Finite Group Databasebeta

← #29648638

Subgroups — #29648638

Order 15616.

950 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic
3#21454447C211yesmaximal, characteristic
4#21444548C214yesmaximal, complemented
5#21444526C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444501C211yesmaximal, characteristic, complemented
8#21444499C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10#16840317—21no—
11#16840317—21no—
12#16840317—21no—
13#16840317—21no—
14#16840317C412yes—
15#16840317C412yes—
16#16840317C2212yes—
17#16840317C2212yes—
18C2×#14519777C2211yescharacteristic
19C2×#14519768—21no—
20C2×#14519757C412yes—
21C2×#14519753—21no—
22C2×Q8×Dic61—21no—
23C2×#14519742C412yes—
24C2×#14519735—21no—
25C2×C42×D61C2211yescharacteristic
26#16839818—21no—
27#16839818C2212yes—
28#16839818C2212yes—
29#16839816—21no—
30#16839815C2211yescharacteristic
31#16839814—21no—
32#16839814C2212yes—
33#16839813—21no—
34#16839786C2211yescharacteristic
35#16839784C2212yes—
36#16839725—21no—
37#16839724—21no—
38M16×Dic61—21no—
39C4×#11758189—21no—
40#16839510C2214yescomplemented
41#16839510C2214yescomplemented
42#16839508C2214yescomplemented
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#431C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C61×#332C2214yescomplemented
47C22×Dic122C2×C412yes—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C2×#11758194—21no—
52C2×#11758189—22no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2×C412yes—
56#14519777C2312yes—
57#14519768—22no—
58#14519768—22no—
59#14519768—22no—
60#14519768—22no—
61#14519757—22no—
62#14519757—22no—
63#14519757C2×C414yes—
64#14519757C2×C414yes—
65#14519753—22no—
66#14519753—22no—
67#14519753—22no—
68#14519753—22no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73#14519742—22no—
74#14519742—22no—
75#14519742C2×C414yes—
76#14519742C2×C414yes—
77#14519735—22no—
78#14519735—22no—
79#14519735—22no—
80#14519735—22no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—22no—
84C2×#11758196—22no—
85C2×#11758198—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C42×D61C2×C411yescharacteristic
91C42×D61C2×C411yescharacteristic
92C42×D61D411yescharacteristic
93C42×D61D411yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61C2311yescharacteristic
97C42×D61C2311yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418C2312yes—
100C4⋊#92418C2312yes—
101#14519706—22no—
102C22⋊#92418—21no—
103#14519689—22no—
104#14519689—22no—
105#14519689—22no—
106#14519687—22no—
107#14519687—22no—
108C4×#92418—21no—
109C8×Dic61—22no—
110C22×Q8×C61—21no—
111C2×M16×C61—21no—
112C4⋊Q8×C61—21no—
113C4⋊Q8×C61—21no—
114C4⋊Q8×C61C2×C412yes—
115C4⋊Q8×C61C2312yes—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2×C412yes—
118C2×C42×C61C2311yescharacteristic
119C4⋊C8×C61—21no—
120C4⋊C8×C61C2312yes—
121C4⋊C8×C61C2312yes—
122C22⋊C8×C61—21no—
123C4×C8×C61—21no—
124C23×D61C22⋊C411yescharacteristic
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C22×C4×C61—22no—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×Dic61—22no—
133C22×Dic61C22⋊C411yescharacteristic
134C22×Dic61C22⋊C411yescharacteristic
135C22×Dic61C22⋊C411yescharacteristic
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61C22⋊C411yescharacteristic
141C2×C4×D61C22⋊C411yescharacteristic
142C2×C4×D61C22⋊C411yescharacteristic
143C2×C4×D61C22⋊C411yescharacteristic
144C2×C4×D61C22⋊C411yescharacteristic
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22×C411yescharacteristic
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×Dic122—22no—
153C2×Dic122—44no—
154C2×Dic122C22×C414yes—
155M16×C61—42no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61C22×C414yes—
162C42×C61C22×C411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2411yescharacteristic
166#11758198—24no—
167#11758198—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—24no—
173C4⋊Dic61—24no—
174#11758196—24no—
175#11758196—24no—
176C4×Dic61—24no—
177C4×Dic61—24no—
178C4×Dic61C22⋊C411yescharacteristic
179C4×Dic61C22⋊C411yescharacteristic
180C4×Dic61C22⋊C411yescharacteristic
181C4×Dic61C22⋊C411yescharacteristic
182C4×Dic61C22⋊C411yescharacteristic
183C4×Dic61C22⋊C411yescharacteristic
184C4×Dic61C22×C411yescharacteristic
185C4×Dic61C22×C411yescharacteristic
186C4×Dic61C2×D411yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C4×Dic61C2×D411yescharacteristic
189C4×Dic61C2×D411yescharacteristic
190#11758194—42no—
191C2×#92418—22no—
192C2×#92418—22no—
193#11758189—44no—
194C23×C61C2×C22⋊C411yescharacteristic
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61C2×C22⋊C411yescharacteristic
199C22×D61C2×C22⋊C411yescharacteristic
200Q8×C61—44no—
201Q8×C61—44no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—24no—
205C2×C4×C61—44no—
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×C4×C61C2×C22⋊C411yescharacteristic
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61#13712yes—
210C2×C4×C61#13712yes—
211C2×C4×C61C23×C411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×Dic61—22no—
215C2×Dic61—24no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—44no—
219C2×Dic61C2×C22⋊C411yescharacteristic
220C2×Dic61C2×C22⋊C411yescharacteristic
221C2×Dic61C2×C22⋊C411yescharacteristic
222C2×Dic61C2×C22⋊C411yescharacteristic
223C2×Dic61C2×C22⋊C411yescharacteristic
224C2×Dic61C2×C22⋊C411yescharacteristic
225C4×D61—21no—
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—21no—
229C4×D61—22no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—41no—
236C4×D61—41no—
237C4×D61—41no—
238C4×D61—41no—
239Dic122—44no—
240Dic122—44no—
241C8×C61—42no—
242C8×C61—42no—
243#92418—42no—
244#92418—42no—
245#17747—611nomaximal, Sylow
246C22×C61—21no—
247C22×C61—21no—
248C22×C61#41912yes—
249C22×C61C22×C22⋊C411yescharacteristic
250C2×D61—21no—
251C2×D61—21no—
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—44no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—44no—
260C4×C61#41312yes—
261C4×C61#41912yes—
262C4×C61C2×#13711yescharacteristic
263C4×C61C2×#13711yescharacteristic
264C4×C61#57512yes—
265Dic61—21no—
266Dic61—21no—
267Dic61—21no—
268Dic61—21no—
269Dic61—21no—
270Dic61—21no—
271Dic61—44no—
272C2×#533—611no—
273#2920—611no—
274#2920—611no—
275#1559—614no—
276#1554—614no—
277#1549—611no—
278#1549D6111yescharacteristic, complemented
279#1548—611no—
280#1548—611no—
281C2×C61—22no—
282C2×C61C2×#41311yescharacteristic
283C2×C61C2×#41911yescharacteristic
284C2×C61#289611yescharacteristic
285D61—41no—
286D61—41no—
287#533—612no—
288#533—612no—
289#533—612no—
290#533—612no—
291#533—1221no—
292#533—1221no—
293#533—1221no—
294#533—1221no—
295C2×C4⋊Q8—611no—
296C2×C4⋊Q8C2×D6111yescharacteristic
297C2×#125—1221no—
298C2×C22⋊Q8—612no—
299C2×C22⋊Q8—1221no—
300C2×C4×Q8—612no—
301C2×C4×Q8—1221no—
302C2×#117—1221no—
303C22×C42—611no—
304#432—612no—
305#432—612no—
306#432—1221no—
307#432—1221no—
308#431—611no—
309#431—612no—
310#431—1221no—
311#431—1221no—
312#431C2×D6111yescharacteristic
313#423—611no—
314#423—612no—
315#423C2×D6111yescharacteristic
316#407—1221no—
317#407—1221no—
318C4×M16—1221no—
319C4×M16—1221no—
320#333—614no—
321#333—614no—
322#332—614no—
323#332C2×D6114yescomplemented
324C61#1774711yescharacteristic, complemented, Sylow
325C22×Q8—21no—
326C22×Q8—612no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—611no—
330C2×M16—21no—
331C2×M16—1222no—
332C2×M16—1221no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—612no—
336C4⋊Q8—612no—
337C4⋊Q8—1221no—
338C4⋊Q8—1221no—
339C4⋊Q8C4×D6112yes—
340C4⋊Q8C22×D6112yes—
341#125—1222no—
342#125—1222no—
343#125—1222no—
344#125—1222no—
345C22⋊Q8—614no—
346C22⋊Q8—614no—
347C22⋊Q8—1222no—
348C22⋊Q8—1222no—
349C22⋊Q8—1222no—
350C22⋊Q8—1222no—
351C22⋊Q8—1222no—
352C22⋊Q8—1222no—
353C4×Q8—614no—
354C4×Q8—614no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×Q8—1222no—
358C4×Q8—1222no—
359C4×Q8—1222no—
360C4×Q8—1222no—
361#117—1222no—
362#117—1222no—
363#117—1222no—
364#117—1222no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C4×D6112yes—
371C2×C22⋊C4—1222no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—1222no—
384C2×C42C22×D6111yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—612no—
387C4⋊C8—612no—
388C4⋊C8—1222no—
389C4⋊C8—1221no—
390C4⋊C8C22×D6112yes—
391C4⋊C8C22×D6112yes—
392C22⋊C8—21no—
393C22⋊C8—1222no—
394C22⋊C8—1222no—
395C22⋊C8—1222no—
396C22⋊C8—1221no—
397#97—1222no—
398#97—1222no—
399C4×C8—21no—
400C4×C8—1222no—
401C4×C8—1221no—
402C24—611no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—614no—
407C2×Q8—1222no—
408C2×Q8—2444no—
409C22×C4—22no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4C2×C4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4D4×D6111yescharacteristic
433M16—42no—
434M16—2444no—
435M16—2442no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—1222no—
439C2×C8—1222no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—1224no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1224no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4C2×C4×D6114yes—
452C22⋊C4—1224no—
453C22⋊C4—1224no—
454C42—611no—
455C42—611no—
456C42—611no—
457C42—611no—
458C42—611no—
459C42—611no—
460C42—611no—
461C42—611no—
462C42—611no—
463C42—611no—
464C42—611no—
465C42—611no—
466C42—1224no—
467C42—1224no—
468C42C2×C4×D6111yescharacteristic
469C42D4×D6111yescharacteristic
470C42D4×D6111yescharacteristic
471C42C23×D6111yescharacteristic
472C23—611no—
473C23—611no—
474C23—1222no—
475C23—2441no—
476C23—2441no—
477C23C22⋊C4×D6111yescharacteristic
478Q8—44no—
479Q8—44no—
480Q8—2444no—
481Q8—2444no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—611no—
489C2×C4—611no—
490C2×C4—611no—
491C2×C4—611no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1224no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2444no—
511C2×C4C22⋊C4×D6111yescharacteristic
512C2×C4C22⋊C4×D6111yescharacteristic
513C2×C4C22⋊C4×D6111yescharacteristic
514C2×C4#1451980612yes—
515C2×C4#1451980912yes—
516C2×C4C22×C4×D6111yescharacteristic
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C2×D4×D6111yescharacteristic
519C8—42no—
520C8—42no—
521C8—2442no—
522C8—2442no—
523C22—21no—
524C22—21no—
525C22—1221no—
526C22—1221no—
527C22—1221no—
528C22—1221no—
529C22—1221no—
530C22—1221no—
531C22—2444no—
532C22#1683976912yes—
533C22C2×C22⋊C4×D6111yescharacteristic
534C4—21no—
535C4—21no—
536C4—44no—
537C4—1221no—
538C4—1221no—
539C4—1221no—
540C4—1221no—
541C4—1221no—
542C4—1221no—
543C4—2444no—
544C4#1683974512yes—
545C4#1683976912yes—
546C4C2×#1451980611yescharacteristic
547C4C2×#1451980911yescharacteristic
548C4#1684055812yes—
549C2—22no—
550C2—2441no—
551C2—2441no—
552C2C2×#1683974511yescharacteristic
553C2C2×#1683976911yescharacteristic
554C2#2145433311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1549D61
hypocenterC61#17747
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. #21444548C2semidirect
5. #21444526C2semidirect
6. #21444504C2semidirect
7. #21444501C2semidirect
8. #21444499C2semidirect
9. C61×#1549C2semidirect
40. #16839510C22semidirect
41. #16839510C22semidirect
42. #16839508C22semidirect
46. C61×#332C22semidirect
278. #1549D61semidirect
323. #332C2×D61semidirect
324. C61#17747semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214544512
#214544472
#214445482
#214445262
#214445042
#214445012
#214444992
C61×#15492
#1774761