Finite Group Databasebeta

← #29734337

Subgroups — #29734337

Order 15616.

1038 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840132C211yesmaximal, characteristic, complemented
3#21454124C211yesmaximal, characteristic
4#21447024C212yesmaximal
5#21447017C212yesmaximal
6#21446793C211yesmaximal, characteristic, complemented
7#21446639C211yesmaximal, characteristic, complemented
8#21446084C212yesmaximal, complemented
9#21446070C212yesmaximal, complemented
10#21445582C211yesmaximal, characteristic
11C2×C61×#501C211yesmaximal, characteristic, complemented
12#16840317—24no—
13#16840317—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768—21no—
16C2×#14519757C2212yes—
17C2×#14519753—21no—
18C2×Q8×Dic61—21no—
19C2×#14519742C2212yes—
20C2×#14519735—21no—
21C2×C42×D61C2211yescharacteristic
22#16840132—24no—
23C2×#14519702C2212yes—
24C2×#14519702C2212yes—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519689C2212yes—
28C2×#14519688C2211yescharacteristic
29#16839658C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839643—21no—
32#16839643—21no—
33#16839633—21no—
34#16839633—21no—
35#16839628C2211yescharacteristic
36#16839619C2211yescharacteristic
37#16839619C2211yescharacteristic
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#125×C61C2211yescharacteristic
43C61×#501—24no—
44C2×#107×C61C2212yes—
45C2×#106×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777D414yes—
54#14519768—28no—
55#14519757—24no—
56#14519757—24no—
57#14519753—28no—
58Q8×Dic61—28no—
59#14519742—24no—
60#14519742—24no—
61#14519735—28no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—22no—
68C2×#11758196—21no—
69C2×#11758196—21no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61D411yescharacteristic
76C2×C4×Dic61C2311yescharacteristic
77C42×D61—22no—
78C42×D61—22no—
79#14519702—24no—
80#14519702—24no—
81#14519700—24no—
82#14519700—24no—
83#14519689—24no—
84#14519688—22no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C22×C8×C61C2312yes—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yes—
95#125×C61—22no—
96#125×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C42×C61C2311yescharacteristic
104#107×C61—24no—
105#106×C61—24no—
106#97×C61—22no—
107C23×D61C2×D411yescharacteristic
108C2×Q8×C61—24no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
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—22no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—41no—
132C2×C4×D61—41no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122C2×D414yes—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61#4114yes—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61C2×D414yes—
151C42×C61C2×D412yes—
152C42×C61C2×D412yes—
153#11758198—28no—
154#11758198—42no—
155#11758198—42no—
156C4⋊Dic61—28no—
157C4⋊Dic61—24no—
158C4⋊Dic61—24no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161#11758196—28no—
162#11758196—42no—
163#11758196—42no—
164C4×Dic61—28no—
165C4×Dic61—22no—
166C4×Dic61—22no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61C22×D411yescharacteristic
173C22×D61—21no—
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61#13712yes—
180Q8×C61—48no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—24no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61#12012yes—
195C2×C4×C61#12012yes—
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4112yes—
199C2×C4×C612−1+412yes—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—24no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61#13712yes—
218C4×D61—28no—
219C4×D61—44no—
220C4×D61—42no—
221C4×D61—42no—
222Dic122—24no—
223Dic122—24no—
224Dic122—42no—
225Dic122—42no—
226C8×C61—24no—
227#92418—42no—
228#25677—611nomaximal, Sylow
229C22×C61C2×#12011yescharacteristic
230C22×C61#53211yescharacteristic
231C22×C61#53611yescharacteristic
232C22×C61C2×#13711yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C22×C61#57511yescharacteristic
235C22×C61#57511yescharacteristic
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—41no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61—41no—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—44no—
246C4×C61—81no—
247C4×C61—81no—
248C4×C61#54712yes—
249C4×C61#54812yes—
250Dic61—22no—
251Dic61—44no—
252Dic61—41no—
253Dic61—41no—
254Dic61—81no—
255Dic61—81no—
256C2×#533—611no—
257C2×#501D6111yescharacteristic, complemented
258C2×#491—611no—
259#2851—611no—
260#2036—612no—
261#2035—612no—
262#1993—611no—
263#1960—611no—
264#1865—612no—
265#1863—612no—
266#1749—611no—
267C2×C61#243211yescharacteristic
268C2×C61#300411yescharacteristic
269C2×C61#300511yescharacteristic
270C2×C61#317211yescharacteristic
271C2×C61#318111yescharacteristic
272C2×C61#323611yescharacteristic
273C2×C61#323711yescharacteristic
274D61—42no—
275#533—1224no—
276#533—1224no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×#125—1221no—
280C2×#125C2×D6111yescharacteristic
281C2×C22⋊Q8—612no—
282C2×C22⋊Q8—1221no—
283C2×C4×Q8—612no—
284C2×C4×Q8—1221no—
285C2×#117—1221no—
286C22×C42—611no—
287#501—24no—
288#491—1224no—
289C2×#107C2×D6112yes—
290C2×#106C2×D6112yes—
291C2×#103—612no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103—612no—
295C2×C22⋊C8—612no—
296C2×#97—611no—
297C2×#97C2×D6111yescharacteristic
298#389—611no—
299#389—611no—
300#387—1221no—
301#387—1221no—
302#385—1221no—
303#385—1221no—
304#382—611no—
305#380—611no—
306#380—611no—
307#340—612no—
308#340—612no—
309#336—612no—
310C61#2567711yescharacteristic, complemented, Sylow
311C22×Q8—21no—
312C22×Q8—612no—
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—611no—
316C22×C8—612no—
317C22×C8C22×D6112yes—
318C4⋊Q8—22no—
319C4⋊Q8—614no—
320C4⋊Q8—1222no—
321C4⋊Q8C22⋊D6114yes—
322#125—22no—
323#125—22no—
324#125—1228no—
325C22⋊Q8—1224no—
326C22⋊Q8—1228no—
327C22⋊Q8—1224no—
328C4×Q8—1224no—
329C4×Q8—1228no—
330C4×Q8—1224no—
331#117—1228no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C22×D6111yescharacteristic
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C4⋊C4C22×D6112yes—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—1222no—
353C2×C42—1222no—
354C2×C42—1222no—
355C2×C42C22×D6111yescharacteristic
356#107—24no—
357#106—24no—
358#103—1224no—
359#103—1224no—
360#103—1224no—
361#103—1224no—
362C22⋊C8—1224no—
363#97—22no—
364#97—1222no—
365#95—1221no—
366#95—1221no—
367#95—1221no—
368#95—1221no—
369#95—1221no—
370#95—1221no—
371C24—611no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—614no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—1222no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—2441no—
399C22×C4—2441no—
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8#1175822614yes—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—1228no—
418C4⋊C4—1228no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—2442no—
424C4⋊C4—2442no—
425C4⋊C4C2×C22⋊D6114yes—
426C22⋊C4—1228no—
427C22⋊C4—2442no—
428C22⋊C4—2442no—
429C42—1228no—
430C42—1221no—
431C42—1221no—
432C42—1222no—
433C42—1222no—
434C42D4×D6112yes—
435C42C2×C22⋊D6112yes—
436C23—612no—
437C23—1221no—
438C23—1221no—
439C23—1221no—
440C23—1221no—
441C23—1221no—
442C23—1221no—
443C23C2×D4×D6111yescharacteristic
444Q8—48no—
445Q8—1224no—
446Q8—1224no—
447Q8—2442no—
448Q8—2442no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—612no—
463C2×C4—1228no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1224no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—2444no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4#1451974512yes—
485C2×C4#1451974512yes—
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4C22×C22⋊D6111yescharacteristic
488C2×C4C2×#1175822612yes—
489C2×C4#1451982512yes—
490C8—24no—
491C8—2442no—
492C22—1222no—
493C22—1222no—
494C22—1222no—
495C22—2441no—
496C22—2441no—
497C22—2441no—
498C22—2441no—
499C22C2×#1451974511yescharacteristic
500C22#1684031311yescharacteristic
501C22#1684033711yescharacteristic
502C22#137×D6111yescharacteristic
503C22#137×D6111yescharacteristic
504C22#1684055811yescharacteristic
505C22#1684055811yescharacteristic
506C4—22no—
507C4—22no—
508C4—44no—
509C4—81no—
510C4—81no—
511C4—1222no—
512C4—4881no—
513C4—4881no—
514C4—2441no—
515C4—2441no—
516C4—2444no—
517C4#1684041512yes—
518C4#1684042312yes—
519C2—2442no—
520C2#2144944411yescharacteristic
521C2#2145505111yescharacteristic
522C2#2145505811yescharacteristic
523C2#2145642711yescharacteristic
524C2#2145653011yescharacteristic
525C2#2145702611yescharacteristic
526C2#2145703411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#501D61
hypocenterC61#25677
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#16840132C2semidirect
6. #21446793C2semidirect
7. #21446639C2semidirect
8. #21446084C2semidirect
9. #21446070C2semidirect
11. C2×C61×#501C2semidirect
257. C2×#501D61semidirect
310. C61#25677semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168401322
#214541242
#214470242
#214470172
#214467932
#214466392
#214460842
#214460702
#214455822
C2×C61×#5012
#2567761