Finite Group Databasebeta

← #29681430

Subgroups — #29681430

Order 15616.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic
2C2×#16840242C211yesmaximal, characteristic
3C2×#16839756C211yesmaximal, characteristic
4C22×#14519701C211yesmaximal, characteristic, complemented
5#21446163C214yesmaximal, complemented
6#21446156C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C61×#2310C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519808—24no—
12C2×#14519808—24no—
13C2×#14519808C418yescomplemented
14C22×#11758202C2212yes—
15C22×#11758201C2212yes—
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18#16840242—24no—
19#16840242—24no—
20C2×C4×Dic122C2212yes—
21C2×C4×D244C2212yes—
22C2×#14519737—22no—
23C2×#14519735—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25C2×C4⋊C4×D61—21no—
26#16839756—24no—
27C2×#14519701—24no—
28C2×#14519701C2214yes—
29C2×#14519700C2212yes—
30C2×#14519697C2212yes—
31#16839524C2214yes—
32#16839517C2214yes—
33C23×Q8×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#391—21no—
37C61×#391—21no—
38C61×#391C2214yes—
39C61×#381—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222C2×C414yes—
44C2×#11758222D414yes—
45C2×#11758222D414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C23×Dic61—21no—
50C22×C4×D61—21no—
51C22×C4×D61—22no—
52#14519808—232no—
53C2×#11758202—24no—
54C2×#11758202—24no—
55C2×#11758202C2×C418yescomplemented
56C2×#11758201—24no—
57C2×#11758201—24no—
58C2×#11758201C2×C418yescomplemented
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194C2×C414yescomplemented
62C22×#92418—22no—
63C22×#92418—22no—
64C22×#92418—22no—
65C22×#92418C2312yes—
66C22×#92418C2312yes—
67C4×Dic122—24no—
68C4×Dic122—24no—
69C4×D244—24no—
70C4×D244—24no—
71#14519737—216no—
72#14519735—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61D414yes—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—22no—
78C4⋊C4×D61—28no—
79C2×#11758198—22no—
80C2×C4×Dic61—22no—
81#14519701—28no—
82#14519700—24no—
83#14519697—24no—
84C22×Q8×C61—21no—
85C22×Q8×C61—22no—
86C22×Q8×C61—22no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61C2×C414yescomplemented
91C22×Q8×C61C2312yes—
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94#117×C61—22no—
95#117×C61—22no—
96#117×C61—22no—
97#117×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61D412yes—
103C2×C42×C61—21no—
104C2×C42×C61C2312yes—
105#95×C61—24no—
106#95×C61—24no—
107C24×C61C2×D411yescharacteristic
108C23×D61—21no—
109C2×Q8×C61—24no—
110C2×Q8×C61—22no—
111C2×Q8×C61—28no—
112C2×Q8×C61—28no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—42no—
118C2×Q8×C61—42no—
119C2×Q8×C61—44no—
120C2×Q8×C61—44no—
121C2×Q8×C61C22×C414yes—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—24no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61—82no—
144#11758222—28no—
145#11758222—28no—
146#11758222—28no—
147#11758222C22⋊C4116yes—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×D244C2×D412yes—
153C2×D244C2×D412yes—
154C2×C4×D61—24no—
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—28no—
158C2×C4×D61—84no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C22×C412yes—
163C2×Dic122C2×D412yes—
164C2×Dic122C2×D412yes—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—42no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—42no—
174#11758202—216no—
175#11758201—216no—
176#11758198—28no—
177C4⋊Dic61—24no—
178C4⋊Dic61—22no—
179C4⋊Dic61C2×D414yes—
180#11758196—28no—
181C4×Dic61—28no—
182#11758194—28no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—44no—
189C2×#92418C22×C414yes—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61C4×D412yes—
193C23×C61#12012yes—
194C23×C61#12112yes—
195C23×C61C22×D411yescharacteristic
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—42no—
200C22×D61—42no—
201Q8×C61—24no—
202Q8×C61—24no—
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—48no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—48no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61C2×C22⋊C414yes—
218C2×C4×C61C4×D414yes—
219C2×C4×C61C4×D412yes—
220C2×C4×C61#12012yes—
221C2×C4×C61#12114yes—
222C2×C4×C61#12112yes—
223C2×C4×C61#12314yes—
224C2×C4×C61C23×C411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—216no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—44no—
236C2×Dic61—84no—
237D244—22no—
238D244—22no—
239D244—22no—
240D244C2×C22⋊C414yes—
241C4×D61—216no—
242C4×D61—88no—
243Dic122—22no—
244Dic122—22no—
245Dic122—22no—
246Dic122C2×C22⋊C414yes—
247#92418—24no—
248#92418—42no—
249#20931—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—24no—
254C22×C61#38614yes—
255C22×C61C2×C4×D411yescharacteristic
256C22×C61C2×#12011yescharacteristic
257C22×C61C2×#12111yescharacteristic
258C22×C61#57712yes—
259C22×C61#57812yes—
260C2×D61—24no—
261C2×D61—44no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—44no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—44no—
268C4×C61—82no—
269C4×C61#38614yes—
270C4×C61C22×C22⋊C411yescharacteristic
271C4×C61C2×C4×D411yescharacteristic
272C4×C61C2×#12111yescharacteristic
273C4×C61C2×#12311yescharacteristic
274Dic61—24no—
275Dic61—44no—
276C22×#137—611no—
277C2×#520—611no—
278C2×#416—611no—
279C22×#103—611no—
280#2310D6111yescharacteristic, complemented
281#1878—614no—
282#1877—614no—
283#1736—611no—
284#1729—611no—
285C2×C61—24no—
286C2×C61C2×#38611yescharacteristic
287C2×C61#294712yes—
288C2×C61#300911yescharacteristic
289C2×C61#301111yescharacteristic
290C2×C61#304612yes—
291D61—44no—
292C22×#41—611no—
293C23×Q8C2×D6111yescharacteristic
294C2×#137—618no—
295C2×#137—1224no—
296C2×#137—1224no—
297C22×Q16—612no—
298C22×SD16—612no—
299C22×M16—611no—
300C23×C8—611no—
301#520—1224no—
302#520—1224no—
303C2×C4×Q8—612no—
304C2×C4×D4—612no—
305C2×C4×D4—1222no—
306C2×#117—1221no—
307C2×#117C2×D6111yescharacteristic
308C2×#117C2×D6111yescharacteristic
309C22×C4⋊C4—611no—
310C22×C4⋊C4—1221no—
311#416—1224no—
312C2×#103—614no—
313C2×#103—612no—
314C2×#103—1224no—
315C2×#102—612no—
316#391—21no—
317#391—21no—
318#391C2×D6114yes—
319#381—22no—
320#340—614no—
321#336—614no—
322C61#2093111yescharacteristic, complemented, Sylow
323C2×#41—614no—
324C2×#41—614no—
325C2×#41—614no—
326C2×#41—1222no—
327C2×#41—1222no—
328C2×#41—1222no—
329C22×Q8—22no—
330C22×Q8—21no—
331C22×Q8—22no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—611no—
336C22×Q8C4×D6114yescomplemented
337C22×Q8C22×D6112yes—
338C22×D4—611no—
339C22×D4—1221no—
340C23×C4—21no—
341C23×C4—1221no—
342C23×C4—1221no—
343C23×C4—1222no—
344C23×C4C22×D6111yescharacteristic
345#137—12232no—
346C2×Q16—618no—
347C2×Q16—1224no—
348C2×Q16—1224no—
349C2×SD16—618no—
350C2×SD16—1224no—
351C2×SD16—1224no—
352C2×M16—614no—
353C2×M16—1222no—
354C2×M16—1222no—
355C22×C8—612no—
356C22×C8—612no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8—1222no—
360C4×Q8—1224no—
361C4×Q8—1224no—
362C4×D4—1224no—
363C4×D4—1224no—
364C4×D4—12216no—
365#117—22no—
366#117—22no—
367#117—22no—
368#117—22no—
369#117—1228no—
370C2×C4⋊C4—614no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—1228no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22⋊D6111yescharacteristic
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4C22⋊D6112yes—
382C2×C42—21no—
383C2×C42—1222no—
384C2×C42C22×D6112yes—
385#103—1224no—
386#103—1228no—
387#102—1224no—
388#95—24no—
389#95—24no—
390C24—1221no—
391C24C2×C22⋊D6111yescharacteristic
392#41—6116no—
393#41—1228no—
394#41—1228no—
395#41—1228no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—612no—
409C2×Q8—612no—
410C2×Q8—612no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8C2×C4×D6114yes—
415C2×D4—612no—
416C2×D4—612no—
417C2×D4—612no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—1224no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—1228no—
430C22×C4—1221no—
431C22×C4—1224no—
432C22×C4—1224no—
433C22×C4—1224no—
434C22×C4—1224no—
435C22×C4—1222no—
436C22×C4—4884no—
437C22×C4—4882no—
438C22×C4C2×C4×D6112yes—
439C22×C4#1175822212yes—
440C22×C4D4×D6112yes—
441C22×C4D4×D6112yes—
442C22×C4D4×D6112yes—
443C22×C4#1175822412yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448Q16—12216no—
449SD16—12216no—
450M16—1228no—
451C2×C8—614no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—2444no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—614no—
463C4⋊C4—1228no—
464C4⋊C4—1222no—
465C4⋊C4—1224no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—1228no—
469C42—22no—
470C42—22no—
471C42—42no—
472C42—1228no—
473C23—22no—
474C23—22no—
475C23—1222no—
476C23—1222no—
477C23—1222no—
478C23—2442no—
479C23—2442no—
480C23C4×C22⋊D6112yes—
481C23#1451974512yes—
482C23#1451974812yes—
483C23C22×C22⋊D6111yescharacteristic
484Q8—24no—
485Q8—24no—
486Q8—42no—
487Q8—42no—
488Q8—48no—
489Q8—614no—
490Q8—1222no—
491Q8—1222no—
492Q8—1222no—
493D4—614no—
494D4—12216no—
495D4—1222no—
496D4—1222no—
497D4—1222no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—12216no—
513C2×C4—1224no—
514C2×C4—4888no—
515C2×C4—4884no—
516C2×C4—2444no—
517C2×C4C22⋊C4×D6114yes—
518C2×C4#1451973714yes—
519C2×C4C4×C22⋊D6112yes—
520C2×C4#1451974512yes—
521C2×C4#1451974812yes—
522C2×C4#1451975014yes—
523C2×C4#1451975914yes—
524C2×C4C22×C4×D6111yescharacteristic
525C2×C4C2×D4×D6111yescharacteristic
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C2×#1175822411yescharacteristic
529C2×C4C2×#1175822211yescharacteristic
530C8—1224no—
531C8—2442no—
532C22—24no—
533C22—22no—
534C22—24no—
535C22—22no—
536C22—1224no—
537C22—2444no—
538C22—2442no—
539C22—2444no—
540C22—2442no—
541C22#1683963814yes—
542C22C2×C4×C22⋊D6111yescharacteristic
543C22C2×#1451974511yescharacteristic
544C22C2×#1451974811yescharacteristic
545C22#1684057012yes—
546C22#1684057712yes—
547C4—42no—
548C4—44no—
549C4—42no—
550C4—82no—
551C4—1224no—
552C4—2444no—
553C4#1683963814yes—
554C4C2×C22⋊C4×D6111yescharacteristic
555C4C2×#1451973711yescharacteristic
556C4C2×#1451975011yescharacteristic
557C4C2×#1451975911yescharacteristic
558C2—24no—
559C2—2444no—
560C2C2×#1683963811yescharacteristic
561C2#2145466212yes—
562C2#2145509411yescharacteristic
563C2#2145511211yescharacteristic
564C2#2145536512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2310C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#2310D61
hypocenterC61#20931
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. C22×#14519701C2semidirect
5. #21446163C2semidirect
6. #21446156C2semidirect
7. #21445540C2semidirect
8. #21445516C2semidirect
9. C61×#2310C2semidirect
13. C2×#14519808C4semidirect
55. C2×#11758202C2×C4semidirect
58. C2×#11758201C2×C4semidirect
61. C2×#11758194C2×C4semidirect
90. C22×Q8×C61C2×C4semidirect
280. #2310D61semidirect
322. C61#20931semidirect
336. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168402422
C2×#168397562
C22×#145197012
#214461632
#214461562
#214455402
#214455162
C61×#23102
#2093161