Finite Group Databasebeta

← #29736141

Subgroups — #29736141

Order 15616.

1056 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2C2×#16840171C211yesmaximal, characteristic, complemented
3C2×#16840043C211yesmaximal, characteristic
4#21451024C211yesmaximal, characteristic, complemented
5#21447158C212yesmaximal, complemented
6#21447148C212yesmaximal, complemented
7#21446774C211yesmaximal, characteristic, complemented
8#21446083C212yesmaximal
9#21446069C212yesmaximal
10#21445645C211yesmaximal, characteristic
11C61×#1985C211yesmaximal, characteristic, complemented
12#16840472—24no—
13#16840472—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×#14519753—21no—
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25#16840171—24no—
26#16840043—24no—
27C2×#14519711C2212yes—
28C2×#14519711C2212yes—
29C2×#14519709C2212yes—
30C2×#14519699C2212yes—
31C2×#14519699C2212yes—
32#16839682—21no—
33#16839682—21no—
34#16839669—21no—
35#16839660C2212yes—
36#16839660C2212yes—
37#16839631—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39#16839525C2212yes—
40#16839525C2212yes—
41#16839525C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#97×C61C2211yescharacteristic
45C61×#336C2212yes—
46C61×#336C2212yes—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53#14519777—22no—
54#14519777—22no—
55#14519777—22no—
56#14519777—22no—
57#14519769—22no—
58#14519769—22no—
59#14519769—22no—
60#14519769D414yes—
61#14519755—28no—
62#14519755—28no—
63#14519753—28no—
64C4×D244—22no—
65C4×D244—22no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244D414yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—22no—
82C2×#11758196—22no—
83C4⋊C4×D61—28no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88#14519711—24no—
89#14519711—24no—
90#14519709—24no—
91#14519699—24no—
92#14519699Q818yescomplemented
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61D411yescharacteristic
97C22×C8×C61C2312yes—
98C22×C8×C61C2312yes—
99C4×Q8×C61—22no—
100C4×Q8×C61—22no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108#97×C61—22no—
109C23×D61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—41no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C2×D244—21no—
126C2×D244—22no—
127C2×D244C2×D412yes—
128C2×D244C2×Q814yes—
129C2×C4×D61—24no—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—82no—
135C2×C4×D61—82no—
136C2×Dic122—24no—
137C2×Dic122—42no—
138C2×Dic122—82no—
139C2×Dic122—82no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—24no—
144C2×C8×C61C2×Q814yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61—21no—
151C42×C61—21no—
152C42×C61C2×D412yes—
153C42×C61C2×D412yes—
154#11758198—24no—
155#11758198—24no—
156#11758198—24no—
157C4⋊Dic61—24no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—42no—
165C4⋊Dic61C2×D412yes—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D412yes—
168C4⋊Dic61C2×Q814yes—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172#11758196—24no—
173#11758196—82no—
174#11758196—82no—
175C4×Dic61—42no—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—24no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—42no—
182Q8×C61—44no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61#12012yes—
196C2×C4×C61#12012yes—
197C2×C4×C61#13712yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×Q812yes—
201C2×C4×C612−1+412yes—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—24no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215D244—22no—
216D244—24no—
217D244C4⋊Q814yes—
218C4×D61—48no—
219C4×D61—84no—
220C4×D61—84no—
221Dic122—84no—
222Dic122—84no—
223C8×C61—24no—
224C8×C61—42no—
225#25850—611nomaximal, Sylow
226C22×C61C2×#12011yescharacteristic
227C22×C61D4×Q812yes—
228C22×C61C2×#13711yescharacteristic
229C22×C61#57511yescharacteristic
230C22×C61#57612yes—
231C2×D61—24no—
232C2×D61—44no—
233C2×D61—44no—
234C2×D61—42no—
235C2×D61—42no—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—22no—
239C4×C61—41no—
240C4×C61—41no—
241C4×C61—41no—
242C4×C61C2×C4⋊Q811yescharacteristic
243C4×C61#53611yescharacteristic
244C4×C61D4×Q812yes—
245Dic61—44no—
246Dic61—42no—
247Dic61—42no—
248Dic61—42no—
249Dic61—82no—
250C2×#554—611no—
251C2×#501—611no—
252C2×#474—611no—
253#2567—611no—
254#2059—612no—
255#2057—612no—
256#1989—611no—
257#1985D6111yescharacteristic, complemented
258#1864—612no—
259#1862—612no—
260#1762—611no—
261C2×C61#243411yescharacteristic
262C2×C61#300711yescharacteristic
263C2×C61#300811yescharacteristic
264C2×C61#338512yes—
265C2×C61#338912yes—
266D61—44no—
267#554—1224no—
268#554—1224no—
269C2×C4⋊Q8—611no—
270C2×C4⋊Q8—611no—
271C2×#125—611no—
272C2×#125—611no—
273C2×C22⋊Q8—1221no—
274C2×C22⋊Q8—1221no—
275C2×C22⋊Q8—1221no—
276C2×C4×Q8C2×D6111yescharacteristic
277C2×C4×D4—611no—
278C2×C4×D4—611no—
279C2×C4×D4—611no—
280C22×C4⋊C4—1221no—
281#501—1224no—
282#474—1224no—
283C2×#107—612no—
284C2×#107—612no—
285C2×#106—612no—
286C2×C4⋊C8C2×D6111yescharacteristic
287C2×#102—612no—
288C2×#102—612no—
289C2×#97C2×D6111yescharacteristic
290#395—1221no—
291#395—1221no—
292#391—1221no—
293#389—612no—
294#389—612no—
295#384—1221no—
296C4×C4⋊C4—611no—
297#340—612no—
298#340—612no—
299#340—612no—
300#336C2×D6112yes—
301#336C2×D6112yes—
302C61#2585011yescharacteristic, complemented, Sylow
303C22×Q8—1221no—
304C22×Q8—1221no—
305C22×Q8D24411yescharacteristic
306C22×D4—611no—
307C23×C4—1221no—
308C23×C4—1221no—
309C23×C4—1221no—
310C22×C8C22×D6112yes—
311C22×C8C22×D6112yes—
312C4⋊Q8—1222no—
313C4⋊Q8—1222no—
314C4⋊Q8—1222no—
315C4⋊Q8—1222no—
316#125—614no—
317#125—1222no—
318#125—1222no—
319#125—1222no—
320C22⋊Q8—1228no—
321C22⋊Q8—1228no—
322C22⋊Q8—1228no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×D4—614no—
326C4×D4—1222no—
327C4×D4—1222no—
328C4×D4—1222no—
329C4×D4—1222no—
330C4×D4—1222no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—1228no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4D24411yescharacteristic
346C2×C4⋊C4D24411yescharacteristic
347C2×C4⋊C4D24411yescharacteristic
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C42—1221no—
352C2×C42C22×D6111yescharacteristic
353C2×C42C22×D6111yescharacteristic
354C2×C42C22×D6111yescharacteristic
355#107—1224no—
356#107—1224no—
357#106—1224no—
358C4⋊C8—22no—
359#102—618no—
360#102—1224no—
361#97—22no—
362#95—1221no—
363#95—1222no—
364#95—1222no—
365C24—1221no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—41no—
369C2×Q8—1224no—
370C2×Q8—4882no—
371C2×Q8—4882no—
372C2×Q8—2442no—
373C2×D4—612no—
374C2×D4—614no—
375C2×D4—1221no—
376C2×D4—1222no—
377C22×C4—1224no—
378C22×C4—1224no—
379C22×C4—1222no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1224no—
384C22×C4—1222no—
385C22×C4—4882no—
386C22×C4—4882no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4C2×D24411yescharacteristic
390C22×C4C2×D24411yescharacteristic
391C22×C4C2×D24411yescharacteristic
392C22×C4C2×D24411yescharacteristic
393C22×C4C2×D24411yescharacteristic
394C22×C4C2×D24411yescharacteristic
395C22×C4C23×D6111yescharacteristic
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8Q8×D6114yes—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—612no—
405C4⋊C4—612no—
406C4⋊C4—612no—
407C4⋊C4—614no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—1222no—
415C4⋊C4—1224no—
416C4⋊C4—1224no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—4882no—
420C4⋊C4—4882no—
421C4⋊C4—2442no—
422C22⋊C4—1224no—
423C22⋊C4—1224no—
424C22⋊C4—1224no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—2442no—
430C42D4×D6112yes—
431C42D4×D6112yes—
432C23—1222no—
433C23—1224no—
434C23—2441no—
435C23—2441no—
436C23—2442no—
437C23C22×D24411yescharacteristic
438Q8—44no—
439Q8—4884no—
440Q8—4884no—
441D4—614no—
442D4—1222no—
443D4—1224no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1224no—
460C2×C4—4884no—
461C2×C4—4884no—
462C2×C4—2448no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2441no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4#1451974712yes—
473C2×C4#1451974712yes—
474C2×C4#1451981012yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C2×Q8×D6112yes—
478C2×C4#1451982412yes—
479C8—24no—
480C8—42no—
481C22—1224no—
482C22—2444no—
483C22—2444no—
484C22—2442no—
485C22—2442no—
486C22C2×#1451974711yescharacteristic
487C22Q8×D24412yes—
488C22C2×#1451981011yescharacteristic
489C22#1684056211yescharacteristic
490C22#1684056712yes—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—4882no—
498C4—2444no—
499C4—2442no—
500C4—2442no—
501C4—2442no—
502C4C4⋊Q8×D6111yescharacteristic
503C4#1684033211yescharacteristic
504C4#1684042512yes—
505C2—2444no—
506C2#2144946211yescharacteristic
507C2#2145508111yescharacteristic
508C2#2145508911yescharacteristic
509C2#2145873112yes—
510C2#2145877912yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1985C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1985D61
hypocenterC61#25850
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. C2×#16840171C2semidirect
4. #21451024C2semidirect
5. #21447158C2semidirect
6. #21447148C2semidirect
7. #21446774C2semidirect
11. C61×#1985C2semidirect
92. #14519699Q8semidirect
257. #1985D61semidirect
302. C61#25850semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168401712
C2×#168400432
#214510242
#214471582
#214471482
#214467742
#214460832
#214460692
#214456452
C61×#19852
#2585061