Finite Group Databasebeta

← #29696707

Subgroups — #29696707

Order 15616.

1096 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic122C211yesmaximal, characteristic
2#21454770C211yesmaximal, characteristic, complemented
3C4×#14519742C211yesmaximal, characteristic, complemented
4#21446795C211yesmaximal, characteristic, complemented
5#21446794C212yesmaximal, complemented
6#21446762C211yesmaximal, characteristic, complemented
7#21446761C212yesmaximal, complemented
8#21445837C212yesmaximal
9#104×Dic61C212yesmaximal
10C4×#14519704C211yesmaximal, characteristic
11C61×#1806C211yesmaximal, characteristic
12Q8×Dic122C411yescharacteristic, complemented
13#16840491C411yescharacteristic, complemented
14#16840488C412yescomplemented
15#16840471—22no—
16#16840471C412yescomplemented
17#16840469—21no—
18#16840469C411yescharacteristic, complemented
19D4×Dic122—21no—
20D4×Dic122C411yescharacteristic, complemented
21#16840319—21no—
22#16840319C2211yescharacteristic
23#16840315—22no—
24#16840315C2212yes—
25#41×Dic61C2212yes—
26C2×C4×Dic122—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519742—21no—
29#16839891—22no—
30#16839889—21no—
31#16839886—21no—
32#16839785C2212yes—
33C2×#14519704C2211yescharacteristic
34C2×#14519704C2211yescharacteristic
35C4×#11758194C2211yescharacteristic
36M16×Dic61C2212yes—
37C4⋊C4×Dic61—21no—
38C4×#11758196C2212yes—
39#16839613—22no—
40C42×Dic61C2212yes—
41C42×Dic61C2211yescharacteristic
42#16839523C2212yes—
43#16839523C2212yes—
44#16839522C2212yes—
45#16839522C2212yes—
46#16839521C2211yescharacteristic
47#16839521C2211yescharacteristic
48C4×#41×C61C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C2×#104×C61C2212yes—
51C4×C4⋊C4×C61C2211yescharacteristic
52C61×#339C2212yes—
53C22×Dic122—21no—
54C22×Dic122D411yescharacteristic
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57#14519777—21no—
58#14519777—21no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C411yescharacteristic
61#14519776—22no—
62#14519776—22no—
63#14519776C2×C412yes—
64#14519776C2×C412yes—
65#14519769—22no—
66#14519769C2×C412yes—
67#14519768—22no—
68#14519768—22no—
69#14519768C2×C412yes—
70#14519768C2×C412yes—
71#14519756—21no—
72#14519756—21no—
73#14519756—21no—
74#14519756—21no—
75#14519756—22no—
76#14519754—22no—
77#14519754—22no—
78#14519754—22no—
79#14519754—22no—
80#14519754—22no—
81#14519754—22no—
82Q8×Dic61C2×C412yes—
83Q8×Dic61C2×C412yes—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122C2×C411yescharacteristic
92C4×Dic122C2×C412yes—
93C4×Dic122C2×C411yescharacteristic
94C4×Dic122D411yescharacteristic
95C4×Dic122D412yes—
96C4×Dic122D411yescharacteristic
97#14519742—28no—
98D4×Dic61—22no—
99D4×Dic61—22no—
100D4×Dic61C2×C412yes—
101D4×Dic61C2×C412yes—
102#14519736—22no—
103#14519736C2312yes—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×#11758196—22no—
107C2×#11758196—22no—
108C2×#11758196D412yes—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—22no—
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116C2×C4×Dic61C2312yes—
117#14519706—22no—
118#14519706—22no—
119#14519704—22no—
120#14519704Q814yescomplemented
121#14519688—21no—
122#14519687—22no—
123C4×#92418—21no—
124C8×Dic61—22no—
125#14519684—22no—
126C2×#41×C61C2311yescharacteristic
127C2×M16×C61C2312yes—
128C4×Q8×C61C2×C411yescharacteristic
129C4×Q8×C61C2×C411yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2×C411yescharacteristic
133C4×D4×C61C2×C411yescharacteristic
134#117×C61—21no—
135#117×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137C2×C42×C61—21no—
138C2×C42×C61C2312yes—
139C2×C42×C61C2311yescharacteristic
140C4⋊C8×C61—21no—
141C4⋊C8×C61—21no—
142#104×C61—24no—
143#95×C61—21no—
144#41×C61—21no—
145#41×C61—21no—
146#41×C61C2×Q812yes—
147#41×C61#4112yes—
148C2×Q8×C61C22×C411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C22×C411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—22no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61C2×D412yes—
163C22×Dic61C2×D412yes—
164C2×Dic122—24no—
165C2×Dic122—21no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—22no—
169C2×Dic122—44no—
170C2×Dic122C22⋊C412yes—
171C2×Dic122C22⋊C411yescharacteristic
172C2×Dic122C22⋊C411yescharacteristic
173M16×C61—22no—
174M16×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C411yescharacteristic
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61C22×C411yescharacteristic
190C42×C61C22×C411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22⋊C411yescharacteristic
203C4⋊Dic61C22⋊C412yes—
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22×C412yes—
206C4⋊Dic61C22×C412yes—
207#11758196—28no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196C22⋊C412yes—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218C4×Dic61—28no—
219C4×Dic61—24no—
220C4×Dic61—22no—
221C4×Dic61—21no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61—44no—
227C4×Dic61—42no—
228C4×Dic61C22×C412yes—
229C4×Dic61C22×C412yes—
230C4×Dic61C2×D412yes—
231C4×Dic61C2×D412yes—
232C4×Dic61C2×Q812yescomplemented
233#11758194—21no—
234#11758194—21no—
235#11758194—21no—
236#11758194C2×Q812yescomplemented
237C2×#92418—21no—
238C2×#92418—21no—
239C23×C61—21no—
240C23×C61C22×D411yescharacteristic
241Q8×C61—21no—
242Q8×C61C4×Q812yescomplemented
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—21no—
246D4×C61C4×Q812yescomplemented
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—24no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—81no—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×Q811yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—24no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—42no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—44no—
280C2×Dic61—82no—
281C2×Dic61C2×C22⋊C412yes—
282C2×Dic61C2×C22⋊C412yes—
283C2×Dic61C2×C22⋊C412yes—
284C2×Dic61C2×C22⋊C412yes—
285Dic122—22no—
286Dic122—24no—
287Dic122—42no—
288Dic122—44no—
289Dic122—82no—
290Dic122—81no—
291Dic122#10414yescomplemented
292C8×C61—42no—
293#92418—22no—
294#92418—41no—
295Q8×#104—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61#54811yescharacteristic
303C22×C61D4×Q811yescharacteristic
304C4×C61—22no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—44no—
309C4×C61—41no—
310C4×C61—41no—
311C4×C61C2×#10412yes—
312C4×C61C2×C4×Q811yescharacteristic
313C4×C61#51911yescharacteristic
314C4×C61#54811yescharacteristic
315C4×C61D4×Q811yescharacteristic
316Dic61—24no—
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—42no—
322Dic61—42no—
323Dic61C2×#10414yes—
324Q8×#41—611no—
325Q8×M16—611no—
326C42×Q8—611no—
327#1994—611no—
328#1994—612no—
329#1986—611no—
330#1986—612no—
331#1806—612no—
332#1806D6111yescharacteristic
333C4×#104—612no—
334C4×#104—611no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61—42no—
338C2×C61Q8×C22⋊C411yescharacteristic
339C2×C61C22×#10411yescharacteristic
340C2×C61#290111yescharacteristic
341Q82—611no—
342#557—612no—
343#557—611no—
344#554—611no—
345#554—612no—
346#554—1221no—
347#554—1222no—
348D4×Q8—611no—
349D4×Q8—1221no—
350#533—612no—
351#533—611no—
352#533—1221no—
353#533—1222no—
354C4×#41—612no—
355C4×#41C2×D6111yescharacteristic
356C2×C4×Q8—611no—
357C2×C4×Q8—1221no—
358C2×C4×Q8—1221no—
359#446—1221no—
360#446—1222no—
361C8×Q8—1221no—
362#423—612no—
363#423C2×D6111yescharacteristic
364C2×#104—611no—
365C2×#104—611no—
366C2×#104C2×D6112yes—
367C4×M16—612no—
368C4×M16—611no—
369C4×C4⋊C4—612no—
370C4×C4⋊C4—1222no—
371C4×C4⋊C4—1221no—
372C4×C4⋊C4C2×D6111yescharacteristic
373C43—612no—
374C43—611no—
375#339—612no—
376#339—612no—
377#339—611no—
378#339—611no—
379#339—612no—
380#339—612no—
381#339C2×D6112yes—
382C61Q8×#10411yescharacteristic, complemented, Sylow
383C2×#41C22×D6111yescharacteristic
384C22×Q8—611no—
385C22×Q8—1221no—
386C2×M16—611no—
387C2×M16—611no—
388C2×M16C22×D6112yes—
389C4⋊Q8—612no—
390C4⋊Q8—612no—
391C4⋊Q8—611no—
392C4⋊Q8—611no—
393C4⋊Q8—1221no—
394C4⋊Q8—1221no—
395C4⋊Q8—1222no—
396C4⋊Q8—1222no—
397#125—612no—
398#125—612no—
399#125—612no—
400#125—1222no—
401#125—1222no—
402#125—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1221no—
410C22⋊Q8—1222no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8—1221no—
414C4×Q8—611no—
415C4×Q8—611no—
416C4×Q8—612no—
417C4×Q8—611no—
418C4×Q8—612no—
419C4×Q8—611no—
420C4×Q8—612no—
421C4×Q8—612no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×Q8—1222no—
425C4×Q8—1221no—
426C4×Q8—1221no—
427C4×Q8—1228no—
428C4×Q8—1221no—
429C4×Q8—1221no—
430C4×Q8C4×D6111yescharacteristic
431C4×Q8C4×D6111yescharacteristic
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—612no—
435C4×D4—612no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4C4×D6111yescharacteristic
439C4×D4C4×D6111yescharacteristic
440#117—21no—
441#117—612no—
442#117—1222no—
443#117C22×D6111yescharacteristic
444C2×C4⋊C4—21no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1222no—
450C2×C42—21no—
451C2×C42—612no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—1222no—
455C2×C42—1222no—
456C2×C42—1221no—
457C2×C42—1222no—
458C2×C42—1221no—
459C2×C42C22×D6112yes—
460C2×C42C22×D6111yescharacteristic
461C4⋊C8—21no—
462C4⋊C8—21no—
463C4⋊C8—1222no—
464C4⋊C8—1222no—
465#104—24no—
466#104—614no—
467#104—1222no—
468#97—1222no—
469#97—1221no—
470C4×C8—1222no—
471C4×C8—1221no—
472#95—21no—
473#95—1222no—
474#41—21no—
475#41—21no—
476#41#1175822412yes—
477#41Q8×D6112yes—
478C2×Q8—611no—
479C2×Q8—612no—
480C2×Q8—611no—
481C2×Q8—1222no—
482C2×Q8—1222no—
483C2×Q8—1224no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—2444no—
487C2×Q8C2×C4×D6111yescharacteristic
488C2×D4—21no—
489C2×D4C2×C4×D6111yescharacteristic
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—612no—
495C22×C4—612no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—22no—
504M16—22no—
505M16—612no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509C2×C8—22no—
510C2×C8—1221no—
511C2×C8—1221no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—612no—
516C4⋊C4—611no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—611no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1228no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1222no—
536C4⋊C4—1221no—
537C4⋊C4C2×C4×D6111yescharacteristic
538C4⋊C4C2×C4×D6111yescharacteristic
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—22no—
549C42—22no—
550C42—612no—
551C42—612no—
552C42—612no—
553C42—612no—
554C42—612no—
555C42—1222no—
556C42—1222no—
557C42—1224no—
558C42—1228no—
559C42—1222no—
560C42—1221no—
561C42—1221no—
562C42—1221no—
563C42—2442no—
564C42—2444no—
565C42C2×C4×D6111yescharacteristic
566C42C2×C4×D6111yescharacteristic
567C42D4×D6111yescharacteristic
568C42D4×D6111yescharacteristic
569C23—21no—
570C23C2×D4×D6111yescharacteristic
571Q8—21no—
572Q8—614no—
573Q8—1224no—
574Q8—1222no—
575Q8—4881no—
576Q8—4882no—
577Q8—2444no—
578Q8—2442no—
579Q8#1451974212yescomplemented
580D4—22no—
581D4—22no—
582D4—21no—
583D4#1451974212yescomplemented
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—81no—
596C2×C4—612no—
597C2×C4—612no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—1224no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—4882no—
608C2×C4—2444no—
609C2×C4—2442no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4C22⋊C4×D6111yescharacteristic
613C2×C4C22⋊C4×D6111yescharacteristic
614C2×C4C22⋊C4×D6111yescharacteristic
615C2×C4C22⋊C4×D6111yescharacteristic
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4C2×Q8×D6111yescharacteristic
619C2×C4C2×#1175822411yescharacteristic
620C2×C4#1451982411yescharacteristic
621C2×C4#1451982411yescharacteristic
622C8—42no—
623C8—1222no—
624C8—2441no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22C2×C22⋊C4×D6111yescharacteristic
631C22#1684041811yescharacteristic
632C22#1684042511yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—22no—
637C4—44no—
638C4—41no—
639C4—41no—
640C4—614no—
641C4—1224no—
642C4—1222no—
643C4—1222no—
644C4—1222no—
645C4—2442no—
646C4—2442no—
647C4—2442no—
648C4#104×D6112yes—
649C4C2×#1451974211yescharacteristic
650C4#1684023911yescharacteristic
651C4#1684041811yescharacteristic
652C4#1684042511yescharacteristic
653C2—21no—
654C2—21no—
655C2—42no—
656C2#2144865111yescharacteristic
657C2C2×#104×D6111yescharacteristic
658C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1806C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#1806D61
hypocenterC61Q8×#104
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21454770C2semidirect
3. C4×#14519742C2semidirect
4. #21446795C2semidirect
5. #21446794C2semidirect
6. #21446762C2semidirect
7. #21446761C2semidirect
12. Q8×Dic122C4semidirect
13. #16840491C4semidirect
14. #16840488C4semidirect
16. #16840471C4semidirect
18. #16840469C4semidirect
20. D4×Dic122C4semidirect
120. #14519704Q8semidirect
232. C4×Dic61C2×Q8semidirect
236. #11758194C2×Q8semidirect
242. Q8×C61C4×Q8semidirect
246. D4×C61C4×Q8semidirect
291. Dic122#104semidirect
382. C61Q8×#104semidirect
579. Q8#14519742semidirect
583. D4#14519742semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1222
#214547702
C4×#145197422
#214467952
#214467942
#214467622
#214467612
#214458372
#104×Dic612
C4×#145197042
C61×#18062
Q8×#10461