Finite Group Databasebeta

← #29688514

Subgroups — #29688514

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840133C211yesmaximal, characteristic
2C2×#16839863C211yesmaximal, characteristic, complemented
3C2×#16839861C211yesmaximal, characteristic, complemented
4C2×#16839853C211yesmaximal, characteristic, complemented
5C2×#16839847C211yesmaximal, characteristic, complemented
6#21448543C211yesmaximal, characteristic
7#21448537C211yesmaximal, characteristic, complemented
8#21446700C211yesmaximal, characteristic, complemented
9#21446676C211yesmaximal, characteristic, complemented
10#21446672C211yesmaximal, characteristic, complemented
11#21446641C211yesmaximal, characteristic
12#21446637C211yesmaximal, characteristic, complemented
13#21445982C211yesmaximal, characteristic, complemented
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1973C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519773C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25#16840133C418yescomplemented
26#16839863—24no—
27#16839861—24no—
28#16839853—24no—
29#16839847—24no—
30C2×#14519711C2211yescharacteristic
31C2×#14519708C2211yescharacteristic
32C2×#14519708C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×C4×#92418C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44#16839668C2211yescharacteristic
45#16839664C2211yescharacteristic
46#16839649—22no—
47#16839644—22no—
48#16839639—22no—
49#16839638—22no—
50#16839627C2211yescharacteristic
51#16839627C2211yescharacteristic
52#16839608C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—22no—
63C23×Dic61—21no—
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190C2×C414yescomplemented
68C2×#11758200—22no—
69C2×#11758200—22no—
70C2×#11758200C2×C414yescomplemented
71C2×#11758201—22no—
72C2×#11758201—22no—
73C2×#11758201C2×C414yescomplemented
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519775—22no—
77#14519775C2×C414yescomplemented
78#14519773—22no—
79#14519773C2×C414yes—
80Q8×Dic61—22no—
81Q8×Dic61—22no—
82#14519742—22no—
83#14519742—22no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86#14519739—22no—
87#14519739—22no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—22no—
92C2×#11758196—21no—
93C2×C22⋊Dic61—21no—
94C2×#11758198—21no—
95C2×#11758198—22no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519711—22no—
101#14519708—22no—
102#14519708—22no—
103#14519702—22no—
104#14519701—22no—
105#14519700—22no—
106#14519699—22no—
107#14519698—22no—
108#14519697—22no—
109#14519687—22no—
110#14519687D414yescomplemented
111C4×#92418—22no—
112C8×Dic61C2×C414yescomplemented
113C8×Dic61D414yescomplemented
114#14519684—22no—
115#14519684—22no—
116#14519684—22no—
117#14519684D412yes—
118#14519684D412yes—
119C22×Q8×C61C2311yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C2×SD16×C61—22no—
122C2×SD16×C61—22no—
123C2×SD16×C61C2×C414yescomplemented
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—22no—
129C2×C42×C61C2311yescharacteristic
130#107×C61—22no—
131#103×C61—22no—
132#102×C61—22no—
133#97×C61—22no—
134#95×C61—22no—
135C24×C61—21no—
136C23×D61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C412yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—28no—
155C2×C22⋊D61—82no—
156C2×C22⋊D61—82no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—82no—
162C22×Dic61C2×D412yes—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244C22×C412yes—
171C2×C4×D61—42no—
172C2×C4×D61—82no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C22×C412yes—
179SD16×C61—28no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C22×C412yes—
183C2×C8×C61C2×D412yes—
184C2×C8×C61C2×D412yes—
185C2×C8×C61#4112yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C22⋊C4×C61—82no—
191C42×C61—21no—
192C42×C61—21no—
193C22⋊Dic61—42no—
194#11758201—28no—
195#11758200—28no—
196#11758198—42no—
197#11758198—82no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203#11758196—28no—
204#11758196—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61C22×C412yes—
208C4×Dic61C2×D412yes—
209C4×Dic61C2×D412yes—
210C4×Dic61#4112yes—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C22×C412yes—
214C2×#92418C2×D412yes—
215C2×#92418C2×D412yes—
216C2×#92418#4112yes—
217#11758190—28no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—42no—
222C23×C61—42no—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—42no—
228C22×D61—42no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C22⋊D61—88no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—84no—
266C2×Dic61C4×D414yes—
267C2×Dic61#12712yes—
268C2×Dic61#12712yes—
269D244—22no—
270D244—22no—
271D244—41no—
272D244—41no—
273C4×D61—84no—
274Dic122—22no—
275Dic122—22no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—22no—
279C8×C61C4×D414yes—
280#92418—22no—
281#92418C4×D414yes—
282#21695—611nomaximal, Sylow
283C22×C61—42no—
284C22×C61—44no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—82no—
288C22×C61C2×C4×D411yescharacteristic
289C22×C61C2×#12711yescharacteristic
290C22×C61#53211yescharacteristic
291C22×C61#57712yes—
292C22×C61#57712yes—
293C2×D61—42no—
294C2×D61—44no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—82no—
298C4×C61—42no—
299C4×C61—42no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61C2×C4×D411yescharacteristic
303C4×C61C2×C4×D411yescharacteristic
304C4×C61#53211yescharacteristic
305C4×C61#53211yescharacteristic
306Dic61—22no—
307Dic61—22no—
308Dic61—24no—
309Dic61—42no—
310Dic61—81no—
311Dic61—81no—
312C2×#492—611no—
313C2×#440—611no—
314C2×#440—611no—
315C2×C4×SD16—611no—
316C2×C4×SD16—611no—
317#2328—611no—
318#2327—611no—
319#1973—611no—
320#1973D6111yescharacteristic, complemented
321#1968—611no—
322#1968—611no—
323#1961—611no—
324#1960—611no—
325#1846—611no—
326#1726—611no—
327C2×C61—82no—
328C2×C61C4×#12711yescharacteristic
329C2×C61#294912yes—
330C2×C61#315411yescharacteristic
331C2×C61#315411yescharacteristic
332C2×C61#317412yes—
333D61—82no—
334C22×SD16—611no—
335C22×SD16—611no—
336C22×SD16—611no—
337C22×SD16C2×D6111yescharacteristic
338C2×C4⋊Q8—611no—
339C2×#127—611no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344#492—618no—
345#440—1224no—
346#440—1224no—
347C4×SD16—1224no—
348C4×SD16—1224no—
349C2×#107—611no—
350C2×#107C2×D6111yescharacteristic
351C2×#106—611no—
352C2×#106—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103C2×D6111yescharacteristic
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102C2×D6111yescharacteristic
361C2×#97—611no—
362C2×#97—611no—
363C2×#97C2×D6111yescharacteristic
364C2×C4×C8—611no—
365C2×C4×C8—611no—
366C2×C4×C8—611no—
367#391—611no—
368#391C2×D6111yescharacteristic
369#390—611no—
370#390C2×D6111yescharacteristic
371#387—1222no—
372#387—1222no—
373#386—1222no—
374#386—1222no—
375#382—611no—
376#382—611no—
377#376—611no—
378C61#2169511yescharacteristic, complemented, Sylow
379C22×Q8—611no—
380C22×Q8C22×D6111yescharacteristic
381C22×D4—611no—
382C22×D4—1222no—
383C22×D4C22×D6111yescharacteristic
384C23×C4—1221no—
385C23×C4—1221no—
386C2×SD16—22no—
387C2×SD16—22no—
388C2×SD16—614no—
389C2×SD16—614no—
390C2×SD16—614no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C2×SD16—1222no—
396C2×SD16—1222no—
397C2×SD16C4×D6114yescomplemented
398C22×C8—611no—
399C22×C8—611no—
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402C4⋊Q8—614no—
403C4⋊Q8—1222no—
404#127—614no—
405#127—1222no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—22no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42C22×D6111yescharacteristic
430#107—22no—
431#107—1222no—
432#106—1222no—
433#106—1222no—
434#103—22no—
435#103—1222no—
436#103—1222no—
437#103—1222no—
438#102—22no—
439#102—1222no—
440#102—1222no—
441#102—1222no—
442#97—22no—
443#97—614no—
444#97—1222no—
445C4×C8—614no—
446C4×C8—614no—
447C4×C8—1222no—
448#95—22no—
449#95—612no—
450#95—612no—
451#95—1222no—
452#95—1222no—
453#95—1222no—
454C24—21no—
455C24—1221no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—612no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8C2×C4×D6112yes—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—612no—
474C2×D4—1228no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—4882no—
481C2×D4—4882no—
482C2×D4C2×C4×D6112yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—612no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—4882no—
493C22×C4—4882no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4D4×D6112yes—
497C22×C4C23×D6111yescharacteristic
498SD16—28no—
499SD16—1228no—
500SD16—1228no—
501SD16—1228no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8C2×C4×D6112yes—
511C2×C8D4×D6112yes—
512C2×C8D4×D6112yes—
513C2×C8#1175822412yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—1228no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C22⋊C4—82no—
526C22⋊C4—4882no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C42—21no—
530C42—21no—
531C42—612no—
532C42—612no—
533C42—612no—
534C42—612no—
535C42—2442no—
536C42—2442no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—1222no—
543C23—2441no—
544C23—2441no—
545C23—2442no—
546C23—2442no—
547C23C2×D4×D6111yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—1222no—
553Q8—1222no—
554Q8—2441no—
555Q8—2441no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4—1222no—
561D4—1222no—
562D4—4888no—
563D4—2441no—
564D4—2441no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—614no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—4884no—
584C2×C4—4884no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C22×C4×D6111yescharacteristic
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4#41×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C8—22no—
598C8—614no—
599C8—1222no—
600C8#1451973714yes—
601C22—44no—
602C22—42no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22—4882no—
607C22—4881no—
608C22—4881no—
609C22—2444no—
610C22—2442no—
611C22C4×D4×D6111yescharacteristic
612C22#127×D6111yescharacteristic
613C22#1684030711yescharacteristic
614C22#1684057112yes—
615C22#1684057212yes—
616C4—42no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—1224no—
621C4—1222no—
622C4—1222no—
623C4—4881no—
624C4—4881no—
625C4—2442no—
626C4C4×D4×D6111yescharacteristic
627C4C2×#1451973711yescharacteristic
628C4#1684030711yescharacteristic
629C4#1684031111yescharacteristic
630C2—82no—
631C2—4882no—
632C2#2144838311yescharacteristic
633C2#2145467512yes—
634C2#2145629211yescharacteristic
635C2#2145629411yescharacteristic
636C2#2145644212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1973C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
cosocleC22×C4×C61C24
hypercenter#1973D61
hypocenterC61#21695
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. C2×#16839863C2semidirect
3. C2×#16839861C2semidirect
4. C2×#16839853C2semidirect
5. C2×#16839847C2semidirect
7. #21448537C2semidirect
8. #21446700C2semidirect
9. #21446676C2semidirect
10. #21446672C2semidirect
12. #21446637C2semidirect
13. #21445982C2semidirect
14. #21445507C2semidirect
15. C61×#1973C2semidirect
25. #16840133C4semidirect
67. C2×#11758190C2×C4semidirect
70. C2×#11758200C2×C4semidirect
73. C2×#11758201C2×C4semidirect
77. #14519775C2×C4semidirect
110. #14519687D4semidirect
112. C8×Dic61C2×C4semidirect
113. C8×Dic61D4semidirect
123. C2×SD16×C61C2×C4semidirect
320. #1973D61semidirect
378. C61#21695semidirect
397. C2×SD16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401332
C2×#168398632
C2×#168398612
C2×#168398532
C2×#168398472
#214485432
#214485372
#214467002
#214466762
#214466722
#214466412
#214466372
#214459822
#214455072
C61×#19732
#2169561