Finite Group Databasebeta

← #29840337

Subgroups — #29840337

Order 15616.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic
2C2×#16840239C211yesmaximal, characteristic, complemented
3C22×#14519736C211yesmaximal, characteristic
4#21448844C214yesmaximal, complemented
5#21448822C214yesmaximal, complemented
6#21448683C214yesmaximal, complemented
7#21448670C214yesmaximal
8#21448497C214yesmaximal, complemented
9#21448435C211yesmaximal, characteristic, complemented
10#21448403C211yesmaximal, characteristic, complemented
11#21448347C214yesmaximal, complemented
12#21448175C211yesmaximal, characteristic, complemented
13C61×#2280C211yesmaximal, characteristic
14C23×Dic122C2211yescharacteristic
15C2×#14519756C4116yescomplemented
16C2×#14519756C2212yes—
17C2×#14519756C2212yes—
18C2×#14519756C2212yes—
19C2×#14519756C2212yes—
20#16840239—28no—
21C2×C4×Dic122C2214yescomplemented
22C2×#14519742C2214yescomplemented
23C2×#14519736—28no—
24C2×#14519736C2212yes—
25C2×#14519736C2212yes—
26C2×#14519736C2211yescharacteristic
27C2×#14519736C2212yes—
28C2×#14519736C2211yescharacteristic
29C2×#14519733C2214yes—
30C22×C4⋊Dic61C2212yes—
31C22×C4⋊Dic61C2211yescharacteristic
32C22×#11758196C2212yes—
33C22×C22⋊Dic61C2212yes—
34C22×C22⋊Dic61C2212yes—
35C22×C4×Dic61C2212yes—
36#16839668C2214yescomplemented
37#16839667C2214yescomplemented
38#16839660C2214yescomplemented
39#16839659C2218yescomplemented
40#16839658C2214yescomplemented
41#16839656C2218yes—
42#16839650C2218yes—
43#16839649C2218yescomplemented
44#16839644C2218yescomplemented
45#16839634C2218yes—
46#16839623C2214yes—
47#16839620C2214yes—
48C4⋊C4×Dic61C2214yescomplemented
49C4×C4⋊Dic61C2214yescomplemented
50#16839613C2218yescomplemented
51C22⋊C4×Dic61C2218yes—
52C22⋊C4×Dic61C2214yes—
53C24×C4×C61C2211yescharacteristic
54C2×#117×C61C2211yescharacteristic
55C2×#117×C61C2211yescharacteristic
56C61×#381C2212yes—
57C61×#380C2212yes—
58C61×#380C2212yes—
59C61×#379C2212yes—
60C4×C22⋊C4×C61C2212yes—
61C4×C22⋊C4×C61C2212yes—
62C22×Dic122—24no—
63C22×Dic122—24no—
64C22×Dic122C2×C418yescomplemented
65C22×Dic122C2312yes—
66C22×Dic122C2312yes—
67C23×Dic61C2314yes—
68C23×Dic61C2312yes—
69C23×Dic61C2312yes—
70#14519756C2×C4116yes—
71#14519756C2×C4116yes—
72#14519756C2×C4116yescomplemented
73#14519756C2×C4116yescomplemented
74C4×Dic122—216no—
75#14519742—216no—
76#14519736—24no—
77#14519736—28no—
78#14519736—28no—
79#14519736—28no—
80#14519736D418yes—
81#14519733—216no—
82C2×C4⋊Dic61—28no—
83C2×C4⋊Dic61C2×C418yescomplemented
84C2×C4⋊Dic61C2314yes—
85C2×C4⋊Dic61C2312yes—
86C2×C4⋊Dic61C2312yes—
87C2×C4⋊Dic61C2312yes—
88C2×C4⋊Dic61C2312yes—
89C2×#11758196C2×C4116yescomplemented
90C2×#11758196C2318yes—
91C2×#11758196C2318yes—
92C2×#11758196C2314yes—
93C2×#11758196C2314yes—
94C2×C22⋊Dic61—28no—
95C2×C22⋊Dic61C2×C4116yes—
96C2×C22⋊Dic61C2312yes—
97C2×C22⋊Dic61C2314yes—
98C2×C22⋊Dic61C2314yes—
99C2×C22⋊Dic61C2314yes—
100C2×C22⋊Dic61C2312yes—
101C2×C4×Dic61—28no—
102C2×C4×Dic61C2314yes—
103C2×C4×Dic61C2312yes—
104C2×C4×Dic61C2318yes—
105C2×C4×Dic61C2318yes—
106C2×C4×Dic61C2312yes—
107#14519684C2318yes—
108#14519684C2318yes—
109#14519684C2318yes—
110#14519684C2318yes—
111C25×C61C2311yescharacteristic
112C23×C4×C61—24no—
113C23×C4×C61—24no—
114C23×C4×C61C2×C418yes—
115C23×C4×C61C2311yescharacteristic
116C23×C4×C61C2311yescharacteristic
117C23×C4×C61C2311yescharacteristic
118C23×C4×C61C2311yescharacteristic
119C23×C4×C61C2311yescharacteristic
120C23×C4×C61C2311yescharacteristic
121#117×C61—24no—
122#117×C61—24no—
123C2×C4⋊C4×C61C2312yes—
124C2×C4⋊C4×C61C2312yes—
125C2×C22⋊C4×C61C2312yes—
126C2×C22⋊C4×C61C2314yes—
127C2×C22⋊C4×C61C2312yes—
128C2×C22⋊C4×C61C2314yes—
129C2×C42×C61C2312yes—
130C2×C42×C61C2312yes—
131#95×C61C2314yes—
132#95×C61C2314yes—
133C24×C61—22no—
134C24×C61—22no—
135C24×C61—24no—
136C24×C61—42no—
137C24×C61C22×C414yes—
138C24×C61C2411yescharacteristic
139C24×C61C2411yescharacteristic
140C24×C61C2411yescharacteristic
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—232no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61C22×C414yes—
153C22×C4×C61C22×C414yes—
154C22×C4×C61C22×C414yes—
155C22×C4×C61C22×C414yes—
156C22×C4×C61C22×C414yes—
157C22×C4×C61C22×C414yes—
158C22×C4×C61C2×D414yes—
159C22×C4×C61#4114yes—
160C22×C4×C61C2414yes—
161C22×C4×C61C2414yes—
162C22×C4×C61C2411yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C22×C4×C61C2411yescharacteristic
165C22×C4×C61C2411yescharacteristic
166C22×Dic61—24no—
167C22×Dic61—24no—
168C22×Dic61—28no—
169C22×Dic61—28no—
170C22×Dic61—44no—
171C22×Dic61—44no—
172C22×Dic61C22×C418yes—
173C22×Dic61C22×C418yes—
174C22×Dic61C2414yes—
175C22×Dic61C2414yes—
176C22×Dic61C2414yes—
177C22×Dic61C2414yes—
178C2×Dic122—232no—
179C2×Dic122—24no—
180C2×Dic122—24no—
181C2×Dic122—24no—
182C2×Dic122—24no—
183C2×Dic122C22×C418yes—
184C2×Dic122C22×C418yes—
185C4⋊C4×C61—24no—
186C4⋊C4×C61—24no—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—24no—
189C22⋊C4×C61—28no—
190C22⋊C4×C61—44no—
191C42×C61—24no—
192C42×C61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—28no—
195C22⋊Dic61C22×C4116yes—
196C22⋊Dic61C22×C4116yes—
197C22⋊Dic61C2×D418yes—
198C4⋊Dic61—24no—
199C4⋊Dic61—28no—
200C4⋊Dic61C22×C418yes—
201C4⋊Dic61C22×C418yes—
202C4⋊Dic61C2×D418yes—
203#11758196—216no—
204#11758196—48no—
205#11758196C22×C4116yes—
206#11758196C22×C4116yes—
207C4×Dic61—24no—
208C4×Dic61—28no—
209C4×Dic61—216no—
210C4×Dic61—48no—
211C4×Dic61C2×D418yes—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—28no—
218C23×C61—28no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—44no—
224C23×C61—48no—
225C23×C61—48no—
226C23×C61C23×C412yes—
227C23×C61C23×C412yes—
228C23×C61C23×C412yes—
229C23×C61C22×D412yes—
230C23×C61C2×#4112yes—
231C23×C612−1+414yes—
232C23×C61C2511yescharacteristic
233C2×C4×C61—28no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—28no—
239C2×C4×C61—28no—
240C2×C4×C61—28no—
241C2×C4×C61—28no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61C4×D4116yes—
249C2×C4×C61C23×C412yes—
250C2×C4×C61C23×C412yes—
251C2×C4×C61C23×C412yes—
252C2×C4×C61C23×C412yes—
253C2×C4×C61C22×D412yes—
254C2×C4×C61C22×D412yes—
255C2×C4×C61C2×#4112yes—
256C2×C4×C61C2×#4112yes—
257C2×Dic61—216no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—28no—
263C2×Dic61—24no—
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—48no—
267C2×Dic61—416no—
268C2×Dic61C23×C418yes—
269C2×Dic61C23×C418yes—
270C2×Dic61C22×D418yes—
271Dic122—216no—
272Dic122—216no—
273#34203—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—24no—
281C22×C61—24no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—416no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61C2×C4×D414yes—
289C22×C61#51914yescomplemented
290C22×C61C24×C411yescharacteristic
291C22×C61C23×D411yescharacteristic
292C22×C61C22×#4111yescharacteristic
293C22×C61C2×2−1+412yes—
294C22×C61#58512yes—
295C4×C61—22no—
296C4×C61—22no—
297C4×C61—44no—
298C4×C61—44no—
299C4×C61C2×C4×D414yes—
300C4×C61C2×C4×D414yes—
301Dic61—28no—
302Dic61—28no—
303Dic61—28no—
304Dic61—44no—
305C22×C22⋊Q8—611no—
306C2×#519—611no—
307C22×#117—611no—
308#2362—614no—
309#2360—614no—
310#2344—614no—
311#2342—614no—
312#2323—614no—
313#2316—611no—
314#2310—611no—
315C4×C22⋊Q8—614no—
316#2280D6111yescharacteristic
317#2274—611no—
318C2×C61—24no—
319C2×C61—24no—
320C2×C61—42no—
321C2×C61C22×C4×D411yescharacteristic
322C2×C61C2×#51911yescharacteristic
323C2×C61#342811yescharacteristic
324C2×C61#348912yes—
325C2×C61#349512yes—
326C23×Q8—611no—
327C24×C4C2×D6111yescharacteristic
328C2×C22⋊Q8—6116no—
329C2×C22⋊Q8—612no—
330C2×C22⋊Q8—612no—
331C2×C22⋊Q8—612no—
332C2×C22⋊Q8—612no—
333#519—1228no—
334C2×C4×Q8—614no—
335C2×C4×Q8—614no—
336C2×#117—611no—
337C2×#117—611no—
338C2×#117—612no—
339C2×#117—612no—
340C2×#117—612no—
341C2×#117—614no—
342C2×#117—1228no—
343C2×#117C2×D6111yescharacteristic
344C2×#117C2×D6111yescharacteristic
345C22×C4⋊C4—611no—
346C22×C4⋊C4—612no—
347C22×C4⋊C4—612no—
348C22×C22⋊C4—612no—
349C22×C22⋊C4—612no—
350C22×C42—612no—
351#391—614no—
352#391—614no—
353#389—614no—
354#389—614no—
355#389—618no—
356#388—618no—
357#388—618no—
358#387—618no—
359#387—618no—
360#385—618no—
361#381—614no—
362#381C2×D6112yes—
363#380—614no—
364#380C2×D6112yes—
365#380C2×D6112yes—
366#379C2×D6112yes—
367C4×C4⋊C4—614no—
368C4×C4⋊C4—614no—
369C4×C4⋊C4—618no—
370C4×C22⋊C4—614no—
371C4×C22⋊C4—618no—
372C4×C22⋊C4C2×D6112yes—
373C4×C22⋊C4C2×D6112yes—
374C61#3420311yescharacteristic, complemented, Sylow
375C25C22×D6111yescharacteristic
376C22×Q8—618no—
377C22×Q8—612no—
378C22×Q8—612no—
379C22×Q8—1224no—
380C22×Q8—1224no—
381C23×C4—24no—
382C23×C4—24no—
383C23×C4—612no—
384C23×C4—612no—
385C23×C4—614no—
386C23×C4C4×D6118yes—
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6111yescharacteristic
389C23×C4C22×D6111yescharacteristic
390C23×C4C22×D6111yescharacteristic
391C23×C4C22×D6111yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C22⋊Q8—6116no—
394C22⋊Q8—6116no—
395C22⋊Q8—6116no—
396C22⋊Q8—6116no—
397C4×Q8—12216no—
398C4×Q8—12216no—
399#117—24no—
400#117—24no—
401#117—618no—
402#117—1228no—
403#117—12216no—
404#117—1228no—
405#117—1228no—
406#117—1224no—
407C2×C4⋊C4—6116no—
408C2×C4⋊C4—618no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—614no—
414C2×C4⋊C4—614no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—618no—
417C2×C4⋊C4—618no—
418C2×C4⋊C4—1228no—
419C2×C4⋊C4C22×D6112yes—
420C2×C4⋊C4C22×D6112yes—
421C2×C22⋊C4—6116no—
422C2×C22⋊C4—612no—
423C2×C22⋊C4—614no—
424C2×C22⋊C4—614no—
425C2×C22⋊C4—612no—
426C2×C22⋊C4—614no—
427C2×C22⋊C4—1228no—
428C2×C22⋊C4C22×D6112yes—
429C2×C22⋊C4C22×D6112yes—
430C2×C22⋊C4C22×D6114yes—
431C2×C22⋊C4C22×D6114yes—
432C2×C42—612no—
433C2×C42—614no—
434C2×C42—612no—
435C2×C42—618no—
436C2×C42—618no—
437C2×C42—1228no—
438C2×C42C22×D6112yes—
439C2×C42C22×D6112yes—
440#95—618no—
441#95—618no—
442#95—618no—
443#95—618no—
444#95C22×D6114yes—
445#95C22×D6114yes—
446C24—22no—
447C24—22no—
448C24—24no—
449C24—42no—
450C24C2×C4×D6114yes—
451C24C23×D6111yescharacteristic
452C24C23×D6111yescharacteristic
453C24C23×D6111yescharacteristic
454C2×Q8—618no—
455C2×Q8—618no—
456C2×Q8—1224no—
457C2×Q8—1224no—
458C2×Q8—1224no—
459C2×Q8—1224no—
460C2×Q8—12232no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—232no—
472C22×C4—618no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—614no—
476C22×C4—618no—
477C22×C4—614no—
478C22×C4—1228no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1228no—
482C22×C4—2444no—
483C22×C4—2444no—
484C22×C4C2×C4×D6114yes—
485C22×C4C2×C4×D6114yes—
486C22×C4C2×C4×D6114yes—
487C22×C4C2×C4×D6114yes—
488C22×C4C2×C4×D6114yes—
489C22×C4C2×C4×D6114yes—
490C22×C4D4×D6114yes—
491C22×C4#1175822414yes—
492C22×C4C23×D6114yes—
493C22×C4C23×D6114yes—
494C22×C4C23×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—6116no—
501C4⋊C4—6116no—
502C4⋊C4—618no—
503C4⋊C4—618no—
504C4⋊C4—618no—
505C4⋊C4—1228no—
506C4⋊C4—12216no—
507C4⋊C4—1224no—
508C4⋊C4—2448no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—28no—
512C22⋊C4—44no—
513C22⋊C4—6116no—
514C22⋊C4—6116no—
515C22⋊C4—618no—
516C22⋊C4—1228no—
517C22⋊C4—1224no—
518C42—24no—
519C42—24no—
520C42—618no—
521C42—1228no—
522C42—12216no—
523C42—1224no—
524C42—2448no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—48no—
538C23—48no—
539C23C22×C4×D6112yes—
540C23C22×C4×D6112yes—
541C23C22×C4×D6112yes—
542C23C2×D4×D6112yes—
543C23C2×#1175822412yes—
544C23#1451982414yes—
545C23C24×D6111yescharacteristic
546Q8—12216no—
547Q8—12216no—
548C2×C4—28no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—618no—
564C2×C4—618no—
565C2×C4—618no—
566C2×C4—12216no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1228no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—2448no—
576C2×C4—24416no—
577C2×C4#14519737116yes—
578C2×C4C22×C4×D6112yes—
579C2×C4C22×C4×D6112yes—
580C2×C4C22×C4×D6112yes—
581C2×C4C22×C4×D6112yes—
582C2×C4C2×D4×D6112yes—
583C2×C4C2×D4×D6112yes—
584C2×C4C2×#1175822412yes—
585C2×C4C2×#1175822412yes—
586C22—24no—
587C22—24no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—24no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22C2×#1451973714yes—
601C22#1684023914yescomplemented
602C22C23×C4×D6111yescharacteristic
603C22C22×D4×D6111yescharacteristic
604C22C22×#1175822411yescharacteristic
605C22C2×#1451982412yes—
606C22#1684059812yes—
607C4—22no—
608C4—22no—
609C4—44no—
610C4—44no—
611C4—1228no—
612C4—1228no—
613C4—1228no—
614C4—2444no—
615C4C2×#1451973714yes—
616C4C2×#1451973714yes—
617C2—24no—
618C2—24no—
619C2—42no—
620C2C22×#1451973711yescharacteristic
621C2C2×#1684023911yescharacteristic
622C2#2145902011yescharacteristic
623C2#2145949112yes—
624C2#2145957312yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2280C2
socle, cosocleC23×C61C25
prime radicalC25×C61C23
hypercenter#2280D61
hypocenterC61#34203
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#16840239C2semidirect
4. #21448844C2semidirect
5. #21448822C2semidirect
6. #21448683C2semidirect
8. #21448497C2semidirect
9. #21448435C2semidirect
10. #21448403C2semidirect
11. #21448347C2semidirect
12. #21448175C2semidirect
15. C2×#14519756C4semidirect
21. C2×C4×Dic122C22semidirect
22. C2×#14519742C22semidirect
36. #16839668C22semidirect
37. #16839667C22semidirect
38. #16839660C22semidirect
39. #16839659C22semidirect
40. #16839658C22semidirect
43. #16839649C22semidirect
44. #16839644C22semidirect
48. C4⋊C4×Dic61C22semidirect
49. C4×C4⋊Dic61C22semidirect
50. #16839613C22semidirect
64. C22×Dic122C2×C4semidirect
72. #14519756C2×C4semidirect
73. #14519756C2×C4semidirect
83. C2×C4⋊Dic61C2×C4semidirect
89. C2×#11758196C2×C4semidirect
289. C22×C61#519semidirect
374. C61#34203semidirect
601. C22#16840239semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C2×#168402392
C22×#145197362
#214488442
#214488222
#214486832
#214486702
#214484972
#214484352
#214484032
#214483472
#214481752
C61×#22802
#3420361