Finite Group Databasebeta

← #29695659

Subgroups — #29695659

Order 15616.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic
2C2×#16839840C211yesmaximal, characteristic, complemented
3C2×#16839761C212yesmaximal, complemented
4#21446742C211yesmaximal, characteristic, complemented
5#21446620C211yesmaximal, characteristic, complemented
6#21446250C212yesmaximal, complemented
7#21446126C212yesmaximal, complemented
8#21445816C211yesmaximal, characteristic, complemented
9#21445757C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2212yes—
13C22×#11758194C2212yes—
14#16840417—24no—
15#16840417C418yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519759—22no—
18C2×#14519756C2212yes—
19C2×#14519748—21no—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22#16839840—24no—
23C2×C4⋊#92418C2211yescharacteristic
24#16839761—28no—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2212yes—
31C2×#14519697C2211yescharacteristic
32C2×C22⋊#92418C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2212yes—
37C2×C4×D4×C61—21no—
38C2×C4×D4×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2212yes—
40C61×#389—21no—
41C61×#385—22no—
42C61×#384C2211yescharacteristic
43C61×#380C2212yes—
44C4×C4⋊C4×C61C2211yescharacteristic
45C2×#11758222—24no—
46C2×#11758222—24no—
47C2×#11758222—24no—
48C2×#11758222—42no—
49C2×#11758222—42no—
50C2×#11758222C2×C418yescomplemented
51C22×Dic122C2312yes—
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—22no—
54C22×C4×D61—21no—
55C22×C4×D61—22no—
56C2×#11758194—24no—
57C2×#11758194—24no—
58C2×#11758194—42no—
59C22×#92418C2311yescharacteristic
60C22×#92418C2312yes—
61C22×#92418C2311yescharacteristic
62#14519777—22no—
63#14519777C2×C414yescomplemented
64#14519759—216no—
65#14519756—24no—
66#14519756C2×C418yescomplemented
67#14519748—28no—
68C4×D244—22no—
69C4×D244C2×C414yescomplemented
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—22no—
74C4⋊C4×D61—28no—
75C2×C22⋊Dic61—22no—
76C2×#11758198—21no—
77C4⋊#92418—22no—
78#14519700—24no—
79#14519700—24no—
80#14519700—42no—
81#14519700—42no—
82#14519697—24no—
83#14519697—22no—
84#14519697D414yescomplemented
85C22⋊#92418—24no—
86C4×#92418—22no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—22no—
89C23×C4×C61C2312yes—
90C4×D4×C61—44no—
91C4×D4×C61C2×C414yes—
92C4×D4×C61D414yes—
93C2×C4⋊C4×C61—24no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—42no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#95×C61—21no—
107#95×C61—22no—
108C24×C61C2×D412yes—
109C23×D61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61C22⋊C414yes—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—44no—
121C22×C4×C61—44no—
122C22×C4×C61—82no—
123C22×C4×C61C22×C414yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D414yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—28no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134#11758222—216no—
135#11758222—48no—
136#11758222—48no—
137#11758222—48no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244C22×C412yes—
142C2×D244C2×D412yes—
143C2×D244#4112yes—
144C2×C4×D61—28no—
145C2×C4×D61—24no—
146C2×C4×D61—44no—
147C2×C4×D61—44no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122C22×C414yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—44no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—81no—
164C4⋊C4×C61C22⋊C414yes—
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—44no—
167C22⋊C4×C61C22⋊C418yes—
168C42×C61—42no—
169C42×C61C22×C412yes—
170C42×C61C2×D412yes—
171C22⋊Dic61—28no—
172#11758198—24no—
173C4⋊Dic61—24no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61C22×C412yes—
177#11758196—28no—
178#11758194—48no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—22no—
187C2×#92418C2×D412yes—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61C2×C22⋊C414yes—
195C23×C61C22×D411yescharacteristic
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203D4×C61—28no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—88no—
222C2×C4×C61C2×C22⋊C414yes—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61AGL(1,ℤ/8)12yes—
226C2×C4×C61#13712yes—
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C22⋊D61—416no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240D244—22no—
241D244—22no—
242D244—22no—
243D244C4×D414yescomplemented
244C4×D61—416no—
245C4×D61—48no—
246Dic122—24no—
247Dic122—24no—
248Dic122—42no—
249Dic122—42no—
250#92418—22no—
251#92418—42no—
252#92418—41no—
253#22431—611nomaximal, Sylow
254C22×C61—24no—
255C22×C61—24no—
256C22×C61—24no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61D4211yescharacteristic
263C22×C61#54611yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C61C2×#13711yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57811yescharacteristic
268C2×D61—24no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C4×C61—24no—
276C4×C61—22no—
277C4×C61—42no—
278C4×C61—81no—
279C4×C61—82no—
280C4×C61#41714yescomplemented
281C4×C61C2×C4×D411yescharacteristic
282C4×C61#51811yescharacteristic
283C4×C61D4211yescharacteristic
284C4×C61#54611yescharacteristic
285Dic61—44no—
286Dic61—42no—
287C2×#547—611no—
288C2×#436—611no—
289C2×#417—612no—
290D4×C4⋊C4D6111yescharacteristic, complemented
291#1981—611no—
292#1956—611no—
293#1894—612no—
294#1872—612no—
295#1802—611no—
296#1788—612no—
297C4×#102—611no—
298C2×C61—44no—
299C2×C61D4×C22⋊C411yescharacteristic
300C2×C61C2×#41711yescharacteristic
301C2×C61#289811yescharacteristic
302C2×C61#327211yescharacteristic
303C2×C61#327611yescharacteristic
304C2×C61#333011yescharacteristic
305C2×C61#333111yescharacteristic
306D61—44no—
307C22×#41—612no—
308C22×M16—612no—
309#547—618no—
310#547—1224no—
311C2×C4⋊Q8—611no—
312C2×#123—1222no—
313C2×C22⋊Q8—612no—
314C2×#121—1221no—
315C2×C4×D4—21no—
316C2×C4×D4—611no—
317C2×C4×D4C2×D6111yescharacteristic
318C22×C4⋊C4—1221no—
319C22×C4⋊C4C2×D6112yes—
320#436—1224no—
321C2×C4⋊C8—611no—
322#417—1228no—
323C2×#103—612no—
324C2×#103—612no—
325C2×#103—612no—
326C2×#103—612no—
327C2×#102—611no—
328C2×#102—612no—
329C2×#102—611no—
330C2×C22⋊C8—612no—
331C2×C4×C8—611no—
332#389—21no—
333#385—22no—
334#384C2×D6111yescharacteristic
335#380C2×D6112yes—
336C4×C4⋊C4C2×D6111yescharacteristic
337#340—612no—
338#340—611no—
339#340—611no—
340C61#2243111yescharacteristic, complemented, Sylow
341C2×#41—618no—
342C2×#41—1224no—
343C2×#41—1224no—
344C2×#41—1224no—
345C2×#41—2442no—
346C2×#41—2442no—
347C22×Q8—612no—
348C22×D4—611no—
349C22×D4—1222no—
350C22×D4C22⋊D6111yescharacteristic
351C23×C4—22no—
352C23×C4—1221no—
353C23×C4—1222no—
354C23×C4C22×D6112yes—
355C2×M16—1224no—
356C2×M16—1224no—
357C2×M16—2442no—
358C22×C8—611no—
359C22×C8—612no—
360C22×C8—611no—
361C4⋊Q8—614no—
362C4⋊Q8—1222no—
363#123—12216no—
364C22⋊Q8—618no—
365C22⋊Q8—1224no—
366#121—1228no—
367C4×D4—44no—
368C4×D4—614no—
369C4×D4—1222no—
370C4×D4C4×D6114yes—
371C4×D4D24414yes—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—42no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1228no—
382C2×C4⋊C4C22⋊D6111yescharacteristic
383C2×C4⋊C4C22×D6112yes—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4C22⋊D6112yes—
390C2×C42—21no—
391C2×C42C22×D6111yescharacteristic
392C4⋊C8—1222no—
393#103—1224no—
394#103—1224no—
395#103—2442no—
396#103—2442no—
397#102—614no—
398#102—1222no—
399#102—1224no—
400C22⋊C8—1224no—
401C4×C8—1222no—
402#95—21no—
403#95—22no—
404C24—1221no—
405C24C2×C22⋊D6112yes—
406#41—12216no—
407#41—2448no—
408#41—2448no—
409#41—2448no—
410C2×Q8—614no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—612no—
419C2×D4—612no—
420C2×D4—612no—
421C2×D4—1228no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—2444no—
426C2×D4—2444no—
427C2×D4#1175819814yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—1221no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1228no—
442C22×C4—2444no—
443C22×C4—2444no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4C2×C4×D6114yes—
447C22×C4C2×D24414yes—
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6112yes—
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452M16—2448no—
453C2×C8—612no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1222no—
457C2×C8—1221no—
458C2×C8—1222no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1222no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—612no—
471C4⋊C4—1221no—
472C4⋊C4—1228no—
473C4⋊C4—1224no—
474C4⋊C4—1224no—
475C4⋊C4#1175819814yes—
476C4⋊C4D4×D6112yes—
477C22⋊C4—44no—
478C22⋊C4—1228no—
479C22⋊C4—1224no—
480C22⋊C4#1175819818yes—
481C42—42no—
482C42C2×C4×D6112yes—
483C42C2×D24412yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—1222no—
491C23—1222no—
492C23—1222no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23C2×#1175819814yes—
498C23C22×C22⋊D6111yescharacteristic
499Q8—1224no—
500Q8—1224no—
501Q8—2442no—
502Q8—2442no—
503D4—28no—
504D4—614no—
505D4—1222no—
506D4—1222no—
507D4—1222no—
508D4—24416no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—88no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—24416no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2448no—
535C2×C4C2×#1175819812yes—
536C2×C4C2×#1175819814yes—
537C2×C4C2×#1175819812yes—
538C2×C4#1451980312yes—
539C2×C4#1451980812yes—
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×D24411yescharacteristic
543C2×C4#41×D6111yescharacteristic
544C2×C4#1451982211yescharacteristic
545C2×C4#1451982311yescharacteristic
546C8—1222no—
547C8—2442no—
548C8—2441no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—1224no—
557C22—2442no—
558C22—2442no—
559C22—2442no—
560C22—2442no—
561C22—2442no—
562C22—2442no—
563C22C22×#1175819811yescharacteristic
564C22D4×C22⋊D6111yescharacteristic
565C22#1684040011yescharacteristic
566C22C2×#1451980311yescharacteristic
567C22C2×#1451980811yescharacteristic
568C22#1684057311yescharacteristic
569C22#1684057711yescharacteristic
570C4—22no—
571C4—24no—
572C4—42no—
573C4—82no—
574C4—81no—
575C4—2444no—
576C4—2442no—
577C4#1683976114yescomplemented
578C4C4×D4×D6111yescharacteristic
579C4#1684023411yescharacteristic
580C4D4×D24411yescharacteristic
581C4#1684040711yescharacteristic
582C2—44no—
583C2—2444no—
584C2D4×#1175819811yescharacteristic
585C2C2×#1683976111yescharacteristic
586C2#2145434911yescharacteristic
587C2#2145741511yescharacteristic
588C2#2145746511yescharacteristic
589C2#2145811811yescharacteristic
590C2#2145812911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingD4×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840417C2
cosocleC22×C4×C61C24
hypercenterD4×C4⋊C4D61
hypocenterC61#22431
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839840C2semidirect
3. C2×#16839761C2semidirect
4. #21446742C2semidirect
5. #21446620C2semidirect
6. #21446250C2semidirect
7. #21446126C2semidirect
8. #21445816C2semidirect
9. #21445757C2semidirect
10. C4×#14519697C2semidirect
11. D4×C4⋊C4×C61C2semidirect
15. #16840417C4semidirect
50. C2×#11758222C2×C4semidirect
63. #14519777C2×C4semidirect
66. #14519756C2×C4semidirect
69. C4×D244C2×C4semidirect
84. #14519697D4semidirect
243. D244C4×D4semidirect
280. C4×C61#417semidirect
290. D4×C4⋊C4D61semidirect
340. C61#22431semidirect
577. C4#16839761semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168398402
C2×#168397612
#214467422
#214466202
#214462502
#214461262
#214458162
#214457572
C4×#145196972
D4×C4⋊C4×C612
#2243161