Finite Group Databasebeta

← #29716435

Subgroups — #29716435

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic
3#21451189C211yesmaximal, characteristic, complemented
4#21447155C212yesmaximal, complemented
5#21446968C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446754C211yesmaximal, characteristic, complemented
8#21446150C212yesmaximal
9#21446063C212yesmaximal
10#21445820C211yesmaximal, characteristic
11C2×C61×#477C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840473—24no—
14#16840473—24no—
15C2×#14519768C2212yes—
16C2×#14519757—21no—
17C2×#14519753—22no—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739C2212yes—
21C2×C4⋊C4×D61—21no—
22C4×#11758201—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2212yes—
26C2×#14519697C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28#16839695—22no—
29#16839669C2212yes—
30#16839668—21no—
31#16839660—21no—
32#16839658C2212yes—
33#16839632C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#477—24no—
42C2×#107×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C2×#11758201Q814yescomplemented
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519768—24no—
55#14519768—24no—
56#14519757—28no—
57#14519753—216no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60C4×D244—22no—
61C4×D244D414yes—
62#14519739—24no—
63#14519739—24no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758196D412yes—
73C4⋊C4×D61—28no—
74C2×#11758198—21no—
75C2×#11758198—22no—
76C2×C4×Dic61C2312yes—
77#14519708—22no—
78#14519701—22no—
79#14519699—24no—
80#14519697—22no—
81C4×#92418—22no—
82#14519684—22no—
83#14519684—22no—
84#14519684—22no—
85C22×Q8×C61—21no—
86C22×Q8×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88C4⋊Q8×C61—22no—
89C4⋊Q8×C61D414yes—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C42×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#107×C61—24no—
99C4⋊C8×C61—22no—
100#103×C61—24no—
101C23×D61—21no—
102C2×Q8×C61—24no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—81no—
109C2×Q8×C61—81no—
110C2×Q8×C61C2×Q812yes—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61C2×D412yes—
120C22×Dic61C2×D412yes—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244C2×D412yes—
125C2×D244C2×Q812yes—
126C2×D244#4112yes—
127C2×C4×D61—24no—
128C2×C4×D61—28no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61—82no—
132C2×C4×D61—82no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61C2×D414yes—
145C42×C61—42no—
146C42×C61C2×D412yes—
147C42×C61C2×D412yes—
148#11758201—28no—
149#11758198—28no—
150#11758198—24no—
151C4⋊Dic61—28no—
152C4⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157C4⋊Dic61—81no—
158C4⋊Dic61C2×D412yes—
159#11758196—28no—
160#11758196—24no—
161#11758196—82no—
162C4×Dic61—22no—
163C4×Dic61—22no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418C2×Q812yes—
170C23×C61C22×D411yescharacteristic
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178Q8×C61—22no—
179Q8×C61—22no—
180Q8×C61—41no—
181Q8×C61—41no—
182Q8×C61—84no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—24no—
186C2×C4×C61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61#12012yes—
195C2×C4×C61#12012yes—
196C2×C4×C61#13512yes—
197C2×C4×C61#13712yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×Q811yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C2×C4×C612−1+411yescharacteristic
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216D244—22no—
217D244—22no—
218D244—22no—
219D244C22⋊Q814yes—
220C4×D61—48no—
221C4×D61—88no—
222Dic122—44no—
223C8×C61—42no—
224#92418—22no—
225#92418—41no—
226#24331—611nomaximal, Sylow
227C22×C61C2×#12011yescharacteristic
228C22×C61#54811yescharacteristic
229C22×C61D4×Q811yescharacteristic
230C22×C61C2×#13511yescharacteristic
231C22×C61C2×#13711yescharacteristic
232C22×C61#57611yescharacteristic
233C22×C61#57711yescharacteristic
234C2×D61—24no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C4×C61—22no—
242C4×C61—42no—
243C4×C61—44no—
244C4×C61—81no—
245C4×C61#45214yes—
246C4×C61C2×C22⋊Q811yescharacteristic
247C4×C61#53611yescharacteristic
248C4×C61#54811yescharacteristic
249C4×C61D4×Q811yescharacteristic
250Dic61—42no—
251Dic61—42no—
252Dic61—42no—
253Dic61—82no—
254C2×#554—611no—
255C2×#477D6111yescharacteristic, complemented
256C2×C4×SD16—611no—
257#2582—611no—
258#2059—612no—
259#2026—612no—
260#1990—611no—
261#1984—611no—
262#1877—612no—
263#1862—612no—
264#1802—611no—
265C2×C61#240911yescharacteristic
266C2×C61C2×#45211yescharacteristic
267C2×C61#301211yescharacteristic
268C2×C61#336111yescharacteristic
269C2×C61#337211yescharacteristic
270C2×C61#338111yescharacteristic
271C2×C61#338511yescharacteristic
272D61—44no—
273C22×SD16—611no—
274#554—1224no—
275#554—1224no—
276C2×C4⋊Q8C2×D6111yescharacteristic
277C2×#125—612no—
278C2×C22⋊Q8—1221no—
279C2×C22⋊Q8—1222no—
280C2×C4×Q8—611no—
281C2×C4×Q8C2×D6111yescharacteristic
282C2×C4×D4—612no—
283C2×C4×D4—611no—
284C22×C4⋊C4—1221no—
285#477—24no—
286C4×SD16—1224no—
287C2×#107C2×D6112yes—
288C2×#106—611no—
289C2×C4⋊C8C2×D6111yescharacteristic
290C2×#103—611no—
291C2×#103C2×D6112yes—
292C2×#102—612no—
293C2×#102—611no—
294C2×C4×C8—611no—
295#398—1222no—
296#391—612no—
297#391—1221no—
298#389—612no—
299#389—1221no—
300#384—611no—
301#340—612no—
302#340—612no—
303#340—612no—
304#336—612no—
305#336—612no—
306C61#2433111yescharacteristic, complemented, Sylow
307C22×Q8—21no—
308C22×Q8—611no—
309C22×Q8C22×D6111yescharacteristic
310C22×D4—611no—
311C23×C4—1222no—
312C23×C4—1221no—
313C2×SD16—614no—
314C2×SD16—1222no—
315C2×SD16—1222no—
316C22×C8—611no—
317C22×C8—611no—
318C22×C8C22×D6112yes—
319C4⋊Q8—22no—
320C4⋊Q8C22⋊D6114yes—
321#125—1224no—
322#125—1224no—
323C22⋊Q8—1228no—
324C22⋊Q8—12216no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—1222no—
328C4×Q8—1222no—
329C4×D4—614no—
330C4×D4—1224no—
331C4×D4—1224no—
332C4×D4—1222no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1228no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C4⋊C4C22×D6112yes—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1221no—
349C2×C42—21no—
350C2×C42—612no—
351C2×C42C22×D6111yescharacteristic
352#107—24no—
353#106—1222no—
354C4⋊C8—22no—
355#103—24no—
356#103—1222no—
357#102—1222no—
358#102—1224no—
359C4×C8—1222no—
360#95—1222no—
361#95—1222no—
362#95—1222no—
363C24—1221no—
364C2×Q8—21no—
365C2×Q8—24no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—81no—
371C2×Q8—81no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8Q8×D6112yes—
376C2×D4—612no—
377C2×D4—612no—
378C2×D4—612no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—612no—
386C22×C4—612no—
387C22×C4—1228no—
388C22×C4—1221no—
389C22×C4—1224no—
390C22×C4—1222no—
391C22×C4—4882no—
392C22×C4—4882no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4D4×D6111yescharacteristic
396C22×C4D4×D6111yescharacteristic
397C22×C4C23×D6111yescharacteristic
398SD16—1228no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—612no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8—1221no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—42no—
413C4⋊C4—612no—
414C4⋊C4—1228no—
415C4⋊C4—1228no—
416C4⋊C4—1221no—
417C4⋊C4—1222no—
418C4⋊C4—1224no—
419C4⋊C4—1222no—
420C4⋊C4—1222no—
421C4⋊C4—4881no—
422C4⋊C4—4882no—
423C4⋊C4—2442no—
424C4⋊C4C2×C22⋊D6114yes—
425C22⋊C4—1228no—
426C22⋊C4—1224no—
427C42—42no—
428C42—1222no—
429C42—1222no—
430C42D4×D6112yes—
431C42C2×C22⋊D6112yes—
432C23—1222no—
433C23—1222no—
434C23—1222no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23C2×D4×D6111yescharacteristic
440Q8—22no—
441Q8—22no—
442Q8—41no—
443Q8—41no—
444Q8—84no—
445Q8—2444no—
446D4—614no—
447D4—1222no—
448D4—1222no—
449D4—1222no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—4888no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2448no—
475C2×C4#1451974512yes—
476C2×C4#1451974512yes—
477C2×C4#1451980012yes—
478C2×C4#1451980912yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4C2×Q8×D6111yescharacteristic
482C2×C4#41×D6111yescharacteristic
483C2×C4#1451982411yescharacteristic
484C2×C4#1451982511yescharacteristic
485C8—42no—
486C8—1222no—
487C8—2441no—
488C22—1224no—
489C22—2442no—
490C22—2442no—
491C22—2442no—
492C22—2442no—
493C22—2442no—
494C22—2442no—
495C22C2×#1451974511yescharacteristic
496C22#1684042211yescharacteristic
497C22#1684042511yescharacteristic
498C22C2×#1451980011yescharacteristic
499C22C2×#1451980911yescharacteristic
500C22#1684056611yescharacteristic
501C22#1684057211yescharacteristic
502C4—22no—
503C4—44no—
504C4—42no—
505C4—81no—
506C4—4882no—
507C4—2442no—
508C4—2442no—
509C4—2442no—
510C4#1683992314yes—
511C4C22⋊Q8×D6111yescharacteristic
512C4#1684033511yescharacteristic
513C4#1684041911yescharacteristic
514C4Q8×C22⋊D6111yescharacteristic
515C2—2444no—
516C2#2144918011yescharacteristic
517C2C2×#1683992311yescharacteristic
518C2#2145512211yescharacteristic
519C2#2145847811yescharacteristic
520C2#2145859911yescharacteristic
521C2#2145869311yescharacteristic
522C2#2145872811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#477D61
hypocenterC61#24331
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #21451189C2semidirect
4. #21447155C2semidirect
5. #21446968C2semidirect
6. #21446781C2semidirect
7. #21446754C2semidirect
11. C2×C61×#477C2semidirect
51. C2×#11758201Q8semidirect
255. C2×#477D61semidirect
306. C61#24331semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×C4×#117582012
#214511892
#214471552
#214469682
#214467812
#214467542
#214461502
#214460632
#214458202
C2×C61×#4772
#2433161