Finite Group Databasebeta

← #29716441

Subgroups — #29716441

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic
4#21447157C212yesmaximal, complemented
5#21446970C212yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446756C211yesmaximal, characteristic, complemented
8#21446156C212yesmaximal
9#21446065C212yesmaximal
10#21445816C211yesmaximal, characteristic
11C61×#2582C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519769C2212yes—
17C2×#14519755—22no—
18C2×#14519753—21no—
19C2×C4×D244C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22#16840057—24no—
23C4×#11758201—24no—
24C2×#14519710C2212yes—
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2212yes—
28C2×#14519701C2211yescharacteristic
29C2×#14519697C2212yes—
30C2×#14519697C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839517C2212yes—
36#16839517C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#398—22no—
40C61×#391—21no—
41C61×#391C2212yes—
42C61×#389—21no—
43C61×#389C2212yes—
44C61×#384C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C2×#11758201Q814yescomplemented
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yes—
57#14519769—24no—
58#14519769—24no—
59#14519755—216no—
60#14519753—28no—
61C4×D244—24no—
62C4×D244—22no—
63C4×D244—24no—
64C4×D244D414yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C4⋊C4×D61—28no—
71C2×#11758198—22no—
72C2×#11758198—21no—
73#14519710—24no—
74#14519708—22no—
75C4⋊#92418—22no—
76#14519701—24no—
77#14519701—22no—
78#14519697—22no—
79#14519697—24no—
80C4×#92418—22no—
81C22×Q8×C61D411yescharacteristic
82C22×Q8×C61C2311yescharacteristic
83C4×Q8×C61—22no—
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61D411yescharacteristic
92C2×C4⋊C4×C61D412yes—
93C2×C4⋊C4×C61C2312yes—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C42×C61—21no—
96C2×C42×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#95×C61—22no—
99#95×C61—22no—
100#95×C61—22no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61C2×Q812yes—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244C2×D412yes—
125C2×D244C2×Q812yes—
126C2×D244#4112yes—
127C2×C4×D61—24no—
128C2×C4×D61—28no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61—82no—
132C2×C4×D61—82no—
133C2×Dic122—24no—
134C2×Dic122—81no—
135C2×Dic122—81no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61—81no—
145C4⋊C4×C61—82no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61—42no—
149C42×C61C2×D412yes—
150C42×C61C2×D412yes—
151#11758201—28no—
152#11758198—28no—
153#11758198—24no—
154C4⋊Dic61—24no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61C2×D412yes—
158C4⋊Dic61C2×D414yes—
159#11758196—28no—
160#11758196—28no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418C2×Q812yes—
170C23×C61C22×D411yescharacteristic
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178Q8×C61—22no—
179Q8×C61—22no—
180Q8×C61—41no—
181Q8×C61—41no—
182Q8×C61—44no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61#12012yes—
201C2×C4×C61#12012yes—
202C2×C4×C61#13512yes—
203C2×C4×C61#13712yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×Q811yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C2×Dic61—22no—
211C2×Dic61—24no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216D244—22no—
217D244—22no—
218D244—22no—
219D244C22⋊Q814yes—
220C4×D61—48no—
221C4×D61—88no—
222Dic122—84no—
223#92418—22no—
224#92418—41no—
225#92418—42no—
226#24331—611nomaximal, Sylow
227C22×C61C2×#12011yescharacteristic
228C22×C61#54811yescharacteristic
229C22×C61D4×Q811yescharacteristic
230C22×C61C2×#13511yescharacteristic
231C22×C61C2×#13711yescharacteristic
232C22×C61#57611yescharacteristic
233C22×C61#57711yescharacteristic
234C2×D61—24no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C4×C61—22no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—82no—
246C4×C61—81no—
247C4×C61#45214yes—
248C4×C61C2×C22⋊Q811yescharacteristic
249C4×C61#53611yescharacteristic
250C4×C61#54811yescharacteristic
251C4×C61D4×Q811yescharacteristic
252Dic61—42no—
253Dic61—44no—
254C2×#554—611no—
255C2×#477—611no—
256C2×C4×SD16—611no—
257#2582D6111yescharacteristic, complemented
258#2059—612no—
259#2026—612no—
260#1990—611no—
261#1984—611no—
262#1877—612no—
263#1862—612no—
264#1802—611no—
265C2×C61#240911yescharacteristic
266C2×C61C2×#45211yescharacteristic
267C2×C61#301211yescharacteristic
268C2×C61#336111yescharacteristic
269C2×C61#337211yescharacteristic
270C2×C61#338111yescharacteristic
271C2×C61#338511yescharacteristic
272D61—44no—
273C22×SD16—611no—
274#554—1224no—
275#554—1224no—
276C2×C4⋊Q8—611no—
277C2×#125—612no—
278C2×C22⋊Q8—1221no—
279C2×C22⋊Q8—1222no—
280C2×C4×Q8C2×D6111yescharacteristic
281C2×C4×Q8C2×D6111yescharacteristic
282C2×C4×D4—611no—
283C2×C4×D4—612no—
284C22×C4⋊C4—1221no—
285#477—1224no—
286C4×SD16—1224no—
287C2×#107—612no—
288C2×#106—611no—
289C2×C4⋊C8—611no—
290C2×#103—612no—
291C2×#103—611no—
292C2×#102—612no—
293C2×#102—611no—
294C2×C4×C8—611no—
295#398—22no—
296#391—21no—
297#391C2×D6112yes—
298#389—21no—
299#389C2×D6112yes—
300#384C2×D6111yescharacteristic
301#340—612no—
302#340—612no—
303#340—612no—
304#336—612no—
305#336—612no—
306C61#2433111yescharacteristic, complemented, Sylow
307C22×Q8—1221no—
308C22×Q8C22⋊D6111yescharacteristic
309C22×Q8C22×D6111yescharacteristic
310C22×D4—611no—
311C23×C4—1222no—
312C23×C4—1221no—
313C2×SD16—614no—
314C2×SD16—1222no—
315C2×SD16—1222no—
316C22×C8—611no—
317C22×C8—611no—
318C22×C8—612no—
319C4⋊Q8—614no—
320C4⋊Q8—1222no—
321#125—1224no—
322#125—1224no—
323C22⋊Q8—12216no—
324C22⋊Q8—1228no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×D4—614no—
330C4×D4—1224no—
331C4×D4—1224no—
332C4×D4—1222no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1228no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C4⋊C4C22⋊D6112yes—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C4⋊C4C22×D6112yes—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1221no—
349C2×C42—21no—
350C2×C42C22×D6112yes—
351C2×C42C22×D6111yescharacteristic
352#107—1224no—
353#106—1222no—
354C4⋊C8—1222no—
355#103—1222no—
356#103—1224no—
357#102—1224no—
358#102—1222no—
359C4×C8—1222no—
360#95—22no—
361#95—22no—
362#95—22no—
363C24—1221no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—1224no—
373C2×Q8—4881no—
374C2×Q8—4881no—
375C2×Q8Q8×D6112yes—
376C2×D4—612no—
377C2×D4—612no—
378C2×D4—612no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—1228no—
386C22×C4—1221no—
387C22×C4—1222no—
388C22×C4—1224no—
389C22×C4—4882no—
390C22×C4—4882no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4C2×C22⋊D6112yes—
394C22×C4C2×C22⋊D6112yes—
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C23×D6111yescharacteristic
398SD16—1228no—
399C2×C8—612no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8—1222no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—82no—
418C4⋊C4—612no—
419C4⋊C4—614no—
420C4⋊C4—1228no—
421C4⋊C4—1228no—
422C4⋊C4—1221no—
423C4⋊C4—1222no—
424C4⋊C4—1224no—
425C22⋊C4—1228no—
426C22⋊C4—1224no—
427C42—22no—
428C42—22no—
429C42—42no—
430C42D4×D6112yes—
431C42D4×D6112yes—
432C23—1222no—
433C23—1222no—
434C23—1222no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23C22×C22⋊D6111yescharacteristic
440Q8—22no—
441Q8—22no—
442Q8—41no—
443Q8—41no—
444Q8—44no—
445Q8—4884no—
446D4—614no—
447D4—1222no—
448D4—1222no—
449D4—1222no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—1222no—
468C2×C4—1224no—
469C2×C4—4888no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2448no—
475C2×C4#1451974512yes—
476C2×C4#1451974512yes—
477C2×C4#1451979712yes—
478C2×C4#1451980812yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4C2×Q8×D6111yescharacteristic
482C2×C4#41×D6111yescharacteristic
483C2×C4#1451982411yescharacteristic
484C2×C4#1451982511yescharacteristic
485C8—1222no—
486C8—2441no—
487C8—2442no—
488C22—1224no—
489C22—2442no—
490C22—2442no—
491C22—2442no—
492C22—2442no—
493C22—2442no—
494C22—2442no—
495C22C2×#1451974511yescharacteristic
496C22#1684041911yescharacteristic
497C22Q8×C22⋊D6111yescharacteristic
498C22C2×#1451979711yescharacteristic
499C22C2×#1451980811yescharacteristic
500C22#1684056511yescharacteristic
501C22#1684057011yescharacteristic
502C4—22no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—82no—
508C4—2444no—
509C4—2442no—
510C4#1683992214yes—
511C4C22⋊Q8×D6111yescharacteristic
512C4#1684033511yescharacteristic
513C4#1684042211yescharacteristic
514C4#1684042511yescharacteristic
515C2—2444no—
516C2#2144918011yescharacteristic
517C2C2×#1683992211yescharacteristic
518C2#2145512011yescharacteristic
519C2#2145847511yescharacteristic
520C2#2145859611yescharacteristic
521C2#2145869011yescharacteristic
522C2#2145872511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2582C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2582D61
hypocenterC61#24331
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
4. #21447157C2semidirect
5. #21446970C2semidirect
6. #21446779C2semidirect
7. #21446756C2semidirect
11. C61×#2582C2semidirect
51. C2×#11758201Q8semidirect
257. #2582D61semidirect
306. C61#24331semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168400572
C2×C4×#117582012
#214471572
#214469702
#214467792
#214467562
#214461562
#214460652
#214458162
C61×#25822
#2433161