Finite Group Databasebeta

← #29696022

Subgroups — #29696022

Order 15616.

1312 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840576C211yesmaximal, characteristic
2C2×#16839884C211yesmaximal, characteristic, complemented
3#21454349C212yesmaximal, complemented
4#21446734C211yesmaximal, characteristic, complemented
5#21446688C211yesmaximal, characteristic, complemented
6#21446332C212yesmaximal, complemented
7#21446188C212yesmaximal, complemented
8#21445879C211yesmaximal, characteristic, complemented
9#21445776C212yesmaximal, complemented
10#21445665C211yesmaximal, characteristic, complemented
11C61×#2952C211yesmaximal, 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#16839884—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×#14519687C2211yescharacteristic
36#16839667C2212yes—
37#16839531C2212yes—
38#16839526C2211yescharacteristic
39#16839521C2211yescharacteristic
40C2×#135×C61C2211yescharacteristic
41C61×#520C2212yes—
42C61×#442—21no—
43C61×#441—21no—
44C4×SD16×C61—22no—
45C2×#106×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#14519700D412yes—
91#14519697—22no—
92#14519697D412yes—
93#14519695—42no—
94#14519687—22no—
95#14519684—42no—
96C2×#41×C61C2312yes—
97#135×C61C2×C414yes—
98#135×C61D414yes—
99C2×Q16×C61D411yescharacteristic
100C2×SD16×C61D412yes—
101C2×D8×C61D411yescharacteristic
102C2×M16×C61C2311yescharacteristic
103C22×C8×C61C2311yescharacteristic
104C4×Q8×C61—22no—
105C4×D4×C61—22no—
106C4×D4×C61—42no—
107#117×C61—22no—
108#117×C61C2311yescharacteristic
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61C2312yes—
111#106×C61—21no—
112#106×C61—21no—
113#103×C61—22no—
114#102×C61—22no—
115#97×C61—21no—
116C4×C8×C61—21no—
117#41×C61—22no—
118#41×C61—22no—
119#41×C61C22×C414yes—
120#41×C61C2×D414yes—
121C2×Q8×C61C2×D412yes—
122C2×D4×C61—22no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—42no—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—21no—
129C2×C22⋊D61—42no—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—42no—
133Q8×D61—28no—
134Q8×D61—44no—
135#11758224—28no—
136#11758224—44no—
137#11758224—48no—
138#11758224—48no—
139#11758222—21no—
140#11758222—21no—
141#11758222—28no—
142#11758222—44no—
143#11758222C22×C412yes—
144#11758222#4112yes—
145C2×D244C2×D411yescharacteristic
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—28no—
153C2×Dic122—22no—
154C2×Dic122—44no—
155C2×Dic122—42no—
156C2×Dic122C22×C414yes—
157C2×Dic122C2×D411yescharacteristic
158Q16×C61C22⋊C414yes—
159SD16×C61C22⋊C418yes—
160D8×C61C22⋊C414yes—
161M16×C61—21no—
162M16×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61C22×C412yes—
165C2×C8×C61C2×D412yes—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61C2×D412yes—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—42no—
174C42×C61—21no—
175C42×C61—42no—
176#11758202—28no—
177#11758200—28no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180#11758194—21no—
181#11758194—22no—
182#11758194—22no—
183#11758194—44no—
184#11758194C22×C412yes—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—22no—
189C2×#92418—44no—
190C2×#92418C2×D412yes—
191Dic244—24no—
192Dic244—24no—
193#11758189—24no—
194C8×D61—24no—
195C23×C61—22no—
196C23×C61H(5,𝔽2)11yescharacteristic
197C22×D61—21no—
198Q8×C61—22no—
199Q8×C61C2×C22⋊C414yes—
200D4×C61—24no—
201D4×C61—24no—
202D4×C61—22no—
203D4×C61C2×C22⋊C414yes—
204C2×C4×C61—22no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—42no—
209C2×C4×C61—82no—
210C2×C4×C61—162no—
211C2×C4×C61C23×C411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C22⋊D61—22no—
218C22⋊D61—22no—
219C22⋊D61—48no—
220C2×Dic61—21no—
221C2×Dic61—24no—
222C2×Dic61—48no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227D244—21no—
228D244C4×D412yes—
229C4×D61—22no—
230C4×D61—22no—
231C4×D61—48no—
232Dic122—21no—
233Dic122—24no—
234Dic122—24no—
235Dic122—48no—
236Dic122—42no—
237Dic122—42no—
238Dic122C4×D412yes—
239C8×C61—41no—
240C8×C61C2×C22⋊C412yes—
241C8×C61C2×C22⋊C412yes—
242#92418—22no—
243#92418—22no—
244#92418—42no—
245#22465—611nomaximal, Sylow
246C22×C61—22no—
247C22×C61—24no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61#51811yescharacteristic
251C22×C61D4211yescharacteristic
252C22×C61#54611yescharacteristic
253C2×D61—22no—
254C2×D61—42no—
255C4×C61—24no—
256C4×C61—42no—
257C4×C61—82no—
258C4×C61C22×C22⋊C411yescharacteristic
259C4×C61C2×C4×D411yescharacteristic
260C4×C61D4211yescharacteristic
261C4×C61#54611yescharacteristic
262Dic61—22no—
263Dic61—44no—
264Dic61—81no—
265C2×#578—611no—
266C2×#444—611no—
267#2952D6111yescharacteristic, complemented
268#2898—612no—
269#1979—611no—
270#1971—611no—
271#1908—612no—
272#1882—612no—
273#1815—611no—
274#1793—612no—
275#1765—611no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—44no—
279C2×C61D4×C22⋊C411yescharacteristic
280C2×C61#328912yes—
281D61—42no—
282C2×2−1+4—612no—
283#578—618no—
284#578—1224no—
285C2×#137—612no—
286C2×#137—1222no—
287C2×#135—1221no—
288C2×#135C2×D6111yescharacteristic
289C22×Q16—611no—
290C22×Q16—1221no—
291C2×#131—612no—
292C2×#131—611no—
293#520C2×D6112yes—
294#444—1224no—
295#442—21no—
296#441—21no—
297C4×SD16—22no—
298C2×#108—611no—
299C2×#106C2×D6111yescharacteristic
300C2×#104—612no—
301C2×#104—611no—
302#419—612no—
303#419—1222no—
304#417—612no—
305#417—1222no—
306#417C2×D6112yes—
307#416—611no—
308#416—1222no—
309C2×#103—612no—
310C2×#103—1222no—
311C2×#101—612no—
312#405C2×D6111yescharacteristic
313C2×#97—611no—
314#391—612no—
315#343—612no—
316#341—611no—
317#339—611no—
318C61#2246511yescharacteristic, complemented, Sylow
3192−1+4—618no—
3202−1+4—1224no—
3212−1+4—1224no—
3222−1+4—2442no—
323C2×#41—611no—
324C2×#41—1222no—
325C2×#41—1222no—
326C2×#41—2442no—
327C2×#41C22×D6112yes—
328C22×Q8—612no—
329C22×Q8—1222no—
330C22×Q8—1222no—
331#137—618no—
332#137—1224no—
333#137—12216no—
334#135—1228no—
335#135C4×D6114yes—
336#135D24414yes—
337C2×Q16—614no—
338C2×Q16—1222no—
339C2×Q16—1228no—
340C2×Q16—1222no—
341C2×Q16—1221no—
342C2×Q16—1221no—
343C2×Q16C22⋊D6111yescharacteristic
344C2×SD16—1222no—
345C2×SD16C22⋊D6112yes—
346C2×D8C22⋊D6111yescharacteristic
347#131—614no—
348#131—1222no—
349#131—1224no—
350#131—2442no—
351C2×M16—611no—
352C2×M16—612no—
353C2×M16—1222no—
354C2×M16—1221no—
355C2×M16C22×D6111yescharacteristic
356C22×C8—611no—
357C22×C8—1222no—
358C22×C8—1221no—
359C22×C8C22×D6111yescharacteristic
360C4×Q8—22no—
361C4×D4—22no—
362C4×D4—42no—
363#117—22no—
364#117C22×D6111yescharacteristic
365C2×C4⋊C4—22no—
366C2×C4⋊C4C22×D6112yes—
367C2×C42—611no—
368#108—1222no—
369#106—21no—
370#106—21no—
371#104—1222no—
372#104—1224no—
373#103—22no—
374#103—612no—
375#103—1222no—
376#103—1222no—
377#103—1222no—
378#103—1222no—
379#103—1222no—
380#102—22no—
381#102—612no—
382#102—1222no—
383#101—2442no—
384#97—21no—
385#97—1222no—
386C4×C8—21no—
387#95—2442no—
388#41—22no—
389#41—22no—
390#41—612no—
391#41—612no—
392#41—1228no—
393#41—1221no—
394#41—1221no—
395#41—1228no—
396#41—2444no—
397#41—2448no—
398#41—2444no—
399#41—2448no—
400#41C2×C4×D6114yes—
401#41C2×D24414yes—
402C2×Q8—614no—
403C2×Q8—611no—
404C2×Q8—1228no—
405C2×Q8—1222no—
406C2×Q8—1228no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—2442no—
412C2×Q8—2444no—
413C2×Q8—2444no—
414C2×Q8C2×C22⋊D6112yes—
415C2×D4—22no—
416C2×D4—611no—
417C2×D4—1221no—
418C2×D4—2442no—
419C2×D4C2×C22⋊D6112yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4—1221no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4C23×D6111yescharacteristic
429Q16—1228no—
430Q16—1224no—
431Q16—1224no—
432Q16#1175819814yes—
433SD16—1228no—
434SD16#1175819818yes—
435D8#1175819814yes—
436M16—21no—
437M16—21no—
438M16—612no—
439M16—1221no—
440M16—1224no—
441M16—1222no—
442M16—1222no—
443M16—2444no—
444C2×C8—21no—
445C2×C8—612no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1224no—
450C2×C8—1222no—
451C2×C8—2444no—
452C2×C8C2×C4×D6112yes—
453C2×C8C2×D24412yes—
454C2×C8C2×C22⋊D6111yescharacteristic
455C2×C8C2×C22⋊D6111yescharacteristic
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4D4×D6112yes—
460C22⋊C4—21no—
461C22⋊C4—42no—
462C42—21no—
463C42—42no—
464C42—1221no—
465C42—1221no—
466C23—22no—
467C23—1221no—
468C23#1451982211yescharacteristic
469Q8—22no—
470Q8—612no—
471Q8—1221no—
472Q8—1224no—
473Q8—1224no—
474Q8—2448no—
475Q8—2442no—
476Q8—2442no—
477Q8C2×#1175819814yes—
478D4—24no—
479D4—24no—
480D4—22no—
481D4—612no—
482D4—1222no—
483D4—1222no—
484D4—1221no—
485D4—2448no—
486D4C2×#1175819814yes—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—82no—
493C2×C4—162no—
494C2×C4—1221no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1224no—
498C2×C4—2448no—
499C2×C4—2448no—
500C2×C4—2442no—
501C2×C4—2441no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4C22×C4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4C22×C22⋊D6111yescharacteristic
507C2×C4C22×D24411yescharacteristic
508C2×C4#41×D6111yescharacteristic
509C2×C4#1451982311yescharacteristic
510C8—41no—
511C8—1222no—
512C8—1222no—
513C8—2442no—
514C8C2×#1175819812yes—
515C8C2×#1175819812yes—
516C22—22no—
517C22—24no—
518C22—42no—
519C22—42no—
520C22—1222no—
521C22—2442no—
522C22#1684023411yescharacteristic
523C22D4×C22⋊D6111yescharacteristic
524C22#1684040711yescharacteristic
525C4—24no—
526C4—42no—
527C4—82no—
528C4—1222no—
529C4—4881no—
530C4—2444no—
531C4C22×#1175819811yescharacteristic
532C4C4×D4×D6111yescharacteristic
533C4D4×D24411yescharacteristic
534C4#1684040011yescharacteristic
535C2—21no—
536C2—21no—
537C2—44no—
538C2—2442no—
539C2D4×#1175819811yescharacteristic
540C2#2145759212yes—
subgroupG/H
centerC22D4×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2952C2
socleC22×C61D42
prime radicalC2×#16840576C2
cosocleC22×C4×C61C24
hypercenter#2952D61
hypocenterC61#22465
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2952

Complements

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

NH ≅ G/Ntype
2. C2×#16839884C2semidirect
3. #21454349C2semidirect
4. #21446734C2semidirect
5. #21446688C2semidirect
6. #21446332C2semidirect
7. #21446188C2semidirect
8. #21445879C2semidirect
9. #21445776C2semidirect
10. #21445665C2semidirect
11. C61×#2952C2semidirect
14. #16840576C4semidirect
51. #14519824C2×C4semidirect
60. #14519807C2×C4semidirect
66. C2×Dic244C2×C4semidirect
73. #14519783C2×C4semidirect
267. #2952D61semidirect
318. C61#22465semidirect

Maximal subgroups

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

subgroupindex
C2×#168405762
C2×#168398842
#214543492
#214467342
#214466882
#214463322
#214461882
#214458792
#214457762
#214456652
C61×#29522
#2246561