Finite Group Databasebeta

← #29679710

Subgroups — #29679710

Order 15616.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840528C211yesmaximal, characteristic
2C2×#16839810C211yesmaximal, characteristic, complemented
3C2×#16839802C211yesmaximal, characteristic, complemented
4#21454339C211yesmaximal, characteristic
5C2×#16839761C211yesmaximal, characteristic
6#21445882C214yesmaximal, complemented
7#21445877C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C2×C61×#343C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840528—24no—
12#16840528C418yescomplemented
13C2×#14519785C2214yes—
14C2×#14519783—22no—
15C22×#11758194C2211yescharacteristic
16C22×#11758194C2211yescharacteristic
17C2×M16×D61—22no—
18C2×#14519736C2211yescharacteristic
19#16839810—24no—
20C2×#14519712C2214yescomplemented
21#16839802—22no—
22#16839802—22no—
23C2×#14519710C2212yescomplemented
24C2×#14519708C2212yescomplemented
25#16839772—21no—
26#16839768—21no—
27#16839764C2212yes—
28#16839764C2212yes—
29#16839761—22no—
30#16839761C2214yes—
31#16839759—22no—
32C2×#14519700—21no—
33C2×#14519697—21no—
34#16839530C2212yescomplemented
35#16839530C2212yescomplemented
36C22×M16×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C22×C4⋊C4×C61C2211yescharacteristic
39C61×#343C2214yescomplemented
40C2×#11758222—21no—
41C2×#11758222—22no—
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222C2×C414yes—
45C2×#11758222D412yes—
46C2×#11758222Q814yes—
47C2×#11758222C2312yes—
48C22×Dic122D411yescharacteristic
49C22×D244D411yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52#14519785—28no—
53#14519785C2×C4116yescomplemented
54#14519783—216no—
55C2×#11758194—21no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194C2×C414yes—
59C2×#11758194D412yes—
60C2×#11758194D412yes—
61C2×#11758194D414yes—
62C2×#11758194C2312yes—
63C2×#11758189—24no—
64M16×D61—216no—
65C22×#92418—21no—
66C22×#92418C2312yes—
67C2×C8×D61—24no—
68#14519736—21no—
69#14519736—21no—
70#14519736—21no—
71#14519736—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C22⋊Dic61—21no—
75C2×C4×Dic61—21no—
76#14519712—28no—
77#14519710—24no—
78#14519708—24no—
79#14519702—22no—
80#14519702—22no—
81#14519700—24no—
82#14519699—22no—
83#14519699—22no—
84#14519697—24no—
85C23×C4×C61—21no—
86C23×C4×C61C2311yescharacteristic
87C2×M16×C61—21no—
88C2×M16×C61—22no—
89C2×M16×C61—21no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C2×M16×C61C2×C414yes—
93C2×M16×C61C2312yes—
94C2×M16×C61C2312yes—
95C22×C8×C61—21no—
96C22×C8×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C24×C61#4111yescharacteristic
104C23×D61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61—82no—
109C22×C4×C61C22×C412yes—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×Q812yes—
116C22×C4×C61#4111yescharacteristic
117C22×C4×C61#4112yes—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—24no—
122C2×C22⋊D61—22no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126#11758222—24no—
127#11758222—24no—
128#11758222—24no—
129#11758222—24no—
130#11758222—24no—
131#11758222—22no—
132#11758222—22no—
133#11758222C4⋊C418yes—
134#11758222C22×C414yes—
135#11758222C2×Q814yes—
136C2×D244—22no—
137C2×D244—21no—
138C2×D244C4⋊C414yes—
139C2×D244C2×D412yes—
140C2×C4×D61—24no—
141C2×C4×D61—22no—
142C2×C4×D61—24no—
143C2×C4×D61—22no—
144C2×Dic122—22no—
145C2×Dic122—21no—
146C2×Dic122C4⋊C414yes—
147C2×Dic122C2×D412yes—
148M16×C61—24no—
149M16×C61—22no—
150M16×C61—22no—
151M16×C61—22no—
152M16×C61—42no—
153M16×C61C22×C414yes—
154C2×C8×C61—24no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—42no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—41no—
161C4⋊C4×C61—41no—
162C22⋊Dic61—41no—
163C22⋊Dic61—41no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170#11758194—22no—
171#11758194—22no—
172#11758194—42no—
173#11758194C22⋊C418yes—
174#11758194C22⋊C418yes—
175C2×#92418—22no—
176C2×#92418—42no—
177C2×#92418C22×C414yes—
178C2×#92418C2×D414yes—
179C2×#92418C2×D414yes—
180#11758189—216no—
181C8×D61—216no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61#11712yes—
185C23×C61#12111yescharacteristic
186C23×C61#12111yescharacteristic
187C23×C61C22⋊Q812yes—
188C23×C61C2×#4111yescharacteristic
189C22×D61—22no—
190C22×D61—24no—
191C22×D61—81no—
192C22×D61—81no—
193C2×C4×C61—42no—
194C2×C4×C61—82no—
195C2×C4×C61—82no—
196C2×C4×C61—82no—
197C2×C4×C61C2×C22⋊C412yes—
198C2×C4×C61C2×C22⋊C412yes—
199C2×C4×C61C2×C4⋊C412yes—
200C2×C4×C61C2×C4⋊C412yes—
201C2×C4×C61#11712yes—
202C2×C4×C61#12112yes—
203C2×C4×C61#12112yes—
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61C22⋊Q812yes—
207C2×C4×C61C22⋊Q812yes—
208C2×C4×C61C22⋊Q812yes—
209C2×C4×C61C23×C411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×Q811yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C22⋊D61—24no—
216C22⋊D61—24no—
217C22⋊D61—28no—
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—81no—
223C2×Dic61—81no—
224D244—22no—
225D244—24no—
226D244C2×C4⋊C414yes—
227C4×D61—24no—
228C4×D61—24no—
229C4×D61—28no—
230Dic122—22no—
231Dic122—24no—
232Dic122C2×C4⋊C414yes—
233C8×C61—24no—
234C8×C61—42no—
235#92418—42no—
236#92418C2×C22⋊C418yes—
237#20679—611nomaximal, Sylow
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61#38012yes—
247C22×C61#38012yes—
248C22×C61C2×#11711yescharacteristic
249C22×C61C2×#12111yescharacteristic
250C22×C61C2×C22⋊Q811yescharacteristic
251C2×D61—24no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—84no—
255C4×C61—82no—
256C4×C61#38012yes—
257C4×C61#38012yes—
258C4×C61C22×C22⋊C411yescharacteristic
259C4×C61C22×C4⋊C411yescharacteristic
260C4×C61C2×#12111yescharacteristic
261C4×C61C2×C22⋊Q811yescharacteristic
262Dic61—24no—
263Dic61—81no—
264Dic61—81no—
265C2×#568—611no—
266C2×#430—611no—
267C2×#428—611no—
268#2896—611no—
269C2×#417—611no—
270#1816—614no—
271#1815—614no—
272#1742—611no—
273C2×#343D6111yescharacteristic, complemented
274C2×C61—22no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—22no—
278C2×C61C2×#38011yescharacteristic
279C2×C61#306811yescharacteristic
280C2×C61#306911yescharacteristic
281D61—44no—
282C22×#41—611no—
283#568—618no—
284#568—1224no—
285C2×#131—614no—
286C2×#131—1222no—
287C22×M16—611no—
288C22×M16—611no—
289C22×M16—1222no—
290C22×M16C2×D6111yescharacteristic
291C22×M16C2×D6111yescharacteristic
292C2×#117—611no—
293C22×C4⋊C4C2×D6111yescharacteristic
294#430—1224no—
295C2×#108—614no—
296#428—1222no—
297#428—1222no—
298C2×#107—612no—
299C2×#106—612no—
300#419—1221no—
301#418—1221no—
302#417—612no—
303#417—612no—
304#417—614no—
305#417—1222no—
306#416—1222no—
307C2×#103—1221no—
308C2×#102—1221no—
309#343—612no—
310#343—612no—
311#343C2×D6114yescomplemented
312C61#2067911yescharacteristic, complemented, Sylow
313C2×#41—614no—
314C2×#41—612no—
315C2×#41—614no—
316C2×#41—612no—
317C2×#41—1221no—
318C2×#41—1222no—
319C2×#41—1221no—
320C2×#41—1222no—
321C22×Q8—611no—
322C22×D4—611no—
323C22×D4—1221no—
324C23×C4—21no—
325C23×C4—1221no—
326C23×C4C22×D6111yescharacteristic
327#131—6116no—
328#131—12216no—
329#131—1228no—
330C2×M16—21no—
331C2×M16—22no—
332C2×M16—21no—
333C2×M16—22no—
334C2×M16—22no—
335C2×M16—614no—
336C2×M16—612no—
337C2×M16—612no—
338C2×M16—614no—
339C2×M16—612no—
340C2×M16—1224no—
341C2×M16—12216no—
342C2×M16—1221no—
343C2×M16—1222no—
344C2×M16—1222no—
345C2×M16C4×D6114yes—
346C2×M16C22×D6112yes—
347C2×M16C22×D6112yes—
348C22×C8—21no—
349C22×C8—21no—
350C22×C8—612no—
351C22×C8—1224no—
352C22×C8—1221no—
353#117—1221no—
354#117—1221no—
355#117—1221no—
356#117—1221no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—1221no—
366C2×C42—1221no—
367#108—1228no—
368#107—1224no—
369#106—1224no—
370#103—1224no—
371#103—1222no—
372#103—1222no—
373#102—1224no—
374#102—1222no—
375#102—1222no—
376C24—1221no—
377C24#1175822211yescharacteristic
378#41—614no—
379#41—614no—
380#41—618no—
381#41—1222no—
382#41—1222no—
383#41—1224no—
384#41—1224no—
385#41—1224no—
386#41—1224no—
387#41—1224no—
388C2×Q8—612no—
389C2×Q8—614no—
390C2×Q8—1221no—
391C2×Q8—1222no—
392C2×D4—612no—
393C2×D4—614no—
394C2×D4—1222no—
395C2×D4—1224no—
396C2×D4—1222no—
397C2×D4—1224no—
398C2×D4—1222no—
399C2×D4—1221no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—82no—
404C22×C4—1221no—
405C22×C4—1222no—
406C22×C4—1224no—
407C22×C4—1222no—
408C22×C4—1224no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4C2×C4×D6112yes—
412C22×C4#1175822212yes—
413C22×C4#1175822211yescharacteristic
414C22×C4D4×D6112yes—
415C22×C4D4×D6111yescharacteristic
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4D4×D6111yescharacteristic
419C22×C4Q8×D6112yes—
420C22×C4C23×D6111yescharacteristic
421M16—22no—
422M16—24no—
423M16—22no—
424M16—22no—
425M16—42no—
426M16—618no—
427M16—618no—
428M16—12216no—
429M16—1222no—
430M16—1222no—
431M16—2442no—
432M16C2×C4×D6114yes—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—42no—
437C2×C8—614no—
438C2×C8—614no—
439C2×C8—614no—
440C2×C8—12216no—
441C2×C8—1222no—
442C2×C8—2442no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C22⋊C4—2441no—
452C22⋊C4—2441no—
453C42—2441no—
454C42—2441no—
455C23—41no—
456C23—41no—
457C23—1224no—
458C23—1222no—
459C23—4881no—
460C23—4881no—
461C23#1451973412yes—
462C23#1451975011yescharacteristic
463C23#1451975011yescharacteristic
464C23#1451975512yes—
465C23C2×#1175822211yescharacteristic
466Q8—614no—
467Q8—1222no—
468Q8—1224no—
469D4—614no—
470D4—1224no—
471D4—1224no—
472D4—1228no—
473D4—1222no—
474D4—1224no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—1222no—
480C2×C4—1224no—
481C2×C4—1224no—
482C2×C4—1224no—
483C2×C4—1228no—
484C2×C4—4881no—
485C2×C4—4881no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4C22⋊C4×D6112yes—
489C2×C4C22⋊C4×D6112yes—
490C2×C4C4⋊C4×D6112yes—
491C2×C4C4⋊C4×D6112yes—
492C2×C4#1451973412yes—
493C2×C4#1451975012yes—
494C2×C4#1451975012yes—
495C2×C4#1451975011yescharacteristic
496C2×C4#1451975011yescharacteristic
497C2×C4#1451975512yes—
498C2×C4#1451975512yes—
499C2×C4#1451975512yes—
500C2×C4C22×C4×D6111yescharacteristic
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4C2×D4×D6111yescharacteristic
503C2×C4C2×D4×D6111yescharacteristic
504C2×C4C2×Q8×D6111yescharacteristic
505C2×C4C2×#1175822211yescharacteristic
506C8—24no—
507C8—42no—
508C8—618no—
509C8—2442no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—1224no—
519C22—4884no—
520C22—2442no—
521C22—2442no—
522C22#1683962112yes—
523C22#1683962112yes—
524C22C2×#1451973411yescharacteristic
525C22C2×#1451975011yescharacteristic
526C22C2×#1451975511yescharacteristic
527C4—82no—
528C4—1224no—
529C4—4881no—
530C4—4881no—
531C4#1683962112yes—
532C4#1683962112yes—
533C4C2×C22⋊C4×D6111yescharacteristic
534C4C2×C4⋊C4×D6111yescharacteristic
535C4C2×#1451975011yescharacteristic
536C4C2×#1451975511yescharacteristic
537C2—22no—
538C2—21no—
539C2—21no—
540C2—22no—
541C2—2444no—
542C2C2×#1683962111yescharacteristic
543C2#2145554611yescharacteristic
544C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#343C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#343D61
hypocenterC61#20679
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16839810C2semidirect
3. C2×#16839802C2semidirect
6. #21445882C2semidirect
7. #21445877C2semidirect
8. #21445557C2semidirect
9. C2×C61×#343C2semidirect
12. #16840528C4semidirect
20. C2×#14519712C22semidirect
23. C2×#14519710C22semidirect
24. C2×#14519708C22semidirect
34. #16839530C22semidirect
35. #16839530C22semidirect
39. C61×#343C22semidirect
53. #14519785C2×C4semidirect
273. C2×#343D61semidirect
311. #343C2×D61semidirect
312. C61#20679semidirect

Maximal subgroups

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

subgroupindex
C2×#168405282
C2×#168398102
C2×#168398022
#214543392
C2×#168397612
#214458822
#214458772
#214455572
C2×C61×#3432
#2067961