Finite Group Databasebeta

← #29730147

Subgroups — #29730147

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840054C211yesmaximal, characteristic, complemented
2C2×C8×D244C211yesmaximal, characteristic
3#21450993C211yesmaximal, characteristic
4#21450742C211yesmaximal, characteristic
5#21447251C212yesmaximal, complemented
6#21447025C212yesmaximal, complemented
7#21446575C211yesmaximal, characteristic, complemented
8#21446241C212yesmaximal, complemented
9#21445847C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C61×#1990C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519757C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16#16840054—24no—
17C8×D244—24no—
18C2×#14519710C2212yes—
19C2×#14519710C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519700C2212yes—
24C2×#14519689C2212yes—
25#16839684—22no—
26#16839673—22no—
27#16839663—21no—
28#16839663—21no—
29#16839661—21no—
30#16839627—21no—
31#16839619C2212yes—
32#16839619C2212yes—
33#16839619C2212yes—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839524C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#340C2212yes—
42C61×#340C2212yes—
43C22×Dic122C2311yescharacteristic
44C22×D244D411yescharacteristic
45C22×C4×D61C2312yes—
46C22×#92418C2312yes—
47C2×C8×D61—24no—
48C2×C8×D61—24no—
49C2×C8×D61—42no—
50#14519757—24no—
51#14519757—42no—
52C4×Dic122—22no—
53C4×Dic122—22no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61D411yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198D412yes—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69#14519710—24no—
70#14519710—42no—
71C4⋊#92418D414yes—
72#14519702—22no—
73#14519702—22no—
74#14519700—24no—
75#14519689—24no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684—22no—
80C22×Q8×C61—21no—
81C22×C8×C61C2311yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91C4×C8×C61—22no—
92C23×D61C2×D412yes—
93C2×Q8×C61—42no—
94C2×Q8×C61—81no—
95C2×Q8×C61—81no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2411yescharacteristic
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61C2×D412yes—
106C2×D244—22no—
107C2×D244—22no—
108C2×D244Q1614yes—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×Dic122—21no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×Dic122—21no—
120C2×Dic122#4112yes—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61#4112yes—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—42no—
132C4⋊C4×C61—42no—
133C4⋊C4×C61—81no—
134C42×C61C2×D412yes—
135C42×C61#4112yes—
136#11758198—24no—
137#11758198—44no—
138#11758198—82no—
139#11758198—82no—
140#11758198—82no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—44no—
149#11758196—42no—
150C4×Dic61—42no—
151C4×Dic61—42no—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418C2×D414yes—
155C8×D61—48no—
156C23×C61C22×D411yescharacteristic
157C22×D61—22no—
158C22×D61—22no—
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—22no—
163C22×D61C2×Q1614yes—
164Q8×C61—84no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61#12112yes—
174C2×C4×C61#12112yes—
175C2×C4×C61C2×Q1612yes—
176C2×C4×C61#13512yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C61H(5,𝔽2)11yescharacteristic
182C2×C4×C612−1+411yescharacteristic
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198D244—28no—
199C4×D61—42no—
200C4×D61—42no—
201C4×D61—42no—
202C4×D61—42no—
203Dic122—22no—
204Dic122—22no—
205Dic122—41no—
206Dic122—41no—
207C8×C61—22no—
208C8×C61—41no—
209#92418—24no—
210#25340—611nomaximal, Sylow
211C22×C61C2×#12111yescharacteristic
212C22×C61#54611yescharacteristic
213C22×C61#54711yescharacteristic
214C22×C61C22×Q1611yescharacteristic
215C22×C61C2×#13511yescharacteristic
216C22×C61#57611yescharacteristic
217C22×C61#57711yescharacteristic
218C2×D61—24no—
219C2×D61—24no—
220C2×D61—24no—
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—42no—
224C2×D61—42no—
225C4×C61—22no—
226C4×C61—82no—
227C4×C61—82no—
228C4×C61#46714yes—
229C4×C61#52911yescharacteristic
230C4×C61#54211yescharacteristic
231C4×C61#54611yescharacteristic
232C4×C61#54811yescharacteristic
233Dic61—42no—
234Dic61—42no—
235Dic61—81no—
236Dic61—81no—
237Dic61—81no—
238C2×#477—611no—
239C2×C8×D4—611no—
240#2563—611no—
241#2538—611no—
242#2076—612no—
243#2036—612no—
244#1990D6111yescharacteristic, complemented
245#1947—611no—
246#1892—612no—
247#1808—612no—
248C4×#103—611no—
249C2×C61#242711yescharacteristic
250C2×C61C2×#46711yescharacteristic
251C2×C61#306211yescharacteristic
252C2×C61#329911yescharacteristic
253C2×C61#330511yescharacteristic
254C2×C61#333811yescharacteristic
255C2×C61#334011yescharacteristic
256D61—44no—
257C23×C8—612no—
258C2×C4⋊Q8C2×D6111yescharacteristic
259C2×C22⋊Q8—612no—
260C2×C4×Q8—611no—
261C2×C4×D4—611no—
262#477—1224no—
263C8×D4—1224no—
264C2×#107—612no—
265C2×#107—612no—
266C2×C4⋊C8—611no—
267C2×#103—611no—
268C2×#103—611no—
269C2×#103—612no—
270C2×C22⋊C8—612no—
271C2×C4×C8C2×D6111yescharacteristic
272#396—1222no—
273#393—1222no—
274#390—1221no—
275#390—1221no—
276#389—1221no—
277#384C2×D6111yescharacteristic
278#382—1221no—
279#380—612no—
280#380—612no—
281#380—612no—
282C4×C4⋊C4—611no—
283#340—611no—
284#340—611no—
285#340—612no—
286#340C2×D6112yes—
287#340C2×D6112yes—
288C61#2534011yescharacteristic, complemented, Sylow
289C22×Q8—21no—
290C22×Q8—611no—
291C22×D4—611no—
292C23×C4—612no—
293C22×C8—612no—
294C22×C8—1224no—
295C22×C8—1224no—
296C22×C8—2442no—
297C22×C8C22×D6111yescharacteristic
298C22×C8C22×D6111yescharacteristic
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C22⋊Q8—1224no—
302C22⋊Q8—2442no—
303C4×Q8—1222no—
304C4×Q8—1222no—
305C4×D4—1222no—
306C4×D4—1222no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—611no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4C22×D6112yes—
317C2×C4⋊C4C22×D6112yes—
318C2×C22⋊C4—612no—
319C2×C22⋊C4—1222no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C42—1221no—
324C2×C42—1221no—
325C2×C42C22×D6111yescharacteristic
326#107—1224no—
327#107—2442no—
328C4⋊C8—614no—
329#103—1224no—
330#103—1222no—
331#103—1222no—
332C22⋊C8—1224no—
333C4×C8—22no—
334#95—1222no—
335#95—1222no—
336#95—1222no—
337#95—1221no—
338C24—612no—
339C2×Q8—42no—
340C2×Q8—81no—
341C2×Q8—81no—
342C2×Q8—612no—
343C2×Q8—1221no—
344C2×Q8—1221no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×Q8—1221no—
348C2×D4—614no—
349C2×D4—1222no—
350C2×D4—1222no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—612no—
354C22×C4—1221no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—614no—
373C2×C8—1222no—
374C2×C8—1222no—
375C2×C8—2448no—
376C2×C8#1175822612yes—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—81no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—2444no—
392C4⋊C4—2442no—
393C22⋊C4—1224no—
394C22⋊C4—4882no—
395C22⋊C4—4882no—
396C22⋊C4—4882no—
397C22⋊C4—2444no—
398C42—2442no—
399C42—2442no—
400C42#1175822212yes—
401C42D4×D6112yes—
402C23—614no—
403C23—1222no—
404C23—1222no—
405C23—1222no—
406C23—1222no—
407C23—1222no—
408C23—1222no—
409C23C2×D4×D6111yescharacteristic
410Q8—84no—
411Q8—1222no—
412Q8—1222no—
413Q8—2441no—
414Q8—2441no—
415D4—1228no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—1222no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4#1451975012yes—
444C2×C4#1451975012yes—
445C2×C4Q16×D6112yes—
446C2×C4#1451979812yes—
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4#41×D6111yescharacteristic
449C2×C4C2×#1175822211yescharacteristic
450C2×C4C2×#1175822611yescharacteristic
451C2×C4#1451982311yescharacteristic
452C2×C4#1451982511yescharacteristic
453C8—22no—
454C8—41no—
455C8—1224no—
456C22—1224no—
457C22—1224no—
458C22—1224no—
459C22—2442no—
460C22—2442no—
461C22—2442no—
462C22—2442no—
463C22C2×#1451975011yescharacteristic
464C22#1684040511yescharacteristic
465C22#1684041511yescharacteristic
466C22C2×Q16×D6111yescharacteristic
467C22C2×#1451979811yescharacteristic
468C22#1684056311yescharacteristic
469C22#1684056811yescharacteristic
470C4—22no—
471C4—82no—
472C4—82no—
473C4—4881no—
474C4—4881no—
475C4—4881no—
476C4—2442no—
477C4—2442no—
478C4#1684000214yes—
479C4#1684029911yescharacteristic
480C4#1684038411yescharacteristic
481C4#1684040311yescharacteristic
482C4#1684042211yescharacteristic
483C2—2444no—
484C2#2144939711yescharacteristic
485C2C2×#1684000211yescharacteristic
486C2#2145549711yescharacteristic
487C2#2145769411yescharacteristic
488C2#2145776811yescharacteristic
489C2#2145821111yescharacteristic
490C2#2145823111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1990D61
hypocenterC61#25340
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. C2×#16840054C2semidirect
5. #21447251C2semidirect
6. #21447025C2semidirect
7. #21446575C2semidirect
8. #21446241C2semidirect
9. #21445847C2semidirect
10. C4×#14519702C2semidirect
11. C61×#1990C2semidirect
244. #1990D61semidirect
288. C61#25340semidirect

Maximal subgroups

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

subgroupindex
C2×#168400542
C2×C8×D2442
#214509932
#214507422
#214472512
#214470252
#214465752
#214462412
#214458472
C4×#145197022
C61×#19902
#2534061