Finite Group Databasebeta

← #30620445

Subgroups — #30620445

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#21462621C211yesmaximal, characteristic, complemented
2C4×Q8×#6923C211yesmaximal, characteristic, complemented
3#21462005C212yesmaximal, complemented
4#21462004C211yesmaximal, characteristic, complemented
5#21461996C212yesmaximal, complemented
6#21461994C211yesmaximal, characteristic, complemented
7#21461693C212yesmaximal
8#21461691C211yesmaximal, characteristic
9C4×#14519841C211yesmaximal, characteristic
10C4×#14519841C212yesmaximal
11#21459398C211yesmaximal, characteristic
12C2×Q8×#6923—21no—
13#16840814—22no—
14#16840813—21no—
15#16840811—21no—
16#16840767C2212yes—
17#16840766C2211yescharacteristic
18C2×#14519841C2212yes—
19C2×#14519841C2211yescharacteristic
20C2×#14519841C2211yescharacteristic
21C4×#11758213C2211yescharacteristic
22C4×#11758213C2212yes—
23C4⋊C4×#6923—22no—
24C4⋊C4×#6923—21no—
25C4×C4⋊#6923C2211yescharacteristic
26C4×C4⋊#6923C2212yes—
27C42×#6923C2211yescharacteristic
28C42×#6923C2212yes—
29#16840632C2212yes—
30#16840632C2211yescharacteristic
31#16840632C2212yes—
32#16840632C2211yescharacteristic
33#16840632C2212yes—
34#16840632C2212yes—
35#16840632C2212yes—
36#16840492C411yescharacteristic, complemented
37#16840490C411yescharacteristic, complemented
38#16840489C412yescomplemented
39#16840471C412yescomplemented
40#16840467—21no—
41#16840466—22no—
42D4×Dic122C411yescharacteristic, complemented
43#16840425—21no—
44#16840318C2212yes—
45#16840317C2211yescharacteristic
46#16840314—21no—
47#16840314—22no—
48C4×#11758224C2211yescharacteristic
49C4×#11758224C2212yes—
50C2×#14519742—21no—
51C4×Q8×D61C2211yescharacteristic
52C61×#554C411yescharacteristic, complemented
53C2×#11758213C2312yes—
54C2×#11758213C2311yescharacteristic
55C2×#11758213C2311yescharacteristic
56Q8×#6923—28no—
57C2×C4⋊#6923—22no—
58C2×C4⋊#6923—21no—
59C2×C4×#6923—22no—
60C2×C4×#6923—21no—
61C2×C4×#6923—21no—
62C2×C4×#6923—22no—
63C2×C4×#6923C2312yes—
64C2×C4×#6923C2311yescharacteristic
65C2×C4×#6923C2311yescharacteristic
66#14519844—22no—
67#14519844—22no—
68C4⋊#92420—21no—
69C4⋊#92420—21no—
70#14519841—24no—
71#14519841—22no—
72#14519841Q814yescomplemented
73#14519832—22no—
74#14519832—21no—
75C4×#92420—21no—
76C4×#92420—22no—
77(C2×#6923)⋊C4—21no—
78(C2×#6923)⋊C4—22no—
79C2×#11758224C2311yescharacteristic
80C22×Dic122—21no—
81C2×Q8×D61D411yescharacteristic
82#14519777C2×C411yescharacteristic
83#14519776—22no—
84#14519776—21no—
85#14519776C2×C412yes—
86#14519776C2×C412yes—
87#14519775—21no—
88#14519775—22no—
89#14519769C2×C412yes—
90#14519768C2×C412yes—
91#14519767—22no—
92#14519767—22no—
93#14519767—22no—
94#14519757—21no—
95#14519756—21no—
96#14519756—22no—
97#14519755—22no—
98#14519755—22no—
99#14519754—22no—
100#14519754—22no—
101#14519754—22no—
102#14519753—21no—
103Q8×Dic61—21no—
104Q8×Dic61D411yescharacteristic
105C4×Dic122C2×C412yes—
106C4×Dic122C2×C411yescharacteristic
107C4×Dic122C2×C412yes—
108C4×Dic122C2×C411yescharacteristic
109#14519742—21no—
110#14519742—21no—
111#14519742—22no—
112#14519742—21no—
113#14519742—22no—
114#14519742—22no—
115#14519742C2×C411yescharacteristic
116#14519742C2×C412yes—
117#14519742D411yescharacteristic
118#14519742D412yes—
119D4×Dic61C2×C411yescharacteristic
120D4×Dic61C2×C412yes—
121C4×C22⋊D61—22no—
122C4×C22⋊D61—21no—
123#14519737—21no—
124#14519737—22no—
125#14519735C2312yes—
126#14519735C2311yescharacteristic
127#14519733—21no—
128#14519733—22no—
129C2×#11758196—22no—
130C4⋊C4×D61D412yes—
131C4⋊C4×D61D411yescharacteristic
132C2×C4×Dic61—22no—
133C2×C4×Dic61—21no—
134C42×D61C2311yescharacteristic
135C42×D61C2312yes—
136C4⋊Q8×C61C2×C411yescharacteristic
137#125×C61C2×C412yes—
138C22⋊Q8×C61—21no—
139C22⋊Q8×C61—22no—
140C4×Q8×C61C2×C411yescharacteristic
141C4×D4×C61C2×C412yes—
142C4×D4×C61C2×C411yescharacteristic
143C2×C4⋊C4×C61—21no—
144C22×#6923—21no—
145C22×#6923—22no—
146C22×#6923—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C22⋊C411yescharacteristic
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—22no—
152C2×C22⋊D61—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61—21no—
156Q8×D61—24no—
157#11758224—21no—
158#11758224—21no—
159#11758224C2×Q812yes—
160#11758224#4112yes—
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D412yes—
165C2×C4×D61C2411yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122—44no—
170C2×Dic122C22⋊C412yes—
171C2×Dic122C22⋊C411yescharacteristic
172C2×Dic122C22×C411yescharacteristic
173C2×#92420—21no—
174C2×#92420—22no—
175C2×#92420—21no—
176C4⋊#6923—24no—
177C4⋊#6923—28no—
178C4×#6923—24no—
179C4×#6923—28no—
180C4×#6923—22no—
181C4×#6923—21no—
182C4×#6923—21no—
183C4×#6923—22no—
184C4×#6923—21no—
185C4×#6923—44no—
186C4×#6923—42no—
187C4×#6923C2×Q812yescomplemented
188#11758213—21no—
189#11758213—22no—
190#11758213—21no—
191#11758213—22no—
192#11758213—21no—
193#11758213C2×Q812yescomplemented
194C4⋊C4×C61—22no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61C22⋊C411yescharacteristic
198C4⋊C4×C61C22⋊C412yes—
199C4⋊C4×C61C22×C411yescharacteristic
200C4⋊C4×C61C22×C412yes—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—21no—
203C42×C61C22×C412yes—
204C42×C61C22×C411yescharacteristic
205C22⋊Dic61—21no—
206C22⋊Dic61—22no—
207#11758198—22no—
208#11758198—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—22no—
211C4⋊Dic61C22⋊C412yes—
212C4⋊Dic61C22⋊C411yescharacteristic
213C4⋊Dic61C22×C412yes—
214C4⋊Dic61C22×C411yescharacteristic
215#11758196—22no—
216#11758196—21no—
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220#11758196—22no—
221#11758196—22no—
222#11758196—21no—
223#11758196—22no—
224#11758196C22⋊C412yes—
225#11758196C22⋊C412yes—
226#11758196C22⋊C412yes—
227C4×Dic61—22no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61—22no—
231C4×Dic61—21no—
232C4×Dic61—22no—
233C4×Dic61C22×C411yescharacteristic
234C4×Dic61C22×C412yes—
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D412yes—
237C4×Dic61C2×D411yescharacteristic
238C4×Dic61C2×D412yes—
239C23×C61—21no—
240C22×D61C22×D411yescharacteristic
241C2×#6923—22no—
242C2×#6923—22no—
243C2×#6923—22no—
244C2×#6923—22no—
245C2×#6923—24no—
246C2×#6923—42no—
247C2×#6923—41no—
248C2×#6923—42no—
249C2×#6923—41no—
250Q8×C61—22no—
251Q8×C61—81no—
252D4×C61—21no—
253D4×C61C4×Q812yescomplemented
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—42no—
257C2×C4×C61—82no—
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C2×C22⋊C412yes—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C22⋊D61—22no—
264C22⋊D61—22no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—44no—
271C2×Dic61—81no—
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C22×D411yescharacteristic
277C4×D61—22no—
278C4×D61—24no—
279C4×D61—42no—
280C4×D61—41no—
281C4×D61C22×Q811yescharacteristic
282C4×D61C2×#4111yescharacteristic
283C4×D612−1+411yescharacteristic
284C4×D612−1+411yescharacteristic
285Dic122—21no—
286Dic122—24no—
287Dic122—42no—
288Dic122—44no—
289Dic122—82no—
290Dic122#10414yescomplemented
291Dic122C4×Q812yescomplemented
292#92420—22no—
293#92420—42no—
294#92420—41no—
295Q8×#104—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61C22×C22⋊C411yescharacteristic
300C2×D61—21no—
301C2×D61—21no—
302C2×D61#54811yescharacteristic
303C2×D61D4×Q811yescharacteristic
304#6923—24no—
305#6923—42no—
306#6923—44no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—21no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61C2×#10412yes—
313C4×C61C2×C4×Q811yescharacteristic
314C4×C61#51911yescharacteristic
315Dic61—22no—
316Dic61—21no—
317Dic61—22no—
318Dic61—21no—
319Dic61—42no—
320Dic61—41no—
321Dic61C2×#10414yes—
322Dic61#54811yescharacteristic
323Dic61D4×Q811yescharacteristic
324Q8×#41—611no—
325Q8×M16—611no—
326C42×Q8—611no—
327#1994—611no—
328#1994—612no—
329#1986—611no—
330#1986—612no—
331#1806—611no—
332#1806—612no—
333C4×#104—612no—
334C4×#104—611no—
335C2×C61—42no—
336C2×C61Q8×C22⋊C411yescharacteristic
337C2×C61C22×#10411yescharacteristic
338C2×C61#290111yescharacteristic
339D61—21no—
340D61—21no—
341Q82—611no—
342#557—612no—
343#557—611no—
344#554—612no—
345#554—1221no—
346#554—1222no—
347#554#692311yescharacteristic, complemented
348D4×Q8—611no—
349D4×Q8—1221no—
350#533—611no—
351#533—612no—
352#533—1221no—
353#533—1222no—
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—611no—
363#423—612no—
364C2×#104—611no—
365C2×#104—612no—
366C2×#104—611no—
367C4×M16—611no—
368C4×M16—612no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—612no—
371C4×C4⋊C4—1222no—
372C4×C4⋊C4—1221no—
373C43—611no—
374C43—612no—
375#339—612no—
376#339—612no—
377#339—612no—
378#339—612no—
379#339—611no—
380#339—612no—
381#339—611no—
382C61Q8×#10411yescharacteristic, complemented, Sylow
383C2×#41—611no—
384C22×Q8—611no—
385C22×Q8—1221no—
386C2×M16—611no—
387C2×M16—612no—
388C2×M16—611no—
389C4⋊Q8—612no—
390C4⋊Q8—612no—
391C4⋊Q8—611no—
392C4⋊Q8—1221no—
393C4⋊Q8—1222no—
394C4⋊Q8—1222no—
395C4⋊Q8—1221no—
396C4⋊Q8C2×#692311yescharacteristic
397#125—612no—
398#125—612no—
399#125—1222no—
400#125—1222no—
401#125—1222no—
402#125C2×#692312yes—
403C22⋊Q8—21no—
404C22⋊Q8—22no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1221no—
410C22⋊Q8—1222no—
411C22⋊Q8—1221no—
412C22⋊Q8—1222no—
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—611no—
423C4×Q8—1222no—
424C4×Q8—1221no—
425C4×Q8—1222no—
426C4×Q8—1221no—
427C4×Q8—1222no—
428C4×Q8—1221no—
429C4×Q8—1228no—
430C4×Q8—1221no—
431C4×Q8C2×#692311yescharacteristic
432C4×D4—612no—
433C4×D4—611no—
434C4×D4—1222no—
435C4×D4—1221no—
436C4×D4—1221no—
437C4×D4—1222no—
438C4×D4C2×#692311yescharacteristic
439C4×D4C2×#692312yes—
440#117—612no—
441#117—611no—
442#117—1222no—
443#117—1221no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1221no—
450C2×C42—611no—
451C2×C42—612no—
452C2×C42—611no—
453C2×C42—612no—
454C2×C42—611no—
455C2×C42—1221no—
456C2×C42—1222no—
457C2×C42—1222no—
458C2×C42—1221no—
459C2×C42—1221no—
460C2×C42—1222no—
461C4⋊C8—1221no—
462C4⋊C8—1221no—
463C4⋊C8—1222no—
464C4⋊C8—1222no—
465#104—614no—
466#104—1224no—
467#104—1222no—
468#97—1222no—
469#97—1221no—
470C4×C8—1221no—
471C4×C8—1222no—
472#95—1221no—
473#95—1222no—
474#41—612no—
475#41—612no—
476#41—1221no—
477#41—1221no—
478C2×Q8—21no—
479C2×Q8—611no—
480C2×Q8—612no—
481C2×Q8—611no—
482C2×Q8—1222no—
483C2×Q8—1222no—
484C2×Q8—1224no—
485C2×Q8—1221no—
486C2×Q8—2444no—
487C2×Q8C22⋊#692311yescharacteristic
488C2×D4—1221no—
489C2×D4C22×#692311yescharacteristic
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—611no—
493C22×C4—612no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—612no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1222no—
500C22×C4—1221no—
501C22×C4—1222no—
502C22×C4—1221no—
503M16—612no—
504M16—1221no—
505M16—1222no—
506M16—1221no—
507M16—1222no—
508M16—1221no—
509C2×C8—1221no—
510C2×C8—1222no—
511C2×C8—1221no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—611no—
520C4⋊C4—612no—
521C4⋊C4—611no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1228no—
529C4⋊C4—1224no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1221no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4C22⋊#692311yescharacteristic
536C4⋊C4C22⋊#692312yes—
537C4⋊C4C22×#692312yes—
538C4⋊C4C22×#692311yescharacteristic
539C22⋊C4—21no—
540C22⋊C4—22no—
541C22⋊C4—1221no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1221no—
545C42—612no—
546C42—611no—
547C42—612no—
548C42—611no—
549C42—611no—
550C42—612no—
551C42—612no—
552C42—1221no—
553C42—1222no—
554C42—1221no—
555C42—1222no—
556C42—1221no—
557C42—1222no—
558C42—1221no—
559C42—1222no—
560C42—1221no—
561C42—1228no—
562C42—1224no—
563C42—1222no—
564C42—1221no—
565C42—2444no—
566C42—2442no—
567C42C22×#692312yes—
568C42C22×#692311yescharacteristic
569C23—21no—
570C23—611no—
571Q8—22no—
572Q8—81no—
573Q8—614no—
574Q8—612no—
575Q8—1224no—
576Q8—1221no—
577Q8—4882no—
578Q8—2444no—
579Q8—2442no—
580D4—21no—
581D4—1222no—
582D4—1222no—
583D4Q8×#692312yescomplemented
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—42no—
587C2×C4—82no—
588C2×C4—612no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—611no—
594C2×C4—611no—
595C2×C4—612no—
596C2×C4—611no—
597C2×C4—1224no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1221no—
604C2×C4—1222no—
605C2×C4—1224no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1221no—
609C2×C4—4881no—
610C2×C4—2442no—
611C2×C4—2441no—
612C2×C4—2444no—
613C2×C4—2442no—
614C2×C4—2441no—
615C2×C4—2442no—
616C2×C4—2441no—
617C2×C4C2×C22⋊#692312yes—
618C2×C4C2×C22⋊#692312yes—
619C2×C4C2×C22⋊#692311yescharacteristic
620C2×C4C2×C22⋊#692311yescharacteristic
621C2×C4C23×#692311yescharacteristic
622C8—1222no—
623C8—2442no—
624C8—2441no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—611no—
629C22—611no—
630C22—1221no—
631C22—1221no—
632C22C22×C22⋊#692311yescharacteristic
633C4—22no—
634C4—22no—
635C4—21no—
636C4—41no—
637C4—42no—
638C4—611no—
639C4—611no—
640C4—614no—
641C4—1224no—
642C4—1222no—
643C4—1221no—
644C4—1221no—
645C4—1222no—
646C4—2444no—
647C4—2442no—
648C4—2442no—
649C4—2441no—
650C4D61⋊#10412yes—
651C4C2×Q8×#692311yescharacteristic
652C4#1684083011yescharacteristic
653C2—42no—
654C2—1221no—
655C2—1221no—
656C2Q8×C22⋊#692311yescharacteristic
657C2C2×D61⋊#10411yescharacteristic
658C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#554C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758224C23
cosocleC2×C4×D61C24
hypercenter#554#6923
hypocenterC61Q8×#104
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #21462621C2semidirect
2. C4×Q8×#6923C2semidirect
3. #21462005C2semidirect
4. #21462004C2semidirect
5. #21461996C2semidirect
6. #21461994C2semidirect
36. #16840492C4semidirect
37. #16840490C4semidirect
38. #16840489C4semidirect
39. #16840471C4semidirect
42. D4×Dic122C4semidirect
52. C61×#554C4semidirect
72. #14519841Q8semidirect
187. C4×#6923C2×Q8semidirect
193. #11758213C2×Q8semidirect
253. D4×C61C4×Q8semidirect
290. Dic122#104semidirect
291. Dic122C4×Q8semidirect
347. #554#6923semidirect
382. C61Q8×#104semidirect
583. D4Q8×#6923semidirect

Maximal subgroups

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

subgroupindex
#214626212
C4×Q8×#69232
#214620052
#214620042
#214619962
#214619942
#214616932
#214616912
C4×#145198412
#214593982
Q8×#10461