Finite Group Databasebeta

← #29707748

Subgroups — #29707748

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×#16840336C211yesmaximal, characteristic
2C2×#16839764C211yesmaximal, characteristic
3C2×#16839756C211yesmaximal, characteristic
4C22×#14519701C211yesmaximal, characteristic, complemented
5#21448442C211yesmaximal, characteristic, complemented
6#21447064C214yesmaximal, complemented
7#21447057C214yesmaximal, complemented
8#21445517C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12#16840336—24no—
13#16840336—24no—
14C2×#14519775C2212yes—
15C2×#14519766C2212yes—
16C2×#14519760—22no—
17C2×#14519753—21no—
18C2×#14519753—21no—
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16839764—24no—
22#16839756—24no—
23C2×#14519702C2212yes—
24C2×#14519701—24no—
25C2×#14519701C2214yes—
26C2×#14519700C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29#16839661C2214yes—
30#16839637—22no—
31#16839632—21no—
32#16839628—21no—
33#16839516C2214yes—
34C23×Q8×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#419—24no—
38C2×#103×C61C2214yes—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222D414yes—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61—21no—
49C22×C4×D61—21no—
50C22×#92418—22no—
51C22×#92418—22no—
52C22×#92418—22no—
53C22×#92418C2312yes—
54#14519775—24no—
55#14519775—24no—
56#14519766—24no—
57#14519766—24no—
58#14519760—216no—
59#14519753—28no—
60#14519753—28no—
61#14519736—22no—
62#14519736—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×C22⋊Dic61D412yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61C2312yes—
76#14519702—24no—
77#14519701—28no—
78#14519700—24no—
79#14519699—24no—
80#14519697—24no—
81#14519684—24no—
82C22×Q8×C61—21no—
83C22×Q8×C61—22no—
84C22×Q8×C61—22no—
85C22×Q8×C61—22no—
86C22×Q8×C61—22no—
87C22×Q8×C61—22no—
88C22×Q8×C61D414yes—
89C22×Q8×C61C2312yes—
90C23×C4×C61—21no—
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C22×C8×C61C2312yes—
96#117×C61—22no—
97#117×C61D414yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61—21no—
101#103×C61—28no—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C2×Q8×C61—24no—
105C2×Q8×C61—22no—
106C2×Q8×C61—28no—
107C2×Q8×C61—28no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—42no—
113C2×Q8×C61—44no—
114C2×Q8×C61—44no—
115C2×Q8×C61—42no—
116C2×Q8×C61C2×D414yes—
117C22×C4×C61—24no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61#4112yes—
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—42no—
136C22×Dic61—82no—
137C22×Dic61C2×D412yes—
138C22×Dic61C2×D412yes—
139#11758222—28no—
140#11758222—28no—
141#11758222—44no—
142#11758222—44no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244C2×D412yes—
149C2×C4×D61—24no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122C2×D412yes—
158M16×C61—44no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C2×D414yes—
164C22⋊C4×C61—24no—
165C42×C61—24no—
166C22⋊Dic61—24no—
167#11758198—28no—
168#11758198—28no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—42no—
174C4⋊Dic61—84no—
175#11758196—28no—
176#11758196—28no—
177C4×Dic61—22no—
178C4×Dic61—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—44no—
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—
194Q8×C61—24no—
195Q8×C61—24no—
196Q8×C61—42no—
197Q8×C61—42no—
198Q8×C61—48no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—48no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12114yes—
208C2×C4×C61#12114yes—
209C2×C4×C61#12112yes—
210C2×C4×C61#12314yes—
211C2×C4×C61#12314yes—
212C2×C4×C61#12412yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C22⋊D61—48no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—44no—
229C2×Dic61—84no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234C4×D61—48no—
235Dic122—22no—
236Dic122—22no—
237Dic122—41no—
238Dic122—41no—
239C8×C61—42no—
240#92418—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—44no—
258C4×C61—44no—
259C4×C61#39414yes—
260C4×C61C2×#12111yescharacteristic
261C4×C61C2×#12111yescharacteristic
262C4×C61C2×#12311yescharacteristic
263C4×C61C2×#12311yescharacteristic
264Dic61—42no—
265Dic61—42no—
266Dic61—42no—
267Dic61—82no—
268C2×#536—611no—
269C2×#417—611no—
270C2×#419D6111yescharacteristic, complemented
271C2×#416—611no—
272C22×#103—611no—
273#2317—611no—
274#2043—614no—
275#2042—614no—
276#1729—611no—
277C2×C61—24no—
278C2×C61C2×#39411yescharacteristic
279C2×C61#301011yescharacteristic
280C2×C61#301111yescharacteristic
281C2×C61#304612yes—
282C2×C61#313712yes—
283D61—82no—
284C22×#41—611no—
285C23×Q8C2×D6111yescharacteristic
286C22×M16C2×D6111yescharacteristic
287C23×C8—611no—
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—24no—
299#417—1224no—
300#416—1224no—
301C2×#103—614no—
302C2×#103—612no—
303C2×#103—612no—
304C2×#103—1224no—
305C2×#103C2×D6114yes—
306C2×#102—612no—
307C2×#102—612no—
308#389—614no—
309#385—1222no—
310#384—1221no—
311#382—1221no—
312#336—614no—
313C61#2351011yescharacteristic, complemented, Sylow
314C2×#41—614no—
315C2×#41—1222no—
316C2×#41—1222no—
317C2×#41—1222no—
318C2×#41—1222no—
319C2×#41—1222no—
320C22×Q8—21no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—22no—
325C22×Q8—22no—
326C22×Q8—611no—
327C22×Q8D24414yes—
328C22×Q8C22×D6112yes—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—21no—
332C23×C4—1221no—
333C23×C4—1221no—
334C23×C4C22×D6111yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C22×C8—612no—
339C22×C8—1222no—
340C22×C8—1222no—
341C22×C8—1222no—
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—1222no—
352#117—1222no—
353#117C22⋊D6114yes—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1224no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—612no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C42—21no—
369C2×C42—612no—
370#103—28no—
371#103—1224no—
372#103—1228no—
373#103—1224no—
374#102—1224no—
375#102—1224no—
376#95—1224no—
377C24—1221no—
378C24D4×D6111yescharacteristic
379#41—1228no—
380#41—1228no—
381#41—2444no—
382#41—2444no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—612no—
396C2×Q8—1221no—
397C2×Q8—1221no—
398C2×Q8—1221no—
399C2×Q8—1221no—
400C2×Q8—1221no—
401C2×Q8C2×D24414yes—
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—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—1221no—
419C22×C4—1224no—
420C22×C4—1222no—
421C22×C4—4882no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4C2×D24412yes—
426C22×C4#1175822212yes—
427C22×C4D4×D6112yes—
428C22×C4D4×D6111yescharacteristic
429C22×C4#1175822412yes—
430C22×C4#1175822412yes—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C23×D6111yescharacteristic
433M16—44no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—2444no—
441C4⋊C4—22no—
442C4⋊C4—1228no—
443C4⋊C4—1228no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—4884no—
449C4⋊C4—2442no—
450C4⋊C4C2×C22⋊D6114yes—
451C22⋊C4—24no—
452C22⋊C4—1228no—
453C22⋊C4—1228no—
454C22⋊C4—1224no—
455C42—24no—
456C42—1222no—
457C42—1222no—
458C23—22no—
459C23—22no—
460C23—1222no—
461C23—2441no—
462C23—2441no—
463C23—2442no—
464C23—2442no—
465C23#1451974712yes—
466C23#1451974912yes—
467C23#1451976312yes—
468C23C2×D4×D6111yescharacteristic
469Q8—24no—
470Q8—24no—
471Q8—42no—
472Q8—42no—
473Q8—48no—
474Q8—1222no—
475Q8—1222no—
476Q8—2441no—
477Q8—2441no—
478D4—1222no—
479D4—1222no—
480D4—2448no—
481D4—2441no—
482D4—2441no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1224no—
494C2×C4—1224no—
495C2×C4—4884no—
496C2×C4—2442no—
497C2×C4—2444no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2448no—
501C2×C4#1451974712yes—
502C2×C4#1451974814yes—
503C2×C4#1451974912yes—
504C2×C4#1451975114yes—
505C2×C4#1451975914yes—
506C2×C4#1451976114yes—
507C2×C4#1451976312yes—
508C2×C4C2×D4×D6111yescharacteristic
509C2×C4C22×C22⋊D6111yescharacteristic
510C2×C4C22×D24411yescharacteristic
511C2×C4C2×#1175822411yescharacteristic
512C2×C4C2×#1175822411yescharacteristic
513C2×C4C2×#1175822211yescharacteristic
514C8—42no—
515C8—2442no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—4882no—
521C22—4882no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22#1683967814yes—
526C22C2×#1451974711yescharacteristic
527C22C2×#1451974911yescharacteristic
528C22C2×#1451976311yescharacteristic
529C22#1684057112yes—
530C22#1684057812yes—
531C4—44no—
532C4—44no—
533C4—4882no—
534C4—2442no—
535C4—2442no—
536C4—2442no—
537C4#1683967814yes—
538C4C2×#1451974811yescharacteristic
539C4C2×#1451975111yescharacteristic
540C4C2×#1451975911yescharacteristic
541C4C2×#1451976111yescharacteristic
542C2—24no—
543C2—4882no—
544C2C2×#1683967811yescharacteristic
545C2#2145510611yescharacteristic
546C2#2145511511yescharacteristic
547C2#2145537312yes—
548C2#2145618012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#419D61
hypocenterC61#23510
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
4. C22×#14519701C2semidirect
5. #21448442C2semidirect
6. #21447064C2semidirect
7. #21447057C2semidirect
8. #21445517C2semidirect
9. C2×C61×#419C2semidirect
270. C2×#419D61semidirect
313. C61#23510semidirect

Maximal subgroups

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

subgroupindex
C2×#168403362
C2×#168397642
C2×#168397562
C22×#145197012
#214484422
#214470642
#214470572
#214455172
C2×C61×#4192
#2351061