Finite Group Databasebeta

← #29689853

Subgroups — #29689853

Order 15616.

1044 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic
3#423×D61C211yesmaximal, characteristic
4#21446809C211yesmaximal, characteristic, complemented
5#21446663C211yesmaximal, characteristic, complemented
6#21446662C211yesmaximal, characteristic, complemented
7#21446286C214yesmaximal
8#21445814C214yesmaximal, complemented
9C61×#1997C211yesmaximal, characteristic, complemented
10C2×M16×D61C2212yes—
11#16840317—21no—
12#16840317—21no—
13#16840317—21no—
14#16840317—21no—
15#16840317—21no—
16#16840317—21no—
17#16840317—21no—
18#16840317—21no—
19C2×#14519777C2211yescharacteristic
20C2×#14519768C412yes—
21C2×#14519757C2212yes—
22C2×#14519753C412yes—
23C2×Q8×Dic61C412yes—
24C2×#14519742C2212yes—
25C2×#14519735C412yes—
26C2×C42×D61C2211yescharacteristic
27#16839817—21no—
28#16839817—21no—
29#16839813—21no—
30#16839813—21no—
31#16839786C2211yescharacteristic
32#16839785C2212yes—
33#16839785C2212yes—
34#16839784—21no—
35#16839784—21no—
36C4⋊C8×D61—21no—
37C4⋊C8×D61—21no—
38C2×#14519695C2214yes—
39C2×#14519695C2214yes—
40#16839725C2212yes—
41C4×#11758194C2211yescharacteristic
42#16839531C2214yes—
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#423C2211yescharacteristic
45C4×M16×C61C2211yescharacteristic
46C61×#343C2214yes—
47C22×Dic122C2312yes—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53C2×#11758189—22no—
54M16×D61—44no—
55C2×C8×D61—22no—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519768—22no—
61#14519768—22no—
62#14519768—22no—
63#14519768—22no—
64#14519757—22no—
65#14519757—22no—
66#14519757—22no—
67#14519757—22no—
68#14519753—22no—
69#14519753—22no—
70#14519753—22no—
71#14519753—22no—
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76#14519742—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519735—22no—
81#14519735—22no—
82#14519735—22no—
83#14519735—22no—
84C2×C4⋊Dic61C2×C412yes—
85C2×C4⋊Dic61C2×C412yes—
86C2×C4⋊Dic61C2×C414yes—
87C2×#11758196C2×C414yes—
88C2×#11758198C2×C414yes—
89C2×C4×Dic61C2×C414yes—
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C42×D61D411yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C4⋊#92418—21no—
102C4⋊#92418—21no—
103#14519706—22no—
104#14519706—22no—
105#14519706—22no—
106#14519706—22no—
107#14519695—44no—
108#14519695—44no—
109#14519689—22no—
110#14519689—22no—
111#14519688—21no—
112C4×#92418—21no—
113C22×Q8×C61C2×C412yes—
114C2×M16×C61C2312yes—
115C2×M16×C61C2312yes—
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C2×C4⋊C4×C61C2×C412yes—
121C2×C4⋊C4×C61C2312yes—
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—21no—
124C4⋊C8×C61—21no—
125#97×C61—21no—
126C4×C8×C61—21no—
127C23×D61C22×C411yescharacteristic
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—44no—
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C414yes—
134C22×C4×C61C2411yescharacteristic
135C22×Dic61C22×C414yes—
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61C22⋊C411yescharacteristic
144C2×C4×D61C22⋊C411yescharacteristic
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—24no—
158C2×Dic122—24no—
159M16×C61—22no—
160M16×C61—24no—
161M16×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C42×C61C2×D411yescharacteristic
169C42×C61C2×D411yescharacteristic
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172#11758198—24no—
173#11758198—24no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—24no—
179C4⋊Dic61—24no—
180#11758196—24no—
181#11758196—24no—
182C4×Dic61—24no—
183C4×Dic61—24no—
184C4×Dic61C22⋊C411yescharacteristic
185C4×Dic61C22⋊C411yescharacteristic
186C4×Dic61C22⋊C411yescharacteristic
187C4×Dic61C22⋊C411yescharacteristic
188C4×Dic61C22⋊C411yescharacteristic
189C4×Dic61C22⋊C411yescharacteristic
190C4×Dic61C22⋊C411yescharacteristic
191C4×Dic61C22⋊C411yescharacteristic
192C4×Dic61C2×D411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196#11758194—22no—
197#11758194—24no—
198#11758194—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201#11758189—44no—
202C8×D61—44no—
203C23×C61C23×C411yescharacteristic
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—41no—
207C22×D61C2×C22⋊C411yescharacteristic
208C22×D61C2×C22⋊C411yescharacteristic
209Q8×C61—44no—
210Q8×C61—44no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—44no—
215C2×C4×C61C2×C22⋊C411yescharacteristic
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61C2×C22⋊C411yescharacteristic
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61H(5,𝔽2)12yes—
222C2×C4×C612−1+412yes—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—28no—
227C2×Dic61—41no—
228C2×Dic61C2×C22⋊C411yescharacteristic
229C2×Dic61C2×C22⋊C411yescharacteristic
230C2×Dic61C2×C22⋊C411yescharacteristic
231C2×Dic61C2×C22⋊C411yescharacteristic
232C2×Dic61C2×C22⋊C411yescharacteristic
233C2×Dic61C2×C22⋊C411yescharacteristic
234C4×D61—22no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—41no—
246C4×D61—41no—
247C4×D61#10114yescomplemented
248Dic122—42no—
249Dic122—42no—
250Dic122—44no—
251C8×C61—42no—
252C8×C61—42no—
253#92418—42no—
254#92418—42no—
255#21879—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61#52012yes—
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—44no—
266C2×D61C2×#10112yes—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—44no—
270C4×C61C2×#10112yes—
271C4×C61#41312yes—
272C4×C61#53511yescharacteristic
273C4×C61#53611yescharacteristic
274C4×C61#53712yes—
275Dic61—21no—
276Dic61—21no—
277Dic61—21no—
278Dic61—21no—
279Dic61—21no—
280Dic61—44no—
281Dic61C2×#10112yes—
282C2×#533—611no—
283#2920—611no—
284C2×#423—611no—
285#1997—611no—
286#1997D6111yescharacteristic, complemented
287#1966—611no—
288#1966—611no—
289#1901—614no—
290#1801—614no—
291C2×C61—22no—
292C2×C61#231111yescharacteristic
293C2×C61C22×#10111yescharacteristic
294C2×C61C2×#41311yescharacteristic
295D61—22no—
296D61—41no—
297C22×M16—612no—
298#533—1221no—
299#533—1221no—
300#533—1221no—
301#533—1221no—
302#533—1221no—
303#533—1221no—
304#533—1221no—
305#533—1221no—
306C2×C4⋊Q8—611no—
307C2×C4⋊Q8C2×D6111yescharacteristic
308C2×#125—612no—
309C2×C22⋊Q8—612no—
310C2×C22⋊Q8—612no—
311C2×C4×Q8—612no—
312C2×C4×Q8—612no—
313C2×#117—612no—
314C22×C42—611no—
315#432—1221no—
316#432—1221no—
317#431—1221no—
318#431—1221no—
319#423—612no—
320#423—612no—
321#423—611no—
322#423—1221no—
323#423—1221no—
324#423C2×D6111yescharacteristic
325C2×C4⋊C8—1221no—
326C2×C4⋊C8—1221no—
327C2×#101—614no—
328C2×#101—614no—
329#407—612no—
330C4×M16—611no—
331C4×M16C2×D6111yescharacteristic
332#343—614no—
333#343C2×D6114yes—
334C61#2187911yescharacteristic, complemented, Sylow
335C22×Q8—612no—
336C22×Q8C4×D6112yes—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C2×M16—612no—
341C2×M16—612no—
342C2×M16—1222no—
343C2×M16—2444no—
344C2×M16C22×D6112yes—
345C2×M16C22×D6112yes—
346C22×C8—1222no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355#125—1222no—
356#125—1222no—
357#125—1222no—
358#125—1222no—
359C22⋊Q8—1222no—
360C22⋊Q8—1222no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C22⋊Q8—1222no—
365C22⋊Q8—1222no—
366C22⋊Q8—1222no—
367C4×Q8—1222no—
368C4×Q8—1222no—
369C4×Q8—1222no—
370C4×Q8—1222no—
371C4×Q8—1222no—
372C4×Q8—1222no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375#117—1222no—
376#117—1222no—
377#117—1222no—
378#117—1222no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—614no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4C4×D6112yes—
384C2×C4⋊C4C22×D6112yes—
385C2×C22⋊C4—614no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—614no—
398C2×C42C22×D6111yescharacteristic
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—1222no—
402C4⋊C8—1222no—
403C4⋊C8—1222no—
404C4⋊C8—1222no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407#101—2444no—
408#101—2444no—
409C22⋊C8—1222no—
410C22⋊C8—1222no—
411#97—21no—
412#97—1221no—
413C4×C8—21no—
414C4×C8—1221no—
415C24—611no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—44no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1224no—
422C2×Q8—1224no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—614no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—2442no—
443C22×C4C2×C4×D6111yescharacteristic
444C22×C4C2×C4×D6111yescharacteristic
445C22×C4C2×C4×D6114yes—
446C22×C4C23×D6111yescharacteristic
447M16—22no—
448M16—24no—
449M16—22no—
450M16—1222no—
451M16—1224no—
452M16—1222no—
453M16—2444no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—2444no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C42—611no—
474C42—611no—
475C42—611no—
476C42—611no—
477C42—611no—
478C42—611no—
479C42—611no—
480C42—611no—
481C42—611no—
482C42—611no—
483C42—611no—
484C42—611no—
485C42—1224no—
486C42—1224no—
487C42D4×D6111yescharacteristic
488C42D4×D6111yescharacteristic
489C42D4×D6111yescharacteristic
490C42D4×D6111yescharacteristic
491C23—611no—
492C23—611no—
493C23—1222no—
494C23—1222no—
495C23—2441no—
496C23C22×C4×D6111yescharacteristic
497Q8—44no—
498Q8—44no—
499Q8—2442no—
500Q8—2442no—
501Q8—2444no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—611no—
507C2×C4—614no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1224no—
527C2×C4—1228no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4C22⋊C4×D6111yescharacteristic
532C2×C4C22⋊C4×D6111yescharacteristic
533C2×C4C22⋊C4×D6111yescharacteristic
534C2×C4C22⋊C4×D6111yescharacteristic
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4#1451982312yes—
538C2×C4#1451982512yes—
539C8—42no—
540C8—42no—
541C8—2442no—
542C8—2442no—
543C22—21no—
544C22—21no—
545C22—612no—
546C22—1221no—
547C22—1221no—
548C22—1221no—
549C22—1221no—
550C22—1221no—
551C22—2444no—
552C22C2×C22⋊C4×D6111yescharacteristic
553C22#1684024412yes—
554C4—21no—
555C4—21no—
556C4—44no—
557C4—612no—
558C4—1221no—
559C4—1221no—
560C4—1221no—
561C4—1221no—
562C4—1221no—
563C4—2444no—
564C4#101×D6112yes—
565C4#1683974312yes—
566C4#1684033011yescharacteristic
567C4#1684033711yescharacteristic
568C4#1684034212yes—
569C2—22no—
570C2—1222no—
571C2—2441no—
572C2#2144841011yescharacteristic
573C2C2×#101×D6111yescharacteristic
574C2C2×#1683974311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22×C4
cosocleC22×C4×C61C24
hypercenter#1997D61
hypocenterC61#21879
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
4. #21446809C2semidirect
5. #21446663C2semidirect
6. #21446662C2semidirect
8. #21445814C2semidirect
9. C61×#1997C2semidirect
247. C4×D61#101semidirect
286. #1997D61semidirect
334. C61#21879semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214544502
#423×D612
#214468092
#214466632
#214466622
#214462862
#214458142
C61×#19972
#2187961