Finite Group Databasebeta

← #29707750

Subgroups — #29707750

Order 15616.

1414 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840338C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic
3C2×#16839772C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic
5C22×#14519702C211yesmaximal, characteristic, complemented
6#21448439C211yesmaximal, characteristic, complemented
7#21447066C214yesmaximal, complemented
8#21447061C214yesmaximal, complemented
9C61×#1729C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×Dic122C2211yescharacteristic
12#16840338—24no—
13#16840338—24no—
14C2×#14519777C2212yes—
15C2×#14519765C2212yes—
16C2×#14519761—22no—
17C2×#14519757—21no—
18C2×#14519757—21no—
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16839772—24no—
22#16839764—24no—
23#16839759—24no—
24C2×#14519702—24no—
25C2×#14519702C2214yes—
26C2×#14519702C2214yes—
27C2×#14519702C2212yes—
28C2×#14519702C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31#16839660C2214yes—
32#16839634—22no—
33#16839631—21no—
34#16839627—21no—
35C22×M16×C61C2211yescharacteristic
36C23×C8×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#336C2214yes—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222D414yes—
45C22×Dic122—22no—
46C22×Dic122—22no—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122—21no—
50C22×Dic122—22no—
51C22×Dic122D414yes—
52C22×Dic122C2312yes—
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C23×Dic61—21no—
58C22×C4×D61—21no—
59#14519777—24no—
60#14519777—24no—
61#14519765—24no—
62#14519765—24no—
63#14519761—216no—
64#14519757—28no—
65#14519757—28no—
66#14519736—22no—
67#14519736D414yes—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—24no—
76C2×C22⋊Dic61—21no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×C4×Dic61—21no—
80#14519702—28no—
81#14519702—24no—
82#14519702—24no—
83#14519702—28no—
84#14519699—24no—
85#14519699—24no—
86#14519684—24no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—22no—
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C22×C8×C61—22no—
92C22×C8×C61—22no—
93C22×C8×C61—22no—
94C22×C8×C61C2312yes—
95C22×C8×C61C2312yes—
96#117×C61—22no—
97#117×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61D412yes—
101C2×C42×C61C2312yes—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D412yes—
110C22×C4×C61#4112yes—
111C22×C4×C61#4112yes—
112C22×C4×C61#4112yes—
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—24no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C22×Dic61—24no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—42no—
123C22×Dic61—42no—
124C22×Dic61—42no—
125C22×Dic61—82no—
126#11758222—28no—
127#11758222—28no—
128#11758222—44no—
129#11758222—44no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244C2×D412yes—
136C2×C4×D61—24no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×Dic122—28no—
140C2×Dic122—28no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—24no—
150C2×Dic122—22no—
151C2×Dic122—21no—
152C2×Dic122—44no—
153C2×Dic122—42no—
154C2×Dic122—44no—
155C2×Dic122—42no—
156C2×Dic122C2×D414yes—
157C2×Dic122C2×D412yes—
158M16×C61—44no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—44no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C22⋊C4×C61—24no—
169C42×C61—22no—
170C42×C61—22no—
171C22⋊Dic61—24no—
172#11758198—28no—
173#11758198—28no—
174C4⋊Dic61—28no—
175C4⋊Dic61—28no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C4⋊Dic61C2×D414yes—
181#11758196—84no—
182C4×Dic61—24no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61#12012yes—
186C23×C61#12112yes—
187C23×C61#12412yes—
188C23×C61C22×D411yescharacteristic
189C22×D61—22no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—42no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61#12012yes—
199C2×C4×C61#12114yes—
200C2×C4×C61#12114yes—
201C2×C4×C61#12112yes—
202C2×C4×C61#12314yes—
203C2×C4×C61#12314yes—
204C2×C4×C61#12412yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C22⋊D61—48no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—24no—
215C2×Dic61—42no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—48no—
222C2×Dic61—42no—
223C2×Dic61—44no—
224C2×Dic61—84no—
225D244—22no—
226D244—22no—
227D244—41no—
228D244—41no—
229C4×D61—48no—
230Dic122—24no—
231Dic122—22no—
232Dic122—24no—
233Dic122—22no—
234Dic122—42no—
235Dic122—41no—
236Dic122—42no—
237Dic122—41no—
238Dic122—48no—
239C8×C61—42no—
240C8×C61—42no—
241#23510—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61#39414yes—
247C22×C61C2×#12011yescharacteristic
248C22×C61C2×#12111yescharacteristic
249C22×C61C2×#12411yescharacteristic
250C22×C61#57712yes—
251C22×C61#57812yes—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—82no—
256C2×D61—82no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61#39414yes—
260C4×C61C2×#12111yescharacteristic
261C4×C61C2×#12111yescharacteristic
262C4×C61C2×#12311yescharacteristic
263C4×C61C2×#12311yescharacteristic
264Dic61—44no—
265Dic61—42no—
266Dic61—44no—
267Dic61—82no—
268C2×#536—611no—
269C2×#417—611no—
270C2×#419—611no—
271C2×#416—611no—
272C22×#103—611no—
273#2317—611no—
274#2043—614no—
275#2042—614no—
276#1729D6111yescharacteristic, complemented
277C2×C61—24no—
278C2×C61C2×#39411yescharacteristic
279C2×C61#301011yescharacteristic
280C2×C61#301111yescharacteristic
281C2×C61#304612yes—
282C2×C61#313712yes—
283D61—82no—
284C22×#41—611no—
285C23×Q8—611no—
286C22×M16C2×D6111yescharacteristic
287C23×C8C2×D6111yescharacteristic
288#536—1224no—
289#536—1224no—
290C2×C4⋊Q8—612no—
291C2×#124—612no—
292C2×#123—1222no—
293C2×C22⋊Q8—1221no—
294C2×C22⋊Q8—1221no—
295C2×#117—611no—
296C2×#117C2×D6111yescharacteristic
297C22×C4⋊C4—611no—
298#419—1224no—
299#417—1224no—
300#416—1224no—
301C2×#103—614no—
302C2×#103—612no—
303C2×#103—614no—
304C2×#103—612no—
305C2×#103—1224no—
306C2×#102—612no—
307C2×#102—612no—
308#389—614no—
309#385—1222no—
310#384—1221no—
311#382—1221no—
312#336C2×D6114yes—
313C61#2351011yescharacteristic, complemented, Sylow
314C2×#41—614no—
315C2×#41—1222no—
316C2×#41—1222no—
317C2×#41—1222no—
318C2×#41—1222no—
319C2×#41—1222no—
320C22×Q8—614no—
321C22×Q8—612no—
322C22×Q8—611no—
323C22×Q8—1221no—
324C22×Q8—1222no—
325C22×Q8—1222no—
326C22×Q8—1222no—
327C22×Q8—1222no—
328C22×Q8—1222no—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—1221no—
332C23×C4—1221no—
333C23×C4—1221no—
334C23×C4C22×D6111yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C22×C8—22no—
339C22×C8—22no—
340C22×C8—22no—
341C22×C8C22×D6112yes—
342C22×C8C22×D6112yes—
343C4⋊Q8—1224no—
344C4⋊Q8—1224no—
345#124—1224no—
346#124—1224no—
347#123—12216no—
348C22⋊Q8—1228no—
349C22⋊Q8—1228no—
350#117—22no—
351#117—22no—
352#117—614no—
353#117—1222no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4D24411yescharacteristic
363C2×C4⋊C4D24411yescharacteristic
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4D24412yes—
368C2×C42—1221no—
369C2×C42C22×D6112yes—
370#103—1224no—
371#103—1228no—
372#103—1228no—
373#103—1224no—
374#102—1224no—
375#102—1224no—
376#95—1224no—
377C24—1221no—
378C24C2×D24411yescharacteristic
379#41—1228no—
380#41—1228no—
381#41—2444no—
382#41—2444no—
383C2×Q8—614no—
384C2×Q8—612no—
385C2×Q8—1222no—
386C2×Q8—1221no—
387C2×Q8—1224no—
388C2×Q8—1222no—
389C2×Q8—1221no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1222no—
395C2×Q8—1221no—
396C2×Q8—1228no—
397C2×Q8—1228no—
398C2×Q8—2444no—
399C2×Q8—2442no—
400C2×Q8—2444no—
401C2×Q8—2442no—
402C2×D4—612no—
403C2×D4—1224no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—2442no—
410C2×D4—2442no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1221no—
414C22×C4—1224no—
415C22×C4—1224no—
416C22×C4—1222no—
417C22×C4—4882no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4C2×D24412yes—
424C22×C4C2×D24412yes—
425C22×C4C2×D24411yescharacteristic
426C22×C4#1175822212yes—
427C22×C4#1175822212yes—
428C22×C4D4×D6112yes—
429C22×C4D4×D6112yes—
430C22×C4D4×D6112yes—
431C22×C4#1175822612yes—
432C22×C4C23×D6111yescharacteristic
433M16—44no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—44no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—614no—
444C4⋊C4—1228no—
445C4⋊C4—1228no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—4884no—
450C4⋊C4—2442no—
451C22⋊C4—24no—
452C22⋊C4—1228no—
453C22⋊C4—1228no—
454C22⋊C4—1224no—
455C42—22no—
456C42—22no—
457C42—1224no—
458C23—22no—
459C23—22no—
460C23—1222no—
461C23—2441no—
462C23—2441no—
463C23—2442no—
464C23—2442no—
465C23#1451974712yes—
466C23#1451975212yes—
467C23#1451976512yes—
468C23C22×D24411yescharacteristic
469Q8—1224no—
470Q8—1222no—
471Q8—1224no—
472Q8—1222no—
473Q8—2448no—
474Q8—2442no—
475Q8—2441no—
476Q8—2442no—
477Q8—2441no—
478D4—1222no—
479D4—1222no—
480D4—2448no—
481D4—2441no—
482D4—2441no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—1224no—
488C2×C4—1222no—
489C2×C4—1224no—
490C2×C4—4884no—
491C2×C4—2442no—
492C2×C4—2444no—
493C2×C4—2448no—
494C2×C4—2441no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2448no—
501C2×C4#1451974712yes—
502C2×C4#1451975014yes—
503C2×C4#1451975014yes—
504C2×C4#1451975212yes—
505C2×C4#1451976214yes—
506C2×C4#1451976214yes—
507C2×C4#1451976512yes—
508C2×C4C2×D4×D6111yescharacteristic
509C2×C4C2×D4×D6111yescharacteristic
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C2×#1175822211yescharacteristic
512C2×C4C2×#1175822211yescharacteristic
513C2×C4C2×#1175822611yescharacteristic
514C8—42no—
515C8—42no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—24no—
520C22—4882no—
521C22—4882no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22#1683968014yes—
526C22C2×#1451974711yescharacteristic
527C22C2×#1451975211yescharacteristic
528C22C2×#1451976511yescharacteristic
529C22#1684057412yes—
530C22#1684057912yes—
531C4—42no—
532C4—42no—
533C4—4882no—
534C4—2444no—
535C4—2442no—
536C4—2444no—
537C4#1683968014yes—
538C4C2×#1451975011yescharacteristic
539C4C2×#1451975011yescharacteristic
540C4C2×#1451976211yescharacteristic
541C4C2×#1451976211yescharacteristic
542C2—24no—
543C2—4882no—
544C2C2×#1683968011yescharacteristic
545C2#2145510711yescharacteristic
546C2#2145511611yescharacteristic
547C2#2145537112yes—
548C2#2145618612yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1729C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1729D61
hypocenterC61#23510
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1729

Complements

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

NH ≅ G/Ntype
3. C2×#16839772C2semidirect
5. C22×#14519702C2semidirect
6. #21448439C2semidirect
7. #21447066C2semidirect
8. #21447061C2semidirect
9. C61×#1729C2semidirect
276. #1729D61semidirect
313. C61#23510semidirect

Maximal subgroups

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

subgroupindex
C2×#168403382
C2×#168397642
C2×#168397722
C2×#168397592
C22×#145197022
#214484392
#214470662
#214470612
C61×#17292
#2351061