Finite Group Databasebeta

← #29714085

Subgroups — #29714085

Order 15616.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C2×#16839864C211yesmaximal, characteristic, complemented
4#21450720C211yesmaximal, characteristic
5#21447016C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446792C211yesmaximal, characteristic, complemented
8#21446139C212yesmaximal, complemented
9#21446020C212yesmaximal, complemented
10#21445835C211yesmaximal, characteristic, complemented
11C61×#1955C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13#16840482—24no—
14#16840482—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519770—22no—
17C2×#14519764—21no—
18C2×#14519756C2212yes—
19C2×#14519751C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519742—21no—
22C2×C4×D244C2211yescharacteristic
23C2×#14519736C2212yes—
24#16840053—24no—
25#16839864—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519709C2212yes—
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2212yes—
32#16839688—21no—
33#16839670—21no—
34#16839638—22no—
35#16839628C2211yescharacteristic
36#16839622C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C22⋊C8×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—22no—
49C22×C4×D61—21no—
50C2×#11758190—22no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53#14519777—22no—
54#14519777D414yescomplemented
55#14519770—216no—
56#14519764—28no—
57#14519756—24no—
58#14519756—24no—
59#14519751—24no—
60#14519751—42no—
61C4×Dic122—22no—
62C4×Dic122D414yescomplemented
63#14519742—28no—
64C4×D244—22no—
65C4×D244—22no—
66#14519736—24no—
67#14519736—24no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—22no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—21no—
82#14519711—22no—
83#14519709—24no—
84#14519702—22no—
85#14519702—42no—
86#14519699—24no—
87#14519699—22no—
88#14519684—22no—
89#14519684—21no—
90#14519684—21no—
91C22×D4×C61D411yescharacteristic
92C23×C4×C61C2312yes—
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C22×C8×C61C2311yescharacteristic
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C22⋊C4×C61D412yes—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102C22⋊C8×C61—24no—
103C22⋊C8×C61—42no—
104#97×C61—22no—
105C24×C61C2×D412yes—
106C23×D61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—42no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—44no—
120C2×C22⋊D61—44no—
121C2×C22⋊D61—82no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244#4112yes—
133C2×C4×D61—42no—
134C2×C4×D61—82no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—24no—
140C2×Dic122—21no—
141C2×Dic122—44no—
142C2×Dic122C2×D412yes—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—21no—
149C2×C8×C61—22no—
150C2×C8×C61—21no—
151C2×C8×C61#4112yes—
152C4⋊C4×C61—22no—
153C22⋊C4×C61—24no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—28no—
157C22⋊Dic61—28no—
158C22⋊Dic61—82no—
159#11758198—44no—
160#11758198—42no—
161C4⋊Dic61—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D414yes—
169C4⋊Dic61C2×D412yes—
170#11758196—28no—
171#11758196—24no—
172C4×Dic61—28no—
173C4×Dic61—24no—
174#11758190—28no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—42no—
187C22×D61—42no—
188D4×C61—44no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61#12012yes—
199C2×C4×C61#12012yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61H(5,𝔽2)11yescharacteristic
205C2×C4×C612−1+411yescharacteristic
206C22⋊D61—88no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—24no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—44no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219D244—22no—
220D244—22no—
221D244—41no—
222D244—41no—
223C4×D61—84no—
224Dic122—22no—
225Dic122—22no—
226Dic122—48no—
227Dic122—41no—
228Dic122—41no—
229C8×C61—22no—
230C8×C61—41no—
231C8×C61—42no—
232#24131—611nomaximal, Sylow
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61C2×#12011yescharacteristic
241C22×C61#54611yescharacteristic
242C22×C61#54711yescharacteristic
243C22×C61#57512yes—
244C22×C61#57712yes—
245C2×D61—42no—
246C2×D61—44no—
247C2×D61—82no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61#52511yescharacteristic
254C4×C61#53711yescharacteristic
255C4×C61#54811yescharacteristic
256C4×C61#55011yescharacteristic
257Dic61—42no—
258Dic61—44no—
259Dic61—42no—
260Dic61—81no—
261Dic61—82no—
262C2×#556—611no—
263C2×#476—611no—
264C2×#440—611no—
265#2535—611no—
266#2035—612no—
267#2000—612no—
268#1993—611no—
269#1955D6111yescharacteristic, complemented
270#1875—612no—
271#1854—612no—
272#1805—611no—
273C2×C61—82no—
274C2×C61#240711yescharacteristic
275C2×C61#300511yescharacteristic
276C2×C61#301011yescharacteristic
277C2×C61#331112yes—
278C2×C61#335012yes—
279D61—82no—
280C22×SD16—611no—
281#556—1224no—
282#556—1224no—
283C2×C4⋊Q8—611no—
284C2×#126—1222no—
285C2×#124—1221no—
286C2×C22⋊Q8—612no—
287C2×#121—612no—
288C2×C4×Q8—611no—
289C2×C4×Q8—1221no—
290C2×C4×D4—611no—
291C2×C4×D4C2×D6111yescharacteristic
292C2×#117—612no—
293#476—1224no—
294#440—1224no—
295C2×#107—611no—
296C2×#106—612no—
297C2×C4⋊C8C2×D6111yescharacteristic
298C2×#103—611no—
299C2×#103—612no—
300C2×#102—611no—
301C2×#102—612no—
302C2×C22⋊C8C2×D6112yes—
303C2×C22⋊C8C2×D6112yes—
304C2×#97C2×D6111yescharacteristic
305#397—1221no—
306#392—1221no—
307#386—1222no—
308#382—611no—
309#380—612no—
310#340—612no—
311#340—612no—
312#340—612no—
313C61#2413111yescharacteristic, complemented, Sylow
314C22×Q8—611no—
315C22×Q8—1221no—
316C22×D4—611no—
317C22×D4—1222no—
318C22×D4D24411yescharacteristic
319C23×C4—1221no—
320C23×C4C22×D6112yes—
321C2×SD16—1222no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C22×C8C22×D6111yescharacteristic
325C22×C8C22×D6112yes—
326C22×C8C22×D6111yescharacteristic
327C4⋊Q8—614no—
328C4⋊Q8—1222no—
329#126—12216no—
330#124—1228no—
331C22⋊Q8—1224no—
332C22⋊Q8—1224no—
333#121—1224no—
334#121—2442no—
335C4×Q8—614no—
336C4×Q8—1228no—
337C4×Q8—1222no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—1222no—
341C4×D4—1222no—
342#117—1224no—
343#117—1224no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4D24411yescharacteristic
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4D24412yes—
358C2×C42—1222no—
359C2×C42—1221no—
360C2×C42C22×D6111yescharacteristic
361#107—1222no—
362#106—1224no—
363C4⋊C8—22no—
364#103—1222no—
365#103—2442no—
366#102—1222no—
367#102—1224no—
368C22⋊C8—24no—
369C22⋊C8—42no—
370#97—22no—
371#95—1222no—
372#95—1221no—
373#95—1221no—
374C24—1221no—
375C24C2×D24412yes—
376C2×Q8—612no—
377C2×Q8—1221no—
378C2×Q8—1224no—
379C2×Q8—1221no—
380C2×Q8—1221no—
381C2×Q8—1221no—
382C2×Q8—1221no—
383C2×Q8—2444no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—612no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—4882no—
393C2×D4—2444no—
394C2×D4—2444no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—1221no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—4882no—
407C22×C4—2442no—
408C22×C4C2×D24412yes—
409C22×C4C2×D24411yescharacteristic
410C22×C4C2×D24411yescharacteristic
411C22×C4C23×D6111yescharacteristic
412SD16—1228no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8#1175822412yes—
422C4⋊C4—22no—
423C4⋊C4—614no—
424C4⋊C4—612no—
425C4⋊C4—1228no—
426C4⋊C4—1221no—
427C4⋊C4—1222no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C22⋊C4—24no—
435C22⋊C4—1228no—
436C22⋊C4—1228no—
437C22⋊C4—4882no—
438C22⋊C4—2444no—
439C22⋊C4—2442no—
440C42—1228no—
441C42—1224no—
442C42D4×D6112yes—
443C42D4×D6112yes—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—1222no—
452C23—2441no—
453C23—2442no—
454C23—2442no—
455C23—2441no—
456C23C22×D24411yescharacteristic
457Q8—1222no—
458Q8—1222no—
459Q8—2448no—
460Q8—2441no—
461Q8—2441no—
462D4—44no—
463D4—1222no—
464D4—1222no—
465D4—4888no—
466D4—2441no—
467D4—2441no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—1222no—
478C2×C4—1224no—
479C2×C4—1222no—
480C2×C4—4884no—
481C2×C4—2441no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2444no—
487C2×C4—2442no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4#1451974712yes—
491C2×C4#1451974712yes—
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4C2×#1175822411yescharacteristic
496C2×C4#1451982211yescharacteristic
497C2×C4#1451982411yescharacteristic
498C8—22no—
499C8—41no—
500C8—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—4881no—
509C22—4882no—
510C22—4881no—
511C22—2444no—
512C22—2442no—
513C22C2×#1451974711yescharacteristic
514C22#1684040711yescharacteristic
515C22#1684041611yescharacteristic
516C22#1684056212yes—
517C22#1684057412yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—4881no—
522C4—4882no—
523C4—2442no—
524C4—2442no—
525C4—2444no—
526C4#1684027211yescharacteristic
527C4#1684034111yescharacteristic
528C4#1684042011yescharacteristic
529C4#1684043211yescharacteristic
530C2—82no—
531C2—4882no—
532C2#2144916211yescharacteristic
533C2#2145506411yescharacteristic
534C2#2145510711yescharacteristic
535C2#2145785712yes—
536C2#2145836612yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1955C2
socleC23×C61C22×D4
prime radical#21450720C2
hypercenter#1955D61
hypocenterC61#24131
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#16840482C2semidirect
2. C2×#16840053C2semidirect
3. C2×#16839864C2semidirect
5. #21447016C2semidirect
6. #21446824C2semidirect
7. #21446792C2semidirect
8. #21446139C2semidirect
9. #21446020C2semidirect
10. #21445835C2semidirect
11. C61×#1955C2semidirect
54. #14519777D4semidirect
62. C4×Dic122D4semidirect
269. #1955D61semidirect
313. C61#24131semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
C2×#168400532
C2×#168398642
#214507202
#214470162
#214468242
#214467922
#214461392
#214460202
#214458352
C61×#19552
#2413161