Finite Group Databasebeta

← #29696055

Subgroups — #29696055

Order 15616.

1316 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840579C211yesmaximal, characteristic
2C2×#16839878C211yesmaximal, characteristic, complemented
3#21454693C211yesmaximal, characteristic, complemented
4#21454350C212yesmaximal, complemented
5#21446732C211yesmaximal, characteristic, complemented
6#21446678C211yesmaximal, characteristic, complemented
7#21446335C212yesmaximal, complemented
8#21446187C212yesmaximal, complemented
9#21445776C212yesmaximal, complemented
10#21445654C211yesmaximal, characteristic, complemented
11C61×#1816C211yesmaximal, characteristic, complemented
12C2×#14519824C2212yes—
13#16840579—24no—
14#16840579C418yescomplemented
15C2×#14519810—22no—
16C2×#14519810C2212yes—
17C2×#14519799—21no—
18C2×#14519799C2211yescharacteristic
19C22×Dic244—21no—
20C22×Dic244C2211yescharacteristic
21C2×#14519785C2212yes—
22#16840243C2212yes—
23#16839878—24no—
24#16839861—22no—
25#16839856—21no—
26#16839844—21no—
27C2×#14519704C2211yescharacteristic
28C2×#14519703C2212yes—
29#16839772C2212yes—
30#16839769—22no—
31#16839761C2212yes—
32#16839760—22no—
33#16839758C2211yescharacteristic
34#16839755—22no—
35C2×#14519700C2212yes—
36#103×D61—22no—
37C2×#14519695C2212yes—
38#16839717C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40#16839667C2212yes—
41#16839526C2211yescharacteristic
42#16839523C2211yescharacteristic
43C2×#131×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C2×#107×C61C2211yescharacteristic
46C61×#417C2212yes—
47C61×#343C2212yes—
48#14519824—24no—
49#14519824—24no—
50#14519824—42no—
51#14519824C2×C418yescomplemented
52C2×#11758222—22no—
53C2×#11758222C2312yes—
54C2×#11758224—22no—
55C2×#11758224—42no—
56C22×Dic122—22no—
57C22×Dic122C2312yes—
58C2×Q8×D61—22no—
59#14519810—24no—
60#14519810—216no—
61#14519810C2×C418yescomplemented
62#14519799—28no—
63#14519799C2×C414yes—
64#14519799D414yes—
65C2×Dic244—22no—
66C2×Dic244—21no—
67C2×Dic244—21no—
68C2×Dic244—22no—
69C2×Dic244—28no—
70C2×Dic244C2×C414yescomplemented
71C2×Dic244D411yescharacteristic
72C2×#11758190—22no—
73C2×#11758190D412yes—
74C2×D488D411yescharacteristic
75#14519785—24no—
76#14519785—42no—
77C2×#11758194C2311yescharacteristic
78C2×#11758189—22no—
79C22×#92418C2311yescharacteristic
80C2×C8×D61—22no—
81#14519742—22no—
82#14519739—22no—
83C4×C22⋊D61—42no—
84#14519736C2311yescharacteristic
85#14519735—22no—
86C4⋊C4×D61—22no—
87C2×C4×Dic61C2311yescharacteristic
88#14519704—22no—
89#14519703—24no—
90#14519702—22no—
91#14519702—22no—
92#14519701D411yescharacteristic
93#14519701D411yescharacteristic
94#14519700—22no—
95#14519700—22no—
96#14519700—22no—
97#14519698D411yescharacteristic
98#14519698D411yescharacteristic
99#14519697—22no—
100#14519695—42no—
101#14519687—21no—
102C8×Dic61—22no—
103C8×Dic61—21no—
104#14519684—42no—
105C2×#41×C61C2311yescharacteristic
106#131×C61—22no—
107#131×C61C2×C414yescomplemented
108C2×M16×C61—21no—
109C2×M16×C61C2312yes—
110C2×M16×C61C2311yescharacteristic
111C22×C8×C61—21no—
112C22×C8×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2312yes—
114#108×C61—22no—
115#107×C61—21no—
116#107×C61—21no—
117#103×C61—22no—
118#102×C61—22no—
119#41×C61—21no—
120#41×C61—21no—
121#41×C61C22×C412yes—
122#41×C61#4112yes—
123C2×Q8×C61C2×D411yescharacteristic
124C2×D4×C61—21no—
125C2×D4×C61C2×D411yescharacteristic
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—22no—
130C2×C22⋊D61—42no—
131C22×Dic61—22no—
132C22×Dic61—21no—
133C22×Dic61—42no—
134Q8×D61—28no—
135Q8×D61—44no—
136#11758224—28no—
137#11758224—44no—
138#11758224—48no—
139#11758224—48no—
140#11758222—22no—
141#11758222—22no—
142#11758222—28no—
143#11758222—44no—
144#11758222C22×C414yes—
145#11758222C2×D414yes—
146C2×D244C2×D412yes—
147C2×C4×D61—22no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—28no—
155C2×Dic122—22no—
156C2×Dic122—44no—
157C2×Dic122—42no—
158C2×Dic122C22×C414yes—
159C2×Dic122C2×D412yes—
160M16×C61—21no—
161M16×C61—24no—
162M16×C61—22no—
163M16×C61—22no—
164M16×C61C22×C412yes—
165C2×C8×C61—21no—
166C2×C8×C61—24no—
167C2×C8×C61—22no—
168C2×C8×C61C22×C412yes—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D412yes—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C22⋊Dic61—21no—
175#11758198—42no—
176C4⋊Dic61C2×D411yescharacteristic
177C4⋊Dic61C2×D411yescharacteristic
178#11758196—42no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—42no—
183#11758194—21no—
184#11758194—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—28no—
191Dic244—24no—
192Dic244—24no—
193Dic244C22⋊C414yes—
194D488C22⋊C414yes—
195#11758190—28no—
196#11758190C22⋊C418yes—
197#11758189—44no—
198C8×D61—44no—
199C23×C61—21no—
200C23×C61H(5,𝔽2)11yescharacteristic
201C22×D61—22no—
202Q8×C61—21no—
203Q8×C61C4×D412yes—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—21no—
207D4×C61C4×D412yes—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—42no—
212C2×C4×C61—82no—
213C2×C4×C61C23×C411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61H(5,𝔽2)11yescharacteristic
219C22⋊D61—24no—
220C22⋊D61—24no—
221C22⋊D61—48no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—24no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—48no—
228C2×Dic61—41no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232D244—22no—
233D244C2×C22⋊C414yes—
234C4×D61—24no—
235C4×D61—24no—
236C4×D61—48no—
237C4×D61—162no—
238Dic122—22no—
239Dic122—24no—
240Dic122—24no—
241Dic122—48no—
242Dic122—42no—
243Dic122—42no—
244Dic122C2×C22⋊C414yes—
245C8×C61—22no—
246C8×C61—42no—
247C8×C61C2×C22⋊C412yes—
248C8×C61C2×C22⋊C412yes—
249#92418—21no—
250#92418—21no—
251#92418—41no—
252#22468—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—22no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61#51811yescharacteristic
259C22×C61D4211yescharacteristic
260C22×C61#54611yescharacteristic
261C2×D61—24no—
262C2×D61—42no—
263C2×D61—42no—
264C4×C61—22no—
265C4×C61—82no—
266C4×C61C22×C22⋊C411yescharacteristic
267C4×C61C2×C4×D411yescharacteristic
268C4×C61D4211yescharacteristic
269C4×C61#54611yescharacteristic
270Dic61—24no—
271Dic61—42no—
272Dic61—44no—
273Dic61—41no—
274Dic61—41no—
275C2×#578—611no—
276C2×#443—611no—
277#2951—611no—
278#2898—612no—
279#1979—611no—
280#1969—611no—
281#1908—612no—
282#1882—612no—
283#1816D6111yescharacteristic, complemented
284#1793—612no—
285#1764—611no—
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—42no—
289C2×C61D4×C22⋊C411yescharacteristic
290C2×C61#329011yescharacteristic
291C2×C61#329311yescharacteristic
292D61—44no—
293C2×2−1+4—612no—
294#578—618no—
295#578—1224no—
296C2×#137—612no—
297C2×#137—1222no—
298C2×#135—611no—
299C2×#135—1221no—
300C22×Q16—611no—
301C22×Q16—1221no—
302C2×#131—612no—
303C2×#131C2×D6111yescharacteristic
304#520—612no—
305#443—1224no—
306#440—1222no—
307C4×Q16—1221no—
308C4×D8—1221no—
309C2×#108C2×D6111yescharacteristic
310C2×#107C2×D6111yescharacteristic
311C2×#104—611no—
312C2×#104—612no—
313#419—612no—
314#419—1222no—
315#417—612no—
316#417—1222no—
317#417C2×D6112yes—
318#416—611no—
319#416—1222no—
320C2×#103—612no—
321C2×#103—1222no—
322C2×#101—612no—
323#405—611no—
324C2×C4×C8—611no—
325#391—612no—
326#343C2×D6112yes—
327#341—611no—
328#339—611no—
329C61#2246811yescharacteristic, complemented, Sylow
3302−1+4—618no—
3312−1+4—1224no—
3322−1+4—1224no—
3332−1+4—2442no—
334C2×#41—612no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—2442no—
338C2×#41C22×D6111yescharacteristic
339C22×Q8—612no—
340C22×Q8—1222no—
341C22×Q8—1222no—
342#137—618no—
343#137—12216no—
344#137—1224no—
345#135—614no—
346#135—614no—
347#135—1228no—
348C2×Q16—614no—
349C2×Q16—611no—
350C2×Q16—1228no—
351C2×Q16—1222no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16—1222no—
355C2×SD16—612no—
356C2×SD16—1222no—
357C2×D8—611no—
358#131—22no—
359#131—1224no—
360#131—2442no—
361#131C4×D6114yescomplemented
362C2×M16—21no—
363C2×M16—611no—
364C2×M16—1222no—
365C2×M16C22×D6112yes—
366C2×M16C22×D6111yescharacteristic
367C22×C8—21no—
368C22×C8—611no—
369C22×C8—1222no—
370C22×C8C22×D6111yescharacteristic
371C4×Q8—1222no—
372C4×D4—1222no—
373C4×D4—2442no—
374#117—611no—
375#117—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4C22×D6112yes—
378C2×C42—611no—
379#108—22no—
380#107—21no—
381#107—21no—
382#104—1224no—
383#104—1222no—
384#103—22no—
385#103—611no—
386#103—611no—
387#103—1222no—
388#103—1222no—
389#103—1222no—
390#103—1222no—
391#103—1222no—
392#102—22no—
393#102—611no—
394#102—611no—
395#102—1222no—
396#101—2442no—
397#97—1221no—
398C4×C8—1222no—
399C4×C8—1221no—
400#95—2442no—
401#41—21no—
402#41—21no—
403#41—614no—
404#41—614no—
405#41—1228no—
406#41—1222no—
407#41—1222no—
408#41—1228no—
409#41—2444no—
410#41—2448no—
411#41—2444no—
412#41—2448no—
413#41C2×C4×D6112yes—
414#41#1175822612yes—
415C2×Q8—614no—
416C2×Q8—612no—
417C2×Q8—1228no—
418C2×Q8—1222no—
419C2×Q8—1228no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8—2442no—
425C2×Q8—2444no—
426C2×Q8—2444no—
427C2×Q8D4×D6111yescharacteristic
428C2×D4—21no—
429C2×D4—612no—
430C2×D4—1222no—
431C2×D4—2442no—
432C2×D4D4×D6111yescharacteristic
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—1222no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4C23×D6111yescharacteristic
442Q16—614no—
443Q16—1228no—
444Q16—1224no—
445Q16—1224no—
446SD16—618no—
447SD16—1228no—
448D8—614no—
449M16—21no—
450M16—24no—
451M16—22no—
452M16—22no—
453M16—1221no—
454M16—1221no—
455M16—2444no—
456M16C2×C4×D6112yes—
457C2×C8—21no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—611no—
461C2×C8—611no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—2444no—
466C2×C8C2×C4×D6112yes—
467C2×C8D4×D6111yescharacteristic
468C2×C8D4×D6112yes—
469C2×C8D4×D6111yescharacteristic
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—2442no—
475C22⋊C4—1221no—
476C22⋊C4—2442no—
477C42—1221no—
478C42—1221no—
479C42—1221no—
480C42—2442no—
481C23—21no—
482C23—1222no—
483C23#1451982311yescharacteristic
484Q8—21no—
485Q8—614no—
486Q8—1222no—
487Q8—1224no—
488Q8—1224no—
489Q8—2448no—
490Q8—2442no—
491Q8—2442no—
492Q8#1451973912yes—
493D4—22no—
494D4—22no—
495D4—21no—
496D4—614no—
497D4—1224no—
498D4—1224no—
499D4—1222no—
500D4—2448no—
501D4#1451973912yes—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—1222no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1222no—
511C2×C4—1224no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—9762no—
515C2×C4—2448no—
516C2×C4—2448no—
517C2×C4—2441no—
518C2×C4—2442no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4C22×C4×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4C2×#1175822611yescharacteristic
526C2×C4#1451982311yescharacteristic
527C8—22no—
528C8—42no—
529C8—1221no—
530C8—1221no—
531C8—2441no—
532C8C22⋊C4×D6112yes—
533C8C22⋊C4×D6112yes—
534C22—21no—
535C22—21no—
536C22—22no—
537C22—41no—
538C22—41no—
539C22—1224no—
540C22—2442no—
541C22—2442no—
542C22#1684023611yescharacteristic
543C22#1684039611yescharacteristic
544C22#1684040311yescharacteristic
545C4—22no—
546C4—82no—
547C4—1224no—
548C4—2442no—
549C4—2444no—
550C4—2441no—
551C4—2441no—
552C4C2×C22⋊C4×D6111yescharacteristic
553C4C2×#1451973911yescharacteristic
554C4#1684039611yescharacteristic
555C4#1684040311yescharacteristic
556C2—21no—
557C2—21no—
558C2—42no—
559C2—2444no—
560C2#2144863911yescharacteristic
561C2#2145760011yescharacteristic
562C2#2145762211yescharacteristic
subgroupG/H
centerC22#16840396
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1816C2
socleC22×C61D42
prime radicalC2×#16840579C2
cosocleC22×C4×C61C24
hypercenter#1816D61
hypocenterC61#22468
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
2. C2×#16839878C2semidirect
3. #21454693C2semidirect
4. #21454350C2semidirect
5. #21446732C2semidirect
6. #21446678C2semidirect
7. #21446335C2semidirect
8. #21446187C2semidirect
9. #21445776C2semidirect
10. #21445654C2semidirect
11. C61×#1816C2semidirect
14. #16840579C4semidirect
51. #14519824C2×C4semidirect
61. #14519810C2×C4semidirect
70. C2×Dic244C2×C4semidirect
107. #131×C61C2×C4semidirect
283. #1816D61semidirect
329. C61#22468semidirect
361. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405792
C2×#168398782
#214546932
#214543502
#214467322
#214466782
#214463352
#214461872
#214457762
#214456542
C61×#18162
#2246861