Finite Group Databasebeta

← #29729861

Subgroups — #29729861

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840112C211yesmaximal, characteristic, complemented
2C2×#16839964C211yesmaximal, characteristic
3#21450988C211yesmaximal, characteristic
4#21447237C212yesmaximal, complemented
5#21447134C212yesmaximal
6#21446767C211yesmaximal, characteristic, complemented
7#21446538C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal
9#21445834C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic
11C61×#2365C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840112—24no—
16#16839964—24no—
17C2×C4⋊#92418C2211yescharacteristic
18C2×C4⋊#92418C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2211yescharacteristic
24C2×C4×#92418C2211yescharacteristic
25#16839675—22no—
26#16839664C2212yes—
27#16839628C2212yes—
28#16839619—21no—
29#16839619—21no—
30#16839524C2211yescharacteristic
31#16839524C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2212yes—
37C61×#391C2212yes—
38C61×#389—21no—
39C61×#387—22no—
40C4×C4⋊C4×C61—21no—
41C4×C4⋊C4×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C2×#11758201—24no—
47C2×#11758201—24no—
48C2×#11758201—42no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777—22no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2312yes—
59C2×#11758198—21no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C4⋊#92418—22no—
63C4⋊#92418—42no—
64#14519701—24no—
65#14519701—24no—
66#14519701—24no—
67#14519697—22no—
68#14519697—22no—
69C4×#92418—22no—
70#14519684—22no—
71#14519684—22no—
72C22×Q8×C61C2312yes—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61D414yes—
75C4×Q8×C61—24no—
76C4×Q8×C61—42no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61D411yescharacteristic
81C2×C4⋊C4×C61D411yescharacteristic
82C2×C4⋊C4×C61C2311yescharacteristic
83C2×C4⋊C4×C61C2311yescharacteristic
84C2×C42×C61—21no—
85C2×C42×C61—22no—
86C2×C42×C61C2312yes—
87C2×C42×C61C2311yescharacteristic
88#95×C61—22no—
89#95×C61—21no—
90#95×C61—22no—
91#95×C61D412yes—
92C23×D61—21no—
93C2×Q8×C61—22no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61C2×D414yes—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—22no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244#4112yes—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61—42no—
118C2×Dic122—42no—
119C2×Dic122—81no—
120C2×Dic122—81no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—44no—
129C4⋊C4×C61#4112yes—
130C42×C61—22no—
131C42×C61—22no—
132C42×C61—44no—
133C42×C61—42no—
134C42×C61C2×D412yes—
135C42×C61#4112yes—
136#11758201—48no—
137#11758198—42no—
138#11758198—82no—
139#11758198—82no—
140C4⋊Dic61—21no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—42no—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—22no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—22no—
151C2×#92418—21no—
152C2×#92418—22no—
153C2×#92418—21no—
154C23×C61C22×D411yescharacteristic
155C22×D61—22no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162Q8×C61—24no—
163Q8×C61—24no—
164Q8×C61—42no—
165Q8×C61—42no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—24no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61#12112yes—
184C2×C4×C61#12112yes—
185C2×C4×C61#13512yes—
186C2×C4×C61#13712yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61H(5,𝔽2)11yescharacteristic
192C2×C4×C612−1+411yescharacteristic
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199D244—22no—
200D244—22no—
201D244—41no—
202D244—41no—
203C4×D61—84no—
204Dic122—84no—
205#92418—41no—
206#92418—42no—
207#92418—41no—
208#25319—611nomaximal, Sylow
209C22×C61C2×#12111yescharacteristic
210C22×C61#54611yescharacteristic
211C22×C61#54811yescharacteristic
212C22×C61C2×#13511yescharacteristic
213C22×C61C2×#13711yescharacteristic
214C22×C61#57711yescharacteristic
215C22×C61#57711yescharacteristic
216C2×D61—42no—
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—22no—
224C4×C61—42no—
225C4×C61—44no—
226C4×C61—42no—
227C4×C61—42no—
228C4×C61—81no—
229C4×C61#46414yes—
230C4×C61#52911yescharacteristic
231C4×C61#54211yescharacteristic
232C4×C61#54611yescharacteristic
233C4×C61#54711yescharacteristic
234Dic61—81no—
235Dic61—82no—
236C2×#487—611no—
237C2×#461—611no—
238#2563—611no—
239#2365D6111yescharacteristic, complemented
240#2074—612no—
241#2055—612no—
242#1987—611no—
243#1939—611no—
244#1880—612no—
245#1805—612no—
246C4×#102—611no—
247C2×C61#242711yescharacteristic
248C2×C61C2×#46411yescharacteristic
249C2×C61#304311yescharacteristic
250C2×C61#330311yescharacteristic
251C2×C61#332911yescharacteristic
252C2×C61#336011yescharacteristic
253C2×C61#337211yescharacteristic
254D61—82no—
255C22×SD16—612no—
256C2×C4⋊Q8—611no—
257C2×C4⋊Q8C2×D6111yescharacteristic
258C2×C4×Q8C2×D6112yes—
259C2×C4×D4—611no—
260#487—1224no—
261#461—1224no—
262C2×C4⋊C8—611no—
263C2×C4⋊C8—612no—
264C2×#103—612no—
265C2×#103—612no—
266C2×#103—612no—
267C2×#102—611no—
268C2×#102—611no—
269C2×C4×C8—611no—
270#393—1222no—
271#391C2×D6112yes—
272#390—612no—
273#389—21no—
274#387—22no—
275#382—612no—
276#380—1221no—
277#380—1221no—
278C4×C4⋊C4—21no—
279C4×C4⋊C4C2×D6111yescharacteristic
280C4×C4⋊C4C2×D6111yescharacteristic
281#340—612no—
282#340—611no—
283#340—612no—
284#340—611no—
285#336—612no—
286C61#2531911yescharacteristic, complemented, Sylow
287C22×Q8—1221no—
288C22×Q8C22×D6112yes—
289C22×D4—611no—
290C23×C4—1221no—
291C2×SD16—1224no—
292C2×SD16—1224no—
293C2×SD16—2442no—
294C22×C8—612no—
295C22×C8—611no—
296C22×C8—611no—
297C4⋊Q8—22no—
298C4⋊Q8—1222no—
299C4⋊Q8—1222no—
300C4⋊Q8D24414yes—
301C4×Q8—24no—
302C4×Q8—42no—
303C4×D4—1222no—
304C4×D4—1222no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—612no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4C22⋊D6111yescharacteristic
312C2×C4⋊C4C22⋊D6111yescharacteristic
313C2×C4⋊C4C22×D6111yescharacteristic
314C2×C4⋊C4C22×D6111yescharacteristic
315C2×C22⋊C4—1222no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1221no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42C22×D6112yes—
321C2×C42C22×D6111yescharacteristic
322C4⋊C8—1222no—
323C4⋊C8—2442no—
324#103—1224no—
325#103—1224no—
326#103—1224no—
327#102—1222no—
328#102—1222no—
329C4×C8—1222no—
330#95—22no—
331#95—21no—
332#95—22no—
333#95—1222no—
334#95—1222no—
335#95C22⋊D6112yes—
336C24—1221no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—4881no—
343C2×Q8—4881no—
344C2×Q8—2442no—
345C2×Q8C2×D24414yes—
346C2×D4—612no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—1221no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4—2442no—
360C22×C4C2×C22⋊D6112yes—
361C22×C4C2×C22⋊D6112yes—
362C22×C4C2×C22⋊D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C23×D6111yescharacteristic
365SD16—2448no—
366C2×C8—1221no—
367C2×C8—1221no—
368C2×C8—1222no—
369C2×C8—1221no—
370C2×C8—1222no—
371C2×C8—1221no—
372C2×C8—1221no—
373C2×C8—1222no—
374C2×C8—1221no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—44no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—2442no—
388C4⋊C4#1175822412yes—
389C22⋊C4—4882no—
390C22⋊C4—4882no—
391C22⋊C4—2442no—
392C42—22no—
393C42—22no—
394C42—44no—
395C42—42no—
396C42C2×D24412yes—
397C42#1175822212yes—
398C23—1222no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23C22×C22⋊D6111yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—42no—
410Q8—4884no—
411D4—1222no—
412D4—1222no—
413D4—2441no—
414D4—2441no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—4884no—
433C2×C4—2441no—
434C2×C4—2442no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4#1451975112yes—
440C2×C4#1451975112yes—
441C2×C4#1451979712yes—
442C2×C4#1451980812yes—
443C2×C4C22×D24411yescharacteristic
444C2×C4#41×D6111yescharacteristic
445C2×C4C2×#1175822411yescharacteristic
446C2×C4C2×#1175822211yescharacteristic
447C2×C4#1451982211yescharacteristic
448C2×C4#1451982411yescharacteristic
449C8—2441no—
450C8—2442no—
451C8—2441no—
452C22—4881no—
453C22—4881no—
454C22—4881no—
455C22—4881no—
456C22—2442no—
457C22—2442no—
458C22—2442no—
459C22C2×#1451975111yescharacteristic
460C22#1684040611yescharacteristic
461C22#1684041911yescharacteristic
462C22C2×#1451979711yescharacteristic
463C22C2×#1451980811yescharacteristic
464C22#1684057011yescharacteristic
465C22#1684057011yescharacteristic
466C4—22no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—42no—
471C4—81no—
472C4—4881no—
473C4—4882no—
474C4#1683998414yes—
475C4#1684029411yescharacteristic
476C4#1684037711yescharacteristic
477C4#1684040711yescharacteristic
478C4#1684041611yescharacteristic
479C2—4882no—
480C2#2144939011yescharacteristic
481C2C2×#1683998411yescharacteristic
482C2#2145534011yescharacteristic
483C2#2145774711yescharacteristic
484C2#2145811311yescharacteristic
485C2#2145846411yescharacteristic
486C2#2145859611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2365C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2365D61
hypocenterC61#25319
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
1. C2×#16840112C2semidirect
4. #21447237C2semidirect
6. #21446767C2semidirect
7. #21446538C2semidirect
9. #21445834C2semidirect
11. C61×#2365C2semidirect
239. #2365D61semidirect
286. C61#25319semidirect

Maximal subgroups

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

subgroupindex
C2×#168401122
C2×#168399642
#214509882
#214472372
#214471342
#214467672
#214465382
#214461742
#214458342
C4×#145196972
C61×#23652
#2531961