Finite Group Databasebeta

← #29688596

Subgroups — #29688596

Order 15616.

1062 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840154C211yesmaximal, characteristic
2C2×#16839866C211yesmaximal, characteristic, complemented
3C2×#16839865C211yesmaximal, characteristic
4C2×#16839854C211yesmaximal, characteristic, complemented
5C2×#16839847C211yesmaximal, characteristic
6#21448539C211yesmaximal, characteristic, complemented
7#21448527C211yesmaximal, characteristic
8#21446697C211yesmaximal, characteristic
9#21446687C211yesmaximal, characteristic, complemented
10#21446677C211yesmaximal, characteristic, complemented
11#21446670C211yesmaximal, characteristic
12#21446630C211yesmaximal, characteristic
13#21445985C211yesmaximal, characteristic, complemented
14#21445505C211yesmaximal, characteristic, complemented
15C61×#1956C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C22×#11758201C2211yescharacteristic
20C2×#14519765C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×#14519742C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25#16840154C418yescomplemented
26#16839866—24no—
27#16839865—24no—
28#16839854—24no—
29#16839847—24no—
30C2×#14519710C2211yescharacteristic
31C2×#14519710C2211yescharacteristic
32C2×#14519708C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519688C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41C2×C8×Dic61C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43#16839669C2211yescharacteristic
44#16839669C2211yescharacteristic
45#16839667C2211yescharacteristic
46#16839663C2211yescharacteristic
47#16839652—22no—
48#16839648—22no—
49#16839648—22no—
50#16839633—22no—
51#16839628C2211yescharacteristic
52#16839608C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C61×#384C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202C2×C414yescomplemented
68C2×#11758202C2×C414yescomplemented
69C2×#11758201—22no—
70C2×#11758201—22no—
71C2×#11758201—22no—
72C2×#11758201—22no—
73C2×#11758201C2×C414yes—
74C2×#11758201C2×C414yes—
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77#14519765—22no—
78#14519765C2×C414yes—
79Q8×Dic61—22no—
80Q8×Dic61—22no—
81Q8×Dic61—22no—
82Q8×Dic61—22no—
83#14519742—22no—
84#14519742—22no—
85#14519739—22no—
86#14519739—22no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758198—22no—
93C2×#11758198—21no—
94C2×#11758198—22no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519710—22no—
101#14519710—22no—
102#14519708—22no—
103#14519708—22no—
104#14519702—22no—
105#14519702—22no—
106#14519699—22no—
107#14519699—22no—
108#14519688C2×C414yescomplemented
109#14519687—22no—
110#14519687D414yes—
111C8×Dic61—22no—
112C8×Dic61D414yes—
113#14519684—22no—
114#14519684—22no—
115#14519684—22no—
116#14519684—22no—
117#14519684D412yes—
118#14519684D412yes—
119C22×Q8×C61C2311yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4⋊Q8×C61—22no—
124C4⋊Q8×C61C2×C414yescomplemented
125C2×C4⋊C4×C61—22no—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130#103×C61—22no—
131#103×C61—22no—
132#103×C61—22no—
133#103×C61—22no—
134C4×C8×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—82no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D412yes—
179C2×C8×C61C2×D412yes—
180C4⋊C4×C61—28no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—82no—
186C42×C61C22×C412yes—
187C42×C61#4112yes—
188#11758202—28no—
189#11758202—28no—
190#11758201—28no—
191#11758201—28no—
192#11758198—28no—
193#11758198—42no—
194#11758198—82no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201#11758196—42no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61C2×D412yes—
208C4×Dic61C2×D412yes—
209C2×#92418C22×C412yes—
210C2×#92418C22×C412yes—
211C2×#92418C2×D412yes—
212C2×#92418C2×D412yes—
213C2×#92418#4112yes—
214C2×#92418#4112yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—42no—
220C22×D61—42no—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61C4×D414yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×Dic61—22no—
249C2×Dic61—22no—
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×Dic61#12712yes—
261C2×Dic61#12712yes—
262D244—22no—
263D244—22no—
264D244—41no—
265D244—41no—
266C4×D61—84no—
267Dic122—22no—
268Dic122—22no—
269Dic122—41no—
270Dic122—41no—
271C8×C61—22no—
272C8×C61—22no—
273#92418C4×D414yes—
274#92418C4×D414yes—
275#21703—611nomaximal, Sylow
276C22×C61C2×C4×D411yescharacteristic
277C22×C61C2×#12711yescharacteristic
278C22×C61#53211yescharacteristic
279C22×C61#57712yes—
280C22×C61#57812yes—
281C2×D61—42no—
282C2×D61—44no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—82no—
286C4×C61—24no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×C4×D411yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293C4×C61#53211yescharacteristic
294C4×C61#53211yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—42no—
298Dic61—42no—
299Dic61—81no—
300Dic61—81no—
301C2×#497—611no—
302C2×#441—611no—
303C2×#440—611no—
304C2×C4×Q16—611no—
305C2×C4×SD16—611no—
306#2327—611no—
307#2326—611no—
308#1973—611no—
309#1971—611no—
310#1969—611no—
311#1968—611no—
312#1958—611no—
313#1956D6111yescharacteristic, complemented
314#1847—611no—
315#1726—611no—
316C2×C61C4×#12711yescharacteristic
317C2×C61#294712yes—
318C2×C61#315411yescharacteristic
319C2×C61#315611yescharacteristic
320C2×C61#317812yes—
321D61—82no—
322C22×Q16—611no—
323C22×Q16—611no—
324C22×SD16—611no—
325C22×SD16—611no—
326C2×C4⋊Q8C2×D6111yescharacteristic
327C2×#124—611no—
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330C2×C4×Q8—611no—
331C2×C4×D4—611no—
332#497—618no—
333#441—1224no—
334#440—1224no—
335C4×Q16—1224no—
336C4×SD16—1224no—
337C2×#107—611no—
338C2×#107—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103C2×D6111yescharacteristic
344C2×#103C2×D6111yescharacteristic
345C2×#103C2×D6111yescharacteristic
346C2×#103C2×D6111yescharacteristic
347C2×#102—611no—
348C2×#102—611no—
349C2×#97—611no—
350C2×#97—611no—
351C2×#97—611no—
352C2×C4×C8—611no—
353C2×C4×C8—611no—
354C2×C4×C8C2×D6111yescharacteristic
355#391—611no—
356#391—611no—
357#391—611no—
358#390—611no—
359#388—1222no—
360#387—1222no—
361#387—1222no—
362#385—1222no—
363#384C2×D6111yescharacteristic
364#382—611no—
365#376—611no—
366C61#2170311yescharacteristic, complemented, Sylow
367C22×Q8—611no—
368C22×Q8C22×D6111yescharacteristic
369C22×Q8C22×D6111yescharacteristic
370C22×D4—611no—
371C23×C4—1221no—
372C2×Q16—614no—
373C2×Q16—614no—
374C2×Q16—1222no—
375C2×Q16—1222no—
376C2×Q16—1222no—
377C2×Q16—1222no—
378C2×SD16—614no—
379C2×SD16—614no—
380C2×SD16—1222no—
381C2×SD16—1222no—
382C2×SD16—1222no—
383C2×SD16—1222no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4⋊Q8—22no—
389C4⋊Q8C4×D6114yescomplemented
390#124—614no—
391#124—1222no—
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1221no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418#107—1222no—
419#107—1222no—
420#106—1222no—
421#106—1222no—
422#103—22no—
423#103—22no—
424#103—22no—
425#103—22no—
426#103—1222no—
427#103—1222no—
428#102—1222no—
429#102—1222no—
430#97—614no—
431#97—614no—
432#97—1222no—
433C4×C8—22no—
434C4×C8—614no—
435C4×C8—1222no—
436#95—612no—
437#95—612no—
438#95—1222no—
439#95—1222no—
440#95—1222no—
441#95—1222no—
442C24—1221no—
443C2×Q8—21no—
444C2×Q8—21no—
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—612no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8C2×C4×D6112yes—
460C2×Q8C2×C4×D6112yes—
461C2×D4—612no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—612no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—4882no—
478C22×C4—2442no—
479C22×C4D4×D6112yes—
480C22×C4C23×D6111yescharacteristic
481Q16—1228no—
482Q16—1228no—
483SD16—1228no—
484SD16—1228no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—612no—
490C2×C8—612no—
491C2×C8—612no—
492C2×C8—612no—
493C2×C8—612no—
494C2×C8—612no—
495C2×C8D4×D6112yes—
496C2×C8D4×D6112yes—
497C4⋊C4—28no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—82no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C22⋊C4—1228no—
511C22⋊C4—4882no—
512C22⋊C4—2442no—
513C42—612no—
514C42—612no—
515C42—1221no—
516C42—1221no—
517C42—2442no—
518C42—2442no—
519C42—2442no—
520C42C2×C4×D6112yes—
521C42#1175822412yes—
522C23—1222no—
523C23—2441no—
524C23—2441no—
525C23—2442no—
526C23—2442no—
527C23C2×D4×D6111yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—1222no—
537Q8—1222no—
538Q8—2441no—
539Q8—2441no—
540D4—1222no—
541D4—1222no—
542D4—2441no—
543D4—2441no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—4884no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4#1451973714yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C8—22no—
579C8—22no—
580C8—614no—
581C8—614no—
582C22—4882no—
583C22—4881no—
584C22—4881no—
585C22—2444no—
586C22—2442no—
587C22C2×#1451973711yescharacteristic
588C22#127×D6111yescharacteristic
589C22#1684031111yescharacteristic
590C22#1684056812yes—
591C22#1684057512yes—
592C4—24no—
593C4—42no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—1222no—
598C4—1222no—
599C4—4881no—
600C4—4881no—
601C4—2442no—
602C4—2442no—
603C4C4×D4×D6111yescharacteristic
604C4C4×D4×D6111yescharacteristic
605C4#1684030711yescharacteristic
606C4#1684030711yescharacteristic
607C2—4882no—
608C2#2144838311yescharacteristic
609C2#2145465612yes—
610C2#2145628811yescharacteristic
611C2#2145630711yescharacteristic
612C2#2145649412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1956D61
hypocenterC61#21703
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#16839866C2semidirect
4. C2×#16839854C2semidirect
6. #21448539C2semidirect
9. #21446687C2semidirect
10. #21446677C2semidirect
13. #21445985C2semidirect
14. #21445505C2semidirect
15. C61×#1956C2semidirect
25. #16840154C4semidirect
67. C2×#11758202C2×C4semidirect
68. C2×#11758202C2×C4semidirect
108. #14519688C2×C4semidirect
124. C4⋊Q8×C61C2×C4semidirect
313. #1956D61semidirect
366. C61#21703semidirect
389. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401542
C2×#168398662
C2×#168398652
C2×#168398542
C2×#168398472
#214485392
#214485272
#214466972
#214466872
#214466772
#214466702
#214466302
#214459852
#214455052
C61×#19562
#2170361