4⋊#6923C2211yescharacteristic
12C22×C4⋊#6923C2211yescharacteristic
13C22×C22⋊#6923C2212yes—
14C22×C22⋊#6923C2212yes—
15C22×C4×#6923C2214yes—
16C22×C4×#6923C2212yes—
17C22×C4×#6923C2212yes—
18C4⋊C4×#6923—28no—
19C4×C22⋊#6923C418yescomplemented
20C2×(C2×#6923)⋊C4C2214yes—
21C2×(C2×#6923)⋊C4C2214yes—
22C2×(C2×#6923)⋊C4C2214yes—
23C2×(C2×#6923)⋊C4C2211yescharacteristic
24C2×(C2×#6923)⋊C4C2211yescharacteristic
25C23×C4×D61C2211yescharacteristic
26C2×C4⋊C4×D61C2212yes—
27C2×C22⋊C4×D61C2212yes—
28#16839622C411yescharacteristic, complemented
29#16839621C412yescomplemented
30#16839620C412yescomplemented
31#16839619C412yescomplemented
32#95×D61C2212yes—
33C61×#380C411yescharacteristic, complemented
34C23×#6923C2312yes—
35C23×#6923C2312yes—
36C23×#6923C2312yes—
37C23×#6923C2312yes—
38#14519856—232no—
39C2×C4⋊#6923—24no—
40C2×C4⋊#6923C2×C414yes—
41C2×C4⋊#6923C2×C414yes—
42C2×C4⋊#6923D418yes—
43C2×C22⋊#6923—28no—
44C2×C22⋊#6923—24no—
45C2×C22⋊#6923—24no—
46C2×C22⋊#6923C2×C418yes—
47C2×C22⋊#6923C2×C418yes—
48C2×C4×#6923—28no—
49C2×C4×#6923—24no—
50C2×C4×#6923—24no—
51C2×C4×#6923—28no—
52C2×C4×#6923—28no—
53C2×C4×#6923—44no—
54C2×C4×#6923C2×C418yes—
55C2×C4×#6923C2×C418yes—
56(C2×#6923)⋊C4—28no—
57(C2×#6923)⋊C4—28no—
58(C2×#6923)⋊C4—44no—
59(C2×#6923)⋊C4C2×C414yes—
60(C2×#6923)⋊C4C2×C414yes—
61C24×D61C2311yescharacteristic
62C23×Dic61C2×C411yescharacteristic
63C22×C4×D61—24no—
64C22×C4×D61—24no—
65C22×C4×D61C2×C411yescharacteristic
66C22×C4×D61C2×C411yescharacteristic
67C22×C4×D61C2×C414yes—
68C22×C4×D61C2×C414yes—
69C22×C4×D61C2314yes—
70C22×C4×D61C2311yescharacteristic
71C22×C4×D61C2311yescharacteristic
72C2×C4⋊Dic61C2×C412yes—
73C2×#11758196C2×C414yes—
74C4⋊C4×D61—24no—
75C4⋊C4×D61—24no—
76C2×C22⋊Dic61C2×C412yes—
77C2×#11758198C2×C414yes—
78C22⋊C4×D61—42no—
79C22⋊C4×D61—42no—
80#14519684C2×C414yes—
81#14519684C2×C412yes—
82C23×C4×C61C2×C411yescharacteristic
83C2×C4⋊C4×C61C2×C412yes—
84C2×C22⋊C4×C61C2×C412yes—
85#95×C61C2×C412yes—
86C24×C61C22×C411yescharacteristic
87C23×D61—22no—
88C23×D61—22no—
89C23×D61—22no—
90C23×D61—22no—
91C23×D61—21no—
92C23×D61—21no—
93C23×D61—21no—
94C23×D61—21no—
95C23×D61C22×C411yescharacteristic
96C23×D61C22×C412yes—
97C23×D61C22×C412yes—
98C23×D61C2411yescharacteristic
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—24no—
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—22no—
107C22×#6923—24no—
108C22×#6923—24no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923C22×C414yes—
117C22×#6923C22×C414yes—
118C22×#6923C22×C414yes—
119C22×#6923C22×C414yes—
120C22×#6923C2×D418yes—
121C22×#6923#4114yes—
122C22×#6923#4114yes—
123C22×C4×C61—24no—
124C22×C4×C61C4214yescomplemented
125C22×C4×C61C22×C414yes—
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×Dic61—24no—
129C22×Dic61C4214yescomplemented
130C22×Dic61C22×C414yes—
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—24no—
136C2×C4×D61—24no—
137C2×C4×D61—28no—
138C2×C4×D61—28no—
139C2×C4×D61—216no—
140C2×C4×D61—24no—
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—24no—
145C2×C4×D61—24no—
146C2×C4×D61C4214yescomplemented
147C2×C4×D61C4214yescomplemented
148C2×C4×D61C22×C412yes—
149C2×C4×D61C22×C412yes—
150C2×C4×D61C22×C412yes—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×D414yes—
153C2×C4×D61#4114yes—
154C22⋊#6923—216no—
155C22⋊#6923—216no—
156C4⋊#6923—28no—
157C4⋊#6923C22⋊C4116yescomplemented
158C4×#6923—216no—
159C4×#6923—216no—
160C4×#6923—416no—
161C4⋊C4×C61—24no—
162C22⋊C4×C61—42no—
163C22⋊Dic61—42no—
164#11758198—44no—
165C4⋊Dic61—24no—
166#11758196—28no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61C2×C4212yes—
172C23×C61C23×C411yescharacteristic
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—22no—
178C22×D61—24no—
179C22×D61—22no—
180C22×D61—24no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61C2×C4212yes—
196C22×D61C23×C411yescharacteristic
197C22×D61C23×C411yescharacteristic
198C22×D61C22×D412yes—
199C22×D61C2×#4112yes—
200C22×D61C2×#4112yes—
201C22×D61C2×#4112yes—
202C22×D61H(5,𝔽2)11yescharacteristic
203C22×D61H(5,𝔽2)11yescharacteristic
204C22×D612−1+411yescharacteristic
205C22×D612−1+411yescharacteristic
206C2×#6923—24no—
207C2×#6923—24no—
208C2×#6923—24no—
209C2×#6923—24no—
210C2×#6923—24no—
211C2×#6923—28no—
212C2×#6923—28no—
213C2×#6923—24no—
214C2×#6923—24no—
215C2×#6923—24no—
216C2×#6923—24no—
217C2×#6923—24no—
218C2×#6923—24no—
219C2×#6923—24no—
220C2×#6923—24no—
221C2×#6923—24no—
222C2×#6923—42no—
223C2×#6923—44no—
224C2×#6923—44no—
225C2×#6923—42no—
226C2×#6923—42no—
227C2×#6923—42no—
228C2×#6923C2×C22⋊C418yes—
229C2×#6923#11718yescomplemented
230C2×C4×C61—22no—
231C2×C4×C61—28no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61C2×C4212yes—
236C2×C4×C61C2×C4212yes—
237C2×C4×C61C4×D414yes—
238C2×Dic61—22no—
239C2×Dic61—28no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61C2×C4212yes—
244C2×Dic61C2×C4212yes—
245C2×Dic61C4×D414yes—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—24no—
249C4×D61—28no—
250C4×D61—28no—
251C4×D61—44no—
252C4×D61—44no—
253C4×D61—44no—
254C4×D61—44no—
255C4×D61C2×C22⋊C414yes—
256C4×D61#11714yes—
257C4×D61C4×D418yes—
258#13801—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61C22×C4211yescharacteristic
265C22×C61C2×C4×D412yes—
266C22×C61C4×#4112yes—
267C22×C61#51811yescharacteristic
268C22×C61#51911yescharacteristic
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—44no—
279C2×D61—44no—
280C2×D61C22×C22⋊C411yescharacteristic
281C2×D61C2×#11711yescharacteristic
282C2×D61C2×#11712yes—
283C2×D61C2×C4×D412yes—
284C2×D61C4×#4111yescharacteristic
285C2×D61C4×#4111yescharacteristic
286C2×D61#52011yescharacteristic
287C2×D61#52011yescharacteristic
288C2×D61#52011yescharacteristic
289C2×D61#52011yescharacteristic
290C2×D61#54612yes—
291C2×D61#54712yes—
292C2×D61#54812yes—
293C2×D61#55012yes—
294C2×D61#55212yes—
295C2×D61#55312yes—
296C2×D61#55512yes—
297C2×D61#55612yes—
298#6923—28no—
299#6923—28no—
300#6923—44no—
301#6923—44no—
302C4×C61—22no—
303C4×C61—44no—
304C4×C61C4×C22⋊C414yes—
305Dic61—22no—
306Dic61—44no—
307Dic61C4×C22⋊C414yes—
308C22×#117—611no—
309C2×#380—611no—
310C2×C4×C4⋊C4—612no—
311C2×C4×C22⋊C4—611no—
312#1437—614no—
313C4⋊#95—612no—
314#1434—612no—
315C4×#95—612no—
316C2×C61—42no—
317C2×C61C2×C4×C22⋊C411yescharacteristic
318C2×C61C4×#11711yescharacteristic
319C2×C61#227411yescharacteristic
320C2×C61#236212yes—
321C2×C61#236312yes—
322D61—42no—
323D61#233911yescharacteristic
324D61#234111yescharacteristic
325D61#234611yescharacteristic
326D61#234711yescharacteristic
327D61#235212yes—
328D61#235412yes—
329C24×C4—611no—
330C2×#117—618no—
331C2×#117—1228no—
332C22×C4⋊C4—611no—
333C22×C4⋊C4—611no—
334C22×C4⋊C4—612no—
335C22×C22⋊C4—612no—
336C22×C22⋊C4—612no—
337C22×C22⋊C4—612no—
338C22×C42—612no—
339C22×C42—612no—
340C22×C42—614no—
341#380—611no—
342#380—612no—
343#380—612no—
344#380—612no—
345#380#692311yescharacteristic, complemented
346C4×C4⋊C4—1228no—
347C4×C22⋊C4—618no—
348C2×#95—611no—
349C2×#95—611no—
350C2×#95—612no—
351C2×#95—614no—
352C2×#95—614no—
353C2×#95—614no—
354C61#1380111yescharacteristic, complemented, Sylow
355C25—611no—
356C23×C4—611no—
357C23×C4—612no—
358C23×C4—612no—
359C23×C4—611no—
360C23×C4—614no—
361C23×C4—614no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—612no—
366C23×C4—612no—
367C23×C4—614no—
368C23×C4—1224no—
369C23×C4—1224no—
370C23×C4C2×#692311yescharacteristic
371#117—12232no—
372C2×C4⋊C4—614no—
373C2×C4⋊C4—618no—
374C2×C4⋊C4—614no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1224no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4C2×#692312yes—
381C2×C22⋊C4—618no—
382C2×C22⋊C4—618no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—614no—
385C2×C22⋊C4—1228no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—2442no—
389C2×C22⋊C4—2442no—
390C2×C22⋊C4C2×#692312yes—
391C2×C42—618no—
392C2×C42—618no—
393C2×C42—1228no—
394C2×C42—1224no—
395C2×C42—1224no—
396C2×C42—1228no—
397C2×C42—1228no—
398C2×C42—2444no—
399#95—614no—
400#95—614no—
401#95—614no—
402#95—612no—
403#95—1228no—
404#95—1228no—
405#95—2444no—
406#95C2×#692312yes—
407C24—611no—
408C24—612no—
409C24—612no—
410C24—611no—
411C24—1222no—
412C24—1222no—
413C24—1222no—
414C24—1221no—
415C24—1221no—
416C24—1222no—
417C24—1221no—
418C24—1221no—
419C24C22×#692311yescharacteristic
420C22×C4—24no—
421C22×C4—611no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—614no—
425C22×C4—614no—
426C22×C4—614no—
427C22×C4—614no—
428C22×C4—618no—
429C22×C4—614no—
430C22×C4—614no—
431C22×C4—614no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—614no—
436C22×C4—614no—
437C22×C4—614no—
438C22×C4—614no—
439C22×C4—614no—
440C22×C4—1228no—
441C22×C4—1224no—
442C22×C4—1228no—
443C22×C4—12216no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1224no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1224no—
461C22×C4—1222no—
462C22×C4—1224no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1224no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4C4×#692314yescomplemented
472C22×C4C22×#692314yes—
473C22×C4C22×#692311yescharacteristic
474C22×C4C22×#692311yescharacteristic
475C4⋊C4—24no—
476C4⋊C4—6116no—
477C4⋊C4—1228no—
478C4⋊C4—1228no—
479C4⋊C4—1224no—
480C22⋊C4—42no—
481C22⋊C4—12216no—
482C22⋊C4—12216no—
483C22⋊C4—2444no—
484C22⋊C4—2442no—
485C42—12216no—
486C42—12216no—
487C42—24416no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—611no—
493C23—611no—
494C23—612no—
495C23—612no—
496C23—612no—
497C23—612no—
498C23—611no—
499C23—611no—
500C23—611no—
501C23—611no—
502C23—612no—
503C23—1222no—
504C23—1224no—
505C23—1224no—
506C23—1222no—
507C23—1222no—
508C23—1222no—
509C23—1222no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—1222no—
514C23—1222no—
515C23—1221no—
516C23—1221no—
517C23—2442no—
518C23—2442no—
519C23—2442no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2442no—
525C23C2×C4×#692312yes—
526C23C23×#692311yescharacteristic
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—612no—
533C2×C4—614no—
534C2×C4—614no—
535C2×C4—614no—
536C2×C4—618no—
537C2×C4—618no—
538C2×C4—618no—
539C2×C4—612no—
540C2×C4—1228no—
541C2×C4—1222no—
542C2×C4—1224no—
543C2×C4—1222no—
544C2×C4—1228no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1228no—
548C2×C4—1228no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—1222no—
556C2×C4—1228no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—2444no—
567C2×C4—2444no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2442no—
571C2×C4—2444no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2444no—
575C2×C4—2442no—
576C2×C4C2×C4×#692312yes—
577C2×C4C2×C4×#692312yes—
578C2×C4D4×#692314yes—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—42no—
584C22—612no—
585C22—612no—
586C22—612no—
587C22—612no—
588C22—612no—
589C22—612no—
590C22—612no—
591C22—612no—
592C22—611no—
593C22—612no—
594C22—611no—
595C22—612no—
596C22—611no—
597C22—611no—
598C22—611no—
599C22—611no—
600C22—611no—
601C22—611no—
602C22—1222no—
603C22—1222no—
604C22—1222no—
605C22—2442no—
606C22—2444no—
607C22—2442no—
608C22—2444no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22—2442no—
613C22C22×C4×#692311yescharacteristic
614C22C2×D4×#692312yes—
615C22#41×#692312yes—
616C22#1684082911yescharacteristic
617C22#1684083011yescharacteristic
618C4—22no—
619C4—44no—
620C4—614no—
621C4—1228no—
622C4—1228no—
623C4—1222no—
624C4—2444no—
625C4—2444no—
626C4—2444no—
627C4C22⋊C4×#692314yes—
628C2—42no—
629C2—611no—
630C2—612no—
631C2—611no—
632C2—612no—
633C2—611no—
634C2—611no—
635C2—2442no—
636C2C2×C22⋊C4×#692311yescharacteristic
637C2#117×#692311yescharacteristic
638C2#2146225711yescharacteristic
639C2#2146234512yes—
640C2#2146234612yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutatorC22×C61C22×C42
FittingC61×#380C4
socleC23×C61C23×C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenter#380#6923
hypocenterC61#13801
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×#6923C2semidirect
5. #21460996C2semidirect
6. #21460994C2semidirect
7. #95×#6923C2semidirect
10. C2×#14519856C4semidirect
19. C4×C22⋊#6923C4semidirect
28. #16839622C4semidirect
29. #16839621C4semidirect
30. #16839620C4semidirect
31. #16839619C4semidirect
33. C61×#380C4semidirect
124. C22×C4×C61C42semidirect
129. C22×Dic61C42semidirect
146. C2×C4×D61C42semidirect
147. C2×C4×D61C42semidirect
157. C4⋊#6923C22⋊C4semidirect
229. C2×#6923#117semidirect
345. #380#6923semidirect
354. C61#13801semidirect
471. C22×C4C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C22×#145198562
C2×C4⋊C4×#69232
C2×C4×C22⋊#69232
#214610002
#214609962
#214609942
#95×#69232
#380×D612
#1380161