Finite Group Databasebeta

← #29736966

Subgroups — #29736966

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic, complemented
2C2×#16839974C211yesmaximal, characteristic
3#21451308C211yesmaximal, characteristic, complemented
4#21446968C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446759C211yesmaximal, characteristic, complemented
8#21446164C212yesmaximal
9#21446150C212yesmaximal, complemented
10#21445645C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C22×#11758199C2211yescharacteristic
14#16840473—24no—
15#16840473—24no—
16C2×#14519768C2212yes—
17C2×#14519757—21no—
18C2×#14519753—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739C2212yes—
22C2×C4⋊C4×D61—21no—
23#16839974—24no—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2212yes—
27C2×#14519698C2211yescharacteristic
28#16839687—21no—
29#16839674—21no—
30#16839669C2212yes—
31#16839665C2212yes—
32#16839656—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#124×C61C2211yescharacteristic
41C61×#489—24no—
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C2×#11758201—22no—
50C2×#11758201—24no—
51C2×#11758199—22no—
52C2×#11758199—24no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519768—24no—
56#14519768—24no—
57#14519757—28no—
58#14519753—216no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×D244—22no—
62C4×D244D414yes—
63#14519739—24no—
64#14519739—24no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—22no—
70C2×#11758196D412yes—
71C4⋊C4×D61—28no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×#11758198—21no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61C2312yes—
78C4⋊#92418—22no—
79#14519701—22no—
80#14519699—24no—
81#14519698—22no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85C22×Q8×C61—21no—
86C22×Q8×C61C2311yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C22×C8×C61C2312yes—
89C4⋊Q8×C61—22no—
90C4⋊Q8×C61D414yes—
91#124×C61—22no—
92#124×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#103×C61—24no—
99#102×C61—24no—
100#97×C61—22no—
101C24×C61—21no—
102C23×D61—21no—
103C2×Q8×C61—24no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—81no—
109C2×Q8×C61—81no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—22no—
121C22×Dic61C2×D412yes—
122C22×Dic61C2×D412yes—
123C2×D244—21no—
124C2×D244—22no—
125C2×D244C2×D412yes—
126C2×D244#4114yes—
127C2×C4×D61—24no—
128C2×C4×D61—28no—
129C2×C4×D61—44no—
130C2×C4×D61—82no—
131C2×C4×D61—82no—
132C2×Dic122—22no—
133C2×Dic122—42no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61C2×D414yes—
140C22⋊C4×C61—42no—
141C22⋊C4×C61—42no—
142C42×C61C2×D412yes—
143C42×C61C2×D412yes—
144C22⋊Dic61—44no—
145C22⋊Dic61—82no—
146#11758201—44no—
147#11758199—44no—
148#11758198—28no—
149#11758198—24no—
150C4⋊Dic61—28no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61C2×D412yes—
156#11758196—28no—
157#11758196—24no—
158C4×Dic61—22no—
159C4×Dic61—22no—
160C4×Dic61—81no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418—22no—
165C23×C61—41no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61C22×D411yescharacteristic
171C22×D61—22no—
172C22×D61—24no—
173C22×D61—42no—
174C22×D61—41no—
175C22×D61—41no—
176Q8×C61—41no—
177Q8×C61—41no—
178Q8×C61—42no—
179Q8×C61—84no—
180D4×C61—41no—
181D4×C61—41no—
182D4×C61—42no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—24no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61#12012yes—
192C2×C4×C61#12012yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4112yes—
196C2×C4×C612−1+412yes—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208D244—22no—
209D244—24no—
210D244#12414yes—
211C4×D61—48no—
212C4×D61—88no—
213Dic122—44no—
214C8×C61—42no—
215#92418—41no—
216#92418—41no—
217#25940—611nomaximal, Sylow
218C22×C61—82no—
219C22×C61—82no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C2×#12011yescharacteristic
224C22×C61#54812yes—
225C22×C61#57512yes—
226C22×C61#57611yescharacteristic
227C22×C61#57711yescharacteristic
228C2×D61—24no—
229C2×D61—44no—
230C2×D61—44no—
231C2×D61—42no—
232C2×D61—42no—
233C4×C61—22no—
234C4×C61—22no—
235C4×C61—44no—
236C4×C61—81no—
237C4×C61C2×#12411yescharacteristic
238C4×C61#53611yescharacteristic
239C4×C61#54812yes—
240Dic61—42no—
241Dic61—42no—
242Dic61—42no—
243Dic61—82no—
244C2×#554—611no—
245C2×#489D6111yescharacteristic, complemented
246C2×#463—611no—
247#2592—611no—
248#2026—612no—
249#2025—612no—
250#1988—611no—
251#1985—611no—
252#1879—612no—
253#1877—612no—
254#1762—611no—
255C2×C61—161no—
256C2×C61#243611yescharacteristic
257C2×C61#300511yescharacteristic
258C2×C61#301211yescharacteristic
259C2×C61#337012yes—
260C2×C61#337612yes—
261D61—44no—
262C22×SD16—611no—
263C22×D8—611no—
264#554—1224no—
265#554—1224no—
266C2×C4⋊Q8C2×D6111yescharacteristic
267C2×#125—612no—
268C2×#124C2×D6111yescharacteristic
269C2×C22⋊Q8—1221no—
270C2×C22⋊Q8—1222no—
271C2×C4×Q8—611no—
272C2×C4×D4—612no—
273C2×C4×D4—611no—
274C22×C4⋊C4—1221no—
275#489—24no—
276#463—1224no—
277C2×C4⋊C8—611no—
278C2×#103—611no—
279C2×#103C2×D6112yes—
280C2×#102—611no—
281C2×#102—612no—
282C2×#102C2×D6112yes—
283C2×#97C2×D6111yescharacteristic
284#397—1221no—
285#393—1221no—
286#391—612no—
287#390—612no—
288#388—1222no—
289C4×C4⋊C4—611no—
290#340—612no—
291#340—612no—
292#340—612no—
293#336—612no—
294#336—612no—
295C61#2594011yescharacteristic, complemented, Sylow
296C22×Q8—21no—
297C22×Q8—611no—
298C22×Q8C22×D6111yescharacteristic
299C22×D4—611no—
300C22×D4C22×D6111yescharacteristic
301C23×C4—1222no—
302C23×C4—1221no—
303C2×SD16—1222no—
304C2×SD16—1224no—
305C2×D8—1222no—
306C2×D8—1224no—
307C22×C8—611no—
308C22×C8—611no—
309C22×C8C22×D6112yes—
310C4⋊Q8—22no—
311C4⋊Q8C22⋊D6114yes—
312#125—1224no—
313#125—1224no—
314#124—22no—
315#124—22no—
316C22⋊Q8—1228no—
317C22⋊Q8—12216no—
318C4×Q8—1222no—
319C4×Q8—1222no—
320C4×D4—614no—
321C4×D4—1224no—
322C4×D4—1224no—
323C4×D4—1222no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4—1228no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4C22×D6112yes—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C42—612no—
340C2×C42—1221no—
341C2×C42C22×D6111yescharacteristic
342C4⋊C8—1222no—
343#103—24no—
344#103—1222no—
345#102—24no—
346#102—1222no—
347#102—1224no—
348#97—22no—
349#95—1222no—
350#95—1222no—
351#95—1221no—
352C24—21no—
353C24—1221no—
354C2×Q8—24no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×Q8—1222no—
362C2×Q8—2442no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—612no—
368C2×D4—614no—
369C2×D4—1221no—
370C2×D4—1222no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—612no—
374C22×C4—612no—
375C22×C4—1228no—
376C22×C4—1221no—
377C22×C4—1224no—
378C22×C4—1222no—
379C22×C4—4882no—
380C22×C4—4882no—
381C22×C4—2444no—
382C22×C4D4×D6111yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4C23×D6111yescharacteristic
385SD16—2444no—
386D8—2444no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—1221no—
391C2×C8—1221no—
392C2×C8—1222no—
393C2×C8—1222no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—612no—
397C4⋊C4—1228no—
398C4⋊C4—1228no—
399C4⋊C4—1221no—
400C4⋊C4—1222no—
401C4⋊C4—1224no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4C2×C22⋊D6114yes—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—1228no—
408C22⋊C4—1224no—
409C22⋊C4—4882no—
410C22⋊C4—2444no—
411C42—1222no—
412C42—1222no—
413C42—4881no—
414C42D4×D6112yes—
415C42C2×C22⋊D6112yes—
416C23—41no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—1222no—
422C23—1224no—
423C23—2442no—
424C23—2441no—
425C23—2441no—
426C23C2×D4×D6111yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—42no—
430Q8—84no—
431Q8—2444no—
432D4—41no—
433D4—41no—
434D4—42no—
435D4—614no—
436D4—1222no—
437D4—1224no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—4888no—
454C2×C4—2448no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4#1451974512yes—
460C2×C4#1451974512yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C22×C22⋊D6111yescharacteristic
463C2×C4#41×D6112yes—
464C2×C4#1451982412yes—
465C8—42no—
466C8—2441no—
467C8—2441no—
468C22—81no—
469C22—81no—
470C22—82no—
471C22—81no—
472C22—82no—
473C22—1224no—
474C22—2442no—
475C22—2444no—
476C22—2442no—
477C22—2444no—
478C22C2×#1451974511yescharacteristic
479C22#1684042012yes—
480C22#1684056112yes—
481C22#1684056611yescharacteristic
482C22#1684057211yescharacteristic
483C4—22no—
484C4—22no—
485C4—44no—
486C4—81no—
487C4—4882no—
488C4—2442no—
489C4—2442no—
490C4—2442no—
491C4#124×D6111yescharacteristic
492C4#1684033211yescharacteristic
493C4#1684041912yes—
494C2—161no—
495C2—2444no—
496C2#2144947411yescharacteristic
497C2#2145506011yescharacteristic
498C2#2145512211yescharacteristic
499C2#2145857112yes—
500C2#2145864512yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#489D61
hypocenterC61#25940
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#16840473C2semidirect
3. #21451308C2semidirect
4. #21446968C2semidirect
5. #21446963C2semidirect
6. #21446772C2semidirect
7. #21446759C2semidirect
9. #21446150C2semidirect
10. #21445645C2semidirect
11. C2×C61×#489C2semidirect
245. C2×#489D61semidirect
295. C61#25940semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×#168399742
#214513082
#214469682
#214469632
#214467722
#214467592
#214461642
#214461502
#214456452
C2×C61×#4892
#2594061