Finite Group Databasebeta

← #29714070

Subgroups — #29714070

Order 15616.

1184 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×#16840039C211yesmaximal, characteristic, complemented
2C2×#16839860C211yesmaximal, characteristic, complemented
3#21450720C211yesmaximal, characteristic
4#21447019C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446136C212yesmaximal, complemented
9#21446007C212yesmaximal, complemented
10#21445834C211yesmaximal, characteristic, complemented
11C2×C61×#554C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16840039—24no—
16#16839860—24no—
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839688—21no—
28#16839670—21no—
29#16839638—22no—
30#16839628C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#554—24no—
36C61×#554—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#125×C61—21no—
39C2×C22⋊Q8×C61—22no—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519751—24no—
55#14519751—42no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×C22⋊Dic61—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63#14519710—24no—
64#14519710—22no—
65C4⋊#92418—22no—
66#14519701—22no—
67#14519701—42no—
68#14519697—24no—
69#14519697—22no—
70C22⋊#92418—24no—
71C22⋊#92418—42no—
72#14519688—22no—
73#14519684—22no—
74#14519684—21no—
75#14519684—21no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61D411yescharacteristic
79C23×C4×C61—22no—
80C23×C4×C61C2312yes—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61D414yescomplemented
83#125×C61—28no—
84C22⋊Q8×C61—216no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yescomplemented
89C4×D4×C61—28no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—24no—
97C2×C4⋊C4×C61—24no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61D412yes—
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D412yes—
106C23×D61—21no—
107C2×Q8×C61—24no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—82no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—24no—
116C2×D4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—28no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—82no—
128C22×C4×C61—82no—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—82no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244#4112yes—
144C2×C4×D61—42no—
145C2×C4×D61—82no—
146C4⋊C4×C61—28no—
147C4⋊C4×C61—28no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D412yes—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—28no—
156C22⋊C4×C61—24no—
157C42×C61—24no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160C22⋊Dic61—82no—
161#11758201—28no—
162#11758198—44no—
163#11758198—42no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418#4112yes—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61#13714yes—
184C23×C61C22×D411yescharacteristic
185C22×D61—22no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—42no—
189C22×D61—42no—
190Q8×C61—22no—
191Q8×C61—22no—
192Q8×C61—41no—
193Q8×C61—41no—
194Q8×C61—84no—
195D4×C61—28no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—88no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C22⋊D61—88no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228D244—22no—
229D244—22no—
230D244—41no—
231D244—41no—
232C4×D61—84no—
233#92418—22no—
234#92418—41no—
235#92418—42no—
236#24130—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—24no—
239C22×C61—24no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61C2×#12011yescharacteristic
245C22×C61#54611yescharacteristic
246C22×C61#54711yescharacteristic
247C22×C61C2×#13712yes—
248C22×C61#57612yes—
249C2×D61—42no—
250C2×D61—44no—
251C2×D61—82no—
252C2×D61—81no—
253C2×D61—81no—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61—44no—
260C4×C61#52511yescharacteristic
261C4×C61#53711yescharacteristic
262C4×C61#54811yescharacteristic
263C4×C61#55011yescharacteristic
264Dic61—81no—
265Dic61—82no—
266C2×#554D6111yescharacteristic, complemented
267C2×#474—611no—
268C2×#440—611no—
269#2535—611no—
270#2036—612no—
271#2000—612no—
272#1993—611no—
273#1955—611no—
274#1874—612no—
275#1852—612no—
276#1805—611no—
277C2×C61—44no—
278C2×C61#240711yescharacteristic
279C2×C61#300411yescharacteristic
280C2×C61#300811yescharacteristic
281C2×C61#331412yes—
282C2×C61#334612yes—
283D61—82no—
284C22×SD16—611no—
285#554—24no—
286#554—24no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×#125—21no—
289C2×C22⋊Q8—22no—
290C2×C22⋊Q8C2×D6112yes—
291C2×#121—612no—
292C2×C4×Q8C2×D6111yescharacteristic
293C2×C4×D4—21no—
294C2×C4×D4—611no—
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4C2×D6112yes—
297#474—1224no—
298#440—1224no—
299C2×#107—611no—
300C2×#107—612no—
301C2×C4⋊C8—611no—
302C2×#103—611no—
303C2×#103—612no—
304C2×#102—611no—
305C2×#102—612no—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97—611no—
309#397—1221no—
310#392—1221no—
311#386—1222no—
312#382—611no—
313#380—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317C61#2413011yescharacteristic, complemented, Sylow
318C22×Q8—21no—
319C22×Q8C22×D6111yescharacteristic
320C22×D4—611no—
321C22×D4—1222no—
322C22×D4C22⋊D6111yescharacteristic
323C23×C4—22no—
324C23×C4—1221no—
325C23×C4C22×D6112yes—
326C2×SD16—1222no—
327C2×SD16—1222no—
328C2×SD16—1222no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8—612no—
332C4⋊Q8—22no—
333C4⋊Q8C22⋊D6114yescomplemented
334#125—28no—
335C22⋊Q8—216no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338#121—1224no—
339#121—2442no—
340C4×Q8—22no—
341C4×Q8C22⋊D6114yescomplemented
342C4×D4—28no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—1222no—
346C4×D4—1222no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4C22⋊D6111yescharacteristic
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C4⋊C4C22×D6112yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4C22⋊D6112yes—
363C2×C42—21no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366#107—1224no—
367C4⋊C8—1222no—
368#103—1222no—
369#103—2442no—
370#102—1222no—
371#102—1224no—
372C22⋊C8—1224no—
373C22⋊C8—2442no—
374#97—1222no—
375#95—1222no—
376#95—1221no—
377#95—1221no—
378C24—1221no—
379C24C2×C22⋊D6112yes—
380C2×Q8—21no—
381C2×Q8—24no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—82no—
387C2×Q8C2×C22⋊D6112yes—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—612no—
391C2×D4—1221no—
392C2×D4—1221no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—4882no—
397C2×D4—2444no—
398C2×D4—2444no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—1222no—
412C22×C4—1221no—
413C22×C4—4882no—
414C22×C4—2442no—
415C22×C4C2×C22⋊D6112yes—
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419SD16—1228no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1222no—
428C2×C8—1221no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4C2×C22⋊D6112yes—
441C4⋊C4C2×C22⋊D6114yes—
442C22⋊C4—28no—
443C22⋊C4—24no—
444C22⋊C4—4882no—
445C22⋊C4—2444no—
446C22⋊C4—2442no—
447C42—24no—
448C42C2×C22⋊D6112yes—
449C42C2×C22⋊D6112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—1222no—
457C23—2441no—
458C23—2442no—
459C23—2442no—
460C23—2441no—
461C23#1451980814yes—
462C23C22×C22⋊D6111yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—41no—
466Q8—41no—
467Q8—84no—
468D4—28no—
469D4—1222no—
470D4—1222no—
471D4—4888no—
472D4—2441no—
473D4—2441no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—88no—
492C2×C4—4884no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4#1451974612yes—
499C2×C4#1451974612yes—
500C2×C4C22×C22⋊D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C2×C4#41×D6111yescharacteristic
503C2×C4#41×D6111yescharacteristic
504C2×C4#1451982311yescharacteristic
505C2×C4#1451982411yescharacteristic
506C8—1222no—
507C8—2441no—
508C8—2442no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—4882no—
517C22—4881no—
518C22—4881no—
519C22—2442no—
520C22—2444no—
521C22C2×#1451974611yescharacteristic
522C22#1684040011yescharacteristic
523C22#1684041111yescharacteristic
524C22C2×#1451980812yes—
525C22#1684056512yes—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—4882no—
533C4—4881no—
534C4#1684026911yescharacteristic
535C4#1684034011yescharacteristic
536C4#1684041911yescharacteristic
537C4#1684043111yescharacteristic
538C2—44no—
539C2—4882no—
540C2#2144915811yescharacteristic
541C2#2145505311yescharacteristic
542C2#2145508711yescharacteristic
543C2#2145789412yes—
544C2#2145829812yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radical#21450720C2
hypercenterC2×#554D61
hypocenterC61#24130
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
2. C2×#16839860C2semidirect
4. #21447019C2semidirect
5. #21446820C2semidirect
6. #21446793C2semidirect
7. #21446618C2semidirect
8. #21446136C2semidirect
9. #21446007C2semidirect
10. #21445834C2semidirect
11. C2×C61×#554C2semidirect
82. C4⋊Q8×C61D4semidirect
88. C4×Q8×C61D4semidirect
266. C2×#554D61semidirect
317. C61#24130semidirect
333. C4⋊Q8C22⋊D61semidirect
341. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C2×#168398602
#214507202
#214470192
#214468202
#214467932
#214466182
#214461362
#214460072
#214458342
C2×C61×#5542
#2413061