Finite Group Databasebeta

← #29736994

Subgroups — #29736994

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic122C211yesmaximal, characteristic, complemented
2C2×#16840122C211yesmaximal, characteristic
3C2×#16839985C211yesmaximal, characteristic
4#21451307C211yesmaximal, characteristic
5#21446961C212yesmaximal, complemented
6#21446956C212yesmaximal
7#21446770C211yesmaximal, characteristic, complemented
8#21446173C212yesmaximal
9#21446159C212yesmaximal, complemented
10#21445650C211yesmaximal, characteristic, complemented
11C61×#1985C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13C22×#11758190C2211yescharacteristic
14Q8×Dic122—24no—
15Q8×Dic122—24no—
16C2×#14519777C2212yes—
17C2×#14519777C2211yescharacteristic
18C2×#14519776—21no—
19C2×#14519776—22no—
20C2×#14519765C2211yescharacteristic
21C2×Q8×Dic61—21no—
22C2×C4×Dic122C2212yes—
23C2×C4×Dic122C2211yescharacteristic
24#16840122—24no—
25#16839985—24no—
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519702C2211yescharacteristic
29C2×#14519699C2212yes—
30C2×#14519699C2211yescharacteristic
31#16839688—21no—
32#16839673—21no—
33#16839668C2212yes—
34#16839664C2212yes—
35#16839652—22no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#97×C61C2211yescharacteristic
43C61×#336C2212yes—
44C61×#336C2212yes—
45C22×Dic122—22no—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×Dic122C2311yescharacteristic
49C22×D244C2311yescharacteristic
50C2×Dic244—22no—
51C2×Dic244—24no—
52C2×#11758190—22no—
53C2×#11758190—24no—
54#14519777—22no—
55#14519777—24no—
56#14519777—24no—
57#14519777D414yes—
58#14519776—216no—
59#14519776—28no—
60#14519765—22no—
61#14519765—22no—
62Q8×Dic61—28no—
63C4×Dic122—24no—
64C4×Dic122—22no—
65C4×Dic122—24no—
66C4×Dic122D414yes—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519702—24no—
82#14519702—22no—
83#14519702—24no—
84#14519699—22no—
85#14519699—24no—
86#14519684—21no—
87#14519684—22no—
88#14519684—22no—
89C22×Q8×C61D411yescharacteristic
90C22×C8×C61C2312yes—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61D412yes—
97C2×C42×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#97×C61—22no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—24no—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—22no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—22no—
117C2×D244—22no—
118C2×Dic122—21no—
119C2×Dic122—22no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—22no—
123C2×Dic122—22no—
124C2×Dic122—28no—
125C2×Dic122—24no—
126C2×Dic122—81no—
127C2×Dic122—81no—
128C2×Dic122—82no—
129C2×Dic122—82no—
130C2×Dic122C2×D412yes—
131C2×Dic122#4114yes—
132C2×C8×C61—22no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—22no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D412yes—
144C42×C61C2×D412yes—
145#11758198—42no—
146#11758198—42no—
147#11758198—44no—
148#11758198—82no—
149C4⋊Dic61—28no—
150C4⋊Dic61—24no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61C2×D414yes—
156C4⋊Dic61C2×D412yes—
157#11758196—28no—
158#11758196—24no—
159C4×Dic61—28no—
160C4×Dic61—24no—
161C4×Dic61—81no—
162Dic244—44no—
163#11758190—44no—
164C23×C61C22×D411yescharacteristic
165C22×D61—41no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—41no—
169C22×D61—41no—
170Q8×C61—28no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61#12012yes—
180C2×C4×C61#12012yes—
181C2×C4×C61#13714yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4112yes—
185C2×C4×C612−1+412yes—
186C2×Dic61—22no—
187C2×Dic61—24no—
188C2×Dic61—24no—
189C2×Dic61—22no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199D244—41no—
200D244—41no—
201D244—42no—
202Dic122—22no—
203Dic122—24no—
204Dic122—41no—
205Dic122—41no—
206Dic122—42no—
207Dic122—84no—
208Dic122—88no—
209Dic122#12414yes—
210C8×C61—41no—
211C8×C61—41no—
212C8×C61—42no—
213#25942—611nomaximal, Sylow
214C22×C61C2×#12011yescharacteristic
215C22×C61#54812yes—
216C22×C61C2×#13712yes—
217C22×C61#57611yescharacteristic
218C22×C61#57711yescharacteristic
219C2×D61—82no—
220C2×D61—82no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C4×C61—22no—
225C4×C61—24no—
226C4×C61—22no—
227C4×C61—42no—
228C4×C61C2×#12411yescharacteristic
229C4×C61#53611yescharacteristic
230C4×C61#54812yes—
231Dic61—24no—
232Dic61—44no—
233Dic61—42no—
234Dic61—82no—
235Dic61—81no—
236C2×Q82—611no—
237C2×#489—611no—
238C2×#464—611no—
239#2592—611no—
240#2024—612no—
241#2023—612no—
242#1988—611no—
243#1985D6111yescharacteristic, complemented
244#1880—612no—
245#1878—612no—
246#1763—611no—
247C2×C61#243611yescharacteristic
248C2×C61#300411yescharacteristic
249C2×C61#301211yescharacteristic
250C2×C61#336912yes—
251C2×C61#337512yes—
252D61—161no—
253C22×Q16—611no—
254C22×SD16—611no—
255Q82—1224no—
256Q82—1224no—
257C2×C4⋊Q8—611no—
258C2×C4⋊Q8—612no—
259C2×C4⋊Q8—1222no—
260C2×C4⋊Q8—1221no—
261C2×#124—611no—
262C2×C4×Q8—611no—
263C2×C4×Q8—612no—
264C2×C4×Q8—1221no—
265C2×C4×Q8C2×D6111yescharacteristic
266#489—1224no—
267#464—1224no—
268C2×C4⋊C8C2×D6111yescharacteristic
269C2×#103—612no—
270C2×#103—612no—
271C2×#103—611no—
272C2×#102—612no—
273C2×#102—611no—
274C2×#97C2×D6111yescharacteristic
275#397—1221no—
276#393—1221no—
277#391—612no—
278#390—612no—
279#388—1222no—
280C4×C4⋊C4—611no—
281#340—612no—
282#340—612no—
283#340—612no—
284#336C2×D6112yes—
285#336C2×D6112yes—
286C61#2594211yescharacteristic, complemented, Sylow
287C22×Q8—611no—
288C22×Q8—611no—
289C22×Q8—1222no—
290C22×Q8—1221no—
291C22×Q8D24411yescharacteristic
292C22×D4—611no—
293C2×Q16—1222no—
294C2×Q16—1224no—
295C2×SD16—1222no—
296C2×SD16—1224no—
297C22×C8C22×D6111yescharacteristic
298C22×C8C22×D6111yescharacteristic
299C22×C8C22×D6112yes—
300C4⋊Q8—614no—
301C4⋊Q8—1224no—
302C4⋊Q8—1224no—
303C4⋊Q8—12216no—
304C4⋊Q8—1228no—
305C4⋊Q8—1222no—
306#124—1222no—
307#124—1222no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—614no—
311C4×Q8—1224no—
312C4×Q8—1224no—
313C4×Q8—1228no—
314C4×Q8—1222no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—612no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4D24412yes—
323C2×C4⋊C4D24411yescharacteristic
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1222no—
328C2×C42—1222no—
329C2×C42—1221no—
330C2×C42—1221no—
331C2×C42C22×D6112yes—
332C2×C42C22×D6111yescharacteristic
333C4⋊C8—22no—
334#103—1222no—
335#103—1224no—
336#103—1224no—
337#102—1222no—
338#102—1224no—
339#97—22no—
340#95—1221no—
341#95—1222no—
342#95—1222no—
343C24—1221no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—612no—
347C2×Q8—614no—
348C2×Q8—1228no—
349C2×Q8—1221no—
350C2×Q8—1222no—
351C2×Q8—1224no—
352C2×Q8—1221no—
353C2×Q8—1221no—
354C2×Q8—1222no—
355C2×Q8—1222no—
356C2×Q8—4881no—
357C2×Q8—4881no—
358C2×Q8—4882no—
359C2×Q8—4882no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1222no—
363C2×D4—1222no—
364C22×C4—1221no—
365C22×C4—1222no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4C2×D24412yes—
370C22×C4C2×D24412yes—
371C22×C4C2×D24411yescharacteristic
372C22×C4C2×D24411yescharacteristic
373C22×C4C23×D6111yescharacteristic
374Q16—2444no—
375SD16—2444no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—612no—
386C4⋊C4—614no—
387C4⋊C4—1228no—
388C4⋊C4—1228no—
389C4⋊C4—1222no—
390C4⋊C4—1221no—
391C4⋊C4—1224no—
392C4⋊C4—1224no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C22⋊C4—4882no—
396C22⋊C4—2444no—
397C22⋊C4—2442no—
398C22⋊C4—2442no—
399C42—22no—
400C42—22no—
401C42—1228no—
402C42—1224no—
403C42—4881no—
404C42D4×D6112yes—
405C42D4×D6112yes—
406C23—2441no—
407C23—2442no—
408C23—2441no—
409C23—2441no—
410C23—2442no—
411C23C22×D24411yescharacteristic
412Q8—28no—
413Q8—614no—
414Q8—1222no—
415Q8—1224no—
416Q8—4884no—
417Q8—4888no—
418Q8—2441no—
419Q8—2441no—
420Q8—2442no—
421D4—2441no—
422D4—2441no—
423D4—2442no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—1222no—
433C2×C4—1224no—
434C2×C4—1224no—
435C2×C4—1222no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2441no—
443C2×C4—2442no—
444C2×C4—2441no—
445C2×C4#1451974712yes—
446C2×C4#1451974712yes—
447C2×C4#1451981014yes—
448C2×C4C2×D4×D6111yescharacteristic
449C2×C4C2×D4×D6111yescharacteristic
450C2×C4#41×D6112yes—
451C2×C4#1451982512yes—
452C8—41no—
453C8—41no—
454C8—42no—
455C22—4881no—
456C22—4881no—
457C22—4882no—
458C22—4882no—
459C22—4881no—
460C22C2×#1451974711yescharacteristic
461C22#1684042412yes—
462C22C2×#1451981012yes—
463C22#1684056711yescharacteristic
464C22#1684057411yescharacteristic
465C4—22no—
466C4—24no—
467C4—22no—
468C4—42no—
469C4—1224no—
470C4—4881no—
471C4—4882no—
472C4—2444no—
473C4—2442no—
474C4#124×D6111yescharacteristic
475C4#1684033711yescharacteristic
476C4#1684042212yes—
477C2—9761no—
478C2#2144947611yescharacteristic
479C2#2145505611yescharacteristic
480C2#2145512611yescharacteristic
481C2#2145856112yes—
482C2#2145863512yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1985C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1985D61
hypocenterC61#25942
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
1. C2×Q8×Dic122C2semidirect
5. #21446961C2semidirect
7. #21446770C2semidirect
9. #21446159C2semidirect
10. #21445650C2semidirect
11. C61×#1985C2semidirect
243. #1985D61semidirect
286. C61#25942semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1222
C2×#168401222
C2×#168399852
#214513072
#214469612
#214469562
#214467702
#214461732
#214461592
#214456502
C61×#19852
#2594261