Finite Group Databasebeta

← #29714077

Subgroups — #29714077

Order 15616.

1184 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×#16840472C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3C2×#16839860C211yesmaximal, characteristic, complemented
4#21450721C211yesmaximal, characteristic
5#21447021C212yesmaximal, complemented
6#21446821C212yesmaximal, complemented
7#21446791C211yesmaximal, characteristic, complemented
8#21446616C211yesmaximal, characteristic, complemented
9#21446135C212yesmaximal, complemented
10#21446008C212yesmaximal, complemented
11C61×#1805C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519755—22no—
18C2×#14519753C2212yes—
19C2×#14519752C2212yes—
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24#16840043—24no—
25#16839860—24no—
26C2×#14519711C2212yes—
27C2×#14519710C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519699C2212yes—
30C2×#14519697C2211yescharacteristic
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839689—21no—
35#16839672—21no—
36#16839640—22no—
37#16839619C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#382C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—21no—
46C22×D244—22no—
47C22×D244D411yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C22×C4×D61C2312yes—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C22×#92418C2311yescharacteristic
56C22×#92418C2311yescharacteristic
57#14519777—22no—
58#14519777D414yescomplemented
59#14519769—28no—
60#14519755—216no—
61#14519753—24no—
62#14519753—24no—
63#14519752—24no—
64#14519752—42no—
65C4×D244—28no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—24no—
78C2×#11758198—22no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83#14519711—24no—
84#14519710—22no—
85#14519701—22no—
86#14519699—24no—
87#14519697—22no—
88#14519689—24no—
89#14519689—42no—
90#14519688—22no—
91#14519684—22no—
92C22×Q8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4×Q8×C61—22no—
95C4×Q8×C61D414yescomplemented
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102#103×C61—42no—
103#95×C61—21no—
104#95×C61—21no—
105C23×D61—21no—
106C23×D61C2×D412yes—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61C2×D412yes—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61C2×D412yes—
122C2×D244—24no—
123C2×D244—22no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—44no—
130C2×D244—44no—
131C2×D244—82no—
132C2×D244#4112yes—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—28no—
140C2×C4×D61—42no—
141C2×C4×D61—82no—
142C2×C4×D61—82no—
143C2×C4×D61—82no—
144C2×Dic122—24no—
145C2×Dic122—82no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D412yes—
154C42×C61—24no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157#11758201—28no—
158#11758198—28no—
159#11758198—24no—
160#11758198—44no—
161#11758198—42no—
162#11758198—82no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61C2×D414yes—
169#11758196—28no—
170#11758196—28no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418#4112yes—
177C23×C61C22×D411yescharacteristic
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61#13714yes—
190Q8×C61—22no—
191Q8×C61—22no—
192Q8×C61—41no—
193Q8×C61—41no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61#12012yes—
205C2×C4×C61#12012yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220D244—28no—
221D244—22no—
222D244—22no—
223D244—41no—
224D244—41no—
225D244—88no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—88no—
231C4×D61—84no—
232Dic122—84no—
233C8×C61—42no—
234#92418—22no—
235#92418—41no—
236#24130—611nomaximal, Sylow
237C22×C61C2×#12011yescharacteristic
238C22×C61#54611yescharacteristic
239C22×C61#54711yescharacteristic
240C22×C61C2×#13712yes—
241C22×C61#57612yes—
242C2×D61—24no—
243C2×D61—24no—
244C2×D61—24no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—44no—
251C2×D61—82no—
252C2×D61—81no—
253C2×D61—81no—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—82no—
259C4×C61#52511yescharacteristic
260C4×C61#53711yescharacteristic
261C4×C61#54811yescharacteristic
262C4×C61#55011yescharacteristic
263Dic61—42no—
264Dic61—44no—
265Dic61—81no—
266C2×#554—611no—
267C2×#474—611no—
268C2×#440—611no—
269#2535—611no—
270#2036—612no—
271#2000—612no—
272#1993—611no—
273#1955—611no—
274#1874—612no—
275#1852—612no—
276#1805D6111yescharacteristic, complemented
277C2×C61#240711yescharacteristic
278C2×C61#300411yescharacteristic
279C2×C61#300811yescharacteristic
280C2×C61#331412yes—
281C2×C61#334612yes—
282D61—44no—
283D61—82no—
284C22×SD16—611no—
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8—611no—
288C2×#125—1221no—
289C2×C22⋊Q8—612no—
290C2×C22⋊Q8—1222no—
291C2×#121—612no—
292C2×C4×Q8C2×D6111yescharacteristic
293C2×C4×D4—611no—
294C2×C4×D4—611no—
295C2×C4×D4—1221no—
296C22×C4⋊C4—612no—
297#474—1224no—
298#440—1224no—
299C2×#107—612no—
300C2×#107—611no—
301C2×C4⋊C8C2×D6111yescharacteristic
302C2×#103—611no—
303C2×#103C2×D6112yes—
304C2×#102—612no—
305C2×#102—611no—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97—611no—
309#397—1221no—
310#392—1221no—
311#386—1222no—
312#382C2×D6111yescharacteristic
313#380—612no—
314#340—612no—
315#340—612no—
316#340C2×D6112yes—
317C61#2413011yescharacteristic, complemented, Sylow
318C22×Q8—1221no—
319C22×Q8C22×D6111yescharacteristic
320C22×D4—611no—
321C22×D4—611no—
322C22×D4—1222no—
323C23×C4—612no—
324C23×C4—1222no—
325C23×C4—1221no—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×SD16—1222no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8C22×D6112yes—
332C4⋊Q8—614no—
333C4⋊Q8—1222no—
334#125—1228no—
335C22⋊Q8—12216no—
336C22⋊Q8—1224no—
337C22⋊Q8—1224no—
338#121—1224no—
339#121—2442no—
340C4×Q8—22no—
341C4×Q8D24414yescomplemented
342C4×D4—1228no—
343C4×D4—1222no—
344C4×D4—1222no—
345C4×D4—1222no—
346C4×D4—1222no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4C22×D6112yes—
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—612no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1221no—
363C2×C42—21no—
364C2×C42C22×D6111yescharacteristic
365#107—1224no—
366#107—1222no—
367C4⋊C8—22no—
368#103—42no—
369#103—1222no—
370#102—1224no—
371#102—1222no—
372C22⋊C8—1224no—
373C22⋊C8—2442no—
374#97—1222no—
375#95—21no—
376#95—21no—
377#95—1222no—
378C24—612no—
379C24—1221no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—1224no—
386C2×Q8—4882no—
387C2×Q8C2×D24412yes—
388C2×D4—612no—
389C2×D4—1224no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1222no—
396C2×D4—4882no—
397C2×D4—2444no—
398C2×D4—2444no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—612no—
403C22×C4—1228no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—4882no—
413C22×C4—4882no—
414C22×C4—4882no—
415C22×C4—2442no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419SD16—1228no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—612no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C2×C8—1221no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—614no—
434C4⋊C4—1228no—
435C4⋊C4—1228no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—1224no—
439C4⋊C4—1221no—
440C4⋊C4—1221no—
441C4⋊C4C2×D24412yes—
442C22⋊C4—1228no—
443C22⋊C4—1224no—
444C22⋊C4—4882no—
445C22⋊C4—2444no—
446C22⋊C4—2442no—
447C42—24no—
448C42C2×D24412yes—
449C42D4×D6112yes—
450C23—614no—
451C23—1222no—
452C23—1222no—
453C23—1222no—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23—2441no—
459C23—2442no—
460C23—2442no—
461C23—2441no—
462C23C2×D4×D6111yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—4884no—
468D4—1228no—
469D4—1222no—
470D4—1222no—
471D4—4888no—
472D4—2441no—
473D4—2441no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—1224no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—4884no—
489C2×C4—4888no—
490C2×C4—2441no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4#1451974712yes—
499C2×C4#1451974712yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C22×D24411yescharacteristic
502C2×C4#41×D6111yescharacteristic
503C2×C4#41×D6111yescharacteristic
504C2×C4#1451982211yescharacteristic
505C2×C4#1451982511yescharacteristic
506C8—42no—
507C8—1222no—
508C8—2441no—
509C22—1224no—
510C22—1224no—
511C22—1224no—
512C22—4882no—
513C22—4881no—
514C22—4881no—
515C22—2442no—
516C22—2444no—
517C22—2442no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22C2×#1451974711yescharacteristic
522C22#1684040911yescharacteristic
523C22#1684041411yescharacteristic
524C22#137×D6112yes—
525C22#1684056412yes—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—82no—
531C4—4881no—
532C4—2442no—
533C4—2444no—
534C4#1684027411yescharacteristic
535C4#1684034311yescharacteristic
536C4#1684042211yescharacteristic
537C4#1684043011yescharacteristic
538C2—4882no—
539C2—2444no—
540C2#2144916511yescharacteristic
541C2#2145505511yescharacteristic
542C2#2145509011yescharacteristic
543C2#2145790712yes—
544C2#2145831112yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radical#21450721C2
hypercenter#1805D61
hypocenterC61#24130
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. C2×#16840043C2semidirect
3. C2×#16839860C2semidirect
5. #21447021C2semidirect
6. #21446821C2semidirect
7. #21446791C2semidirect
8. #21446616C2semidirect
9. #21446135C2semidirect
10. #21446008C2semidirect
11. C61×#1805C2semidirect
58. #14519777D4semidirect
95. C4×Q8×C61D4semidirect
276. #1805D61semidirect
317. C61#24130semidirect
341. C4×Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168400432
C2×#168398602
#214507212
#214470212
#214468212
#214467912
#214466162
#214461352
#214460082
C61×#18052
#2413061