Finite Group Databasebeta

← #29688600

Subgroups — #29688600

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×#16839869C211yesmaximal, characteristic, complemented
2C2×#16839863C211yesmaximal, characteristic
3C2×Q16×Dic61C211yesmaximal, characteristic, complemented
4C2×SD16×Dic61C211yesmaximal, characteristic
5#21448539C211yesmaximal, characteristic, complemented
6#21448532C211yesmaximal, characteristic
7#21446702C211yesmaximal, characteristic
8#21446691C211yesmaximal, characteristic, complemented
9#21446681C211yesmaximal, characteristic, complemented
10#21446675C211yesmaximal, characteristic
11#21446631C211yesmaximal, characteristic
12#21446622C211yesmaximal, characteristic, complemented
13#21445987C211yesmaximal, characteristic, complemented
14#21445507C211yesmaximal, characteristic, complemented
15C2×C61×#497C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16839869—24no—
21#16839863—24no—
22Q16×Dic61—24no—
23SD16×Dic61—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519701C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37C2×#14519687C2211yescharacteristic
38C2×C4×#92418C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40C2×C8×Dic61C2211yescharacteristic
41#16839669C2211yescharacteristic
42#16839669C2211yescharacteristic
43#16839669C2211yescharacteristic
44#16839665C2211yescharacteristic
45#16839655—22no—
46#16839648—22no—
47#16839648—22no—
48#16839637—22no—
49#16839632C2211yescharacteristic
50#16839628C2211yescharacteristic
51#16839608C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C22×SD16×C61C2211yescharacteristic
56C2×C4⋊Q8×C61C2211yescharacteristic
57C2×#124×C61C2211yescharacteristic
58C61×#497C418yescomplemented
59C2×#97×C61C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519711—22no—
88#14519709—22no—
89#14519709—22no—
90#14519701—22no—
91#14519701—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701—22no—
95#14519701—22no—
96#14519698—22no—
97#14519698—22no—
98#14519687—22no—
99#14519687D414yes—
100C4×#92418—22no—
101C8×Dic61—22no—
102C8×Dic61D414yes—
103#14519684—22no—
104#14519684—22no—
105#14519684—22no—
106#14519684—22no—
107#14519684D412yes—
108#14519684D412yes—
109C22×Q8×C61C2311yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C22×D4×C61C2311yescharacteristic
113C2×Q16×C61—22no—
114C2×Q16×C61—22no—
115C2×Q16×C61—22no—
116C2×Q16×C61—22no—
117C2×Q16×C61C2×C414yescomplemented
118C2×Q16×C61C2×C414yescomplemented
119C2×SD16×C61—22no—
120C2×SD16×C61—22no—
121C2×SD16×C61—22no—
122C2×SD16×C61—22no—
123C2×SD16×C61C2×C414yes—
124C2×SD16×C61C2×C414yes—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C4⋊Q8×C61—22no—
128C4⋊Q8×C61C2×C414yescomplemented
129#124×C61—22no—
130#124×C61C2×C414yes—
131C2×C4⋊C4×C61—22no—
132C2×C22⋊C4×C61—22no—
133C2×C42×C61C2311yescharacteristic
134#97×C61C2×C414yescomplemented
135C24×C61—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×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61C22×C412yes—
152C2×Q8×C61C22×C412yes—
153C2×Q8×C61C22×C412yes—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61C22×C412yes—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—42no—
170C22×Dic61—82no—
171C22×Dic61C2×D412yes—
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174Q16×C61—28no—
175Q16×C61—28no—
176SD16×C61—28no—
177SD16×C61—28no—
178C2×C8×C61C22×C412yes—
179C2×C8×C61C22×C412yes—
180C2×C8×C61C2×D412yes—
181C2×C8×C61C2×D412yes—
182C2×C8×C61#4112yes—
183C2×C8×C61#4112yes—
184C4⋊C4×C61—28no—
185C22⋊C4×C61—28no—
186C42×C61C22×C412yes—
187C42×C61#4112yes—
188C22⋊Dic61—42no—
189C22⋊Dic61—82no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—42no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61—82no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61C2×D412yes—
208C4×Dic61C2×D412yes—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D412yes—
214C2×#92418C2×D412yes—
215C23×C61—22no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61C22×D411yescharacteristic
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
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—22no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61C4×D414yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—84no—
269C2×Dic61#12712yes—
270C2×Dic61#12712yes—
271C8×C61C4×D414yes—
272C8×C61C4×D414yes—
273#92418—22no—
274#92418—22no—
275#21703—611nomaximal, Sylow
276C22×C61—42no—
277C22×C61—44no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—82no—
281C22×C61C2×C4×D411yescharacteristic
282C22×C61C2×#12711yescharacteristic
283C22×C61#53211yescharacteristic
284C22×C61#57712yes—
285C22×C61#57812yes—
286C4×C61—24no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61C2×C4×D411yescharacteristic
291C4×C61C2×C4×D411yescharacteristic
292C4×C61#53211yescharacteristic
293C4×C61#53211yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—42no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300Dic61—81no—
301C2×#497D6111yescharacteristic
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#1956—611no—
314#1847—611no—
315#1726—611no—
316C2×C61—82no—
317C2×C61C4×#12711yescharacteristic
318C2×C61#294712yes—
319C2×C61#315411yescharacteristic
320C2×C61#315611yescharacteristic
321C2×C61#317812yes—
322C22×Q16C2×D6111yescharacteristic
323C22×Q16C2×D6111yescharacteristic
324C22×SD16C2×D6111yescharacteristic
325C22×SD16C2×D6111yescharacteristic
326C2×C4⋊Q8C2×D6111yescharacteristic
327C2×#124C2×D6111yescharacteristic
328C2×C4×Q8—611no—
329C2×C4×Q8—611no—
330C2×C4×Q8—611no—
331C2×C4×D4—611no—
332#497Dic6118yescomplemented
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×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#97—611no—
350C2×#97—611no—
351C2×#97C2×D6111yescharacteristic
352C2×C4×C8—611no—
353C2×C4×C8—611no—
354C2×C4×C8—611no—
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#384—611no—
364#382—611no—
365#376—611no—
366C61#2170311yescharacteristic, complemented, Sylow
367C22×Q8C22×D6111yescharacteristic
368C22×Q8C22×D6111yescharacteristic
369C22×Q8C22×D6111yescharacteristic
370C22×D4C22×D6111yescharacteristic
371C23×C4—1221no—
372C2×Q16—22no—
373C2×Q16—22no—
374C2×Q16—22no—
375C2×Q16—22no—
376C2×Q16C2×Dic6114yescomplemented
377C2×Q16C2×Dic6114yescomplemented
378C2×SD16—22no—
379C2×SD16—22no—
380C2×SD16—22no—
381C2×SD16—22no—
382C2×SD16C2×Dic6114yes—
383C2×SD16C2×Dic6114yes—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4⋊Q8—22no—
389C4⋊Q8C2×Dic6114yescomplemented
390#124—22no—
391#124C2×Dic6114yes—
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—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C22⋊C4—22no—
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—1222no—
423#103—1222no—
424#103—1222no—
425#103—1222no—
426#103—1222no—
427#103—1222no—
428#102—1222no—
429#102—1222no—
430#97—614no—
431#97—1222no—
432#97C2×Dic6114yescomplemented
433C4×C8—614no—
434C4×C8—1222no—
435C4×C8—1222no—
436#95—612no—
437#95—612no—
438#95—1222no—
439#95—1222no—
440#95—1222no—
441#95—1222no—
442C24—21no—
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—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8C22×Dic6112yes—
459C2×Q8C22×Dic6112yes—
460C2×Q8C22×Dic6112yes—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C22×Dic6112yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—612no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—4882no—
478C22×C4—2442no—
479C22×C4D4×D6112yes—
480C22×C4C23×D6111yescharacteristic
481Q16—28no—
482Q16—28no—
483SD16—28no—
484SD16—28no—
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8D4×D6112yes—
492C2×C8D4×D6112yes—
493C2×C8#1175822412yes—
494C2×C8#1175822412yes—
495C2×C8C22×Dic6112yes—
496C2×C8C22×Dic6112yes—
497C4⋊C4—28no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—4882no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C22⋊C4—28no—
511C22⋊C4—4882no—
512C22⋊C4—2442no—
513C42—612no—
514C42—612no—
515C42—1221no—
516C42—1221no—
517C42—2442no—
518C42—2442no—
519C42—2442no—
520C42#1175822412yes—
521C42C22×Dic6112yes—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23C2×D4×D6111yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—41no—
543D4—41no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—4884no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4D4×Dic6114yes—
572C2×C4C23×Dic6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4C2×#1175822411yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C8—1222no—
579C8—1222no—
580C8D4×Dic6114yes—
581C8D4×Dic6114yes—
582C22—44no—
583C22—42no—
584C22—82no—
585C22—81no—
586C22—81no—
587C22C2×D4×Dic6111yescharacteristic
588C22#127×D6111yescharacteristic
589C22#1684031011yescharacteristic
590C22#1684057112yes—
591C22#1684057812yes—
592C4—24no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—1222no—
597C4—1222no—
598C4—4881no—
599C4—4881no—
600C4—4881no—
601C4—4881no—
602C4—2442no—
603C4C2×D4×Dic6111yescharacteristic
604C4C2×D4×Dic6111yescharacteristic
605C4#1684031011yescharacteristic
606C4#1684031011yescharacteristic
607C2—82no—
608C2#127×Dic6111yescharacteristic
609C2#2145466312yes—
610C2#2145629411yescharacteristic
611C2#2145631111yescharacteristic
612C2#2145648412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#497C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#497D61
hypocenterC61#21703
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
1. C2×#16839869C2semidirect
3. C2×Q16×Dic61C2semidirect
5. #21448539C2semidirect
8. #21446691C2semidirect
9. #21446681C2semidirect
12. #21446622C2semidirect
13. #21445987C2semidirect
14. #21445507C2semidirect
58. C61×#497C4semidirect
117. C2×Q16×C61C2×C4semidirect
118. C2×Q16×C61C2×C4semidirect
128. C4⋊Q8×C61C2×C4semidirect
134. #97×C61C2×C4semidirect
332. #497Dic61semidirect
366. C61#21703semidirect
376. C2×Q16C2×Dic61semidirect
377. C2×Q16C2×Dic61semidirect
389. C4⋊Q8C2×Dic61semidirect
432. #97C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398692
C2×#168398632
C2×Q16×Dic612
C2×SD16×Dic612
#214485392
#214485322
#214467022
#214466912
#214466812
#214466752
#214466312
#214466222
#214459872
#214455072
C2×C61×#4972
#2170361