Finite Group Databasebeta

← #104×Dic122 #29696710

Subgroups — #104×Dic122 #29696710

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
2M16×Dic122C211yesmaximal, characteristic, complemented
3C42×Dic122C211yesmaximal, characteristic, complemented
4#21446796C211yesmaximal, characteristic, complemented
5#21446794C212yesmaximal, complemented
6#21446763C211yesmaximal, characteristic, complemented
7#21446761C212yesmaximal, complemented
8#21445839C211yesmaximal, characteristic
9#21445837C212yesmaximal
10#104×Dic61C212yesmaximal
11C4×#104×C61C211yesmaximal, 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×Dic122—21no—
28C2×C4×Dic122C2211yescharacteristic
29#16839892—21no—
30#16839890—22no—
31C8×Dic122—21no—
32#16839786C2211yescharacteristic
33#16839785C2212yes—
34C2×#14519704C2212yes—
35M16×Dic61C2212yes—
36C4×#11758196—22no—
37C4×#11758196C2212yes—
38C4×C4⋊Dic61—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40C42×Dic61C2212yes—
41#16839523C2212yes—
42#16839523C2212yes—
43#16839522C2212yes—
44#16839522C2212yes—
45#16839521C2212yes—
46C4×#41×C61C2211yescharacteristic
47C2×#104×C61C2211yescharacteristic
48C2×#104×C61C2211yescharacteristic
49C4×M16×C61C2211yescharacteristic
50C43×C61C2211yescharacteristic
51C61×#339C2211yescharacteristic
52C61×#339C2211yescharacteristic
53C22×Dic122—21no—
54C22×Dic122D411yescharacteristic
55C2×#11758194C2312yes—
56#14519777—21no—
57#14519777—21no—
58#14519777C2×C411yescharacteristic
59#14519777C2×C411yescharacteristic
60#14519776—22no—
61#14519776—22no—
62#14519776C2×C412yes—
63#14519776C2×C412yes—
64#14519769—22no—
65#14519769C2×C412yes—
66#14519768—22no—
67#14519768—22no—
68#14519768C2×C412yes—
69#14519768C2×C412yes—
70#14519756—21no—
71#14519756—21no—
72#14519756—21no—
73#14519756—21no—
74#14519756—22no—
75#14519754—22no—
76#14519754—22no—
77#14519754—22no—
78#14519754—22no—
79#14519754—22no—
80#14519754—22no—
81Q8×Dic61C2×C412yes—
82Q8×Dic61C2×C412yes—
83C4×Dic122—28no—
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
97D4×Dic61—22no—
98D4×Dic61—22no—
99D4×Dic61C2×C412yes—
100D4×Dic61C2×C412yes—
101#14519736—22no—
102#14519736C2312yes—
103C2×C4⋊Dic61—21no—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61D411yescharacteristic
106C2×#11758196—22no—
107C2×#11758196—22no—
108C2×#11758196D412yes—
109C2×C4×Dic61—22no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61C2312yes—
113C2×C4×Dic61C2312yes—
114C4⋊#92418—21no—
115C4⋊#92418—21no—
116#14519706—22no—
117#14519706—22no—
118#14519704—24no—
119#14519687—22no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—22no—
123C2×#41×C61C2311yescharacteristic
124C2×M16×C61C2311yescharacteristic
125C2×M16×C61C2311yescharacteristic
126C4×Q8×C61C2×C411yescharacteristic
127C4×Q8×C61C2×C411yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2×C411yescharacteristic
131C4×D4×C61C2×C411yescharacteristic
132#117×C61—21no—
133#117×C61C2311yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61—21no—
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C2×C42×C61C2311yescharacteristic
139C2×C42×C61C2311yescharacteristic
140#104×C61—22no—
141#104×C61Q814yescomplemented
142#97×C61—21no—
143C4×C8×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—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—22no—
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—21no—
174M16×C61—21no—
175M16×C61—21no—
176M16×C61C2×Q812yescomplemented
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C22×C411yescharacteristic
182C4⋊C4×C61C22×C411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C42×C61—24no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—42no—
194C42×C61C22×C411yescharacteristic
195C42×C61C22×C411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×Q812yescomplemented
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C4⋊Dic61—24no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61—22no—
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22⋊C412yes—
211C4⋊Dic61C22⋊C411yescharacteristic
212C4⋊Dic61C22×C412yes—
213C4⋊Dic61C22×C412yes—
214#11758196—28no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220#11758196—22no—
221#11758196C22⋊C412yes—
222#11758196C22⋊C412yes—
223#11758196C22⋊C412yes—
224#11758196C22⋊C412yes—
225C4×Dic61—28no—
226C4×Dic61—22no—
227C4×Dic61—22no—
228C4×Dic61—22no—
229C4×Dic61—22no—
230C4×Dic61—22no—
231C4×Dic61—44no—
232C4×Dic61C22×C412yes—
233C4×Dic61C22×C412yes—
234C4×Dic61C2×D412yes—
235C4×Dic61C2×D412yes—
236#11758194—22no—
237#11758194—22no—
238C2×#92418—22no—
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—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—42no—
261C2×C4×C61—81no—
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C23×C411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×Q811yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C612−1+411yescharacteristic
271C2×C4×C612−1+411yescharacteristic
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
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#10414yesdirect product factor
292C8×C61—22no—
293C8×C61—41no—
294#92418—42no—
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—24no—
305C4×C61—22no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61C2×#10412yes—
313C4×C61C2×C4×Q811yescharacteristic
314C4×C61#51911yescharacteristic
315C4×C61#54811yescharacteristic
316C4×C61D4×Q811yescharacteristic
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—44no—
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#1806—611no—
333C4×#104—612no—
334C4×#104D6111yescharacteristic
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#423—611no—
364C2×#104—612no—
365C2×#104C2×D6111yescharacteristic
366C2×#104C2×D6111yescharacteristic
367C4×M16—612no—
368C4×M16C2×D6111yescharacteristic
369C4×C4⋊C4—612no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—1221no—
372C4×C4⋊C4—1222no—
373C43—612no—
374C43C2×D6111yescharacteristic
375#339—612no—
376#339—612no—
377#339—612no—
378#339—612no—
379#339—612no—
380#339C2×D6111yescharacteristic
381#339C2×D6111yescharacteristic
382C61Q8×#10411yescharacteristic, complemented, Sylow
383C2×#41C22×D6111yescharacteristic
384C22×Q8—611no—
385C22×Q8—1221no—
386C2×M16—612no—
387C2×M16C22×D6111yescharacteristic
388C2×M16C22×D6111yescharacteristic
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—612no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1221no—
450C2×C42—21no—
451C2×C42—21no—
452C2×C42—21no—
453C2×C42—612no—
454C2×C42—612no—
455C2×C42—1222no—
456C2×C42—1222no—
457C2×C42—1222no—
458C2×C42C22×D6111yescharacteristic
459C2×C42C22×D6111yescharacteristic
460C2×C42C22×D6111yescharacteristic
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463C4⋊C8—1221no—
464C4⋊C8—1221no—
465#104—22no—
466#104—1224no—
467#104Dic12214yesdirect product factor
468#97—21no—
469#97—1222no—
470C4×C8—21no—
471C4×C8—1222no—
472#95—1222no—
473#95—1221no—
474#41—21no—
475#41—21no—
476#41C2×Dic12212yes—
477#41#1175822212yes—
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—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—612no—
496C22×C4—612no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—21no—
504M16—21no—
505M16—21no—
506M16—1222no—
507M16—1222no—
508M16C2×Dic12212yescomplemented
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—1222no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—612no—
515C4⋊C4—611no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—612no—
520C4⋊C4—611no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1228no—
532C4⋊C4—1224no—
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—24no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—42no—
553C42—612no—
554C42—612no—
555C42—612no—
556C42—612no—
557C42—1222no—
558C42—1222no—
559C42—1228no—
560C42—1222no—
561C42—1222no—
562C42—1222no—
563C42—2444no—
564C42C2×Dic12212yescomplemented
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—
579Q8C4×Dic12212yescomplemented
580D4—22no—
581D4—22no—
582D4—21no—
583D4C4×Dic12212yescomplemented
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—81no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—1224no—
604C2×C4—1224no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—4882no—
608C2×C4—2444no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
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×C4C22×Dic12211yescharacteristic
619C2×C4C2×#1175822211yescharacteristic
620C2×C4#1451982411yescharacteristic
621C2×C4#1451982411yescharacteristic
622C8—22no—
623C8—41no—
624C8—2442no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22C2×C22⋊C4×D6111yescharacteristic
631C22#1684042111yescharacteristic
632C22D4×Dic12211yescharacteristic
633C4—24no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—22no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—614no—
642C4—1222no—
643C4—1222no—
644C4—1222no—
645C4—2444no—
646C4—2442no—
647C4—2442no—
648C4#104×D6112yes—
649C4C2×C4×Dic12211yescharacteristic
650C4#1684023911yescharacteristic
651C4#1684042111yescharacteristic
652C4D4×Dic12211yescharacteristic
653C2—21no—
654C2—21no—
655C2—42no—
656C2C22⋊C4×Dic12211yescharacteristic
657C2C2×#104×D6111yescharacteristic
658C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#104×C61C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenterC4×#104D61
hypocenterC61Q8×#104
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. M16×Dic122C2semidirect
3. C42×Dic122C2semidirect
4. #21446796C2semidirect
5. #21446794C2semidirect
6. #21446763C2semidirect
7. #21446761C2semidirect
12. Q8×Dic122C4semidirect
13. #16840491C4semidirect
14. #16840488C4semidirect
16. #16840471C4semidirect
18. #16840469C4semidirect
20. D4×Dic122C4semidirect
141. #104×C61Q8semidirect
176. M16×C61C2×Q8semidirect
198. C42×C61C2×Q8semidirect
242. Q8×C61C4×Q8semidirect
246. D4×C61C4×Q8semidirect
291. Dic122#104direct (N × H)
382. C61Q8×#104semidirect
467. #104Dic122direct (N × H)
508. M16C2×Dic122semidirect
564. C42C2×Dic122semidirect
579. Q8C4×Dic122semidirect
583. D4C4×Dic122semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1222
M16×Dic1222
C42×Dic1222
#214467962
#214467942
#214467632
#214467612
#214458392
#214458372
#104×Dic612
C4×#104×C612
Q8×#10461