Finite Group Databasebeta

← #29707916

Subgroups — #29707916

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic
2C2×#16839810C211yesmaximal, characteristic, complemented
3#21454336C211yesmaximal, characteristic
4C2×#16839770C211yesmaximal, characteristic, complemented
5C22×#14519695C211yesmaximal, characteristic, complemented
6#21454263C211yesmaximal, characteristic
7#21447121C218yesmaximal, complemented
8C61×#1742C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519810—24no—
12C2×#14519810—22no—
13C2×#14519810—24no—
14C2×#14519810C2214yes—
15C22×Dic244—21no—
16C22×#11758190—21no—
17C22×#11758194C2211yescharacteristic
18C22×#11758194C2211yescharacteristic
19#16839810—24no—
20C2×#14519712C2214yescomplemented
21#16839770—22no—
22#16839770—22no—
23#16839770—21no—
24#16839766—21no—
25#16839761C2214yes—
26#16839756—22no—
27C2×#14519700C2214yescomplemented
28C2×#14519695—24no—
29C2×#14519695C2214yescomplemented
30#16839730C2214yes—
31#16839728—22no—
32#16839725—22no—
33C22×M16×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#343C2214yescomplemented
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222D412yes—
44C2×#11758222C2312yes—
45C22×Dic122—21no—
46C22×Dic122—21no—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×Dic122—22no—
50C22×Dic122—22no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122D414yes—
54C22×Dic122D411yescharacteristic
55C22×Dic122C2312yes—
56C22×D244D411yescharacteristic
57C22×C22⋊D61—21no—
58C23×Dic61—21no—
59C22×C4×D61—21no—
60#14519810—216no—
61#14519810—28no—
62#14519810—28no—
63C2×Dic244—28no—
64C2×Dic244—24no—
65C2×#11758190—28no—
66C2×#11758190—24no—
67C2×#11758194—22no—
68C2×#11758194—22no—
69C2×#11758194—21no—
70C2×#11758194—22no—
71C2×#11758194—22no—
72C2×#11758194D412yes—
73C2×#11758194D412yes—
74C2×#11758194C2312yes—
75C22×#92418—21no—
76C22×#92418C2312yes—
77#14519712—28no—
78#14519700—24no—
79#14519700—28no—
80#14519697—24no—
81#14519695—28no—
82#14519689—28no—
83C23×C4×C61C2311yescharacteristic
84C2×M16×C61—22no—
85C2×M16×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61—22no—
88C2×M16×C61—22no—
89C2×M16×C61D414yes—
90C2×M16×C61C2312yes—
91C2×M16×C61C2312yes—
92C22×C8×C61—21no—
93C22×C8×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—21no—
97#117×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61—21no—
101C24×C61#4111yescharacteristic
102C23×D61—21no—
103C22×C4×C61—22no—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61#4112yes—
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4112yes—
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—22no—
115C2×C22⋊D61—22no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C22×Dic61—21no—
119C22×Dic61—24no—
120C22×Dic61—22no—
121C22×Dic61—41no—
122C22×Dic61—41no—
123C22×Dic61—42no—
124#11758222—24no—
125#11758222—24no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—42no—
131#11758222—42no—
132#11758222—42no—
133#11758222—42no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—21no—
137C2×D244C2×D412yes—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×Dic122—24no—
143C2×Dic122—24no—
144C2×Dic122—22no—
145C2×Dic122—24no—
146C2×Dic122—24no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—21no—
153C2×Dic122—24no—
154C2×Dic122—22no—
155C2×Dic122—42no—
156C2×Dic122—42no—
157C2×Dic122—41no—
158C2×Dic122—41no—
159C2×Dic122—42no—
160C2×Dic122—42no—
161C2×Dic122—42no—
162C2×Dic122C2×D414yes—
163C2×Dic122C2×D412yes—
164M16×C61—24no—
165M16×C61—22no—
166M16×C61—22no—
167M16×C61—22no—
168M16×C61—42no—
169M16×C61C2×D414yes—
170C2×C8×C61—24no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—42no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C42×C61—41no—
179C42×C61—41no—
180#11758194—24no—
181#11758194—24no—
182#11758194—22no—
183#11758194—22no—
184#11758194—42no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—42no—
189C2×#92418C2×D414yes—
190Dic244—216no—
191#11758190—216no—
192C23×C61—42no—
193C23×C61#12111yescharacteristic
194C23×C61#12111yescharacteristic
195C23×C61#12312yes—
196C23×C61#12412yes—
197C23×C61C2×#4111yescharacteristic
198C22×D61—22no—
199C22×D61—42no—
200C22×D61—82no—
201C2×C4×C61—42no—
202C2×C4×C61—82no—
203C2×C4×C61#12012yes—
204C2×C4×C61#12012yes—
205C2×C4×C61#12112yes—
206C2×C4×C61#12112yes—
207C2×C4×C61#12112yes—
208C2×C4×C61#12112yes—
209C2×C4×C61#12111yescharacteristic
210C2×C4×C61#12111yescharacteristic
211C2×C4×C61#12312yes—
212C2×C4×C61#12312yes—
213C2×C4×C61#12312yes—
214C2×C4×C61#12412yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C22⋊D61—42no—
222C22⋊D61—42no—
223C22⋊D61—44no—
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—44no—
229C2×Dic61—44no—
230C2×Dic61—42no—
231D244—22no—
232D244—22no—
233D244—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—44no—
237Dic122—24no—
238Dic122—22no—
239Dic122—24no—
240Dic122—22no—
241Dic122—42no—
242Dic122—42no—
243Dic122—44no—
244Dic122—44no—
245Dic122—42no—
246C8×C61—24no—
247C8×C61—42no—
248#92418—24no—
249#92418—42no—
250#23532—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61#39412yes—
257C22×C61#39412yes—
258C22×C61C2×#12111yescharacteristic
259C22×C61C2×#12311yescharacteristic
260C22×C61C2×#12411yescharacteristic
261C2×D61—42no—
262C2×D61—44no—
263C2×D61—84no—
264C4×C61—82no—
265C4×C61#39412yes—
266C4×C61#39412yes—
267C4×C61C2×#12011yescharacteristic
268C4×C61C2×#12111yescharacteristic
269C4×C61C2×#12111yescharacteristic
270C4×C61C2×#12311yescharacteristic
271Dic61—44no—
272Dic61—42no—
273C22×#137—611no—
274C2×#430—611no—
275#2896—611no—
276C2×#419—611no—
277C22×#101—611no—
278#2879—611no—
279#2052—618no—
280#1742D6111yescharacteristic, complemented
281C2×C61—22no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—22no—
285C2×C61C2×#39411yescharacteristic
286C2×C61#306912yes—
287D61—82no—
288C22×#41—611no—
289C23×Q8—611no—
290C2×#137—614no—
291C2×#137—1224no—
292C2×#137—1222no—
293C2×#137—1224no—
294C22×Q16—1221no—
295C22×SD16—1221no—
296C22×M16—611no—
297C22×M16—611no—
298C22×M16C2×D6111yescharacteristic
299C22×M16C2×D6111yescharacteristic
300C2×#117C2×D6111yescharacteristic
301#430—1224no—
302C2×#108—614no—
303#419—1222no—
304#419—1222no—
305#419—1221no—
306#418—1221no—
307#417—614no—
308#416—1222no—
309C2×#103—614no—
310C2×#101—614no—
311C2×#101—1224no—
312#408—614no—
313#408—1222no—
314#407—1222no—
315#343C2×D6114yescomplemented
316C61#2353211yescharacteristic, complemented, Sylow
317C2×#41—612no—
318C2×#41—612no—
319C2×#41—1221no—
320C2×#41—1221no—
321C2×#41—1222no—
322C2×#41—1222no—
323C2×#41—1222no—
324C2×#41—1222no—
325C22×Q8—611no—
326C22×Q8—614no—
327C22×Q8—612no—
328C22×Q8—1221no—
329C22×Q8—1222no—
330C22×Q8—1222no—
331C22×Q8—1222no—
332C22×Q8—1221no—
333C22×Q8—1221no—
334C22×Q8—1221no—
335C22×Q8—1221no—
336C22×D4—611no—
337C22×D4—1221no—
338C23×C4—1221no—
339C23×C4—1221no—
340C23×C4C22×D6111yescharacteristic
341#137—1228no—
342#137—12216no—
343#137—1228no—
344C2×Q16—1224no—
345C2×Q16—1228no—
346C2×SD16—1224no—
347C2×SD16—1228no—
348C2×M16—21no—
349C2×M16—22no—
350C2×M16—21no—
351C2×M16—22no—
352C2×M16—22no—
353C2×M16—612no—
354C2×M16—612no—
355C2×M16—612no—
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1221no—
359C2×M16—1222no—
360C2×M16—1222no—
361C2×M16D24414yes—
362C2×M16C22×D6112yes—
363C2×M16C22×D6112yes—
364C22×C8—21no—
365C22×C8—21no—
366C22×C8—612no—
367C22×C8—1221no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—21no—
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—21no—
374C2×C42—21no—
375#108—1228no—
376#103—1228no—
377#103—1224no—
378#102—1224no—
379#101—1228no—
380C22⋊C8—1228no—
381C24—1221no—
382C24#1175822211yescharacteristic
383#41—1222no—
384#41—1222no—
385#41—1222no—
386#41—1222no—
387#41—1224no—
388#41—1224no—
389#41—2442no—
390#41—2442no—
391#41—2442no—
392#41—2442no—
393C2×Q8—614no—
394C2×Q8—612no—
395C2×Q8—1222no—
396C2×Q8—1221no—
397C2×Q8—1224no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8—1224no—
403C2×Q8—1222no—
404C2×Q8—1224no—
405C2×Q8—1224no—
406C2×Q8—1224no—
407C2×Q8—1222no—
408C2×Q8—2441no—
409C2×Q8—2441no—
410C2×Q8—2442no—
411C2×Q8—2442no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C2×Q8—2442no—
415C2×D4—612no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1221no—
420C2×D4—1222no—
421C2×D4—2442no—
422C2×D4—2442no—
423C22×C4—22no—
424C22×C4—1222no—
425C22×C4—1221no—
426C22×C4—1224no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4C2×D24412yes—
435C22×C4#1175822212yes—
436C22×C4#1175822211yescharacteristic
437C22×C4D4×D6112yes—
438C22×C4D4×D6111yescharacteristic
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4D4×D6111yescharacteristic
442C22×C4#1175822612yes—
443C22×C4C23×D6111yescharacteristic
444Q16—12216no—
445SD16—12216no—
446M16—22no—
447M16—24no—
448M16—22no—
449M16—22no—
450M16—42no—
451M16—1224no—
452M16—1224no—
453M16—1222no—
454M16—1222no—
455M16—2442no—
456M16C2×D24414yes—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—42no—
461C2×C8—614no—
462C2×C8—1222no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—2442no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C42—41no—
471C42—41no—
472C23—42no—
473C23—1222no—
474C23—4882no—
475C23—2442no—
476C23#1451975011yescharacteristic
477C23#1451975011yescharacteristic
478C23#1451976212yes—
479C23#1451976512yes—
480C23C2×#1175822211yescharacteristic
481Q8—1224no—
482Q8—1222no—
483Q8—1224no—
484Q8—1222no—
485Q8—2444no—
486Q8—2444no—
487Q8—2442no—
488Q8—2442no—
489Q8—2442no—
490D4—1222no—
491D4—1222no—
492D4—2444no—
493D4—2442no—
494D4—2442no—
495D4—2442no—
496C2×C4—42no—
497C2×C4—82no—
498C2×C4—1222no—
499C2×C4—1224no—
500C2×C4—2442no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2444no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4#1451974712yes—
509C2×C4#1451974712yes—
510C2×C4#1451975011yescharacteristic
511C2×C4#1451975011yescharacteristic
512C2×C4#1451975012yes—
513C2×C4#1451975012yes—
514C2×C4#1451975212yes—
515C2×C4#1451975212yes—
516C2×C4#1451976212yes—
517C2×C4#1451976212yes—
518C2×C4#1451976212yes—
519C2×C4#1451976512yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C2×D4×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×D24411yescharacteristic
524C2×C4C2×#1175822211yescharacteristic
525C2×C4C2×#1175822611yescharacteristic
526C8—24no—
527C8—42no—
528C8—1224no—
529C8—2442no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—4884no—
536C22—2442no—
537C22—2444no—
538C22#1683968012yes—
539C22#1683968012yes—
540C22C2×#1451975011yescharacteristic
541C22C2×#1451976211yescharacteristic
542C22C2×#1451976511yescharacteristic
543C4—82no—
544C4—2442no—
545C4—2444no—
546C4#1683968012yes—
547C4#1683968012yes—
548C4C2×#1451974711yescharacteristic
549C4C2×#1451975011yescharacteristic
550C4C2×#1451975211yescharacteristic
551C4C2×#1451976211yescharacteristic
552C2—22no—
553C2—22no—
554C2—21no—
555C2—21no—
556C2—4882no—
557C2C2×#1683968011yescharacteristic
558C2#2145555912yes—
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
hypercenter#1742D61
hypocenterC61#23532
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C2×#16839810C2semidirect
4. C2×#16839770C2semidirect
5. C22×#14519695C2semidirect
7. #21447121C2semidirect
8. C61×#1742C2semidirect
20. C2×#14519712C22semidirect
27. C2×#14519700C22semidirect
29. C2×#14519695C22semidirect
36. C61×#343C22semidirect
280. #1742D61semidirect
315. #343C2×D61semidirect
316. C61#23532semidirect

Maximal subgroups

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

subgroupindex
C22×#145198102
C2×#168398102
#214543362
C2×#168397702
C22×#145196952
#214542632
#214471212
C61×#17422
#2353261