Finite Group Databasebeta

← #29688599

Subgroups — #29688599

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×#16840156C211yesmaximal, characteristic
2C2×#16839870C211yesmaximal, characteristic, complemented
3C2×#16839864C211yesmaximal, characteristic
4C2×C4×Dic244C211yesmaximal, characteristic, complemented
5C2×C4×#11758190C211yesmaximal, characteristic
6#21448538C211yesmaximal, characteristic, complemented
7#21448530C211yesmaximal, characteristic
8#21446700C211yesmaximal, characteristic
9#21446690C211yesmaximal, characteristic, complemented
10#21446680C211yesmaximal, characteristic, complemented
11#21446673C211yesmaximal, characteristic
12#21446630C211yesmaximal, characteristic
13#21446621C211yesmaximal, characteristic, complemented
14#21445987C211yesmaximal, characteristic, complemented
15C61×#1726C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×#11758190C2211yescharacteristic
19C22×#11758190C2211yescharacteristic
20C2×#14519777C2211yescharacteristic
21C2×#14519765C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×Dic122C2211yescharacteristic
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26#16840156C418yescomplemented
27#16839870—24no—
28#16839864—24no—
29C4×Dic244—24no—
30C4×#11758190—24no—
31C2×#14519711C2211yescharacteristic
32C2×#14519711C2211yescharacteristic
33C2×#14519709C2211yescharacteristic
34C2×#14519709C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519702C2211yescharacteristic
39C2×#14519702C2211yescharacteristic
40C2×#14519702C2211yescharacteristic
41C2×#14519699C2211yescharacteristic
42C2×#14519699C2211yescharacteristic
43#16839668C2211yescharacteristic
44#16839668C2211yescharacteristic
45#16839668C2211yescharacteristic
46#16839664C2211yescharacteristic
47#16839654—22no—
48#16839647—22no—
49#16839647—22no—
50#16839636—22no—
51#16839632C2211yescharacteristic
52#16839628C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C61×#376C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×Dic244—22no—
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×Dic244—22no—
69C2×Dic244C2×C414yescomplemented
70C2×Dic244C2×C414yescomplemented
71C2×#11758190—22no—
72C2×#11758190—22no—
73C2×#11758190—22no—
74C2×#11758190—22no—
75C2×#11758190C2×C414yes—
76C2×#11758190C2×C414yes—
77#14519777—22no—
78#14519777C2×C414yescomplemented
79#14519765—22no—
80#14519765C2×C414yes—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85C4×Dic122—22no—
86C4×Dic122—22no—
87C4×D244—22no—
88C4×D244—22no—
89C2×C4⋊Dic61—22no—
90C2×C4⋊Dic61—22no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758198—22no—
99C2×#11758198—21no—
100C2×#11758198—22no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104#14519711—22no—
105#14519711—22no—
106#14519709—22no—
107#14519709—22no—
108#14519702—22no—
109#14519702—22no—
110#14519702—22no—
111#14519702—22no—
112#14519702—22no—
113#14519702—22no—
114#14519699—22no—
115#14519699—22no—
116#14519684—22no—
117#14519684—22no—
118#14519684—22no—
119#14519684—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#97×C61—22no—
128#97×C61C2×C414yescomplemented
129#97×C61D414yes—
130C4×C8×C61—22no—
131C4×C8×C61—22no—
132C4×C8×C61D414yes—
133#95×C61D412yes—
134#95×C61D412yes—
135C23×D61—21no—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×C4×D61—42no—
155C2×C4×D61—82no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122C22×C412yes—
172C2×Dic122C22×C412yes—
173C2×Dic122C22×C412yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C22×C412yes—
179C2×C8×C61C22×C412yes—
180C2×C8×C61C2×D412yes—
181C2×C8×C61C2×D412yes—
182C2×C8×C61C2×D412yes—
183C2×C8×C61C2×D412yes—
184C2×C8×C61#4112yes—
185C2×C8×C61#4112yes—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C22×C412yes—
189C42×C61C2×D412yes—
190C42×C61C2×D412yes—
191C42×C61#4112yes—
192#11758198—28no—
193#11758198—42no—
194#11758198—82no—
195C4⋊Dic61—28no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—82no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C4×Dic61—42no—
211Dic244—28no—
212Dic244—28no—
213#11758190—28no—
214#11758190—28no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—42no—
220C22×D61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61C4×D414yes—
229C2×C4×C61#12712yes—
230C2×C4×C61#12712yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254D244—22no—
255D244—22no—
256D244—41no—
257D244—41no—
258C4×D61—84no—
259Dic122—22no—
260Dic122—22no—
261Dic122—22no—
262Dic122—22no—
263Dic122—22no—
264Dic122—22no—
265Dic122—41no—
266Dic122—41no—
267Dic122—41no—
268Dic122—41no—
269Dic122—41no—
270Dic122—41no—
271C8×C61—22no—
272C8×C61—22no—
273C8×C61C4×D414yes—
274C8×C61C4×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—22no—
287C4×C61—22no—
288C4×C61—24no—
289C4×C61—42no—
290C4×C61C2×C4×D411yescharacteristic
291C4×C61C2×C4×D411yescharacteristic
292C4×C61#53211yescharacteristic
293C4×C61#53211yescharacteristic
294Dic61—42no—
295Dic61—42no—
296Dic61—42no—
297Dic61—81no—
298Dic61—81no—
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#1956—611no—
314#1847—611no—
315#1726D6111yescharacteristic, complemented
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⋊Q8—611no—
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×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#97C2×D6111yescharacteristic
350C2×#97C2×D6111yescharacteristic
351C2×#97C2×D6111yescharacteristic
352C2×C4×C8C2×D6111yescharacteristic
353C2×C4×C8C2×D6111yescharacteristic
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#384—611no—
364#382—611no—
365#376C2×D6111yescharacteristic
366C61#2170311yescharacteristic, complemented, Sylow
367C22×Q8—611no—
368C22×Q8—611no—
369C22×Q8—611no—
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×C8C22×D6111yescharacteristic
385C22×C8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4⋊Q8—614no—
389C4⋊Q8—1222no—
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—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1221no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42C22×D6111yescharacteristic
416C2×C42C22×D6111yescharacteristic
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—22no—
431#97C4×D6114yescomplemented
432#97D24414yes—
433C4×C8—22no—
434C4×C8—22no—
435C4×C8D24414yes—
436#95—1222no—
437#95—1222no—
438#95—1222no—
439#95—1222no—
440#95D24412yes—
441#95D24412yes—
442C24—1221no—
443C2×Q8—612no—
444C2×Q8—612no—
445C2×Q8—612no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×D4—612no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—4882no—
475C22×C4—2442no—
476C22×C4C2×D24412yes—
477C22×C4C2×D24412yes—
478C22×C4C2×D24411yescharacteristic
479C22×C4C2×D24411yescharacteristic
480C22×C4C23×D6111yescharacteristic
481Q16—1228no—
482Q16—1228no—
483SD16—1228no—
484SD16—1228no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8C2×C4×D6112yes—
490C2×C8C2×C4×D6112yes—
491C2×C8C2×D24412yes—
492C2×C8C2×D24412yes—
493C2×C8C2×D24412yes—
494C2×C8C2×D24412yes—
495C2×C8#1175822212yes—
496C2×C8#1175822212yes—
497C4⋊C4—1228no—
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—1228no—
511C22⋊C4—4882no—
512C22⋊C4—2442no—
513C42—21no—
514C42—21no—
515C42—2442no—
516C42—2442no—
517C42—2442no—
518C42C2×C4×D6112yes—
519C42C2×D24412yes—
520C42C2×D24412yes—
521C42#1175822212yes—
522C23—1222no—
523C23—2441no—
524C23—2441no—
525C23—2442no—
526C23—2442no—
527C23C22×D24411yescharacteristic
528Q8—1222no—
529Q8—1222no—
530Q8—1222no—
531Q8—1222no—
532Q8—1222no—
533Q8—1222no—
534Q8—2441no—
535Q8—2441no—
536Q8—2441no—
537Q8—2441no—
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—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—4884no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4C4×D24414yes—
570C2×C4#1451977412yes—
571C2×C4#1451977412yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4C22×D24411yescharacteristic
574C2×C4C22×D24411yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C2×C4C2×#1175822211yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C8—22no—
579C8—22no—
580C8C4×D24414yes—
581C8C4×D24414yes—
582C22—4882no—
583C22—4881no—
584C22—4881no—
585C22—2444no—
586C22—2442no—
587C22C2×C4×D24411yescharacteristic
588C22C2×#1451977411yescharacteristic
589C22#1684031211yescharacteristic
590C22#1684057412yes—
591C22#1684057912yes—
592C4—24no—
593C4—22no—
594C4—22no—
595C4—42no—
596C4—4881no—
597C4—4881no—
598C4—4881no—
599C4—4881no—
600C4—2442no—
601C4—2442no—
602C4—2442no—
603C4C2×C4×D24411yescharacteristic
604C4C2×C4×D24411yescharacteristic
605C4#1684031211yescharacteristic
606C4#1684031211yescharacteristic
607C2—4882no—
608C2C4×#1451977411yescharacteristic
609C2#2145466412yes—
610C2#2145629511yescharacteristic
611C2#2145631311yescharacteristic
612C2#2145648712yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1726C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1726D61
hypocenterC61#21703
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

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

NH ≅ G/Ntype
2. C2×#16839870C2semidirect
4. C2×C4×Dic244C2semidirect
6. #21448538C2semidirect
9. #21446690C2semidirect
10. #21446680C2semidirect
13. #21446621C2semidirect
14. #21445987C2semidirect
15. C61×#1726C2semidirect
26. #16840156C4semidirect
69. C2×Dic244C2×C4semidirect
70. C2×Dic244C2×C4semidirect
78. #14519777C2×C4semidirect
128. #97×C61C2×C4semidirect
315. #1726D61semidirect
366. C61#21703semidirect
431. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401562
C2×#168398702
C2×#168398642
C2×C4×Dic2442
C2×C4×#117581902
#214485382
#214485302
#214467002
#214466902
#214466802
#214466732
#214466302
#214466212
#214459872
C61×#17262
#2170361