Finite Group Databasebeta

← #29736986

Subgroups — #29736986

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×#16840124C211yesmaximal, characteristic
2C2×#16839984C211yesmaximal, characteristic
3#21451307C211yesmaximal, characteristic
4#21446957C212yesmaximal, complemented
5#21446952C212yesmaximal
6#21446772C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal
9#21446160C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×Q82×C61C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13C22×#11758201C2211yescharacteristic
14C2×#14519765C2211yescharacteristic
15#16840124—24no—
16#16839984—24no—
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701C2212yes—
19C2×#14519700C2212yes—
20C2×#14519700C2211yescharacteristic
21C2×#14519697C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519688C2211yescharacteristic
24#16839688—21no—
25#16839673—21no—
26#16839668C2212yes—
27#16839664C2212yes—
28#16839652—22no—
29C4×C4⋊Dic61C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35Q82×C61—24no—
36Q82×C61—24no—
37C2×C4⋊Q8×C61—21no—
38C2×C4⋊Q8×C61—22no—
39C2×C4⋊Q8×C61C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×Q8×C61—21no—
42C2×C4×Q8×C61C2212yes—
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C2×#11758202—22no—
48C2×#11758202—24no—
49C2×#11758201—22no—
50C2×#11758201—24no—
51C22×#92418C2312yes—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519765—22no—
55#14519765—22no—
56C2×C4⋊Dic61C2312yes—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×C4×Dic61—21no—
62C4⋊#92418—22no—
63#14519701—24no—
64#14519700—22no—
65#14519700—24no—
66#14519697—22no—
67#14519697—24no—
68#14519688—22no—
69#14519684—21no—
70#14519684—22no—
71#14519684—22no—
72C22×Q8×C61—22no—
73C22×Q8×C61—21no—
74C22×Q8×C61D411yescharacteristic
75C22×Q8×C61C2311yescharacteristic
76C4⋊Q8×C61—216no—
77C4⋊Q8×C61—28no—
78C4⋊Q8×C61—24no—
79C4⋊Q8×C61—24no—
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61D414yes—
82C4×Q8×C61—28no—
83C4×Q8×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yes—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D412yes—
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C42×C61—22no—
98C2×C42×C61—21no—
99C2×C42×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101C23×D61—21no—
102C2×Q8×C61—28no—
103C2×Q8×C61—22no—
104C2×Q8×C61—24no—
105C2×Q8×C61—24no—
106C2×Q8×C61—21no—
107C2×Q8×C61—22no—
108C2×Q8×C61—81no—
109C2×Q8×C61—81no—
110C2×Q8×C61—82no—
111C2×Q8×C61—82no—
112C2×Q8×C61C2×D412yes—
113C2×Q8×C61#4114yes—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—22no—
127C2×D244—22no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C4⋊C4×C61—28no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61C2×D414yes—
141C4⋊C4×C61C2×D412yes—
142C42×C61—28no—
143C42×C61—24no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61C2×D412yes—
147C42×C61C2×D412yes—
148#11758202—44no—
149#11758201—44no—
150#11758198—42no—
151#11758198—42no—
152#11758198—44no—
153#11758198—82no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4×Dic61—81no—
157C2×#92418—22no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C23×C61C22×D411yescharacteristic
165C22×D61—41no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—41no—
169C22×D61—41no—
170Q8×C61—28no—
171Q8×C61—22no—
172Q8×C61—24no—
173Q8×C61—84no—
174Q8×C61—88no—
175Q8×C61#12414yes—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—24no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—24no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61#12012yes—
193C2×C4×C61#12012yes—
194C2×C4×C61#13714yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4112yes—
198C2×C4×C612−1+412yes—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204D244—41no—
205D244—41no—
206D244—42no—
207Dic122—41no—
208Dic122—41no—
209Dic122—42no—
210#92418—41no—
211#92418—41no—
212#92418—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—24no—
225C4×C61—22no—
226C4×C61—24no—
227C4×C61—22no—
228C4×C61—42no—
229C4×C61—44no—
230C4×C61—42no—
231C4×C61C2×#12411yescharacteristic
232C4×C61#53611yescharacteristic
233C4×C61#54812yes—
234Dic61—82no—
235Dic61—81no—
236C2×Q82D6111yescharacteristic, complemented
237C2×#489—611no—
238C2×#464—611no—
239#2592—611no—
240#2024—612no—
241#2023—612no—
242#1988—611no—
243#1985—611no—
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—24no—
256Q82—24no—
257C2×C4⋊Q8—22no—
258C2×C4⋊Q8—21no—
259C2×C4⋊Q8C2×D6112yes—
260C2×C4⋊Q8C2×D6111yescharacteristic
261C2×#124—611no—
262C2×C4×Q8—21no—
263C2×C4×Q8C2×D6112yes—
264C2×C4×Q8C2×D6111yescharacteristic
265C2×C4×Q8C2×D6111yescharacteristic
266#489—1224no—
267#464—1224no—
268C2×C4⋊C8—611no—
269C2×#103—612no—
270C2×#103—612no—
271C2×#103—611no—
272C2×#102—612no—
273C2×#102—611no—
274C2×#97—611no—
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#336—612no—
285#336—612no—
286C61#2594211yescharacteristic, complemented, Sylow
287C22×Q8—22no—
288C22×Q8—21no—
289C22×Q8—611no—
290C22×Q8C22⋊D6111yescharacteristic
291C22×Q8C22×D6111yescharacteristic
292C22×D4—611no—
293C2×Q16—1224no—
294C2×Q16—1222no—
295C2×SD16—1224no—
296C2×SD16—1222no—
297C22×C8—611no—
298C22×C8—611no—
299C22×C8—612no—
300C4⋊Q8—216no—
301C4⋊Q8—28no—
302C4⋊Q8—24no—
303C4⋊Q8—24no—
304C4⋊Q8—22no—
305C4⋊Q8C22⋊D6114yes—
306#124—1222no—
307#124—1222no—
308C4×Q8—28no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8C22⋊D6114yes—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—612no—
320C2×C4⋊C4C22⋊D6112yes—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C4⋊C4C22×D6112yes—
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1222no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—1221no—
331C2×C42C22×D6112yes—
332C2×C42C22×D6111yescharacteristic
333C4⋊C8—1222no—
334#103—1224no—
335#103—1222no—
336#103—1224no—
337#102—1222no—
338#102—1224no—
339#97—1222no—
340#95—1221no—
341#95—1222no—
342#95—1222no—
343C24—1221no—
344C2×Q8—28no—
345C2×Q8—24no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8—82no—
353C2×Q8—82no—
354C2×Q8—1221no—
355C2×Q8—1221no—
356C2×Q8—1222no—
357C2×Q8—1222no—
358C2×Q8#1175822614yes—
359C2×Q8C2×C22⋊D6112yes—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1222no—
363C2×D4—1222no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4C2×C22⋊D6112yes—
370C22×C4C2×C22⋊D6112yes—
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373C22×C4C23×D6111yescharacteristic
374Q16—2444no—
375SD16—2444no—
376C2×C8—1222no—
377C2×C8—1221no—
378C2×C8—1221no—
379C2×C8—1222no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C4⋊C4C2×C22⋊D6114yes—
394C4⋊C4C2×C22⋊D6112yes—
395C22⋊C4—4882no—
396C22⋊C4—2444no—
397C22⋊C4—2442no—
398C22⋊C4—2442no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—4881no—
404C42C2×C22⋊D6112yes—
405C42C2×C22⋊D6112yes—
406C23—2441no—
407C23—2442no—
408C23—2441no—
409C23—2441no—
410C23—2442no—
411C23C22×C22⋊D6111yescharacteristic
412Q8—28no—
413Q8—22no—
414Q8—24no—
415Q8—84no—
416Q8—88no—
417Q8—2441no—
418Q8—2441no—
419Q8—2442no—
420Q8#1451976614yes—
421D4—2441no—
422D4—2441no—
423D4—2442no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4#1451974612yes—
446C2×C4#1451974612yes—
447C2×C4#1451980814yes—
448C2×C4C22×C22⋊D6111yescharacteristic
449C2×C4C22×C22⋊D6111yescharacteristic
450C2×C4C2×#1175822612yes—
451C2×C4#1451982512yes—
452C8—2441no—
453C8—2441no—
454C8—2442no—
455C22—4881no—
456C22—4881no—
457C22—4882no—
458C22—4882no—
459C22—4881no—
460C22C2×#1451974611yescharacteristic
461C22#1684042312yes—
462C22C2×#1451980812yes—
463C22#1684056511yescharacteristic
464C22#1684057311yescharacteristic
465C4—24no—
466C4—22no—
467C4—24no—
468C4—22no—
469C4—42no—
470C4—44no—
471C4—42no—
472C4—4881no—
473C4—4882no—
474C4C2×#1451976611yescharacteristic
475C4#1684033611yescharacteristic
476C4#1684042312yes—
477C2—9761no—
478C2#2144947311yescharacteristic
479C2#2145505311yescharacteristic
480C2#2145512311yescharacteristic
481C2#2145855912yes—
482C2#2145863312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×Q82×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×Q82D61
hypocenterC61#25942
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
4. #21446957C2semidirect
6. #21446772C2semidirect
7. #21446760C2semidirect
9. #21446160C2semidirect
10. #21445649C2semidirect
11. C2×Q82×C61C2semidirect
236. C2×Q82D61semidirect
286. C61#25942semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168399842
#214513072
#214469572
#214469522
#214467722
#214467602
#214461742
#214461602
#214456492
C2×Q82×C612
#2594261