Finite Group Databasebeta

← #29650165

Subgroups — #29650165

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455856C211yesmaximal, characteristic
2#21454915C211yesmaximal, characteristic
3#21454898C211yesmaximal, characteristic
4#21444690C211yesmaximal, characteristic, complemented
5#21444688C211yesmaximal, characteristic, complemented
6#21444671C211yesmaximal, characteristic, complemented
7#21444635C211yesmaximal, characteristic, complemented
8#21444635C211yesmaximal, characteristic, complemented
9Q16×#92418C211yesmaximal, characteristic, complemented
10#21444625C211yesmaximal, characteristic, complemented
11#21444598C211yesmaximal, characteristic
12#21444264C211yesmaximal, characteristic
13#21444238C211yesmaximal, characteristic
14#21444115C211yesmaximal, characteristic
15C61×#1584C211yesmaximal, characteristic, complemented
16#16840316C2211yescharacteristic
17C4×Q8×D61C2211yescharacteristic
18#16840099C411yescharacteristic
19#16840096C411yescharacteristic
20#16840056C411yescharacteristic
21#16840055C411yescharacteristic
22#16839971C411yescharacteristic
23#16839970C411yescharacteristic
24#16839919C411yescharacteristic
25#16839916C411yescharacteristic
26#16839892C2211yescharacteristic
27#16839890—21no—
28#16839889—21no—
29Q8×#92418C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31C8×Dic122—21no—
32C8×Dic122C2211yescharacteristic
33#16839886—21no—
34#16839886C2211yescharacteristic
35#16839886C2211yescharacteristic
36C4×Dic244C2211yescharacteristic
37C4×#11758202C2211yescharacteristic
38C4×#11758202C2211yescharacteristic
39#16839824—21no—
40#16839822—21no—
41#16839818C2211yescharacteristic
42#16839814—21no—
43#16839814C2211yescharacteristic
44#16839813—21no—
45#16839813C2211yescharacteristic
46#16839813C2211yescharacteristic
47C4×#11758189C2211yescharacteristic
48#16839515C2211yescharacteristic
49#16839512C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839494C2211yescharacteristic
52#16839494C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54Q8⋊#92418C2211yescharacteristic
55#16839492C2211yescharacteristic
56#16839492C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839484C2211yescharacteristic
59C8×#92418C2211yescharacteristic
60C61×#446C2211yescharacteristic
61C8×Q8×C61C2211yescharacteristic
62C4×Q16×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×Q8⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic
67C2×Q8×D61C2×C411yescharacteristic
68C2×Dic244C2×C411yescharacteristic
69C2×#11758202C2×C411yescharacteristic
70C2×#11758202C2×C411yescharacteristic
71C2×#11758189—21no—
72#14519776C2×C411yescharacteristic
73#14519775C2×C411yescharacteristic
74#14519767—21no—
75#14519757C2×C411yescharacteristic
76#14519755—21no—
77#14519753C2×C411yescharacteristic
78Q8×Dic61C2×C411yescharacteristic
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82C4×Dic122C2311yescharacteristic
83#14519742—21no—
84#14519742C2×C411yescharacteristic
85#14519734—21no—
86C4⋊C4×D61—21no—
87C4⋊C4×D61C2×C411yescharacteristic
88C42×D61—21no—
89C42×D61C2311yescharacteristic
90#14519711C2×C411yescharacteristic
91#14519710C2×C411yescharacteristic
92#14519710C2×C411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519706—21no—
98#14519706—21no—
99#14519706—21no—
100#14519706—21no—
101#14519706C2×C411yescharacteristic
102#14519706C2×C411yescharacteristic
103#14519706D411yescharacteristic
104#14519706D411yescharacteristic
105#14519702C2×C411yescharacteristic
106#14519702C2×C411yescharacteristic
107#14519701C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519700C2×C411yescharacteristic
110#14519700C2×C411yescharacteristic
111#14519689—21no—
112#14519689—21no—
113#14519689C2×C411yescharacteristic
114#14519689C2×C411yescharacteristic
115#14519689D411yescharacteristic
116#14519689D411yescharacteristic
117#14519687—21no—
118#14519687—21no—
119#14519687—21no—
120C4×#92418—21no—
121C4×#92418—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×Q16×C61C2×C411yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132#107×C61C2×C411yescharacteristic
133C4⋊C8×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61C2×C411yescharacteristic
138#103×C61C2×C411yescharacteristic
139#97×C61—21no—
140C4×C8×C61—21no—
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146Q8×D61—22no—
147Q8×D61—22no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61C22×C411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61#4111yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C22×C411yescharacteristic
156C2×Dic122C22×C411yescharacteristic
157Q16×C61—22no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C22×C411yescharacteristic
162C2×C8×C61C22×C411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D411yescharacteristic
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C22×C411yescharacteristic
168C4⋊C4×C61C22×C411yescharacteristic
169C42×C61—21no—
170C42×C61—21no—
171C42×C61C2411yescharacteristic
172#11758202—22no—
173#11758202—22no—
174#11758198—21no—
175#11758198—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C22×C411yescharacteristic
179C4⋊Dic61C22×C411yescharacteristic
180#11758196—21no—
181#11758196—21no—
182#11758196—21no—
183#11758196—21no—
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61C22×C411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61#4111yescharacteristic
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C411yescharacteristic
199C2×#92418C22×C411yescharacteristic
200C2×#92418C2×D411yescharacteristic
201C2×#92418C2×D411yescharacteristic
202Dic244—22no—
203#11758189—42no—
204C22×D61C4×D411yescharacteristic
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61#13112yes—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61C23×C411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61C4×D411yescharacteristic
221C2×Dic61C4×D411yescharacteristic
222C2×Dic61C4×D411yescharacteristic
223C4×D61—24no—
224C4×D61—21no—
225C4×D61—21no—
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—41no—
229C4×D61—81no—
230Dic122—24no—
231Dic122—21no—
232Dic122—21no—
233Dic122—21no—
234Dic122—81no—
235Dic122#13112yes—
236C8×C61—21no—
237C8×C61—21no—
238C8×C61—21no—
239C8×C61—21no—
240C8×C61—41no—
241C8×C61—41no—
242C8×C61—41no—
243#92418—22no—
244#92418—21no—
245#92418—21no—
246#92418—21no—
247#92418—21no—
248#92418—41no—
249#92418—41no—
250#17925—611nomaximal, Sylow
251C22×C61C2×C4×D411yescharacteristic
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—21no—
255C4×C61—22no—
256C4×C61—21no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61C2×#13111yescharacteristic
261C4×C61C2×#13111yescharacteristic
262C4×C61#57811yescharacteristic
263C4×C61#57811yescharacteristic
264Dic61—21no—
265Dic61—22no—
266Dic61—21no—
267Dic61—21no—
268Dic61—41no—
269Dic61—41no—
270Dic61—41no—
271#3104—611no—
272#2985—611no—
273#2982—611no—
274#1587—611no—
275#1587—611no—
276#1584—611no—
277#1584D6111yescharacteristic, complemented
278#1579—611no—
279#1579—611no—
280C8×Q16—611no—
281C8×Q16—611no—
282#1572—611no—
283#1491—611no—
284#1485—611no—
285#1453—611no—
286C2×C61#292911yescharacteristic
287C2×C61#295011yescharacteristic
288C2×C61#295611yescharacteristic
289D61—41no—
290#533—611no—
291C2×C4×Q8—611no—
292#484—611no—
293#484—611no—
294#477—611no—
295#477—611no—
296#462—611no—
297#462—611no—
298#451—611no—
299#451—611no—
300#446—611no—
301#446—1221no—
302#446—1221no—
303#446C2×D6111yescharacteristic
304C8×Q8—611no—
305C8×Q8—611no—
306C8×Q8—611no—
307C8×Q8—611no—
308C8×Q8—611no—
309C8×Q8—1221no—
310C8×Q8—1221no—
311C8×Q8C2×D6111yescharacteristic
312C4×Q16—611no—
313C4×Q16—611no—
314C4×Q16—611no—
315C4×Q16C2×D6111yescharacteristic
316#433—1221no—
317#433—1221no—
318#432—611no—
319#431—611no—
320#431—611no—
321#431—611no—
322#431—1221no—
323#431—1221no—
324C4×M16—611no—
325#335—611no—
326#335C2×D6111yescharacteristic
327#334—611no—
328#334—611no—
329Q8⋊C8—611no—
330Q8⋊C8—611no—
331Q8⋊C8—611no—
332Q8⋊C8—611no—
333Q8⋊C8—611no—
334Q8⋊C8—611no—
335Q8⋊C8C2×D6111yescharacteristic
336Q8⋊C8C2×D6111yescharacteristic
337#322—611no—
338#322—611no—
339#322C2×D6111yescharacteristic
340C82—611no—
341C61#1792511yescharacteristic, complemented, Sylow
342C22×Q8—611no—
343C2×Q16—611no—
344C2×Q16—611no—
345C2×Q16—611no—
346C2×Q16C4×D6111yescharacteristic
347C2×M16—1221no—
348C4⋊Q8—611no—
349C4⋊Q8—611no—
350#125—1221no—
351C22⋊Q8—611no—
352C22⋊Q8—611no—
353C22⋊Q8—1221no—
354C4×Q8—611no—
355C4×Q8—611no—
356C4×Q8—611no—
357C4×Q8—611no—
358C4×Q8—1221no—
359C4×Q8—1221no—
360C4×Q8—1221no—
361C4×Q8C22×D6111yescharacteristic
362C4×Q8C22×D6111yescharacteristic
363#117—1221no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—1221no—
366C2×C42—611no—
367C2×C42—1221no—
368#107—611no—
369#107—611no—
370#107—611no—
371#107C4×D6111yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—611no—
375C4⋊C8—611no—
376C4⋊C8—611no—
377C4⋊C8—611no—
378C4⋊C8—611no—
379C4⋊C8—611no—
380C4⋊C8—1221no—
381C4⋊C8—1221no—
382C4⋊C8—1221no—
383C4⋊C8—1221no—
384C4⋊C8—1221no—
385C4⋊C8—1221no—
386C4⋊C8C22×D6111yescharacteristic
387C4⋊C8C22×D6111yescharacteristic
388#103—611no—
389#103—611no—
390#103—611no—
391#103—611no—
392#103—611no—
393#103—611no—
394#103C4×D6111yescharacteristic
395#103C4×D6111yescharacteristic
396C22⋊C8—611no—
397C22⋊C8—611no—
398C22⋊C8—611no—
399C22⋊C8—611no—
400C22⋊C8—1221no—
401C22⋊C8—1221no—
402#97—21no—
403#97—1221no—
404#97—1221no—
405#97—1221no—
406C4×C8—21no—
407C4×C8—611no—
408C4×C8—611no—
409C4×C8—611no—
410C4×C8—1221no—
411C4×C8—1221no—
412C4×C8—1221no—
413C4×C8—1221no—
414C4×C8—1221no—
415C4×C8—1221no—
416C4×C8C22×D6111yescharacteristic
417C4×C8C22×D6111yescharacteristic
418C4×C8C22×D6111yescharacteristic
419C2×Q8—611no—
420C2×Q8—611no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1222no—
424C2×Q8—1222no—
425C2×Q8C2×C4×D6111yescharacteristic
426C2×Q8C2×C4×D6111yescharacteristic
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—1221no—
431C22×C4—1221no—
432Q16—22no—
433Q16—1222no—
434Q16—1222no—
435Q16—1222no—
436M16—2442no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—611no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—611no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8C2×C4×D6111yescharacteristic
448C2×C8C2×C4×D6111yescharacteristic
449C2×C8D4×D6111yescharacteristic
450C2×C8D4×D6111yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—611no—
454C4⋊C4—611no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4C2×C4×D6111yescharacteristic
466C4⋊C4C2×C4×D6111yescharacteristic
467C22⋊C4—1221no—
468C22⋊C4—1221no—
469C42—21no—
470C42—21no—
471C42—611no—
472C42—611no—
473C42—611no—
474C42—1221no—
475C42—1221no—
476C42—1221no—
477C42—1221no—
478C42C23×D6111yescharacteristic
479C23—611no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—612no—
484Q8—1224no—
485Q8—1221no—
486Q8—1221no—
487Q8—1221no—
488Q8—4881no—
489Q8#1451978512yes—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—611no—
497C2×C4—1224no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—4881no—
507C2×C4—2441no—
508C2×C4C22×C4×D6111yescharacteristic
509C2×C4C2×D4×D6111yescharacteristic
510C2×C4#41×D6111yescharacteristic
511C8—21no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—1222no—
519C8—1221no—
520C8—1221no—
521C8—1221no—
522C8—1221no—
523C8—2441no—
524C8—2441no—
525C22—1221no—
526C22—1221no—
527C22—1221no—
528C22C4×D4×D6111yescharacteristic
529C4—21no—
530C4—22no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—1221no—
535C4—1221no—
536C4—1221no—
537C4—1222no—
538C4—2441no—
539C4—2441no—
540C4—2441no—
541C4#131×D6111yescharacteristic
542C4C2×#1451978511yescharacteristic
543C4#1684057511yescharacteristic
544C4#1684057611yescharacteristic
545C2—2441no—
546C2#2145451011yescharacteristic
547C2#2145468211yescharacteristic
548C2#2145472711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1584C2
socleC22×C61C2×C4×D4
prime radicalC22×D61C4×D4
cosocleC42×C61C24
hypercenter#1584D61
hypocenterC61#17925
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #21444690C2semidirect
5. #21444688C2semidirect
6. #21444671C2semidirect
7. #21444635C2semidirect
8. #21444635C2semidirect
9. Q16×#92418C2semidirect
10. #21444625C2semidirect
15. C61×#1584C2semidirect
277. #1584D61semidirect
341. C61#17925semidirect

Maximal subgroups

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

subgroupindex
#214558562
#214549152
#214548982
#214446902
#214446882
#214446712
#214446352
Q16×#924182
#214446252
#214445982
#214442642
#214442382
#214441152
C61×#15842
#1792561