Finite Group Databasebeta

← #29779757

Subgroups — #29779757

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519765C211yesmaximal, characteristic
2#21447387C212yesmaximal, complemented
3#21447060C212yesmaximal
4#21446515C212yesmaximal
5#21446508C212yesmaximal, complemented
6#21445821C211yesmaximal, characteristic, complemented
7#21445821C211yesmaximal, characteristic, complemented
8#21445816C211yesmaximal, characteristic
9#21445816C211yesmaximal, characteristic
10#21445672C211yesmaximal, characteristic, complemented
11C61×#2859C211yesmaximal, characteristic, complemented
12C2×#14519765C2211yescharacteristic
13C2×C4×Dic122C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15C2×C4⋊#92418C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519700C2212yes—
18C2×#14519700C2212yes—
19C2×#14519697C2212yes—
20C2×#14519697C2212yes—
21C2×#14519688C2212yes—
22#16839668C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839653—21no—
25#16839653—21no—
26#16839628C2211yescharacteristic
27C4×#11758198—21no—
28C4×#11758198—21no—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C2×#125×C61—21no—
35C61×#389—21no—
36C61×#389—21no—
37C61×#389C2212yes—
38C61×#389C2212yes—
39C61×#384C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#382—21no—
42C43×C61C2211yescharacteristic
43C22×Dic122C2311yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×#92418C2312yes—
47C22×#92418C2312yes—
48#14519765—22no—
49#14519765—22no—
50C4×Dic122—22no—
51C4×Dic122—22no—
52C4×D244—22no—
53C4×D244—22no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×C4×Dic61—21no—
63C4⋊#92418—22no—
64C4⋊#92418—22no—
65#14519700—24no—
66#14519700—42no—
67#14519697—24no—
68#14519697—42no—
69#14519688—42no—
70#14519684—21no—
71#14519684—21no—
72#14519684—21no—
73#14519684—21no—
74#125×C61—44no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61C2312yes—
82C2×C4⋊C4×C61C2312yes—
83C2×C42×C61D411yescharacteristic
84C2×C42×C61D411yescharacteristic
85C2×C42×C61D411yescharacteristic
86C2×C42×C61D411yescharacteristic
87C2×C42×C61C2311yescharacteristic
88C2×C42×C61C2311yescharacteristic
89C2×C42×C61C2311yescharacteristic
90#95×C61—22no—
91#95×C61—22no—
92C23×D61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244#4112yes—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×Dic122—21no—
115C2×Dic122—21no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×Dic122#4112yes—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—42no—
125C4⋊C4×C61—42no—
126C4⋊C4×C61—44no—
127C4⋊C4×C61—44no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C42×C61—22no—
131C42×C61—22no—
132C42×C61—21no—
133C42×C61—21no—
134C42×C61—21no—
135C42×C61—21no—
136C42×C61—41no—
137C42×C61—41no—
138C42×C61#4112yes—
139C42×C61#4112yes—
140#11758198—24no—
141#11758198—24no—
142#11758198—42no—
143#11758198—42no—
144#11758198—42no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149#11758196—42no—
150C4×Dic61—42no—
151C2×#92418—22no—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C23×C61C22×D411yescharacteristic
158C22×D61—22no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61#12412yes—
186C2×C4×C61#12412yes—
187C2×C4×C61AGL(1,ℤ/8)12yes—
188C2×C4×C61#13712yes—
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61H(5,𝔽2)11yescharacteristic
194C2×C4×C612−1+411yescharacteristic
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204D244—22no—
205D244—22no—
206D244—41no—
207D244—41no—
208C4×D61—84no—
209Dic122—22no—
210Dic122—22no—
211Dic122—41no—
212Dic122—41no—
213#92418—42no—
214#92418—42no—
215#29131—611nomaximal, Sylow
216C22×C61#52511yescharacteristic
217C22×C61C2×#12411yescharacteristic
218C22×C61#53711yescharacteristic
219C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
220C22×C61C2×#13711yescharacteristic
221C22×C61#57511yescharacteristic
222C22×C61#57511yescharacteristic
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—42no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—22no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61—41no—
234C4×C61—41no—
235C4×C61—82no—
236C4×C61—82no—
237C4×C61#48914yes—
238C4×C61#55211yescharacteristic
239C4×C61#55211yescharacteristic
240C4×C61#55311yescharacteristic
241C4×C61#55311yescharacteristic
242Dic61—42no—
243Dic61—81no—
244Dic61—81no—
245#2859D6111yescharacteristic, complemented
246C4×#124—611no—
247#2101—612no—
248#2042—612no—
249#1936—612no—
250#1935—612no—
251#1803—611no—
252#1803—611no—
253#1802—611no—
254#1802—611no—
255#1767—611no—
256C2×C61#256211yescharacteristic
257C2×C61#311511yescharacteristic
258C2×C61#311611yescharacteristic
259C2×C61C2×#48911yescharacteristic
260C2×C61#314211yescharacteristic
261C2×C61#325111yescharacteristic
262C2×C61#325211yescharacteristic
263D61—82no—
264C2×#125—21no—
265C2×#124—611no—
266C2×C4×Q8—611no—
267C2×C4×D4—611no—
268C2×C4⋊C8—611no—
269C2×C4⋊C8—611no—
270C2×#103—612no—
271C2×#103—612no—
272C2×#102—612no—
273C2×#102—612no—
274C2×#97—612no—
275#391—611no—
276#390—611no—
277#389—21no—
278#389—21no—
279#389C2×D6112yes—
280#389C2×D6112yes—
281#388—1221no—
282#388—1221no—
283#384C2×D6111yescharacteristic
284#384C2×D6111yescharacteristic
285#382—21no—
286#382—611no—
287C4×C22⋊C4—1221no—
288C4×C22⋊C4—1221no—
289C43C2×D6111yescharacteristic
290#340—612no—
291#340—612no—
292#340—612no—
293#340—612no—
294#336—612no—
295C61#2913111yescharacteristic, complemented, Sylow
296C22×Q8—611no—
297C22×D4—611no—
298C23×C4—1221no—
299C22×C8—612no—
300C22×C8—612no—
301#125—44no—
302#124—1222no—
303#124—1222no—
304C4×Q8—1222no—
305C4×Q8—1222no—
306C4×D4—1222no—
307C4×D4—1222no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4C22×D6112yes—
318C2×C4⋊C4C22×D6112yes—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C42—1221no—
325C2×C42C22⋊D6111yescharacteristic
326C2×C42C22⋊D6111yescharacteristic
327C2×C42C22⋊D6111yescharacteristic
328C2×C42C22⋊D6111yescharacteristic
329C2×C42C22×D6111yescharacteristic
330C2×C42C22×D6111yescharacteristic
331C2×C42C22×D6111yescharacteristic
332C4⋊C8—1222no—
333C4⋊C8—1222no—
334#103—1224no—
335#103—2442no—
336#102—1224no—
337#102—2442no—
338#97—2442no—
339#95—22no—
340#95—22no—
341#95—1221no—
342#95—1221no—
343#95—1221no—
344#95—1221no—
345C24—1221no—
346C2×Q8—612no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—1221no—
352C2×D4—612no—
353C2×D4—1221no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—1221no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—2442no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4C2×C22⋊D6111yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C23×D6111yescharacteristic
373C2×C8—1222no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8—1222no—
377C2×C8—1222no—
378C2×C8—1222no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—2442no—
394C22⋊C4—1224no—
395C22⋊C4—1224no—
396C22⋊C4—2442no—
397C22⋊C4—2442no—
398C22⋊C4—2442no—
399C42—22no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—22no—
405C42—41no—
406C42—41no—
407C42—2442no—
408C42#1175822612yes—
409C42#1175822612yes—
410C23—1222no—
411C23—2441no—
412C23—2441no—
413C23—2441no—
414C23—2441no—
415C23—2441no—
416C23—2441no—
417C23C22×C22⋊D6111yescharacteristic
418Q8—1222no—
419Q8—1222no—
420Q8—2441no—
421Q8—2441no—
422D4—1222no—
423D4—1222no—
424D4—2441no—
425D4—2441no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—1222no—
447C2×C4—4884no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4#1451976612yes—
457C2×C4#1451976612yes—
458C2×C4#1451980312yes—
459C2×C4#1451980812yes—
460C2×C4#41×D6111yescharacteristic
461C2×C4#41×D6111yescharacteristic
462C2×C4C2×#1175822611yescharacteristic
463C2×C4C2×#1175822611yescharacteristic
464C2×C4#1451982311yescharacteristic
465C2×C4#1451982411yescharacteristic
466C8—2442no—
467C8—2442no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22—2442no—
473C22—2442no—
474C22—2442no—
475C22#1684026911yescharacteristic
476C22C2×#1451976611yescharacteristic
477C22#1684034011yescharacteristic
478C22C2×#1451980311yescharacteristic
479C22C2×#1451980811yescharacteristic
480C22#1684056011yescharacteristic
481C22#1684056011yescharacteristic
482C4—22no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—82no—
489C4—4881no—
490C4—4881no—
491C4—2442no—
492C4#1684012414yes—
493C4#1684045011yescharacteristic
494C4#1684045011yescharacteristic
495C4#1684046511yescharacteristic
496C4#1684046511yescharacteristic
497C2—4882no—
498C2#2145098611yescharacteristic
499C2#2145596411yescharacteristic
500C2#2145597511yescharacteristic
501C2C2×#1684012411yescharacteristic
502C2#2145622211yescharacteristic
503C2#2145717511yescharacteristic
504C2#2145718611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2859C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2859D61
hypocenterC61#29131
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #2859

Complements

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

NH ≅ G/Ntype
2. #21447387C2semidirect
5. #21446508C2semidirect
6. #21445821C2semidirect
7. #21445821C2semidirect
10. #21445672C2semidirect
11. C61×#2859C2semidirect
245. #2859D61semidirect
295. C61#29131semidirect

Maximal subgroups

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

subgroupindex
C4×#145197652
#214473872
#214470602
#214465152
#214465082
#214458212
#214458162
#214456722
C61×#28592
#2913161