Finite Group Databasebeta

← #29689839

Subgroups — #29689839

Order 15616.

1098 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×#16840361C211yesmaximal, characteristic
2#21448436C211yesmaximal, characteristic, complemented
3#21448419C211yesmaximal, characteristic, complemented
4#21446805C211yesmaximal, characteristic, complemented
5#21446655C211yesmaximal, characteristic
6#21446654C211yesmaximal, characteristic
7#21446276C214yesmaximal, complemented
8#21445806C214yesmaximal, complemented
9C61×#1996C211yesmaximal, characteristic, complemented
10#16840361—22no—
11#16840361—22no—
12#16840361—22no—
13#16840361—22no—
14C2×#14519772C412yescomplemented
15C2×#14519768C412yescomplemented
16C2×#14519762C412yescomplemented
17C2×#14519761C2212yes—
18C2×#14519753C412yescomplemented
19C2×#14519751C2211yescharacteristic
20C2×#14519737C2212yes—
21C2×#14519733C2212yes—
22C22×#11758196C2211yescharacteristic
23C2×#14519691C2214yes—
24C2×#14519691C2214yes—
25#16839659—22no—
26#16839657—22no—
27#16839646—22no—
28#16839644—22no—
29#16839623C2212yes—
30#16839622C2211yescharacteristic
31#16839620C2212yes—
32#16839620C2212yes—
33C4×C22⋊Dic61C2211yescharacteristic
34#16839529C2214yes—
35C2×C22⋊Q8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C4×C22⋊C4×C61C2211yescharacteristic
38C61×#342C2214yes—
39C22×D244C2×C411yescharacteristic
40C22×C22⋊D61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C22×C4×D61C2×C412yes—
44#14519772—24no—
45#14519772—24no—
46#14519768—24no—
47#14519768—24no—
48#14519762—24no—
49#14519762—24no—
50#14519761—24no—
51#14519761—24no—
52#14519753—24no—
53#14519753—24no—
54#14519751—21no—
55#14519751—21no—
56#14519751—21no—
57#14519751—21no—
58#14519737—24no—
59#14519737—24no—
60#14519733—44no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2×C411yescharacteristic
63C2×C4⋊Dic61C2×C412yes—
64C2×C4⋊Dic61C2×C412yes—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×#11758196C2×C412yes—
69C2×#11758196C2×C412yes—
70C2×#11758196C2×C412yes—
71C2×#11758196D412yes—
72C2×#11758196D412yes—
73C2×#11758196D412yes—
74C2×#11758196D412yes—
75C2×C22⋊Dic61C2312yes—
76C2×C22⋊Dic61C2312yes—
77C2×#11758198C2×C412yes—
78C2×#11758198C2×C412yes—
79C2×#11758198C2×C412yes—
80C2×#11758198C2×C412yes—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2×C412yes—
84#14519691—44no—
85#14519691—44no—
86#14519684—21no—
87#14519684—21no—
88#14519684—22no—
89#14519684—22no—
90#14519684—24no—
91C22×Q8×C61C2×C411yescharacteristic
92C23×C4×C61C2311yescharacteristic
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2×C412yes—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C22⋊C4×C61C2312yes—
101C2×C22⋊C4×C61C2312yes—
102C2×C22⋊C4×C61C2312yes—
103C2×C42×C61—21no—
104#95×C61—21no—
105#95×C61—21no—
106C24×C61C2411yescharacteristic
107C23×D61C22×C412yes—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—81no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—81no—
114C22×C4×C61C22×C412yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—81no—
131C22×Dic61C22×C412yes—
132C22×Dic61C22×C412yes—
133C22×Dic61C22×C412yes—
134C22×Dic61C2×D412yes—
135C22×Dic61C2×D412yes—
136C22×Dic61C2×D412yes—
137C22×Dic61C2×D412yes—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—81no—
141C2×C4×D61—24no—
142C2×C4×D61—82no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—81no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—22no—
153C42×C61—81no—
154C22⋊Dic61—22no—
155C22⋊Dic61—24no—
156C22⋊Dic61—22no—
157#11758198—22no—
158#11758198—22no—
159#11758198—24no—
160#11758198—24no—
161#11758198—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61—24no—
166C4⋊Dic61—81no—
167#11758196—24no—
168#11758196—44no—
169#11758196—82no—
170#11758196—82no—
171#11758196C22⋊C414yes—
172#11758196C22⋊C414yes—
173#11758196C22⋊C414yes—
174#11758196C22⋊C414yes—
175C4×Dic61—24no—
176C4×Dic61—44no—
177C4×Dic61—81no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C23×C411yescharacteristic
181C23×C61C22×D411yescharacteristic
182C23×C61C22×D411yescharacteristic
183C23×C61H(5,𝔽2)12yes—
184C23×C612−1+412yes—
185C22×D61—22no—
186C22×D61—24no—
187C22×D61—42no—
188C22×D61—42no—
189Q8×C61—81no—
190Q8×C61—81no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61C2×C22⋊C411yescharacteristic
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C22⋊D61—42no—
212C22⋊D61—42no—
213C22⋊D61—44no—
214C2×Dic61—24no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61C2×C22⋊C414yes—
230C2×Dic61C2×C22⋊C414yes—
231D244—81no—
232D244—81no—
233C4×D61—84no—
234#21877—611nomaximal, Sylow
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61C22×C22⋊C411yescharacteristic
244C22×C61#52012yes—
245C22×C61#53511yescharacteristic
246C22×C61#53611yescharacteristic
247C22×C61#53712yes—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—84no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—82no—
259C4×C61—82no—
260Dic61—24no—
261Dic61—24no—
262Dic61—42no—
263Dic61—82no—
264Dic61—82no—
265C2×#539—611no—
266#2316—611no—
267#2313—611no—
268#1996—611no—
269#1996D6111yescharacteristic, complemented
270#1964—611no—
271#1964—611no—
272#1899—614no—
273#1799—614no—
274C2×C61—21no—
275C2×C61—22no—
276C2×C61—22no—
277C2×C61—21no—
278C2×C61#231111yescharacteristic
279C2×C61#288311yescharacteristic
280C2×C61#288411yescharacteristic
281D61—82no—
282#539—1222no—
283#539—1222no—
284#539—1222no—
285#539—1222no—
286C2×#126—612no—
287C2×#125—612no—
288C2×#123—612no—
289C2×#123—612no—
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8C2×D6111yescharacteristic
292C2×#121—611no—
293C2×C4×D4—612no—
294C2×#117—612no—
295C22×C4⋊C4—611no—
296C2×#99—614no—
297C2×#99—614no—
298#389—1222no—
299#389—1222no—
300#387—1222no—
301#387—1222no—
302#381—612no—
303#380—611no—
304#380—612no—
305#380—612no—
306#380C2×D6111yescharacteristic
307C4×C22⋊C4—611no—
308C4×C22⋊C4C2×D6111yescharacteristic
309#342—614no—
310#342C2×D6114yes—
311C61#2187711yescharacteristic, complemented, Sylow
312C22×Q8C4×D6111yescharacteristic
313C22×D4—612no—
314C22×D4—611no—
315C23×C4—611no—
316C23×C4—611no—
317C23×C4—612no—
318C23×C4C22×D6111yescharacteristic
319#126—1224no—
320#126—1224no—
321#125—1224no—
322#125—1224no—
323#123—1224no—
324#123—1224no—
325#123—1224no—
326#123—1224no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—1224no—
332C22⋊Q8—1224no—
333#121—1221no—
334#121—1221no—
335#121—1221no—
336#121—1221no—
337C4×D4—1224no—
338C4×D4—1224no—
339#117—2444no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4C4×D6112yes—
356C2×C4⋊C4C4×D6111yescharacteristic
357C2×C22⋊C4—612no—
358C2×C22⋊C4—612no—
359C2×C22⋊C4—612no—
360C2×C22⋊C4—612no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4C22×D6112yes—
364C2×C22⋊C4C22×D6112yes—
365C2×C22⋊C4C22×D6112yes—
366C2×C42—21no—
367C2×C42—612no—
368C2×C42—1221no—
369C2×C42—1222no—
370#99—2444no—
371#99—2444no—
372#95—21no—
373#95—21no—
374#95—1222no—
375#95—1222no—
376#95—1224no—
377#95—1221no—
378#95—1221no—
379C24—612no—
380C24C23×D6111yescharacteristic
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—81no—
384C2×D4—1222no—
385C2×D4—1222no—
386C2×D4—1224no—
387C2×D4—1224no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—4881no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—81no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—1224no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—4882no—
407C22×C4—4881no—
408C22×C4C2×C4×D6112yes—
409C22×C4C2×C4×D6112yes—
410C22×C4C2×C4×D6111yescharacteristic
411C22×C4C2×C4×D6111yescharacteristic
412C22×C4D4×D6111yescharacteristic
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—81no—
421C4⋊C4—614no—
422C4⋊C4—614no—
423C4⋊C4—614no—
424C4⋊C4—614no—
425C4⋊C4—1224no—
426C4⋊C4—1224no—
427C4⋊C4—1224no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—4882no—
431C4⋊C4—4882no—
432C4⋊C4—4881no—
433C4⋊C4—2444no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—1224no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C22⋊C4—1222no—
443C22⋊C4—1222no—
444C22⋊C4—1222no—
445C22⋊C4—1224no—
446C22⋊C4—1222no—
447C42—81no—
448C42—1224no—
449C42—4881no—
450C42—2444no—
451C23—41no—
452C23—41no—
453C23—1224no—
454C23—1222no—
455C23—2442no—
456C23—2442no—
457C23C22×C4×D6111yescharacteristic
458C23C2×D4×D6111yescharacteristic
459C23C2×D4×D6111yescharacteristic
460C23#1451982312yes—
461C23#1451982512yes—
462Q8—81no—
463Q8—81no—
464D4—4881no—
465D4—4881no—
466D4—2442no—
467D4—2442no—
468D4—2444no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—614no—
486C2×C4—614no—
487C2×C4—1224no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1224no—
491C2×C4—1222no—
492C2×C4—1224no—
493C2×C4—4884no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4C22⋊C4×D6111yescharacteristic
504C2×C4C22⋊C4×D6111yescharacteristic
505C2×C4C22⋊C4×D6111yescharacteristic
506C2×C4C22⋊C4×D6111yescharacteristic
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—4884no—
516C22—2442no—
517C22—2442no—
518C22—2442no—
519C22C2×C22⋊C4×D6111yescharacteristic
520C22#1684024412yes—
521C22#1684033011yescharacteristic
522C22#1684033711yescharacteristic
523C22#1684034212yes—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—42no—
529C4—42no—
530C4—82no—
531C4—82no—
532C4—1224no—
533C4—1224no—
534C4—4882no—
535C4—4882no—
536C4—2442no—
537C2—21no—
538C2—22no—
539C2—22no—
540C2—21no—
541C2—4882no—
542C2#2144841011yescharacteristic
543C2#2145427711yescharacteristic
544C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1996C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519751C22
cosocleC24×C61C24
hypercenter#1996D61
hypocenterC61#21877
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. #21448436C2semidirect
3. #21448419C2semidirect
4. #21446805C2semidirect
7. #21446276C2semidirect
8. #21445806C2semidirect
9. C61×#1996C2semidirect
14. C2×#14519772C4semidirect
15. C2×#14519768C4semidirect
16. C2×#14519762C4semidirect
18. C2×#14519753C4semidirect
269. #1996D61semidirect
311. C61#21877semidirect

Maximal subgroups

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

subgroupindex
C2×#168403612
#214484362
#214484192
#214468052
#214466552
#214466542
#214462762
#214458062
C61×#19962
#2187761