Finite Group Databasebeta

← #29707589

Subgroups — #29707589

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×#16840340C211yesmaximal, characteristic
2C22×#14519708C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C2×#16839729C211yesmaximal, characteristic, complemented
6#21447015C214yesmaximal, complemented
7#21447008C214yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C61×#2311C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13#16840340—24no—
14#16840340—24no—
15C2×#14519756C2212yes—
16C2×#14519753—21no—
17C2×#14519751C2212yes—
18C2×#14519749—22no—
19C2×#14519748—21no—
20C22×C4⋊Dic61C2211yescharacteristic
21C2×#14519708—24no—
22C2×#14519708C2214yes—
23#16839761—24no—
24#16839756—24no—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29#16839729—24no—
30C2×C22⋊#92418C2214yes—
31#16839524C2214yes—
32C22×#41×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#391—21no—
36C61×#390—21no—
37C61×#386—22no—
38C61×#380C2214yes—
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—
48C22×C22⋊D61—22no—
49C23×Dic61—21no—
50C22×C4×D61—21no—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C22×#92418—22no—
55C22×#92418—22no—
56C22×#92418D414yescomplemented
57C22×#92418C2312yes—
58C22×#92418C2312yes—
59#14519756—24no—
60#14519756—24no—
61#14519753—28no—
62#14519751—24no—
63#14519751—24no—
64#14519749—216no—
65#14519748—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61D414yes—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×C22⋊Dic61—22no—
72C2×#11758198—22no—
73#14519708—28no—
74#14519700—24no—
75#14519700—24no—
76#14519697—24no—
77#14519697—24no—
78C22⋊#92418—28no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C22×Q8×C61D411yescharacteristic
86C22×D4×C61D411yescharacteristic
87C22×D4×C61D412yes—
88C23×C4×C61—21no—
89C23×C4×C61C2312yes—
90C23×C4×C61C2311yescharacteristic
91#117×C61—22no—
92#117×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2312yes—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—24no—
100C2×C42×C61—21no—
101#95×C61—24no—
102C24×C61C2×D412yes—
103C24×C61C2×D411yescharacteristic
104C23×D61—21no—
105#41×C61—44no—
106#41×C61—44no—
107#41×C61—44no—
108#41×C61—44no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×D4×C61—24no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—42no—
127C22×C4×C61—82no—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—28no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—84no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—82no—
146#11758222—28no—
147#11758222—28no—
148#11758222—44no—
149#11758222—44no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C2×D412yes—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122C2×D412yes—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—42no—
170C22⋊C4×C61—42no—
171C22⋊C4×C61—84no—
172C42×C61—42no—
173C22⋊Dic61—28no—
174#11758198—28no—
175C4⋊Dic61—24no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2×D414yes—
178#11758196—28no—
179#11758194—44no—
180C2×#92418—28no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418C2×D414yes—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—24no—
193C23×C61—24no—
194C23×C61#12012yes—
195C23×C61#12112yes—
196C23×C61#12312yes—
197C23×C61C22×D411yescharacteristic
198C22×D61—22no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—42no—
203Q8×C61—44no—
204D4×C61—48no—
205D4×C61—44no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—44no—
217C2×C4×C61—42no—
218C2×C4×C61—84no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12114yes—
222C2×C4×C61#12112yes—
223C2×C4×C61#12314yes—
224C2×C4×C61#12312yes—
225C2×C4×C61#12414yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—48no—
233C22⋊D61—88no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—44no—
239C2×Dic61—84no—
240D244—22no—
241D244—22no—
242D244—41no—
243D244—41no—
244C4×D61—48no—
245Dic122—22no—
246Dic122—22no—
247Dic122—41no—
248Dic122—41no—
249#92418—24no—
250#92418—42no—
251#23489—611nomaximal, Sylow
252C22×C61—24no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—24no—
256C22×C61—44no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—44no—
261C22×C61#39414yes—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×#12111yescharacteristic
264C22×C61C2×#12311yescharacteristic
265C22×C61#57712yes—
266C22×C61#57712yes—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—82no—
271C2×D61—82no—
272C4×C61—42no—
273C4×C61—82no—
274C4×C61—82no—
275C4×C61#39414yes—
276C4×C61C2×#12111yescharacteristic
277C4×C61C2×#12111yescharacteristic
278C4×C61C2×#12311yescharacteristic
279C4×C61C2×#12411yescharacteristic
280Dic61—42no—
281Dic61—44no—
282C2×#537—611no—
283C22×#106—611no—
284C2×#417—611no—
285C2×#416—611no—
286C2×#408—611no—
287#2311D6111yescharacteristic, complemented
288#2035—614no—
289#2034—614no—
290#1736—611no—
291C2×C61—24no—
292C2×C61—82no—
293C2×C61C2×#39411yescharacteristic
294C2×C61#300911yescharacteristic
295C2×C61#301011yescharacteristic
296C2×C61#306012yes—
297C2×C61#309912yes—
298D61—82no—
299C22×#41—611no—
300C22×#41C2×D6111yescharacteristic
301C22×M16—611no—
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—21no—
324#390—21no—
325#386—22no—
326#380C2×D6114yes—
327#340—614no—
328C61#2348911yescharacteristic, complemented, Sylow
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—614no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—611no—
342C22×Q8C22⋊D6111yescharacteristic
343C22×D4—611no—
344C22×D4—1222no—
345C22×D4—1221no—
346C22×D4C22⋊D6112yes—
347C22×D4C22⋊D6111yescharacteristic
348C23×C4—21no—
349C23×C4—1221no—
350C23×C4—1221no—
351C23×C4C22×D6112yes—
352C23×C4C22×D6111yescharacteristic
353C2×M16—1222no—
354C2×M16—1222no—
355C2×M16—1222no—
356C22×C8—614no—
357C22×C8—612no—
358C22×C8—612no—
359C22×C8—1222no—
360C22×C8—1222no—
361C22⋊Q8—1224no—
362C22⋊Q8—1224no—
363C22⋊Q8—1228no—
364#121—1224no—
365#121—1224no—
366#121—1228no—
367#121—12216no—
368#117—22no—
369#117—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4C22×D6112yes—
379C2×C4⋊C4C22×D6112yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C42—21no—
385#106—1228no—
386#103—1224no—
387#103—1224no—
388#102—1224no—
389#102—1224no—
390C22⋊C8—1228no—
391#95—24no—
392C24—1221no—
393C24C2×C22⋊D6112yes—
394C24C2×C22⋊D6111yescharacteristic
395#41—44no—
396#41—44no—
397#41—44no—
398#41—44no—
399#41—1228no—
400#41—1228no—
401#41—2444no—
402#41—2444no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—612no—
407C2×Q8—1221no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—612no—
419C2×D4—1228no—
420C2×D4—1224no—
421C2×D4—1221no—
422C2×D4—1221no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—4884no—
427C2×D4—2442no—
428C2×D4—2442no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1221no—
442C22×C4—1224no—
443C22×C4—4882no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4#1175822212yes—
447C22×C4#1175822212yes—
448C22×C4D4×D6112yes—
449C22×C4D4×D6112yes—
450C22×C4D4×D6112yes—
451C22×C4#1175822412yes—
452C22×C4C2×C22⋊D6112yes—
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455M16—2444no—
456C2×C8—614no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1228no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1222no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—614no—
469C4⋊C4—1228no—
470C4⋊C4—1222no—
471C4⋊C4—1224no—
472C22⋊C4—42no—
473C22⋊C4—84no—
474C22⋊C4—1228no—
475C22⋊C4—1228no—
476C42—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—1222no—
485C23—2441no—
486C23—2441no—
487C23—2442no—
488C23—2442no—
489C23#1451974512yes—
490C23#1451974912yes—
491C23#1451976012yes—
492C23C22×C22⋊D6111yescharacteristic
493Q8—44no—
494Q8—1222no—
495Q8—1222no—
496Q8—2441no—
497Q8—2441no—
498D4—48no—
499D4—44no—
500D4—1222no—
501D4—1222no—
502D4—4888no—
503D4—2448no—
504D4—2441no—
505D4—2441no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—84no—
519C2×C4—1224no—
520C2×C4—1222no—
521C2×C4—4884no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2444no—
525C2×C4—2448no—
526C2×C4#1451974512yes—
527C2×C4#1451974912yes—
528C2×C4#1451975014yes—
529C2×C4#1451975014yes—
530C2×C4#1451975914yes—
531C2×C4#1451976012yes—
532C2×C4#1451976314yes—
533C2×C4C2×D4×D6111yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4C2×#1175822411yescharacteristic
537C2×C4C2×#1175822211yescharacteristic
538C2×C4C2×#1175822211yescharacteristic
539C8—1224no—
540C8—2442no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—44no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—4882no—
551C22—4882no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22#1683967714yes—
556C22C2×#1451974511yescharacteristic
557C22C2×#1451974911yescharacteristic
558C22C2×#1451976011yescharacteristic
559C22#1684057012yes—
560C22#1684057312yes—
561C4—42no—
562C4—82no—
563C4—82no—
564C4—2442no—
565C4—2444no—
566C4#1683967714yes—
567C4C2×#1451975011yescharacteristic
568C4C2×#1451975011yescharacteristic
569C4C2×#1451975911yescharacteristic
570C4C2×#1451976311yescharacteristic
571C2—24no—
572C2—82no—
573C2—4882no—
574C2C2×#1683967711yescharacteristic
575C2#2145509411yescharacteristic
576C2#2145510311yescharacteristic
577C2#2145547812yes—
578C2#2145580812yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC23×C61C22×D4
prime radicalC2×#16840340C2
hypercenter#2311D61
hypocenterC61#23489
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C22×#14519708C2semidirect
3. C2×#16839761C2semidirect
4. C2×#16839756C2semidirect
5. C2×#16839729C2semidirect
6. #21447015C2semidirect
7. #21447008C2semidirect
8. #21445540C2semidirect
9. C61×#2311C2semidirect
56. C22×#92418D4semidirect
287. #2311D61semidirect
328. C61#23489semidirect

Maximal subgroups

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

subgroupindex
C2×#168403402
C22×#145197082
C2×#168397612
C2×#168397562
C2×#168397292
#214470152
#214470082
#214455402
C61×#23112
#2348961