Finite Group Databasebeta

← #29729865

Subgroups — #29729865

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×#16840113C211yesmaximal, characteristic, complemented
2C2×#16839968C211yesmaximal, characteristic
3#21450988C211yesmaximal, characteristic
4#21448875C211yesmaximal, characteristic, complemented
5#21447239C212yesmaximal, complemented
6#21447137C212yesmaximal
7#21446766C211yesmaximal, characteristic, complemented
8#21446173C212yesmaximal
9#21445835C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic
11C61×#1939C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×C4×Dic122C2212yes—
16C2×C4×D244C2211yescharacteristic
17#16840113—24no—
18#16839968—24no—
19C2×#14519702C2212yes—
20C2×#14519702C2212yes—
21C2×#14519702C2212yes—
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24#16839675—22no—
25#16839668C2212yes—
26#16839664C2212yes—
27#16839658—21no—
28#16839645—22no—
29#16839628C2212yes—
30#16839619—21no—
31#16839619—21no—
32C4⋊C4×Dic61—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2212yes—
37#16839525C2212yes—
38C2×C4⋊C8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2212yes—
40C2×C4×C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#336C2212yes—
43C22×Dic122—21no—
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758190—24no—
48C2×#11758190—24no—
49C2×#11758190—42no—
50#14519777—22no—
51#14519777—22no—
52#14519777—22no—
53#14519777D414yes—
54C4×Dic122—24no—
55C4×Dic122—42no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—22no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—21no—
71#14519702—24no—
72#14519702—24no—
73#14519702—24no—
74#14519699—22no—
75#14519699—22no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684—22no—
80#14519684—22no—
81C22×C8×C61C2311yescharacteristic
82C22×C8×C61C2312yes—
83C22×C8×C61C2311yescharacteristic
84C2×C4⋊C4×C61D411yescharacteristic
85C2×C4⋊C4×C61D411yescharacteristic
86C2×C42×C61C2312yes—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—22no—
89C4⋊C8×C61—42no—
90C4×C8×C61—22no—
91#95×C61D412yes—
92C23×D61—21no—
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—22no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244#4112yes—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C2×C4×D61—42no—
112C2×Dic122—22no—
113C2×Dic122—22no—
114C2×Dic122—22no—
115C2×Dic122—22no—
116C2×Dic122—22no—
117C2×Dic122—42no—
118C2×Dic122—81no—
119C2×Dic122—81no—
120C2×Dic122C2×D414yes—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61—22no—
127C2×C8×C61—21no—
128C2×C8×C61—22no—
129C2×C8×C61—21no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C42×C61—22no—
133C42×C61—22no—
134C42×C61C2×D412yes—
135C42×C61#4112yes—
136#11758198—42no—
137#11758198—82no—
138#11758198—82no—
139C4⋊Dic61—24no—
140C4⋊Dic61—24no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—42no—
149C4⋊Dic61#4112yes—
150#11758196—44no—
151C4×Dic61—44no—
152C4×Dic61—42no—
153#11758190—48no—
154C23×C61C22×D411yescharacteristic
155C22×D61—22no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61#12112yes—
171C2×C4×C61#12112yes—
172C2×C4×C61#13512yes—
173C2×C4×C61#13712yes—
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×C4×C61C2×#4111yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61H(5,𝔽2)11yescharacteristic
179C2×C4×C612−1+411yescharacteristic
180C2×Dic61—24no—
181C2×Dic61—22no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195D244—22no—
196D244—22no—
197D244—41no—
198D244—41no—
199C4×D61—84no—
200Dic122—24no—
201Dic122—24no—
202Dic122—42no—
203Dic122—42no—
204Dic122—84no—
205C8×C61—42no—
206C8×C61—41no—
207C8×C61—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—42no—
226C4×C61#46414yes—
227C4×C61#52911yescharacteristic
228C4×C61#54211yescharacteristic
229C4×C61#54611yescharacteristic
230C4×C61#54711yescharacteristic
231Dic61—44no—
232Dic61—42no—
233Dic61—81no—
234Dic61—81no—
235Dic61—82no—
236C2×#487—611no—
237C2×#461—611no—
238#2563—611no—
239#2365—611no—
240#2074—612no—
241#2055—612no—
242#1987—611no—
243#1939D6111yescharacteristic, complemented
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⋊Q8—611no—
258C2×C4×Q8—612no—
259C2×C4×D4—611no—
260#487—1224no—
261#461—1224no—
262C2×C4⋊C8C2×D6111yescharacteristic
263C2×C4⋊C8C2×D6112yes—
264C2×#103—612no—
265C2×#103—612no—
266C2×#103—612no—
267C2×#102—611no—
268C2×#102—611no—
269C2×C4×C8C2×D6111yescharacteristic
270#393—1222no—
271#391—612no—
272#390—612no—
273#389—1221no—
274#387—1222no—
275#382—612no—
276#380—1221no—
277#380—1221no—
278C4×C4⋊C4—611no—
279C4×C4⋊C4—1221no—
280C4×C4⋊C4C2×D6111yescharacteristic
281#340—612no—
282#340—611no—
283#340—612no—
284#340—611no—
285#336C2×D6112yes—
286C61#2531911yescharacteristic, complemented, Sylow
287C22×Q8—612no—
288C22×Q8—1221no—
289C22×D4—611no—
290C23×C4—1221no—
291C2×SD16—1224no—
292C2×SD16—1224no—
293C2×SD16—2442no—
294C22×C8C22×D6112yes—
295C22×C8C22×D6111yescharacteristic
296C22×C8C22×D6111yescharacteristic
297C4⋊Q8—614no—
298C4⋊Q8—1222no—
299C4⋊Q8—1222no—
300C4⋊Q8—1222no—
301C4×Q8—1224no—
302C4×Q8—2442no—
303C4×D4—1222no—
304C4×D4—1222no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—612no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4D24411yescharacteristic
314C2×C4⋊C4D24411yescharacteristic
315C2×C22⋊C4—1222no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1221no—
318C2×C42—1222no—
319C2×C42—1221no—
320C2×C42C22×D6112yes—
321C2×C42C22×D6111yescharacteristic
322C4⋊C8—22no—
323C4⋊C8—42no—
324#103—1224no—
325#103—1224no—
326#103—1224no—
327#102—1222no—
328#102—1222no—
329C4×C8—22no—
330#95—1221no—
331#95—1222no—
332#95—1222no—
333#95—1222no—
334#95—1222no—
335#95D24412yes—
336C24—1221no—
337C2×Q8—614no—
338C2×Q8—1222no—
339C2×Q8—1222no—
340C2×Q8—1222no—
341C2×Q8—1222no—
342C2×Q8—1222no—
343C2×Q8—4881no—
344C2×Q8—4881no—
345C2×Q8—2442no—
346C2×D4—612no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—1221no—
352C22×C4—1222no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4—2442no—
360C22×C4C2×D24412yes—
361C22×C4C2×D24412yes—
362C22×C4C2×D24411yescharacteristic
363C22×C4C2×D24411yescharacteristic
364C22×C4C23×D6111yescharacteristic
365SD16—2448no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—612no—
378C4⋊C4—1221no—
379C4⋊C4—1224no—
380C4⋊C4—1224no—
381C4⋊C4—1221no—
382C4⋊C4—1222no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—1222no—
386C4⋊C4—1221no—
387C4⋊C4—2444no—
388C4⋊C4—2442no—
389C22⋊C4—4882no—
390C22⋊C4—4882no—
391C22⋊C4—2442no—
392C42—22no—
393C42—22no—
394C42—2444no—
395C42—2442no—
396C42D4×D6112yes—
397C42#1175822612yes—
398C23—1222no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23C22×D24411yescharacteristic
406Q8—1224no—
407Q8—1224no—
408Q8—4884no—
409Q8—2442no—
410Q8—2442no—
411D4—1222no—
412D4—1222no—
413D4—2441no—
414D4—2441no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—1224no—
424C2×C4—1222no—
425C2×C4—4884no—
426C2×C4—2441no—
427C2×C4—2442no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4#1451975212yes—
440C2×C4#1451975212yes—
441C2×C4#1451979912yes—
442C2×C4#1451981012yes—
443C2×C4C2×D4×D6111yescharacteristic
444C2×C4#41×D6111yescharacteristic
445C2×C4#41×D6111yescharacteristic
446C2×C4C2×#1175822611yescharacteristic
447C2×C4#1451982211yescharacteristic
448C2×C4#1451982511yescharacteristic
449C8—41no—
450C8—42no—
451C8—41no—
452C22—4881no—
453C22—4881no—
454C22—4881no—
455C22—4881no—
456C22—2442no—
457C22—2442no—
458C22—2442no—
459C22C2×#1451975211yescharacteristic
460C22#1684040711yescharacteristic
461C22#1684042411yescharacteristic
462C22C2×#1451979911yescharacteristic
463C22C2×#1451981011yescharacteristic
464C22#1684057411yescharacteristic
465C22#1684057411yescharacteristic
466C4—22no—
467C4—42no—
468C4—42no—
469C4—4881no—
470C4—4881no—
471C4—4882no—
472C4—2442no—
473C4—2444no—
474C4#1683998514yes—
475C4#1684029711yescharacteristic
476C4#1684038611yescharacteristic
477C4#1684040911yescharacteristic
478C4#1684041411yescharacteristic
479C2—4882no—
480C2#2144939411yescharacteristic
481C2C2×#1683998511yescharacteristic
482C2#2145534111yescharacteristic
483C2#2145774811yescharacteristic
484C2#2145811411yescharacteristic
485C2#2145847011yescharacteristic
486C2#2145860211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1939C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1939D61
hypocenterC61#25319
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
1. C2×#16840113C2semidirect
4. #21448875C2semidirect
5. #21447239C2semidirect
7. #21446766C2semidirect
9. #21445835C2semidirect
11. C61×#1939C2semidirect
243. #1939D61semidirect
286. C61#25319semidirect

Maximal subgroups

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

subgroupindex
C2×#168401132
C2×#168399682
#214509882
#214488752
#214472392
#214471372
#214467662
#214461732
#214458352
C4×#145196992
C61×#19392
#2531961