Finite Group Databasebeta

← #29648610

Subgroups — #29648610

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3#21454447C211yesmaximal, characteristic, complemented
4#21444538C214yesmaximal, complemented
5#21444521C214yesmaximal, complemented
6#21444492C211yesmaximal, characteristic, complemented
7#21444490C211yesmaximal, characteristic, complemented
8#21444487C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C22×#11758224—21no—
11#16840310—21no—
12#16840310—21no—
13#16840310—21no—
14#16840310—21no—
15#16840310C412yes—
16#16840310C412yes—
17#16840310C2212yes—
18#16840310C2212yes—
19C2×#14519777C2211yescharacteristic
20C2×#14519764—21no—
21C2×#14519748—21no—
22C2×#14519748C412yes—
23C2×D4×Dic61—21no—
24C2×D4×Dic61C412yes—
25C2×C42×D61C2211yescharacteristic
26#16839818—21no—
27#16839818C2212yescomplemented
28#16839818C2212yescomplemented
29#16839816—21no—
30#16839815C2211yescharacteristic
31#16839814—21no—
32#16839814C2212yescomplemented
33#16839813—21no—
34#16839786C2211yescharacteristic
35#16839784C2212yescomplemented
36#16839725—21no—
37#16839724—21no—
38M16×Dic61—21no—
39C4×#11758189—21no—
40#16839509C2214yescomplemented
41#16839507C2214yescomplemented
42#16839506C2214yescomplemented
43C2×#127×C61C2211yescharacteristic
44C61×#431C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C61×#331C2214yescomplemented
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224—44no—
52C2×#11758224—44no—
53C22×Dic122—21no—
54C22×C22⋊D61—22no—
55C23×Dic61—22no—
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C2×#11758194—21no—
60C2×#11758189—22no—
61#14519777—21no—
62#14519777—21no—
63#14519777C2×C412yes—
64#14519777C2312yes—
65#14519764—22no—
66#14519764—22no—
67#14519764—22no—
68#14519764—22no—
69#14519748—22no—
70#14519748—22no—
71#14519748—22no—
72#14519748—22no—
73#14519748—22no—
74#14519748—22no—
75#14519748C2×C414yes—
76#14519748C2×C414yes—
77D4×Dic61—22no—
78D4×Dic61—22no—
79D4×Dic61—22no—
80D4×Dic61—22no—
81D4×Dic61—22no—
82D4×Dic61—22no—
83D4×Dic61C2×C414yes—
84D4×Dic61C2×C414yes—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2×C412yes—
87C2×C22⋊Dic61—22no—
88C2×C22⋊Dic61—22no—
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C42×D61C2×C411yescharacteristic
93C42×D61C2×C411yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61C2311yescharacteristic
99C42×D61C2311yescharacteristic
100C4⋊#92418—21no—
101C4⋊#92418C2312yes—
102C4⋊#92418C2312yes—
103#14519706—22no—
104C22⋊#92418—21no—
105#14519689—22no—
106#14519689—22no—
107#14519689—22no—
108#14519687—22no—
109#14519687—22no—
110C4×#92418—21no—
111C8×Dic61—22no—
112C22×D4×C61—21no—
113C22×D4×C61—21no—
114C22×D4×C61C2×C412yes—
115C2×M16×C61—21no—
116#127×C61—21no—
117#127×C61—21no—
118#127×C61C2×C412yes—
119#127×C61C2312yes—
120C2×C42×C61C2311yescharacteristic
121C4⋊C8×C61—21no—
122C4⋊C8×C61C2312yes—
123C4⋊C8×C61C2312yes—
124C22⋊C8×C61—21no—
125C4×C8×C61—21no—
126C24×C61—22no—
127C23×D61C22⋊C411yescharacteristic
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—44no—
134C2×D4×C61—44no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61C22×C414yes—
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—48no—
144C22×Dic61—22no—
145C22×Dic61—24no—
146C22×Dic61—24no—
147C22×Dic61—48no—
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151#11758224—44no—
152#11758224—44no—
153#11758224—44no—
154#11758224—44no—
155#11758224—48no—
156#11758224—48no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22×C411yescharacteristic
168C2×C4×D61C22×C411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177M16×C61—42no—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C42×C61C22×C411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—24no—
185C22⋊Dic61—24no—
186C22⋊Dic61—24no—
187C22⋊Dic61—24no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C22×C414yes—
192C4×Dic61C22⋊C411yescharacteristic
193C4×Dic61C22⋊C411yescharacteristic
194C4×Dic61C22⋊C411yescharacteristic
195C4×Dic61C22⋊C411yescharacteristic
196C4×Dic61C22⋊C411yescharacteristic
197C4×Dic61C22⋊C411yescharacteristic
198C4×Dic61C22×C411yescharacteristic
199C4×Dic61C22×C411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204#11758194—42no—
205C2×#92418—22no—
206C2×#92418—22no—
207#11758189—44no—
208C23×C61—24no—
209C23×C61—44no—
210C23×C61—44no—
211C23×C61—44no—
212C23×C61C2×C22⋊C411yescharacteristic
213C22×D61—22no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61C2×C22⋊C411yescharacteristic
217C22×D61C2×C22⋊C411yescharacteristic
218D4×C61—44no—
219D4×C61—44no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—44no—
223D4×C61—44no—
224D4×C61—44no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61AGL(1,ℤ/8)12yes—
231C2×C4×C61AGL(1,ℤ/8)12yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C22⋊D61—48no—
236C22⋊D61—48no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—48no—
242C2×Dic61—48no—
243C2×Dic61—44no—
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264Dic122—42no—
265Dic122—42no—
266Dic122—44no—
267C8×C61—42no—
268C8×C61—42no—
269#92418—42no—
270#92418—42no—
271#17743—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—44no—
277C22×C61—88no—
278C22×C61#41812yes—
279C22×C61C22×C22⋊C411yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—21no—
286C2×D61—44no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61#41312yes—
290C4×C61#41812yes—
291C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C4×C61#57512yes—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—21no—
299Dic61—21no—
300Dic61—44no—
301C2×#532—611no—
302#2920—611no—
303#2920—611no—
304#1557—614no—
305#1553—614no—
306#1546—611no—
307#1546—611no—
308#1545—611no—
309#1545D6111yescharacteristic, complemented
310C2×C61—22no—
311C2×C61—84no—
312C2×C61C2×#41311yescharacteristic
313C2×C61C2×#41811yescharacteristic
314C2×C61#289611yescharacteristic
315D61—41no—
316D61—41no—
317C22×#41—1221no—
318#532—612no—
319#532—612no—
320#532—612no—
321#532—612no—
322#532—1221no—
323#532—1221no—
324#532—1221no—
325#532—1221no—
326C2×C4⋊Q8—611no—
327C2×#127C2×D6111yescharacteristic
328C2×#124—1221no—
329C2×#121—612no—
330C2×#121—1221no—
331C2×C4×D4—612no—
332C2×C4×D4—1221no—
333C22×C42—611no—
334#432—612no—
335#432—612no—
336#432—1221no—
337#432—1221no—
338#431—611no—
339#431—612no—
340#431—1221no—
341#431—1221no—
342#431C2×D6111yescharacteristic
343#423—611no—
344#423—612no—
345#423C2×D6111yescharacteristic
346#407—1221no—
347#407—1221no—
348C4×M16—1221no—
349C4×M16—1221no—
350#332—614no—
351#332—614no—
352#331—614no—
353#331C2×D6114yescomplemented
354C61#1774311yescharacteristic, complemented, Sylow
355C2×#41—1222no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1222no—
359C2×#41—2444no—
360C2×#41—2444no—
361C22×Q8—1221no—
362C22×D4—21no—
363C22×D4—21no—
364C22×D4—1222no—
365C22×D4C4×D6112yes—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—1222no—
370C2×M16—21no—
371C2×M16—1222no—
372C2×M16—1221no—
373C4⋊Q8—612no—
374C4⋊Q8—612no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377#127—21no—
378#127—21no—
379#127C4×D6112yes—
380#127C22×D6112yes—
381#124—1222no—
382#124—1222no—
383#124—1222no—
384#124—1222no—
385#121—614no—
386#121—614no—
387#121—1222no—
388#121—1222no—
389#121—1222no—
390#121—1222no—
391#121—1222no—
392#121—1222no—
393C4×D4—614no—
394C4×D4—614no—
395C4×D4—1222no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1221no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1222no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42C22×D6111yescharacteristic
417C4⋊C8—21no—
418C4⋊C8—612no—
419C4⋊C8—612no—
420C4⋊C8—1222no—
421C4⋊C8—1221no—
422C4⋊C8C22×D6112yes—
423C4⋊C8C22×D6112yes—
424C22⋊C8—21no—
425C22⋊C8—1222no—
426C22⋊C8—1222no—
427C22⋊C8—1222no—
428C22⋊C8—1221no—
429#97—1222no—
430#97—1222no—
431C4×C8—21no—
432C4×C8—1222no—
433C4×C8—1221no—
434C24—22no—
435C24—611no—
436#41—2448no—
437#41—2448no—
438#41—2444no—
439#41—2444no—
440#41—2444no—
441#41—2444no—
442C2×Q8—1222no—
443C2×Q8—1222no—
444C2×Q8—2442no—
445C2×Q8—2442no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—1224no—
456C2×D4—1224no—
457C2×D4—2448no—
458C2×D4C2×C4×D6114yes—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—1224no—
475C22×C4—1224no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4—2448no—
482C22×C4C2×C4×D6111yescharacteristic
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485M16—42no—
486M16—2444no—
487M16—2442no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—1222no—
491C2×C8—1222no—
492C4⋊C4—614no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C22⋊C4—1224no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42C2×C4×D6111yescharacteristic
513C42D4×D6111yescharacteristic
514C42D4×D6111yescharacteristic
515C42C23×D6111yescharacteristic
516C23—24no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—611no—
521C23—611no—
522C23—1222no—
523C23—2441no—
524C23—2441no—
525C23C22⋊C4×D6111yescharacteristic
526Q8—2444no—
527Q8—2442no—
528Q8—2442no—
529D4—44no—
530D4—44no—
531D4—44no—
532D4—44no—
533D4—42no—
534D4—42no—
535D4—44no—
536D4—2448no—
537D4—2448no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1224no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2448no—
565C2×C4—2448no—
566C2×C4—2444no—
567C2×C4C22⋊C4×D6111yescharacteristic
568C2×C4C22⋊C4×D6111yescharacteristic
569C2×C4C22⋊C4×D6111yescharacteristic
570C2×C4#1451980112yes—
571C2×C4#1451980412yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C8—42no—
576C8—42no—
577C8—2442no—
578C8—2442no—
579C22—21no—
580C22—21no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—88no—
585C22—1221no—
586C22—1221no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—2444no—
592C22#1683976512yes—
593C22C2×C22⋊C4×D6111yescharacteristic
594C4—21no—
595C4—21no—
596C4—1221no—
597C4—1221no—
598C4—1221no—
599C4—1221no—
600C4—1221no—
601C4—1221no—
602C4—2444no—
603C4#1683974312yes—
604C4#1683976512yes—
605C4C2×#1451980111yescharacteristic
606C4C2×#1451980411yescharacteristic
607C4#1684055812yes—
608C2—22no—
609C2—84no—
610C2—2441no—
611C2—2441no—
612C2C2×#1683974311yescharacteristic
613C2C2×#1683976511yescharacteristic
614C2#2145433311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840310C2
cosocleC42×C61C24
hypercenter#1545D61
hypocenterC61#17743
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. #21454447C2semidirect
4. #21444538C2semidirect
5. #21444521C2semidirect
6. #21444492C2semidirect
7. #21444490C2semidirect
8. #21444487C2semidirect
9. C61×#1545C2semidirect
27. #16839818C22semidirect
28. #16839818C22semidirect
32. #16839814C22semidirect
35. #16839784C22semidirect
40. #16839509C22semidirect
41. #16839507C22semidirect
42. #16839506C22semidirect
46. C61×#331C22semidirect
309. #1545D61semidirect
353. #331C2×D61semidirect
354. C61#17743semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
#214544512
#214544472
#214445382
#214445212
#214444922
#214444902
#214444872
C61×#15452
#1774361