Finite Group Databasebeta

← #29708631

Subgroups — #29708631

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839800C211yesmaximal, characteristic, complemented
2C2×#16839759C211yesmaximal, characteristic
3C2×#16839759C211yesmaximal, characteristic
4#21448309C211yesmaximal, characteristic
5#21448302C211yesmaximal, characteristic, complemented
6#21447213C212yesmaximal, complemented
7#21447206C212yesmaximal, complemented
8#21447199C212yesmaximal, complemented
9#21447192C212yesmaximal, complemented
10#21445536C211yesmaximal, characteristic, complemented
11C22×C4⋊C8×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16839800—24no—
17C2×#14519711C2212yes—
18C2×#14519709C2212yes—
19#16839759—24no—
20#16839759—24no—
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2212yes—
23C2×#14519702C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519699C2211yescharacteristic
27#16839668C2212yes—
28#16839664C2212yes—
29#16839660C2212yes—
30#16839660C2212yes—
31#16839641—21no—
32#16839641—21no—
33#16839635—21no—
34#16839635—21no—
35C4×#11758196—22no—
36C4×#11758198—22no—
37#16839525C2212yes—
38#16839525C2212yes—
39C23×C8×C61C2211yescharacteristic
40C23×C8×C61C2211yescharacteristic
41C22×C42×C61C2211yescharacteristic
42C2×C4⋊C8×C61—24no—
43C2×C4⋊C8×C61C2212yes—
44C2×C4⋊C8×C61C2212yes—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222Q814yes—
51C22×Dic122C2311yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55#14519736—22no—
56#14519736—22no—
57#14519736—22no—
58#14519736—22no—
59#14519736—22no—
60#14519736D414yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61—21no—
75#14519711—24no—
76#14519709—24no—
77#14519702—22no—
78#14519702—22no—
79#14519702—24no—
80#14519699—22no—
81#14519699—22no—
82#14519699—24no—
83#14519684—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684—22no—
87C23×C4×C61D411yescharacteristic
88C23×C4×C61D411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C22×C8×C61—22no—
91C22×C8×C61—22no—
92C22×C8×C61—22no—
93C22×C8×C61—22no—
94C22×C8×C61—22no—
95C22×C8×C61D414yes—
96C22×C8×C61C2312yes—
97C22×C8×C61C2311yescharacteristic
98C22×C8×C61C2311yescharacteristic
99C2×C42×C61—22no—
100C2×C42×C61—22no—
101C2×C42×C61D412yes—
102C2×C42×C61C2312yes—
103C4⋊C8×C61—24no—
104C4⋊C8×C61—24no—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—24no—
109C22×C4×C61—22no—
110C22×C4×C61—42no—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×Q812yes—
117C22×C4×C61#4112yes—
118C22×C4×C61#4112yes—
119C22×C4×C61#4112yes—
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—24no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129#11758222—28no—
130#11758222—28no—
131#11758222—44no—
132#11758222—44no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244C2×Q812yes—
139C2×C4×D61—24no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122C2×Q812yes—
148C2×C8×C61—28no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—44no—
157C2×C8×C61C2×D412yes—
158C2×C8×C61C2×D412yes—
159C42×C61—24no—
160C42×C61—22no—
161C42×C61—22no—
162C22⋊Dic61—24no—
163C22⋊Dic61—42no—
164C22⋊Dic61—42no—
165#11758198—44no—
166#11758198—44no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61C2×D414yes—
175#11758196—44no—
176#11758196—44no—
177C4×Dic61—24no—
178C4×Dic61—42no—
179C4×Dic61—42no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61#12112yes—
183C23×C61#12112yes—
184C23×C61C22⋊Q812yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—42no—
191C2×C4×C61—28no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—44no—
198C2×C4×C61#12114yes—
199C2×C4×C61#12112yes—
200C2×C4×C61#12112yes—
201C2×C4×C61C22⋊Q814yes—
202C2×C4×C61C22⋊Q814yes—
203C2×C4×C61C22⋊Q812yes—
204C2×C4×C61#12614yes—
205C2×C4×C61#13514yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×Q811yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C22⋊D61—48no—
213C2×Dic61—24no—
214C2×Dic61—22no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226D244—22no—
227D244—22no—
228D244—41no—
229D244—41no—
230C4×D61—48no—
231Dic122—22no—
232Dic122—22no—
233Dic122—41no—
234Dic122—41no—
235C8×C61—22no—
236C8×C61—22no—
237C8×C61—42no—
238#23609—611nomaximal, Sylow
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61#39614yes—
244C22×C61C2×#12111yescharacteristic
245C22×C61C2×#12111yescharacteristic
246C22×C61C2×C22⋊Q811yescharacteristic
247C22×C61C2×#13512yes—
248C22×C61#57512yes—
249C2×D61—44no—
250C2×D61—42no—
251C2×D61—82no—
252C2×D61—81no—
253C2×D61—81no—
254C4×C61—24no—
255C4×C61—42no—
256C4×C61#39614yes—
257C4×C61C2×#12111yescharacteristic
258C4×C61C2×C22⋊Q811yescharacteristic
259C4×C61C2×C22⋊Q811yescharacteristic
260C4×C61C2×#12611yescharacteristic
261Dic61—42no—
262Dic61—44no—
263Dic61—82no—
264Dic61—81no—
265Dic61—81no—
266C2×#427—611no—
267C22×C4⋊C8D6111yescharacteristic, complemented
268C2×#416—611no—
269C2×#416—611no—
270#2297—611no—
271#2296—611no—
272#2068—612no—
273#2067—612no—
274#2066—612no—
275#2065—612no—
276#1735—611no—
277C2×C61—24no—
278C2×C61C2×#39611yescharacteristic
279C2×C61#303512yes—
280C2×C61#305011yescharacteristic
281C2×C61#305411yescharacteristic
282C2×C61#307612yes—
283D61—82no—
284C22×#41—611no—
285C23×C8C2×D6111yescharacteristic
286C23×C8C2×D6111yescharacteristic
287C2×#117—611no—
288C2×#117—611no—
289C2×#117—611no—
290C22×C42C2×D6111yescharacteristic
291#427—1224no—
292C2×#107—612no—
293C2×#106—612no—
294C2×C4⋊C8—24no—
295C2×C4⋊C8C2×D6112yes—
296C2×C4⋊C8C2×D6112yes—
297#416—1224no—
298#416—1224no—
299C2×#103—611no—
300C2×#103—612no—
301C2×#103—611no—
302C2×#102—611no—
303C2×#102—612no—
304C2×#102—611no—
305#391—612no—
306#390—612no—
307#389—612no—
308#389—612no—
309#386—1221no—
310#386—1221no—
311#385—1221no—
312#385—1221no—
313C4×C4⋊C4—1222no—
314C4×C22⋊C4—1222no—
315#340—612no—
316#340—612no—
317C61#2360911yescharacteristic, complemented, Sylow
318C2×#41—614no—
319C2×#41—1222no—
320C2×#41—1222no—
321C2×#41—1222no—
322C2×#41—1222no—
323C2×#41—1222no—
324C22×Q8—611no—
325C22×D4—611no—
326C22×D4—1221no—
327C23×C4—1221no—
328C23×C4D24411yescharacteristic
329C23×C4D24411yescharacteristic
330C23×C4C22×D6111yescharacteristic
331C22×C8—22no—
332C22×C8—22no—
333C22×C8—22no—
334C22×C8—22no—
335C22×C8—22no—
336C22×C8C22⋊D6114yes—
337C22×C8C22×D6112yes—
338C22×C8C22×D6111yescharacteristic
339C22×C8C22×D6111yescharacteristic
340#117—614no—
341#117—1222no—
342#117—1222no—
343#117—1222no—
344#117—1222no—
345#117—1222no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1222no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1222no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—1221no—
360C2×C42—1221no—
361C2×C42—1221no—
362C2×C42D24412yes—
363C2×C42C22×D6112yes—
364#107—1224no—
365#106—1224no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368#103—1224no—
369#103—1222no—
370#103—1222no—
371#102—1224no—
372#102—1222no—
373#102—1222no—
374#95—1222no—
375#95—1222no—
376#95—1222no—
377#95—1222no—
378C24—1221no—
379C24C2×D24411yescharacteristic
380#41—1228no—
381#41—1228no—
382#41—2444no—
383#41—2444no—
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×D4—612no—
391C2×D4—1224no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—2442no—
398C2×D4—2442no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4—1224no—
406C22×C4—1222no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4C2×D24412yes—
412C22×C4C2×D24412yes—
413C22×C4C2×D24411yescharacteristic
414C22×C4#1175822212yes—
415C22×C4D4×D6112yes—
416C22×C4#1175822412yes—
417C22×C4Q8×D6112yes—
418C22×C4#1175822612yes—
419C22×C4C2×C22⋊D6112yes—
420C22×C4C23×D6111yescharacteristic
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—28no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—44no—
430C2×C8C2×C22⋊D6112yes—
431C2×C8C2×C22⋊D6112yes—
432C4⋊C4—614no—
433C4⋊C4—1222no—
434C4⋊C4—1222no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—1222no—
440C4⋊C4—2444no—
441C4⋊C4—2444no—
442C22⋊C4—1224no—
443C22⋊C4—2444no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2444no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—1224no—
451C42—2442no—
452C42—2442no—
453C23—22no—
454C23—22no—
455C23—1222no—
456C23—2441no—
457C23—2441no—
458C23—2442no—
459C23—2442no—
460C23#1451975112yes—
461C23#1451975212yes—
462C23#1451975712yes—
463C23C22×D24411yescharacteristic
464Q8—1222no—
465Q8—1222no—
466Q8—2441no—
467Q8—2441no—
468D4—1222no—
469D4—1222no—
470D4—2448no—
471D4—2441no—
472D4—2441no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—44no—
480C2×C4—1224no—
481C2×C4—1222no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2448no—
494C2×C4#1451974914yes—
495C2×C4#1451975112yes—
496C2×C4#1451975212yes—
497C2×C4#1451975314yes—
498C2×C4#1451975514yes—
499C2×C4#1451975712yes—
500C2×C4#1451977214yes—
501C2×C4#1451979914yes—
502C2×C4C2×D4×D6111yescharacteristic
503C2×C4C22×C22⋊D6111yescharacteristic
504C2×C4C2×Q8×D6111yescharacteristic
505C2×C4C2×#1175822411yescharacteristic
506C2×C4C2×#1175822211yescharacteristic
507C2×C4C2×#1175822611yescharacteristic
508C8—22no—
509C8—22no—
510C8—42no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—24no—
515C22—4881no—
516C22—4881no—
517C22—4882no—
518C22—2442no—
519C22—2444no—
520C22#1683968414yes—
521C22C2×#1451975111yescharacteristic
522C22C2×#1451975211yescharacteristic
523C22C2×#1451975711yescharacteristic
524C22C2×#1451979912yes—
525C22#1684056212yes—
526C4—24no—
527C4—42no—
528C4—4881no—
529C4—4881no—
530C4—4882no—
531C4—2442no—
532C4—2444no—
533C4#1683968414yes—
534C4C2×#1451974911yescharacteristic
535C4C2×#1451975311yescharacteristic
536C4C2×#1451975511yescharacteristic
537C4C2×#1451977211yescharacteristic
538C2—24no—
539C2—4882no—
540C2C2×#1683968411yescharacteristic
541C2#2145527612yes—
542C2#2145540011yescharacteristic
543C2#2145542811yescharacteristic
544C2#2145560912yes—
subgroupG/H
centerC22×C4C2×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×C4⋊C8×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC22×C4⋊C8D61
hypocenterC61#23609
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#16839800C2semidirect
5. #21448302C2semidirect
6. #21447213C2semidirect
7. #21447206C2semidirect
8. #21447199C2semidirect
9. #21447192C2semidirect
10. #21445536C2semidirect
11. C22×C4⋊C8×C61C2semidirect
267. C22×C4⋊C8D61semidirect
317. C61#23609semidirect

Maximal subgroups

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

subgroupindex
C2×#168398002
C2×#168397592
#214483092
#214483022
#214472132
#214472062
#214471992
#214471922
#214455362
C22×C4⋊C8×C612
#2360961