Finite Group Databasebeta

← #29696711

Subgroups — #29696711

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#21459397C211yesmaximal, characteristic
2M16×Dic122C211yesmaximal, characteristic, complemented
3C4×#14519742C211yesmaximal, characteristic, complemented
4#21446797C211yesmaximal, characteristic, complemented
5#21446794C212yesmaximal, complemented
6#21446761C212yesmaximal, complemented
7#21445840C211yesmaximal, characteristic
8#21445838C212yesmaximal
9C4×#14519703C211yesmaximal, characteristic
10#21445623C212yesmaximal
11C61×#1986C211yesmaximal, characteristic, complemented
12#16840492C411yescharacteristic, complemented
13#16840489C412yescomplemented
14#16840487C411yescharacteristic, complemented
15#16840473C411yescharacteristic, complemented
16#16840470—21no—
17#16840468C412yescomplemented
18#16840466—22no—
19#16840427C411yescharacteristic, complemented
20#16840425—21no—
21#16840316—21no—
22#16840316—22no—
23#16840315C2212yes—
24C4×#11758222C2211yescharacteristic
25#16840230C2212yes—
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519742—21no—
28C4×Q8×D61—21no—
29#16839892—21no—
30#16839890—22no—
31C8×Dic122—21no—
32#16839786C2211yescharacteristic
33#16839785C2212yes—
34C2×#14519705C2212yes—
35C2×#14519703C2211yescharacteristic
36C2×#14519703C2211yescharacteristic
37M16×Dic61C2212yes—
38C4⋊C4×Dic61—21no—
39C4×#11758196C2212yes—
40#16839613—22no—
41C42×Dic61C2212yes—
42C42×Dic61C2211yescharacteristic
43#16839523C2211yescharacteristic
44#16839523C2211yescharacteristic
45#16839522C2212yes—
46#16839521C2212yes—
47#16839521C2212yes—
48C61×#533C2211yescharacteristic
49C4×M16×C61C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#339C2212yes—
52C61×#339C2212yes—
53C2×#11758222C2311yescharacteristic
54C22×Dic122D411yescharacteristic
55C2×Q8×D61—21no—
56C2×#11758194C2312yes—
57#14519776—22no—
58#14519776—21no—
59#14519776C2×C412yes—
60#14519776C2×C412yes—
61#14519775—21no—
62#14519775—22no—
63#14519768C2×C412yes—
64#14519768C2×C412yes—
65#14519767—22no—
66#14519767—22no—
67#14519767—22no—
68#14519757—21no—
69#14519757—22no—
70#14519755—22no—
71#14519755—22no—
72#14519755—22no—
73#14519754—22no—
74#14519754—22no—
75#14519753—21no—
76#14519753—22no—
77Q8×Dic61—21no—
78C4×Dic122—22no—
79C4×Dic122—22no—
80C4×Dic122C2×C411yescharacteristic
81C4×Dic122C2×C411yescharacteristic
82C4×Dic122C2×C412yes—
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122D411yescharacteristic
86C4×Dic122D412yes—
87C4×Dic122D411yescharacteristic
88#14519742—28no—
89#14519742—21no—
90#14519742—22no—
91#14519742C2×C412yes—
92#14519742C2×C412yes—
93C4×D244C2×C411yescharacteristic
94C4×D244C2×C411yescharacteristic
95#14519739C2×C412yes—
96#14519739C2×C412yes—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—21no—
99#14519737—22no—
100#14519737—22no—
101#14519736C2311yescharacteristic
102#14519734—21no—
103#14519734—22no—
104C2×C4⋊Dic61D411yescharacteristic
105C2×#11758196—22no—
106C2×#11758196D412yes—
107C4⋊C4×D61—21no—
108C4⋊C4×D61—22no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61—22no—
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2312yes—
116C42×D61—21no—
117C42×D61—22no—
118C4⋊#92418—21no—
119C4⋊#92418—21no—
120#14519706—22no—
121#14519706—22no—
122#14519705—24no—
123#14519703—22no—
124#14519703Q814yescomplemented
125#14519687—22no—
126C8×Dic61—22no—
127#14519684—22no—
128C2×M16×C61C2311yescharacteristic
129C2×M16×C61C2311yescharacteristic
130C4⋊Q8×C61C2×C411yescharacteristic
131C4⋊Q8×C61C2×C411yescharacteristic
132#125×C61C2×C412yes—
133C22⋊Q8×C61—21no—
134C22⋊Q8×C61—21no—
135C4×Q8×C61—21no—
136C4×Q8×C61—21no—
137#117×C61C2312yes—
138C2×C4⋊C4×C61—21no—
139C2×C42×C61C2312yes—
140C2×C42×C61C2311yescharacteristic
141#97×C61—21no—
142C4×C8×C61—21no—
143#95×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61C2×D412yes—
154C22×Dic61C2×D412yes—
155Q8×D61—44no—
156#11758222—21no—
157#11758222—21no—
158#11758222C2×Q812yes—
159#11758222#4112yes—
160C2×D244C22×C411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×C4×D61—21no—
165C2×C4×D61—22no—
166C2×Dic122—24no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122C22⋊C412yes—
170C2×Dic122C22⋊C411yescharacteristic
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—24no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C22×C412yes—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61—22no—
190C42×C61C22×C411yescharacteristic
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C22⋊Dic61—21no—
195#11758198—21no—
196#11758198—21no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61C22⋊C411yescharacteristic
202C4⋊Dic61C22⋊C411yescharacteristic
203C4⋊Dic61C22⋊C412yes—
204C4⋊Dic61C22×C411yescharacteristic
205C4⋊Dic61C22×C411yescharacteristic
206#11758196—28no—
207#11758196—21no—
208#11758196—21no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218#11758196C22⋊C412yes—
219C4×Dic61—28no—
220C4×Dic61—24no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—22no—
226C4×Dic61—22no—
227C4×Dic61—21no—
228C4×Dic61—21no—
229C4×Dic61—44no—
230C4×Dic61—42no—
231C4×Dic61C22×C412yes—
232C4×Dic61C22×C412yes—
233C4×Dic61C2×D412yes—
234C4×Dic61C2×D412yes—
235C4×Dic61C2×Q812yescomplemented
236#11758194—22no—
237#11758194—22no—
238C2×#92418—22no—
239C23×C61C22×D411yescharacteristic
240C22×D61—21no—
241Q8×C61—42no—
242Q8×C61—81no—
243C2×C4×C61—22no—
244C2×C4×C61—24no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C22⋊D61—22no—
261C22⋊D61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—24no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—42no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C412yes—
277D244—21no—
278D244C4×Q812yescomplemented
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—42no—
282C4×D61—44no—
283C4×D61—82no—
284C4×D61—81no—
285Dic122—21no—
286Dic122—22no—
287Dic122—24no—
288Dic122—44no—
289Dic122—82no—
290Dic122#10414yescomplemented
291Dic122C4×Q812yescomplemented
292C8×C61—22no—
293C8×C61—41no—
294#92418—42no—
295Q8×#104—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61C22×C22⋊C411yescharacteristic
299C22×C61#54811yescharacteristic
300C22×C61D4×Q811yescharacteristic
301C2×D61—22no—
302C2×D61—41no—
303C2×D61—41no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—44no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61C2×#10412yes—
311C4×C61C2×C4×Q811yescharacteristic
312C4×C61#51911yescharacteristic
313C4×C61#54811yescharacteristic
314C4×C61D4×Q811yescharacteristic
315Dic61—24no—
316Dic61—22no—
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—41no—
322Dic61—41no—
323Dic61C2×#10414yes—
324Q8×#41—611no—
325Q8×M16—611no—
326C42×Q8—611no—
327#1994—612no—
328#1994—611no—
329#1986—612no—
330#1986D6111yescharacteristic, complemented
331#1806—611no—
332#1806—612no—
333C4×#104—611no—
334C4×#104—612no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61Q8×C22⋊C411yescharacteristic
338C2×C61C22×#10411yescharacteristic
339C2×C61#290111yescharacteristic
340D61—42no—
341Q82—611no—
342#557—611no—
343#557—612no—
344#554—611no—
345#554—612no—
346#554—1222no—
347#554—1221no—
348D4×Q8—611no—
349D4×Q8—1221no—
350#533—612no—
351#533—1222no—
352#533—1221no—
353#533C2×D6111yescharacteristic
354C4×#41—612no—
355C4×#41—611no—
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—611no—
365C2×#104—611no—
366C2×#104—612no—
367C4×M16—612no—
368C4×M16C2×D6111yescharacteristic
369C4×C4⋊C4—612no—
370C4×C4⋊C4—1222no—
371C4×C4⋊C4—1221no—
372C4×C4⋊C4C2×D6111yescharacteristic
373C43—612no—
374C43—611no—
375#339—611no—
376#339—611no—
377#339—612no—
378#339—612no—
379#339—612no—
380#339C2×D6112yes—
381#339C2×D6112yes—
382C61Q8×#10411yescharacteristic, complemented, Sylow
383C2×#41—611no—
384C22×Q8—611no—
385C22×Q8—1221no—
386C2×M16—612no—
387C2×M16C22×D6111yescharacteristic
388C2×M16C22×D6111yescharacteristic
389C4⋊Q8—612no—
390C4⋊Q8—612no—
391C4⋊Q8—1221no—
392C4⋊Q8—1222no—
393C4⋊Q8—1222no—
394C4⋊Q8—1221no—
395C4⋊Q8C4×D6111yescharacteristic
396C4⋊Q8C4×D6111yescharacteristic
397#125—612no—
398#125—612no—
399#125—1222no—
400#125—1222no—
401#125—1222no—
402#125C4×D6112yes—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8—1222no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—611no—
417C4×Q8—611no—
418C4×Q8—612no—
419C4×Q8—611no—
420C4×Q8—612no—
421C4×Q8—611no—
422C4×Q8—612no—
423C4×Q8—612no—
424C4×Q8—611no—
425C4×Q8—611no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—1222no—
429C4×Q8—1221no—
430C4×Q8—1228no—
431C4×Q8—1221no—
432C4×D4—612no—
433C4×D4—612no—
434C4×D4—611no—
435C4×D4—611no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4—1221no—
439C4×D4—1221no—
440#117—611no—
441#117—1222no—
442#117—1221no—
443#117C22×D6112yes—
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—612no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—1222no—
454C2×C42—1222no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42—1222no—
458C2×C42—1221no—
459C2×C42C22×D6112yes—
460C2×C42C22×D6111yescharacteristic
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463C4⋊C8—1221no—
464C4⋊C8—1221no—
465#104—614no—
466#104—1224no—
467#104—1222no—
468#97—21no—
469#97—1222no—
470C4×C8—21no—
471C4×C8—1222no—
472#95—21no—
473#95—1222no—
474#41—612no—
475#41—612no—
476#41—1221no—
477#41—1221no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—611no—
481C2×Q8—612no—
482C2×Q8—611no—
483C2×Q8—611no—
484C2×Q8—1222no—
485C2×Q8—1222no—
486C2×Q8—1224no—
487C2×Q8—2444no—
488C2×D4—611no—
489C2×D4—1221no—
490C22×C4—22no—
491C22×C4—612no—
492C22×C4—612no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—21no—
504M16—21no—
505M16—21no—
506M16—1222no—
507M16—1222no—
508M16Q8×D6112yescomplemented
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—1222no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—612no—
518C4⋊C4—611no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—611no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1228no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4C2×C4×D6112yes—
538C4⋊C4C2×C4×D6112yes—
539C22⋊C4—22no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1221no—
543C22⋊C4—1221no—
544C22⋊C4—1221no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—612no—
549C42—612no—
550C42—612no—
551C42—612no—
552C42—612no—
553C42—1222no—
554C42—1222no—
555C42—1224no—
556C42—1228no—
557C42—1221no—
558C42—1221no—
559C42—1221no—
560C42—1221no—
561C42—1221no—
562C42—1221no—
563C42—2442no—
564C42—2444no—
565C42C2×C4×D6111yescharacteristic
566C42C2×C4×D6111yescharacteristic
567C42D4×D6111yescharacteristic
568C42D4×D6111yescharacteristic
569C23—1221no—
570C23C2×D4×D6111yescharacteristic
571Q8—42no—
572Q8—81no—
573Q8—614no—
574Q8—612no—
575Q8—1224no—
576Q8—1221no—
577Q8—1222no—
578Q8—4882no—
579Q8—2444no—
580D4—612no—
581D4—1222no—
582D4—1222no—
583D4—1221no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—1224no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—4881no—
606C2×C4—4882no—
607C2×C4—2444no—
608C2×C4—2442no—
609C2×C4—2441no—
610C2×C4—2441no—
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×C4C2×Q8×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4#1451982411yescharacteristic
621C2×C4#1451982511yescharacteristic
622C8—22no—
623C8—41no—
624C8—2442no—
625C22—21no—
626C22—21no—
627C22—1222no—
628C22—2441no—
629C22—2441no—
630C22C2×C22⋊C4×D6111yescharacteristic
631C22#1684042011yescharacteristic
632C22#1684042711yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—614no—
640C4—1224no—
641C4—1222no—
642C4—1222no—
643C4—1222no—
644C4—1222no—
645C4—2442no—
646C4—2441no—
647C4—2441no—
648C4#104×D6112yes—
649C4C4×Q8×D6111yescharacteristic
650C4#1684023811yescharacteristic
651C4#1684042211yescharacteristic
652C4#1684042511yescharacteristic
653C2—21no—
654C2—21no—
655C2—2442no—
656C2#2144865311yescharacteristic
657C2C2×#104×D6111yescharacteristic
658C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1986C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1986D61
hypocenterC61Q8×#104
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. M16×Dic122C2semidirect
3. C4×#14519742C2semidirect
4. #21446797C2semidirect
5. #21446794C2semidirect
6. #21446761C2semidirect
11. C61×#1986C2semidirect
12. #16840492C4semidirect
13. #16840489C4semidirect
14. #16840487C4semidirect
15. #16840473C4semidirect
17. #16840468C4semidirect
19. #16840427C4semidirect
124. #14519703Q8semidirect
176. M16×C61C2×Q8semidirect
235. C4×Dic61C2×Q8semidirect
278. D244C4×Q8semidirect
290. Dic122#104semidirect
291. Dic122C4×Q8semidirect
330. #1986D61semidirect
382. C61Q8×#104semidirect
508. M16Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#214593972
M16×Dic1222
C4×#145197422
#214467972
#214467942
#214467612
#214458402
#214458382
C4×#145197032
#214456232
C61×#19862
Q8×#10461