Finite Group Databasebeta

← #29696059

Subgroups — #29696059

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×#16840576C211yesmaximal, characteristic
2C2×#16839879C211yesmaximal, characteristic, complemented
3#21454349C212yesmaximal, complemented
4#21446734C211yesmaximal, characteristic, complemented
5#21446678C211yesmaximal, characteristic, complemented
6#21446332C212yesmaximal, complemented
7#21446188C212yesmaximal, complemented
8#21445884C211yesmaximal, characteristic, complemented
9#21445776C212yesmaximal, complemented
10#21445658C211yesmaximal, characteristic, complemented
11C61×#2951C211yesmaximal, characteristic, complemented
12C2×#14519824C2212yes—
13#16840576—24no—
14#16840576C418yescomplemented
15C2×#14519807C2212yes—
16C2×#14519806—22no—
17C2×#14519796—21no—
18C22×Dic244C2211yescharacteristic
19C2×Q16×D61—21no—
20C2×#14519784C2212yes—
21C2×#14519783C2211yescharacteristic
22#16839879—24no—
23C2×#14519713C2211yescharacteristic
24C2×#14519704C2212yes—
25C2×#14519703C2211yescharacteristic
26#16839770—22no—
27#16839770C2212yes—
28#16839761—22no—
29#16839761C2212yes—
30#16839756—22no—
31#16839756C2211yescharacteristic
32C2×#14519700—22no—
33C2×#14519700C2212yes—
34C2×#14519695C2212yes—
35C2×C8×Dic61C2211yescharacteristic
36#16839667C2212yes—
37#16839531C2212yes—
38#16839526C2211yescharacteristic
39#16839521C2211yescharacteristic
40C2×#135×C61C2211yescharacteristic
41C61×#520C2212yes—
42C61×#440—22no—
43C4×Q16×C61—21no—
44C4×D8×C61—21no—
45C2×#107×C61C2211yescharacteristic
46C61×#417C2212yes—
47C61×#405C2211yescharacteristic
48#14519824—24no—
49#14519824—24no—
50#14519824—42no—
51#14519824C2×C418yescomplemented
52C2×#11758222—22no—
53C2×#11758222C2311yescharacteristic
54C2×#11758224—22no—
55C2×#11758224—42no—
56C22×Dic122—22no—
57C22×Dic122C2312yes—
58C2×Q8×D61—22no—
59#14519807—24no—
60#14519807C2×C418yescomplemented
61#14519806—216no—
62#14519796—28no—
63C2×Dic244—21no—
64C2×Dic244—21no—
65C2×Dic244—22no—
66C2×Dic244C2×C414yescomplemented
67C2×#11758202—22no—
68Q16×D61—28no—
69C2×#11758200—22no—
70#14519784—24no—
71#14519784—42no—
72#14519783—22no—
73#14519783C2×C414yescomplemented
74C2×#11758194—22no—
75C2×#11758194C2312yes—
76C2×#11758194C2311yescharacteristic
77C2×#11758189—21no—
78C22×#92418—22no—
79C22×#92418C2311yescharacteristic
80C2×C8×D61—21no—
81C2×C4×Dic61C2311yescharacteristic
82#14519713—22no—
83#14519704—24no—
84#14519703—22no—
85#14519700—22no—
86#14519700—22no—
87#14519700—22no—
88#14519700—22no—
89#14519700—22no—
90#14519700D411yescharacteristic
91#14519700D411yescharacteristic
92#14519697—22no—
93#14519697D411yescharacteristic
94#14519697D411yescharacteristic
95#14519695—42no—
96C8×Dic61—22no—
97#14519684—42no—
98C2×#41×C61C2312yes—
99#135×C61C2×C414yes—
100#135×C61D414yes—
101C2×Q16×C61D411yescharacteristic
102C2×SD16×C61D412yes—
103C2×D8×C61D411yescharacteristic
104C2×M16×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C4×Q8×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61—42no—
109#117×C61—22no—
110#117×C61C2311yescharacteristic
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61C2312yes—
113#107×C61—21no—
114#107×C61—21no—
115#103×C61—22no—
116#102×C61—22no—
117#97×C61—21no—
118C4×C8×C61—21no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61C22×C414yes—
122#41×C61C2×D414yes—
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—22no—
125C2×D4×C61C2×D412yes—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—42no—
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—21no—
131C2×C22⋊D61—42no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—42no—
135Q8×D61—28no—
136Q8×D61—44no—
137#11758224—28no—
138#11758224—44no—
139#11758224—48no—
140#11758224—48no—
141#11758222—21no—
142#11758222—21no—
143#11758222—28no—
144#11758222—44no—
145#11758222C22×C412yes—
146#11758222#4112yes—
147C2×D244C2×D411yescharacteristic
148C2×C4×D61—21no—
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×D411yescharacteristic
160Q16×C61C22⋊C414yes—
161SD16×C61C22⋊C418yes—
162D8×C61C22⋊C414yes—
163M16×C61—21no—
164M16×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C22×C412yes—
167C2×C8×C61C2×D412yes—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—42no—
177C42×C61—21no—
178C42×C61—42no—
179#11758202—28no—
180#11758200—28no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183#11758194—21no—
184#11758194—22no—
185#11758194—22no—
186#11758194—44no—
187#11758194C22×C412yes—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—22no—
192C2×#92418—44no—
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—24no—
196Dic244—24no—
197#11758189—24no—
198C8×D61—24no—
199C23×C61—22no—
200C23×C61H(5,𝔽2)11yescharacteristic
201C22×D61—21no—
202Q8×C61—22no—
203Q8×C61C2×C22⋊C414yes—
204D4×C61—24no—
205D4×C61—24no—
206D4×C61—22no—
207D4×C61C2×C22⋊C414yes—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—42no—
214C2×C4×C61—82no—
215C2×C4×C61—162no—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C22⋊D61—22no—
223C22⋊D61—22no—
224C22⋊D61—48no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—48no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233D244—21no—
234D244C4×D412yes—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—48no—
238Dic122—21no—
239Dic122—24no—
240Dic122—24no—
241Dic122—48no—
242Dic122—42no—
243Dic122—42no—
244Dic122C4×D412yes—
245C8×C61—41no—
246C8×C61C2×C22⋊C412yes—
247C8×C61C2×C22⋊C412yes—
248#92418—22no—
249#92418—21no—
250#92418—21no—
251#92418—42no—
252#22468—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—24no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61#51811yescharacteristic
259C22×C61D4211yescharacteristic
260C22×C61#54611yescharacteristic
261C2×D61—22no—
262C2×D61—41no—
263C2×D61—41no—
264C4×C61—24no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—82no—
268C4×C61C22×C22⋊C411yescharacteristic
269C4×C61C2×C4×D411yescharacteristic
270C4×C61D4211yescharacteristic
271C4×C61#54611yescharacteristic
272Dic61—22no—
273Dic61—42no—
274Dic61—44no—
275C2×#578—611no—
276C2×#443—611no—
277#2951D6111yescharacteristic, complemented
278#2898—612no—
279#1979—611no—
280#1969—611no—
281#1908—612no—
282#1882—612no—
283#1816—611no—
284#1793—612no—
285#1764—611no—
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—44no—
289C2×C61D4×C22⋊C411yescharacteristic
290C2×C61#329011yescharacteristic
291C2×C61#329311yescharacteristic
292D61—42no—
293C2×2−1+4—612no—
294#578—618no—
295#578—1224no—
296C2×#137—612no—
297C2×#137—1222no—
298C2×#135—1221no—
299C2×#135C2×D6111yescharacteristic
300C22×Q16—611no—
301C22×Q16—1221no—
302C2×#131—612no—
303C2×#131—611no—
304#520C2×D6112yes—
305#443—1224no—
306#440—22no—
307C4×Q16—21no—
308C4×D8—21no—
309C2×#108—611no—
310C2×#107C2×D6111yescharacteristic
311C2×#104—612no—
312C2×#104—611no—
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#405C2×D6111yescharacteristic
324C2×C4×C8—611no—
325#391—612no—
326#343—612no—
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—611no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—2442no—
338C2×#41C22×D6112yes—
339C22×Q8—612no—
340C22×Q8—1222no—
341C22×Q8—1222no—
342#137—618no—
343#137—1224no—
344#137—12216no—
345#135—1228no—
346#135C4×D6114yes—
347#135D24414yes—
348C2×Q16—614no—
349C2×Q16—1222no—
350C2×Q16—1228no—
351C2×Q16—1222no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16C22⋊D6111yescharacteristic
355C2×SD16—1222no—
356C2×SD16C22⋊D6112yes—
357C2×D8C22⋊D6111yescharacteristic
358#131—614no—
359#131—1222no—
360#131—1224no—
361#131—2442no—
362C2×M16—611no—
363C2×M16—612no—
364C2×M16—1222no—
365C2×M16—1221no—
366C2×M16C22×D6111yescharacteristic
367C22×C8—611no—
368C22×C8—1222no—
369C22×C8—1221no—
370C22×C8C22×D6111yescharacteristic
371C4×Q8—22no—
372C4×D4—22no—
373C4×D4—42no—
374#117—22no—
375#117C22×D6111yescharacteristic
376C2×C4⋊C4—22no—
377C2×C4⋊C4C22×D6112yes—
378C2×C42—611no—
379#108—1222no—
380#107—21no—
381#107—21no—
382#104—1222no—
383#104—1224no—
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—21no—
398C4×C8—21no—
399C4×C8—1222no—
400#95—2442no—
401#41—22no—
402#41—22no—
403#41—612no—
404#41—612no—
405#41—1228no—
406#41—1221no—
407#41—1221no—
408#41—1228no—
409#41—2444no—
410#41—2448no—
411#41—2444no—
412#41—2448no—
413#41C2×C4×D6114yes—
414#41C2×D24414yes—
415C2×Q8—614no—
416C2×Q8—611no—
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×Q8C2×C22⋊D6112yes—
428C2×D4—22no—
429C2×D4—611no—
430C2×D4—1221no—
431C2×D4—2442no—
432C2×D4C2×C22⋊D6112yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1222no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4C23×D6111yescharacteristic
442Q16—1228no—
443Q16—1224no—
444Q16—1224no—
445Q16#1175819814yes—
446SD16—1228no—
447SD16#1175819818yes—
448D8#1175819814yes—
449M16—21no—
450M16—21no—
451M16—612no—
452M16—1221no—
453M16—1224no—
454M16—1222no—
455M16—1222no—
456M16—2444no—
457C2×C8—21no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1224no—
464C2×C8—1222no—
465C2×C8—2444no—
466C2×C8C2×C4×D6112yes—
467C2×C8C2×D24412yes—
468C2×C8C2×C22⋊D6111yescharacteristic
469C2×C8C2×C22⋊D6111yescharacteristic
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4D4×D6111yescharacteristic
474C4⋊C4D4×D6111yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—42no—
477C42—21no—
478C42—42no—
479C42—1221no—
480C42—1221no—
481C23—22no—
482C23—1221no—
483C23#1451982211yescharacteristic
484Q8—22no—
485Q8—612no—
486Q8—1221no—
487Q8—1224no—
488Q8—1224no—
489Q8—2448no—
490Q8—2442no—
491Q8—2442no—
492Q8C2×#1175819814yes—
493D4—24no—
494D4—24no—
495D4—22no—
496D4—612no—
497D4—1222no—
498D4—1222no—
499D4—1221no—
500D4—2448no—
501D4C2×#1175819814yes—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—162no—
510C2×C4—1221no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1224no—
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×C4C22×C22⋊D6111yescharacteristic
524C2×C4C22×D24411yescharacteristic
525C2×C4#41×D6111yescharacteristic
526C2×C4#1451982311yescharacteristic
527C8—41no—
528C8—1222no—
529C8—1221no—
530C8—1221no—
531C8—2442no—
532C8C2×#1175819812yes—
533C8C2×#1175819812yes—
534C22—21no—
535C22—21no—
536C22—24no—
537C22—42no—
538C22—42no—
539C22—1222no—
540C22—2441no—
541C22—2441no—
542C22#1684023411yescharacteristic
543C22D4×C22⋊D6111yescharacteristic
544C22#1684040711yescharacteristic
545C4—24no—
546C4—41no—
547C4—41no—
548C4—82no—
549C4—1222no—
550C4—2442no—
551C4—2444no—
552C4C22×#1175819811yescharacteristic
553C4C4×D4×D6111yescharacteristic
554C4D4×D24411yescharacteristic
555C4#1684040011yescharacteristic
556C2—21no—
557C2—21no—
558C2—44no—
559C2—2442no—
560C2D4×#1175819811yescharacteristic
561C2#2145760111yescharacteristic
562C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2951C2
socleC22×C61D42
prime radicalC2×#16840576C2
cosocleC22×C4×C61C24
hypercenter#2951D61
hypocenterC61#22468
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. C2×#16839879C2semidirect
3. #21454349C2semidirect
4. #21446734C2semidirect
5. #21446678C2semidirect
6. #21446332C2semidirect
7. #21446188C2semidirect
8. #21445884C2semidirect
9. #21445776C2semidirect
10. #21445658C2semidirect
11. C61×#2951C2semidirect
14. #16840576C4semidirect
51. #14519824C2×C4semidirect
60. #14519807C2×C4semidirect
66. C2×Dic244C2×C4semidirect
73. #14519783C2×C4semidirect
277. #2951D61semidirect
329. C61#22468semidirect

Maximal subgroups

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

subgroupindex
C2×#168405762
C2×#168398792
#214543492
#214467342
#214466782
#214463322
#214461882
#214458842
#214457762
#214456582
C61×#29512
#2246861