Finite Group Databasebeta

← #29688561

Subgroups — #29688561

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840155C211yesmaximal, characteristic
2C2×Q16×Dic61C211yesmaximal, characteristic, complemented
3C2×#16839854C211yesmaximal, characteristic, complemented
4C2×#16839849C211yesmaximal, characteristic
5C2×#16839847C211yesmaximal, characteristic
6#21448388C211yesmaximal, characteristic, complemented
7#21448362C211yesmaximal, characteristic
8#21446703C211yesmaximal, characteristic
9#21446700C211yesmaximal, characteristic
10#21446689C211yesmaximal, characteristic, complemented
11#21446637C211yesmaximal, characteristic, complemented
12#21446632C211yesmaximal, characteristic
13#21445984C211yesmaximal, characteristic, complemented
14#97×Dic61C211yesmaximal, characteristic, complemented
15C61×#1971C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519766C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×#14519742C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25#16840155C418yescomplemented
26Q16×Dic61—24no—
27#16839854—24no—
28#16839849—24no—
29#16839847—24no—
30C2×#14519711C2211yescharacteristic
31C2×#14519710C2211yescharacteristic
32C2×#14519708C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519688C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×#14519687C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44#16839668C2211yescharacteristic
45#16839664C2211yescharacteristic
46#16839659—22no—
47#16839654—22no—
48#16839631C2211yescharacteristic
49#16839627C2211yescharacteristic
50#16839613—22no—
51#16839609—22no—
52C42×Dic61C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202C2×C414yescomplemented
66C2×#11758190—22no—
67C2×#11758190—22no—
68C2×#11758190C2×C414yes—
69C2×#11758201—22no—
70C2×#11758201—22no—
71C2×#11758201C2×C414yes—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519775—22no—
75#14519775C2×C414yescomplemented
76#14519766—22no—
77#14519766C2×C414yes—
78Q8×Dic61—22no—
79Q8×Dic61—22no—
80Q8×Dic61—22no—
81Q8×Dic61—22no—
82#14519742—22no—
83#14519742—22no—
84#14519739—22no—
85#14519739—22no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—22no—
91C2×#11758196—21no—
92C2×#11758196—22no—
93C2×#11758198—22no—
94C2×#11758198—22no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61D412yes—
100C2×C4×Dic61D412yes—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519711—22no—
104#14519710—22no—
105#14519708—22no—
106#14519702—22no—
107#14519701—22no—
108#14519701—22no—
109#14519700—22no—
110#14519699—22no—
111#14519697—22no—
112#14519688—22no—
113#14519687—22no—
114#14519687C2×C414yescomplemented
115C8×Dic61D414yes—
116C8×Dic61D414yes—
117#14519684—22no—
118#14519684—22no—
119C22×Q8×C61C2311yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C2×Q16×C61—22no—
122C2×Q16×C61—22no—
123C2×Q16×C61C2×C414yescomplemented
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#106×C61—22no—
130#103×C61—22no—
131#103×C61—22no—
132#97×C61—22no—
133#95×C61—22no—
134#95×C61—22no—
135C23×D61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C22×C412yes—
147C2×Q8×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61C2×D412yes—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244C22×C412yes—
166C2×C4×D61—42no—
167C2×C4×D61—44no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174Q16×C61—28no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C22×C412yes—
178C2×C8×C61C2×D412yes—
179C2×C8×C61C2×D412yes—
180C2×C8×C61#4112yes—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187#11758202—28no—
188#11758201—28no—
189#11758198—28no—
190#11758198—44no—
191#11758198—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—42no—
197C4⋊Dic61—42no—
198#11758196—28no—
199#11758196—44no—
200#11758196—42no—
201C4×Dic61—24no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61C22×C412yes—
207C4×Dic61C2×D412yes—
208C4×Dic61C2×D412yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C22×C412yes—
213C2×#92418C2×D412yes—
214C2×#92418C2×D412yes—
215C2×#92418#4112yes—
216#11758190—28no—
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—42no—
222C22×D61—42no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61C4×D414yes—
261C2×Dic61#12712yes—
262C2×Dic61#12712yes—
263C2×Dic61#13714yes—
264D244—22no—
265D244—22no—
266D244—41no—
267D244—41no—
268C4×D61—84no—
269Dic122—22no—
270Dic122—22no—
271Dic122—41no—
272Dic122—41no—
273C8×C61—22no—
274C8×C61C4×D414yes—
275#92418—22no—
276#92418C4×D414yes—
277C4×#497—611nomaximal, Sylow
278C22×C61C2×C4×D411yescharacteristic
279C22×C61C2×#12711yescharacteristic
280C22×C61#53211yescharacteristic
281C22×C61C2×#13712yes—
282C22×C61#57512yes—
283C2×D61—42no—
284C2×D61—44no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—82no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61#53211yescharacteristic
296C4×C61#53211yescharacteristic
297Dic61—22no—
298Dic61—24no—
299Dic61—42no—
300Dic61—81no—
301Dic61—81no—
302Dic61#49714yescomplemented
303C2×#497—611no—
304C2×C4×Q16—611no—
305C2×C4×Q16—611no—
306C2×C4×SD16—611no—
307C2×C4×SD16—611no—
308C4×C4⋊Q8—611no—
309C4×#124—611no—
310#1973—611no—
311#1973—611no—
312#1971—611no—
313#1971D6111yescharacteristic, complemented
314#1960—611no—
315#1959—611no—
316#1847—611no—
317C4×#97—611no—
318C2×C61C4×#12711yescharacteristic
319C2×C61C4×#13712yes—
320C2×C61C2×#49711yescharacteristic
321C2×C61#315011yescharacteristic
322C2×C61#317212yes—
323D61—82no—
324C22×Q16—611no—
325C22×Q16C2×D6111yescharacteristic
326C22×SD16—611no—
327C22×SD16—611no—
328C2×C4⋊Q8—611no—
329C2×#124—611no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334#497—618no—
335C4×Q16—1224no—
336C4×Q16—1224no—
337C4×SD16—1224no—
338C4×SD16—1224no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#106—611no—
342C2×#106C2×D6111yescharacteristic
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103C2×D6111yescharacteristic
348C2×#103C2×D6111yescharacteristic
349C2×#102—611no—
350C2×#102—611no—
351C2×#97—611no—
352C2×#97—611no—
353C2×#97—611no—
354C2×#97C2×D6111yescharacteristic
355C2×C4×C8—611no—
356C2×C4×C8—611no—
357#391—611no—
358#391C2×D6111yescharacteristic
359#391C2×D6111yescharacteristic
360#390—611no—
361#389—1222no—
362#388—1222no—
363#384—611no—
364#382—611no—
365C4×C4⋊C4—1222no—
366C4×C22⋊C4—1222no—
367C43—611no—
368C61C4×#49711yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8C22×D6111yescharacteristic
371C22×Q8C22×D6111yescharacteristic
372C22×D4—611no—
373C23×C4—1221no—
374C2×Q16—22no—
375C2×Q16—22no—
376C2×Q16—614no—
377C2×Q16—1222no—
378C2×Q16—1222no—
379C2×Q16C4×D6114yescomplemented
380C2×SD16—614no—
381C2×SD16—614no—
382C2×SD16—1222no—
383C2×SD16—1222no—
384C2×SD16—1222no—
385C2×SD16—1222no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390C4⋊Q8—614no—
391C4⋊Q8—1222no—
392#124—614no—
393#124—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C22⋊C4—1222no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1222no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—612no—
417C2×C42—612no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42C22×D6111yescharacteristic
422#107—1222no—
423#107—1222no—
424#106—22no—
425#106—1222no—
426#103—22no—
427#103—22no—
428#103—1222no—
429#103—1222no—
430#103—1222no—
431#103—1222no—
432#102—1222no—
433#102—1222no—
434#97—22no—
435#97—614no—
436#97—1222no—
437#97—1222no—
438C4×C8—614no—
439C4×C8—614no—
440#95—22no—
441#95—22no—
442#95—1222no—
443#95—1222no—
444C24—1221no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—612no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8C2×C4×D6112yes—
462C2×Q8C2×C4×D6112yes—
463C2×D4—612no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—611no—
474C22×C4—612no—
475C22×C4—611no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—2442no—
480C22×C4—2444no—
481C22×C4D4×D6112yes—
482C22×C4C23×D6111yescharacteristic
483Q16—28no—
484Q16—1228no—
485SD16—1228no—
486SD16—1228no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—612no—
490C2×C8—612no—
491C2×C8—612no—
492C2×C8—612no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8C2×C4×D6112yes—
496C2×C8D4×D6112yes—
497C2×C8D4×D6112yes—
498C2×C8#1175822412yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—1228no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4—2444no—
512C22⋊C4—1228no—
513C22⋊C4—2444no—
514C22⋊C4—2442no—
515C42—21no—
516C42—21no—
517C42—612no—
518C42—612no—
519C42—612no—
520C42—612no—
521C42—1224no—
522C42—2442no—
523C42—2442no—
524C42—2442no—
525C42—2442no—
526C23—1222no—
527C23—2442no—
528C23—2441no—
529C23—2442no—
530C23—2441no—
531C23C2×D4×D6111yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—1222no—
541Q8—1222no—
542Q8—2441no—
543Q8—2441no—
544D4—1222no—
545D4—1222no—
546D4—2441no—
547D4—2441no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—614no—
562C2×C4—612no—
563C2×C4—614no—
564C2×C4—612no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—4884no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4#41×D6111yescharacteristic
581C2×C4C2×#1175822411yescharacteristic
582C8—22no—
583C8—614no—
584C8—1222no—
585C8#1451973714yes—
586C22—4882no—
587C22—4881no—
588C22—4881no—
589C22—2444no—
590C22—2442no—
591C22C4×D4×D6111yescharacteristic
592C22#127×D6111yescharacteristic
593C22#1684030711yescharacteristic
594C22#137×D6112yes—
595C22#1684056112yes—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—614no—
602C4—1224no—
603C4—1222no—
604C4—4881no—
605C4—4881no—
606C4—2442no—
607C4C4×D4×D6111yescharacteristic
608C4C2×#1451973711yescharacteristic
609C4#1684030711yescharacteristic
610C4#1684031111yescharacteristic
611C2—4882no—
612C2#2144838311yescharacteristic
613C2#2145463512yes—
614C2#497×D6111yescharacteristic
615C2#2145626311yescharacteristic
616C2#2145642012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1971C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1971D61
hypocenterC61C4×#497
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1971

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic61C2semidirect
3. C2×#16839854C2semidirect
6. #21448388C2semidirect
10. #21446689C2semidirect
11. #21446637C2semidirect
13. #21445984C2semidirect
14. #97×Dic61C2semidirect
15. C61×#1971C2semidirect
25. #16840155C4semidirect
65. C2×#11758202C2×C4semidirect
75. #14519775C2×C4semidirect
114. #14519687C2×C4semidirect
123. C2×Q16×C61C2×C4semidirect
302. Dic61#497semidirect
313. #1971D61semidirect
368. C61C4×#497semidirect
379. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401552
C2×Q16×Dic612
C2×#168398542
C2×#168398492
C2×#168398472
#214483882
#214483622
#214467032
#214467002
#214466892
#214466372
#214466322
#214459842
#97×Dic612
C61×#19712
C4×#49761