Finite Group Databasebeta

← #29707593

Subgroups — #29707593

Order 15616.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840343C211yesmaximal, characteristic
2C22×#14519708C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C2×#16839730C211yesmaximal, characteristic, complemented
6#21448407C211yesmaximal, characteristic, complemented
7#21447018C214yesmaximal, complemented
8#21447011C214yesmaximal, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758222C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13#16840343—24no—
14#16840343—24no—
15C2×#14519757—21no—
16C2×#14519757C2212yes—
17C2×#14519752—21no—
18C2×#14519752C2212yes—
19C2×#14519751—22no—
20C22×C4⋊Dic61C2211yescharacteristic
21C2×#14519708—24no—
22C2×#14519708C2214yes—
23#16839764—24no—
24#16839756—24no—
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29#16839730—24no—
30C2×#14519689C2214yes—
31#16839668—21no—
32#16839664—21no—
33#16839639—22no—
34#16839619C2214yes—
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C22×C4⋊C4×C61C2211yescharacteristic
38C61×#340C2214yes—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222D414yes—
51C22×Dic122D411yescharacteristic
52C22×Dic122C2311yescharacteristic
53C22×D244—22no—
54C22×D244D411yescharacteristic
55C22×D244C2311yescharacteristic
56C22×C22⋊D61—21no—
57C22×C22⋊D61D412yes—
58C23×Dic61—21no—
59C22×C4×D61—21no—
60C22×C4×D61C2312yes—
61C22×#92418—22no—
62C22×#92418—22no—
63C22×#92418D414yescomplemented
64C22×#92418C2312yes—
65#14519757—28no—
66#14519757—24no—
67#14519757—24no—
68#14519752—28no—
69#14519752—24no—
70#14519752—24no—
71#14519751—216no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2312yes—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198—24no—
80#14519708—28no—
81#14519702—24no—
82#14519700—24no—
83#14519699—24no—
84#14519697—24no—
85#14519689—28no—
86#14519684—24no—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C22×C8×C61C2312yes—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61D414yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2312yes—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61—21no—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C23×D61C2×D412yes—
105C22×C4×C61—22no—
106C22×C4×C61—24no—
107C22×C4×C61—22no—
108C22×C4×C61—82no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D412yes—
113C22×C4×C61#4112yes—
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—24no—
118C2×C22⋊D61—24no—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—42no—
126C22×Dic61—82no—
127C22×Dic61C2×D412yes—
128#11758222—28no—
129#11758222—28no—
130#11758222—44no—
131#11758222—44no—
132#11758222—44no—
133#11758222—44no—
134#11758222—44no—
135#11758222—44no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—21no—
144C2×D244—28no—
145C2×D244—84no—
146C2×D244C2×D412yes—
147C2×C4×D61—24no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—21no—
164C2×Dic122C2×D412yes—
165M16×C61—44no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61C2×D414yes—
174C22⋊C4×C61—42no—
175C42×C61—42no—
176#11758198—28no—
177#11758198—28no—
178#11758198—84no—
179C4⋊Dic61—28no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—42no—
183C2×#92418—28no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418C2×D414yes—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61#12012yes—
190C23×C61#12112yes—
191C23×C61#12312yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—24no—
198C22×D61—24no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—42no—
203C2×C4×C61—24no—
204C2×C4×C61—44no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—84no—
209C2×C4×C61#12012yes—
210C2×C4×C61#12114yes—
211C2×C4×C61#12114yes—
212C2×C4×C61#12112yes—
213C2×C4×C61#12314yes—
214C2×C4×C61#12312yes—
215C2×C4×C61#12414yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C22⋊D61—48no—
223C22⋊D61—48no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—44no—
232C2×Dic61—84no—
233D244—22no—
234D244—22no—
235D244—41no—
236D244—41no—
237D244—44no—
238D244—88no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61—42no—
243C4×D61—48no—
244Dic122—22no—
245Dic122—22no—
246Dic122—41no—
247Dic122—41no—
248Dic122—44no—
249C8×C61—42no—
250#92418—24no—
251#23489—611nomaximal, Sylow
252C22×C61—24no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—24no—
256C22×C61#39414yes—
257C22×C61C2×#12011yescharacteristic
258C22×C61C2×#12111yescharacteristic
259C22×C61C2×#12311yescharacteristic
260C22×C61#57712yes—
261C22×C61#57712yes—
262C2×D61—44no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—44no—
270C2×D61—82no—
271C2×D61—82no—
272C4×C61—44no—
273C4×C61—82no—
274C4×C61#39414yes—
275C4×C61C2×#12111yescharacteristic
276C4×C61C2×#12111yescharacteristic
277C4×C61C2×#12311yescharacteristic
278C4×C61C2×#12411yescharacteristic
279Dic61—42no—
280Dic61—42no—
281Dic61—82no—
282C2×#537—611no—
283C22×#106—611no—
284C2×#417—611no—
285C2×#416—611no—
286C2×#408—611no—
287#2311—611no—
288#2035—614no—
289#2034—614no—
290#1736D6111yescharacteristic, complemented
291C2×C61—24no—
292C2×C61C2×#39411yescharacteristic
293C2×C61#300911yescharacteristic
294C2×C61#301011yescharacteristic
295C2×C61#306012yes—
296C2×C61#309912yes—
297D61—82no—
298D61—82no—
299C22×#41—611no—
300C22×#41—611no—
301C22×M16C2×D6111yescharacteristic
302C23×C8—611no—
303#537—1224no—
304#537—1224no—
305C2×C22⋊Q8—612no—
306C2×C22⋊Q8—1221no—
307C2×#121—612no—
308C2×#121—1221no—
309C2×#121—1222no—
310C2×#117C2×D6111yescharacteristic
311C22×C4⋊C4—611no—
312C22×C4⋊C4C2×D6111yescharacteristic
313C2×#106—614no—
314C2×#106—1224no—
315#417—1224no—
316#416—1224no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#102—612no—
320C2×#102—612no—
321#408—1224no—
322C2×C22⋊C8—614no—
323#391—1221no—
324#390—1221no—
325#386—1222no—
326#380—614no—
327#340C2×D6114yes—
328C61#2348911yescharacteristic, complemented, Sylow
329C2×#41—614no—
330C2×#41—1222no—
331C2×#41—1222no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C2×#41—1222no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—611no—
342C22×Q8—611no—
343C22×D4—612no—
344C22×D4—611no—
345C22×D4—611no—
346C22×D4—1222no—
347C22×D4—1221no—
348C23×C4—21no—
349C23×C4—612no—
350C23×C4—1221no—
351C23×C4—1221no—
352C23×C4C22×D6111yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—22no—
356C22×C8—614no—
357C22×C8—612no—
358C22×C8—1222no—
359C22×C8—1222no—
360C22×C8C22×D6112yes—
361C22⋊Q8—1224no—
362C22⋊Q8—1228no—
363C22⋊Q8—1224no—
364#121—1224no—
365#121—1228no—
366#121—12216no—
367#121—1224no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4D24414yes—
378C2×C4⋊C4C22×D6112yes—
379C2×C4⋊C4C22×D6112yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1224no—
384C2×C42—21no—
385#106—1228no—
386#103—1224no—
387#103—1224no—
388#102—1224no—
389#102—1224no—
390C22⋊C8—1228no—
391#95—1224no—
392C24—612no—
393C24—1221no—
394C24D4×D6111yescharacteristic
395#41—1228no—
396#41—1228no—
397#41—2444no—
398#41—2444no—
399#41—2444no—
400#41—2444no—
401#41—2444no—
402#41—2444no—
403C2×Q8—612no—
404C2×Q8—1221no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×D4—612no—
413C2×D4—1228no—
414C2×D4—1224no—
415C2×D4—1224no—
416C2×D4—1224no—
417C2×D4—1224no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—4884no—
427C2×D4—2442no—
428C2×D4—2442no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—82no—
433C22×C4—612no—
434C22×C4—1221no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—4882no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4C2×D24412yes—
448C22×C4#1175822212yes—
449C22×C4#1175822212yes—
450C22×C4D4×D6112yes—
451C22×C4D4×D6112yes—
452C22×C4D4×D6111yescharacteristic
453C22×C4#1175822612yes—
454C22×C4C23×D6111yescharacteristic
455M16—44no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—614no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1228no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—1228no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—2442no—
471C4⋊C4C2×D24414yes—
472C22⋊C4—42no—
473C22⋊C4—1228no—
474C22⋊C4—1228no—
475C22⋊C4—4884no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1224no—
484C23—1224no—
485C23—2441no—
486C23—2441no—
487C23—2442no—
488C23—2442no—
489C23#1451974712yes—
490C23#1451975012yes—
491C23#1451976212yes—
492C23C2×D4×D6111yescharacteristic
493Q8—1222no—
494Q8—1222no—
495Q8—2444no—
496Q8—2441no—
497Q8—2441no—
498D4—1222no—
499D4—1222no—
500D4—4888no—
501D4—2448no—
502D4—2448no—
503D4—2444no—
504D4—2441no—
505D4—2441no—
506C2×C4—24no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—4884no—
517C2×C4—2442no—
518C2×C4—2444no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2448no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974712yes—
527C2×C4#1451975012yes—
528C2×C4#1451975014yes—
529C2×C4#1451975214yes—
530C2×C4#1451976212yes—
531C2×C4#1451976214yes—
532C2×C4#1451976514yes—
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×D24411yescharacteristic
536C2×C4C2×#1175822211yescharacteristic
537C2×C4C2×#1175822211yescharacteristic
538C2×C4C2×#1175822611yescharacteristic
539C8—42no—
540C8—1224no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—4882no—
546C22—4882no—
547C22—2442no—
548C22—2442no—
549C22—2442no—
550C22—2442no—
551C22—2444no—
552C22—2444no—
553C22—2442no—
554C22—2442no—
555C22#1683968014yes—
556C22C2×#1451974711yescharacteristic
557C22C2×#1451975011yescharacteristic
558C22C2×#1451976211yescharacteristic
559C22#1684057112yes—
560C22#1684057212yes—
561C4—44no—
562C4—82no—
563C4—4882no—
564C4—2442no—
565C4—2442no—
566C4#1683968014yes—
567C4C2×#1451975011yescharacteristic
568C4C2×#1451975211yescharacteristic
569C4C2×#1451976211yescharacteristic
570C4C2×#1451976511yescharacteristic
571C2—24no—
572C2—4882no—
573C2—4882no—
574C2C2×#1683968011yescharacteristic
575C2#2145509711yescharacteristic
576C2#2145510611yescharacteristic
577C2#2145548012yes—
578C2#2145581212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC2×#16840343C2
hypercenter#1736D61
hypocenterC61#23489
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C22×#14519708C2semidirect
3. C2×#16839764C2semidirect
4. C2×#16839756C2semidirect
5. C2×#16839730C2semidirect
6. #21448407C2semidirect
7. #21447018C2semidirect
8. #21447011C2semidirect
9. C61×#1736C2semidirect
63. C22×#92418D4semidirect
290. #1736D61semidirect
328. C61#23489semidirect

Maximal subgroups

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

subgroupindex
C2×#168403432
C22×#145197082
C2×#168397642
C2×#168397562
C2×#168397302
#214484072
#214470182
#214470112
C61×#17362
#2348961