Finite Group Databasebeta

← #29708823

Subgroups — #29708823

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839761C211yesmaximal, characteristic
2C2×#16839756C211yesmaximal, characteristic
3#21448407C211yesmaximal, characteristic
4#21447238C212yesmaximal, complemented
5#21447233C212yesmaximal, complemented
6#21447228C212yesmaximal, complemented
7#21447223C212yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C2×#16839524C211yesmaximal, characteristic, complemented
10#21445516C211yesmaximal, characteristic, complemented
11C61×#2317C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839761—24no—
17#16839756—24no—
18C2×#14519700C2211yescharacteristic
19C2×#14519700C2212yes—
20C2×#14519700C2211yescharacteristic
21C2×#14519697C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2211yescharacteristic
24#16839668C2212yes—
25#16839664C2212yes—
26#16839639—22no—
27#16839619—21no—
28#16839619—21no—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C2×#117×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C22×C4⋊C4×C61C2211yescharacteristic
38C61×#389—21no—
39C61×#389—21no—
40C61×#385—22no—
41C61×#384C2212yes—
42C61×#382C2212yes—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D414yes—
49C22×Dic122C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61—21no—
53C22×C4×D61—21no—
54C2×#11758194—22no—
55C2×#11758194—22no—
56C2×#11758194—22no—
57C22×#92418—22no—
58C22×#92418—22no—
59C22×#92418—22no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2312yes—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2312yes—
67C2×#11758198—22no—
68C2×#11758198—22no—
69#14519700—22no—
70#14519700—22no—
71#14519700—24no—
72#14519697—22no—
73#14519697—22no—
74#14519697—24no—
75#14519684—22no—
76#14519684—22no—
77C23×C4×C61—21no—
78C23×C4×C61C2311yescharacteristic
79#117×C61—22no—
80#117×C61—22no—
81#117×C61—22no—
82#117×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61D411yescharacteristic
89C2×C4⋊C4×C61D411yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61D412yes—
95C2×C42×C61—21no—
96C2×C42×C61C2312yes—
97#95×C61—22no—
98#95×C61—22no—
99C24×C61C2×D411yescharacteristic
100C23×D61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—42no—
105C22×C4×C61—81no—
106C22×C4×C61—81no—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D412yes—
111C22×C4×C61#4112yes—
112C22×C4×C61#4112yes—
113C22×C4×C61#4112yes—
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—24no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61—42no—
123C22×Dic61—81no—
124C22×Dic61—81no—
125#11758222—28no—
126#11758222—28no—
127#11758222—44no—
128#11758222—44no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244C2×D412yes—
135C2×C4×D61—24no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122C2×D412yes—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—82no—
154C4⋊C4×C61—82no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—42no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61—42no—
160#11758198—82no—
161#11758198—82no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—42no—
165#11758194—44no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—44no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61#12112yes—
181C23×C61#12312yes—
182C23×C61#12412yes—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—84no—
207C2×C4×C61#12112yes—
208C2×C4×C61#12314yes—
209C2×C4×C61#12314yes—
210C2×C4×C61#12312yes—
211C2×C4×C61#12412yes—
212C2×C4×C61#12614yes—
213C2×C4×C61#12614yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C22⋊D61—48no—
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—84no—
228D244—22no—
229D244—22no—
230D244—41no—
231D244—41no—
232C4×D61—48no—
233Dic122—22no—
234Dic122—22no—
235Dic122—41no—
236Dic122—41no—
237#92418—41no—
238#92418—41no—
239#92418—42no—
240#23634—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61#39714yes—
248C22×C61C2×#12111yescharacteristic
249C22×C61C2×#12311yescharacteristic
250C22×C61C2×#12411yescharacteristic
251C22×C61#57611yescharacteristic
252C22×C61#57711yescharacteristic
253C22×C61#57711yescharacteristic
254C22×C61#57811yescharacteristic
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—81no—
265C4×C61—82no—
266C4×C61—81no—
267C4×C61#39714yes—
268C4×C61C2×#12311yescharacteristic
269C4×C61C2×#12311yescharacteristic
270C4×C61C2×#12611yescharacteristic
271C4×C61C2×#12611yescharacteristic
272Dic61—42no—
273Dic61—82no—
274C2×#417—611no—
275C2×#416—611no—
276#2317D6111yescharacteristic, complemented
277#2311—611no—
278#2074—612no—
279#2073—612no—
280#2072—612no—
281#2071—612no—
282#1736—611no—
283C2×#340—611no—
284#1729—611no—
285C2×C61—24no—
286C2×C61C2×#39711yescharacteristic
287C2×C61#304511yescharacteristic
288C2×C61#304611yescharacteristic
289C2×C61#309811yescharacteristic
290C2×C61#309911yescharacteristic
291C2×C61#313611yescharacteristic
292C2×C61#313711yescharacteristic
293D61—82no—
294C22×#41—611no—
295C22×M16—611no—
296C23×C8—611no—
297C2×#117C2×D6111yescharacteristic
298C2×#117C2×D6111yescharacteristic
299C22×C4⋊C4—611no—
300C22×C4⋊C4C2×D6111yescharacteristic
301#417—1224no—
302#416—1224no—
303C2×#103—611no—
304C2×#103—612no—
305C2×#103—611no—
306C2×#102—611no—
307C2×#102—612no—
308C2×#102—611no—
309#391—612no—
310#390—612no—
311#389—21no—
312#389—21no—
313#386—1222no—
314#385—22no—
315#384C2×D6112yes—
316#382C2×D6112yes—
317#380—1221no—
318#380—1221no—
319#340—612no—
320#340—612no—
321#340—612no—
322#340—612no—
323#336—612no—
324#336—612no—
325C61#2363411yescharacteristic, complemented, Sylow
326C2×#41—614no—
327C2×#41—1222no—
328C2×#41—1222no—
329C2×#41—1222no—
330C2×#41—1222no—
331C2×#41—1222no—
332C22×Q8—611no—
333C22×D4—611no—
334C22×D4—1221no—
335C23×C4—21no—
336C23×C4—1221no—
337C23×C4—1221no—
338C23×C4C22×D6111yescharacteristic
339C2×M16—1222no—
340C2×M16—1222no—
341C2×M16—1222no—
342C22×C8—611no—
343C22×C8—611no—
344C22×C8—612no—
345C22×C8—1222no—
346C22×C8—1222no—
347C22×C8—1222no—
348#117—22no—
349#117—22no—
350#117—22no—
351#117—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22⋊D6111yescharacteristic
363C2×C4⋊C4C22×D6112yes—
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4C22⋊D6112yes—
370C2×C42—21no—
371C2×C42C22×D6112yes—
372#103—1222no—
373#103—1222no—
374#103—1224no—
375#102—1222no—
376#102—1222no—
377#102—1224no—
378#95—22no—
379#95—22no—
380#95—1222no—
381#95—1222no—
382C24—1221no—
383C24C2×C22⋊D6111yescharacteristic
384#41—1228no—
385#41—1228no—
386#41—2444no—
387#41—2444no—
388C2×Q8—612no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×D4—612no—
395C2×D4—1224no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—2442no—
402C2×D4—2442no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—1221no—
410C22×C4—1224no—
411C22×C4—1222no—
412C22×C4—4881no—
413C22×C4—4881no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4#1175822212yes—
418C22×C4#1175822212yes—
419C22×C4D4×D6112yes—
420C22×C4#1175822412yes—
421C22×C4#1175822612yes—
422C22×C4#1175822612yes—
423C22×C4C2×C22⋊D6112yes—
424C22×C4C2×C22⋊D6112yes—
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C23×D6111yescharacteristic
427M16—2444no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1222no—
431C2×C8—1221no—
432C2×C8—1221no—
433C2×C8—1222no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1222no—
437C2×C8—2444no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—2442no—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—4882no—
455C22⋊C4—4882no—
456C42—22no—
457C42—22no—
458C42—42no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—1222no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23#1451974812yes—
471C23#1451976012yes—
472C23#1451976612yes—
473C23C22×C22⋊D6111yescharacteristic
474Q8—1222no—
475Q8—1222no—
476Q8—2441no—
477Q8—2441no—
478D4—1222no—
479D4—1222no—
480D4—2448no—
481D4—2441no—
482D4—2441no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—84no—
499C2×C4—1222no—
500C2×C4—4884no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2448no—
507C2×C4#1451974812yes—
508C2×C4#1451976012yes—
509C2×C4#1451976214yes—
510C2×C4#1451976214yes—
511C2×C4#1451976612yes—
512C2×C4#1451977214yes—
513C2×C4#1451977214yes—
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4C2×#1175822411yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C2×C4C2×#1175822211yescharacteristic
518C2×C4C2×#1175822611yescharacteristic
519C2×C4C2×#1175822611yescharacteristic
520C8—2441no—
521C8—2441no—
522C8—2442no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—4881no—
530C22—4881no—
531C22—4881no—
532C22—4881no—
533C22—2442no—
534C22—2442no—
535C22—2442no—
536C22#1683968814yes—
537C22C2×#1451974811yescharacteristic
538C22C2×#1451976011yescharacteristic
539C22C2×#1451976611yescharacteristic
540C22#1684056511yescharacteristic
541C22#1684057011yescharacteristic
542C22#1684057311yescharacteristic
543C22#1684057711yescharacteristic
544C4—42no—
545C4—42no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4—4882no—
550C4—2442no—
551C4#1683968814yes—
552C4C2×#1451976211yescharacteristic
553C4C2×#1451976211yescharacteristic
554C4C2×#1451977211yescharacteristic
555C4C2×#1451977211yescharacteristic
556C2—24no—
557C2—4882no—
558C2C2×#1683968811yescharacteristic
559C2#2145535411yescharacteristic
560C2#2145536511yescharacteristic
561C2#2145579711yescharacteristic
562C2#2145580811yescharacteristic
563C2#2145617811yescharacteristic
564C2#2145618911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2317C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#2317D61
hypocenterC61#23634
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. #21447238C2semidirect
5. #21447233C2semidirect
6. #21447228C2semidirect
7. #21447223C2semidirect
8. #21445540C2semidirect
9. C2×#16839524C2semidirect
10. #21445516C2semidirect
11. C61×#2317C2semidirect
276. #2317D61semidirect
325. C61#23634semidirect

Maximal subgroups

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

subgroupindex
C2×#168397612
C2×#168397562
#214484072
#214472382
#214472332
#214472282
#214472232
#214455402
C2×#168395242
#214455162
C61×#23172
#2363461