Finite Group Databasebeta

← #30615916

Subgroups — #30615916

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic
2C22×#14519840C211yesmaximal, characteristic
3#21462293C211yesmaximal, characteristic
4#21462256C211yesmaximal, characteristic
5#21461899C214yesmaximal, complemented
6#21461898C214yesmaximal, complemented
7#21461504C211yesmaximal, characteristic
8#21461487C211yesmaximal, characteristic
9#419×D61C211yesmaximal, characteristic
10C22×D61⋊C8C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C2×#14519856C2211yescharacteristic
13C22×C4⋊#6923C2211yescharacteristic
14#16840773—24no—
15C2×#14519846C2212yes—
16C2×#14519845C2212yes—
17C2×#14519840—24no—
18C2×#14519840C2214yes—
19(C2×#6923)⋊Q8—22no—
20(C2×#6923)⋊Q8C2212yes—
21(C2×#6923)⋊Q8C2212yes—
22#16840705—22no—
23C4⋊C4×#6923C2214yes—
24C4×C4⋊#6923—22no—
25C22⋊C4×#6923—22no—
26#16840635C2212yes—
27#16840635C2212yes—
28(C2×#6923)⋊C8C2214yes—
29C22×Q8×D61C2211yescharacteristic
30C2×M16×D61C2211yescharacteristic
31#117×D61C2211yescharacteristic
32#16839772C411yescharacteristic, complemented
33#16839771C411yescharacteristic, complemented
34#16839770C411yescharacteristic, complemented
35#16839769C414yescomplemented
36#103×D61C2214yes—
37C61×#419C411yescharacteristic, complemented
38C23×#6923—21no—
39C2×D61⋊C8—22no—
40C2×D61⋊C8—22no—
41C2×D61⋊C8—21no—
42C2×D61⋊C8C2312yes—
43C22×#92420—21no—
44#14519856—22no—
45#14519856—22no—
46#14519856—22no—
47#14519856—22no—
48C2×C4⋊#6923—21no—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923D412yes—
51C2×C4⋊#6923D414yes—
52C2×C4⋊#6923C2311yescharacteristic
53C2×C4⋊#6923C2311yescharacteristic
54C2×C4⋊#6923C2312yes—
55C2×C22⋊#6923—21no—
56C2×C22⋊#6923D412yes—
57C2×C4×#6923—21no—
58C2×C4×#6923—24no—
59C2×C4×#6923C2312yes—
60#14519846—24no—
61#14519845—24no—
62#14519840—28no—
63(C2×#6923)⋊C4—24no—
64(C2×#6923)⋊C4—24no—
65(C2×#6923)⋊C4—24no—
66C22×Dic122—21no—
67C22×Dic122C2×C411yescharacteristic
68C2×Q8×D61—21no—
69C2×Q8×D61—21no—
70C2×Q8×D61—22no—
71C2×Q8×D61—22no—
72C2×Q8×D61—22no—
73C2×Q8×D61—22no—
74C2×Q8×D61C2×C412yes—
75C2×Q8×D61C2312yes—
76C22×C4×D61—21no—
77C22×C4×D61C2311yescharacteristic
78C2×#11758194C2×C411yescharacteristic
79C2×#11758189C2×C412yes—
80M16×D61—22no—
81M16×D61—22no—
82C2×C8×D61C2312yes—
83#14519736C2×C411yescharacteristic
84#14519735C2×C412yes—
85#14519734—21no—
86#14519733—21no—
87C4⋊C4×D61C2312yes—
88C22⋊C4×D61—21no—
89C42×D61—21no—
90#14519702C2×C414yescomplemented
91#14519701C2×C414yescomplemented
92#14519700C2×C414yescomplemented
93C22×Q8×C61C2×C411yescharacteristic
94C2×M16×C61C2×C411yescharacteristic
95#117×C61C2×C411yescharacteristic
96#103×C61C2×C414yescomplemented
97C23×D61C2×D411yescharacteristic
98C22×#6923—22no—
99C22×#6923—24no—
100C22×#6923—21no—
101C22×#6923—21no—
102C22×#6923—82no—
103C22×#6923C2×D414yes—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61C22×C412yes—
108C22×C4×C61—21no—
109C22×C4×C61C22×C411yescharacteristic
110C22×Dic61—21no—
111C22×Dic61#4111yescharacteristic
112Q8×D61—24no—
113Q8×D61—24no—
114Q8×D61—22no—
115Q8×D61—22no—
116Q8×D61—22no—
117Q8×D61—22no—
118Q8×D61—42no—
119Q8×D61—42no—
120Q8×D61—42no—
121Q8×D61—42no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61C22×C411yescharacteristic
128C2×C4×D61C2×D412yes—
129C2×C4×D61C2×D411yescharacteristic
130C2×C4×D61#4112yes—
131C2×C4×D61#4112yes—
132C2×C4×D61#4112yes—
133C2×C4×D61#4111yescharacteristic
134C2×C4×D61C2411yescharacteristic
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—42no—
141C2×Dic122—42no—
142C2×Dic122C22×C412yes—
143C22⋊#6923—24no—
144C22⋊#6923—42no—
145C2×#92420—22no—
146C2×#92420—42no—
147C4⋊#6923—24no—
148C4⋊#6923—24no—
149C4⋊#6923—21no—
150C4⋊#6923—21no—
151C4⋊#6923—22no—
152C4⋊#6923—21no—
153C4⋊#6923—21no—
154C4⋊#6923C2×D414yes—
155C4×#6923—22no—
156C4×#6923—22no—
157C4×#6923—42no—
158C4×#6923—84no—
159D61⋊C8—22no—
160D61⋊C8—22no—
161D61⋊C8—42no—
162M16×C61—22no—
163C2×C8×C61C22×C412yes—
164C4⋊C4×C61C22×C412yes—
165C22⋊C4×C61—21no—
166C42×C61—21no—
167C22⋊Dic61—21no—
168#11758198—22no—
169C4⋊Dic61C22×C412yes—
170#11758196—22no—
171C4×Dic61—21no—
172C4×Dic61—22no—
173#11758194—22no—
174C2×#92418C22×C412yes—
175#11758189—24no—
176C8×D61—22no—
177C8×D61—22no—
178C23×C61C4×D411yescharacteristic
179C22×D61—22no—
180C22×D61—22no—
181C22×D61C4×D411yescharacteristic
182C22×D61#12112yes—
183C22×D61#12312yes—
184C22×D61C22×D411yescharacteristic
185C2×#6923—24no—
186C2×#6923—24no—
187C2×#6923—24no—
188C2×#6923—24no—
189C2×#6923—44no—
190C2×#6923—41no—
191C2×#6923—41no—
192C2×#6923—42no—
193C2×#6923—42no—
194C2×#6923—84no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61—42no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—42no—
201C2×C4×C61#11712yes—
202C2×C4×C61C4×D412yes—
203C2×C4×C61C4×D411yescharacteristic
204C2×C4×C61C23×C411yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—42no—
208C2×Dic61C4×D411yescharacteristic
209C2×Dic61#12412yes—
210C2×Dic61#12612yes—
211C2×Dic61C2×#4111yescharacteristic
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—42no—
216C4×D61—44no—
217C4×D61#11712yes—
218C4×D61C4×D412yes—
219C4×D61#12112yes—
220C4×D61#12312yes—
221C4×D61#12412yes—
222C4×D61#12612yes—
223C4×D61C22×D411yescharacteristic
224C4×D61C2×#4111yescharacteristic
225C4×D61C2×#4111yescharacteristic
226C4×D61C2×#4111yescharacteristic
227Dic122—22no—
228Dic122—22no—
229Dic122—42no—
230Dic122—42no—
231Dic122—42no—
232Dic122—44no—
233#92420—42no—
234C8×C61—22no—
235#92418—22no—
236#21064—611nomaximal, Sylow
237C22×C61—22no—
238C22×C61#38812yes—
239C22×C61C2×C4×D411yescharacteristic
240C2×D61—22no—
241C2×D61—24no—
242C2×D61—22no—
243C2×D61—22no—
244C2×D61#38812yes—
245C2×D61C2×#12111yescharacteristic
246C2×D61C2×#12311yescharacteristic
247C2×D61#57712yes—
248C2×D61#57812yes—
249#6923—44no—
250#6923—44no—
251#6923—81no—
252#6923—81no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61#38812yes—
256C4×C61C2×#11711yescharacteristic
257C4×C61C2×C4×D411yescharacteristic
258Dic61—42no—
259Dic61—42no—
260Dic61#38812yes—
261Dic61C2×#12411yescharacteristic
262Dic61C2×#12611yescharacteristic
263C2×#428—611no—
264C2×#419—611no—
265C22×#103—611no—
266#2310—611no—
267#2274—611no—
268#1928—614no—
269#1927—614no—
270#1735—611no—
271#1729—611no—
272C2×C61—22no—
273C2×C61C2×#38811yescharacteristic
274C2×C61#294712yes—
275D61—22no—
276D61#304211yescharacteristic
277D61#304411yescharacteristic
278D61#309712yes—
279C23×Q8—611no—
280C22×M16—611no—
281C23×C8—611no—
282C2×#117—611no—
283C2×#117—611no—
284C2×#117—611no—
285C22×C4⋊C4—611no—
286#428—1224no—
287C2×#107—612no—
288C2×#106—612no—
289#419—611no—
290#419—611no—
291#419—611no—
292#419—614no—
293#419#692311yescharacteristic, complemented
294C2×#103—614no—
295C2×#103—614no—
296C2×#103—1224no—
297#391—612no—
298#391—612no—
299#391—1222no—
300#381—1222no—
301C4×C4⋊C4—614no—
302C4×C4⋊C4—1222no—
303C4×C22⋊C4—1222no—
304#340—612no—
305#340—612no—
306#336—614no—
307C61#2106411yescharacteristic, complemented, Sylow
308C22×Q8—612no—
309C22×Q8—612no—
310C22×Q8—611no—
311C22×Q8—1221no—
312C22×Q8—1221no—
313C22×Q8—1221no—
314C22×Q8—1222no—
315C22×Q8—1222no—
316C22×Q8—1222no—
317C22×Q8—1222no—
318C22×Q8C2×#692311yescharacteristic
319C23×C4—611no—
320C23×C4—1221no—
321C23×C4—1221no—
322C2×M16—611no—
323C2×M16—612no—
324C2×M16—1222no—
325C2×M16—1222no—
326C2×M16C2×#692311yescharacteristic
327C22×C8—612no—
328C22×C8—612no—
329C22×C8—1221no—
330C22×C8—1222no—
331C22×C8—1221no—
332C22×C8—1222no—
333#117—611no—
334#117—612no—
335#117—1221no—
336#117—1221no—
337#117—1222no—
338#117—1222no—
339#117—1222no—
340#117—1222no—
341#117C2×#692311yescharacteristic
342C2×C4⋊C4—614no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1221no—
350C2×C22⋊C4—612no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C42—612no—
354C2×C42—1221no—
355C2×C42—1224no—
356C2×C42—1221no—
357#107—1224no—
358#106—1224no—
359#103—614no—
360#103—614no—
361#103—614no—
362#103—1228no—
363#103C2×#692314yescomplemented
364#95—1224no—
365#95—1224no—
366#95—1224no—
367C24—611no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—612no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1224no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—1224no—
379C2×Q8—1222no—
380C2×Q8—1221no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8—2442no—
384C2×Q8—2442no—
385C2×Q8—2442no—
386C2×Q8—2442no—
387C2×Q8—2442no—
388C2×Q8—2442no—
389C2×Q8C22×#692312yes—
390C22×C4—21no—
391C22×C4—611no—
392C22×C4—612no—
393C22×C4—612no—
394C22×C4—611no—
395C22×C4—612no—
396C22×C4—611no—
397C22×C4—612no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—614no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1224no—
405C22×C4—1222no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—4882no—
410C22×C4—2442no—
411C22×C4—2442no—
412C22×C4C22×#692311yescharacteristic
413M16—22no—
414M16—1224no—
415M16—1222no—
416C2×C8—612no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—2442no—
423C2×C8—2442no—
424C2×C8C22×#692312yes—
425C4⋊C4—614no—
426C4⋊C4—612no—
427C4⋊C4—1222no—
428C4⋊C4—1224no—
429C4⋊C4—1222no—
430C4⋊C4—1224no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4C22×#692312yes—
436C22⋊C4—21no—
437C22⋊C4—1222no—
438C22⋊C4—1224no—
439C22⋊C4—1221no—
440C22⋊C4—2442no—
441C42—21no—
442C42—1222no—
443C42—1222no—
444C42—1222no—
445C42—1221no—
446C42—4884no—
447C42—2442no—
448C23—611no—
449C23—612no—
450C23—612no—
451C23—611no—
452C23—1222no—
453C23—1222no—
454C23D4×#692311yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—42no—
458Q8—1222no—
459Q8—1222no—
460Q8—2442no—
461Q8—2442no—
462Q8—2444no—
463Q8—2442no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—612no—
468C2×C4—612no—
469C2×C4—611no—
470C2×C4—611no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—611no—
477C2×C4—611no—
478C2×C4—612no—
479C2×C4—611no—
480C2×C4—611no—
481C2×C4—1224no—
482C2×C4—1224no—
483C2×C4—1224no—
484C2×C4—1224no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—4884no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2444no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2444no—
498C2×C4—2442no—
499C2×C4#1451985612yes—
500C2×C4D4×#692311yescharacteristic
501C2×C4D4×#692312yes—
502C2×C4C23×#692311yescharacteristic
503C8—22no—
504C8—1222no—
505C8—2442no—
506C22—22no—
507C22—611no—
508C22—611no—
509C22—612no—
510C22—612no—
511C22—612no—
512C22—1222no—
513C22—1222no—
514C22—1224no—
515C22—1222no—
516C22#1684071212yes—
517C22C2×D4×#692311yescharacteristic
518C4—42no—
519C4—42no—
520C4—611no—
521C4—611no—
522C4—612no—
523C4—4881no—
524C4—4881no—
525C4—2444no—
526C4—2444no—
527C4—2442no—
528C4—2442no—
529C4#1684071212yes—
530C4C2×#1451985611yescharacteristic
531C4C2×D4×#692311yescharacteristic
532C2—22no—
533C2—611no—
534C2—611no—
535C2—612no—
536C2—1222no—
537C2C2×#1684071211yescharacteristic
538C2#2146257712yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#419C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#419#6923
hypocenterC61#21064
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
5. #21461899C2semidirect
6. #21461898C2semidirect
32. #16839772C4semidirect
33. #16839771C4semidirect
34. #16839770C4semidirect
35. #16839769C4semidirect
37. C61×#419C4semidirect
90. #14519702C2×C4semidirect
91. #14519701C2×C4semidirect
92. #14519700C2×C4semidirect
96. #103×C61C2×C4semidirect
293. #419#6923semidirect
307. C61#21064semidirect
363. #103C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
C22×#145198402
#214622932
#214622562
#214618992
#214618982
#214615042
#214614872
#419×D612
#2106461