Finite Group Databasebeta

← #29729931

Subgroups — #29729931

Order 15616.

930 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840109C211yesmaximal, characteristic
2C2×#16839963C211yesmaximal, characteristic
3#21451034C211yesmaximal, characteristic, complemented
4#21448883C211yesmaximal, characteristic
5#21447234C212yesmaximal, complemented
6#21447137C212yesmaximal, complemented
7#21446770C211yesmaximal, characteristic, complemented
8#21446152C212yesmaximal, complemented
9#21445830C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C61×#1939C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15C2×C4×Dic122C2211yescharacteristic
16C2×C4×D244C2212yes—
17C2×#14519739—21no—
18#16840109—24no—
19#16839963—24no—
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519699C2212yes—
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25#16839702—22no—
26#16839668C2212yes—
27#16839667—21no—
28#16839667—21no—
29#16839664C2212yes—
30#16839663—21no—
31#16839640—22no—
32#16839628C2212yes—
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×Dic122C2311yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48C22×C4×D61—22no—
49C2×#11758190—24no—
50C2×#11758190—24no—
51C2×#11758190—42no—
52#14519777—22no—
53#14519777—22no—
54#14519774—22no—
55#14519774D414yes—
56C4×Dic122—22no—
57C4×Dic122—22no—
58C4×D244—24no—
59C4×D244—42no—
60#14519739—44no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2312yes—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758196—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61—21no—
72#14519702—22no—
73#14519702—22no—
74#14519699—24no—
75#14519699—24no—
76#14519699—24no—
77#14519684—22no—
78#14519684—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684—21no—
82C22×C8×C61C2311yescharacteristic
83C22×C8×C61C2312yes—
84C22×C8×C61C2311yescharacteristic
85C2×C4⋊C4×C61D411yescharacteristic
86C2×C4⋊C4×C61D411yescharacteristic
87C2×C42×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—22no—
90C4⋊C8×C61—42no—
91C4×C8×C61—22no—
92#95×C61D412yes—
93C23×D61—22no—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—22no—
108C2×D244—24no—
109C2×D244—22no—
110C2×D244—24no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—81no—
114C2×D244—81no—
115C2×D244C2×D414yes—
116C2×C4×D61—44no—
117C2×C4×D61—44no—
118C2×C4×D61—82no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—42no—
125C2×Dic122—42no—
126C2×Dic122—42no—
127C2×Dic122#4112yes—
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61—22no—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C2×C8×C61—22no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C42×C61—22no—
140C42×C61—22no—
141C42×C61C2×D412yes—
142C42×C61#4112yes—
143#11758198—44no—
144#11758198—44no—
145#11758198—82no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—42no—
154C4⋊Dic61#4112yes—
155#11758196—42no—
156C4×Dic61—42no—
157C4×Dic61—42no—
158#11758190—48no—
159C23×C61C22×D411yescharacteristic
160C22×D61—24no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—42no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
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#12112yes—
176C2×C4×C61#12112yes—
177C2×C4×C61#13512yes—
178C2×C4×C61AGL(1,ℤ/8)12yes—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×C4×C61H(5,𝔽2)11yescharacteristic
184C2×C4×C612−1+411yescharacteristic
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—41no—
196C2×Dic61—42no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200D244—24no—
201D244—24no—
202D244—42no—
203D244—42no—
204D244—48no—
205D244—84no—
206C4×D61—88no—
207Dic122—22no—
208Dic122—22no—
209Dic122—41no—
210Dic122—41no—
211Dic122—84no—
212C8×C61—42no—
213C8×C61—41no—
214C8×C61—41no—
215#25325—611nomaximal, Sylow
216C22×C61C2×#12111yescharacteristic
217C22×C61#54611yescharacteristic
218C22×C61#54811yescharacteristic
219C22×C61C2×#13511yescharacteristic
220C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
221C22×C61#57711yescharacteristic
222C22×C61#57711yescharacteristic
223C2×D61—44no—
224C2×D61—44no—
225C2×D61—44no—
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—82no—
230C4×C61—22no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61#46314yes—
234C4×C61#52911yescharacteristic
235C4×C61#54211yescharacteristic
236C4×C61#54611yescharacteristic
237C4×C61#54711yescharacteristic
238Dic61—42no—
239Dic61—42no—
240Dic61—81no—
241Dic61—82no—
242Dic61—81no—
243C2×#486—611no—
244C2×#460—611no—
245#2568—611no—
246#2366—611no—
247#2073—612no—
248#2055—612no—
249#1988—611no—
250#1939D6111yescharacteristic, complemented
251#1877—612no—
252#1804—612no—
253C4×#103—611no—
254C2×C61#242711yescharacteristic
255C2×C61C2×#46311yescharacteristic
256C2×C61#304211yescharacteristic
257C2×C61#330411yescharacteristic
258C2×C61#332411yescharacteristic
259C2×C61#336511yescharacteristic
260C2×C61#336811yescharacteristic
261D61—84no—
262C22×SD16—612no—
263C2×C4⋊Q8—611no—
264C2×#127—611no—
265C2×C4×Q8—611no—
266C2×C4×D4—612no—
267C2×C4×D4—1221no—
268#486—1224no—
269#460—1224no—
270C2×C4⋊C8C2×D6111yescharacteristic
271C2×C4⋊C8C2×D6112yes—
272C2×#103—611no—
273C2×#103—611no—
274C2×#102—612no—
275C2×#102—612no—
276C2×#102—612no—
277C2×C4×C8C2×D6111yescharacteristic
278#400—1222no—
279#391—612no—
280#391—1221no—
281#391—1221no—
282#390—612no—
283#390—1221no—
284#386—1222no—
285#382—612no—
286C4×C4⋊C4—611no—
287C4×C4⋊C4C2×D6111yescharacteristic
288#340—612no—
289#340—611no—
290#340—612no—
291#340—611no—
292#336C2×D6112yes—
293C61#2532511yescharacteristic, complemented, Sylow
294C22×Q8—611no—
295C22×Q8—1221no—
296C22×D4—612no—
297C22×D4—1221no—
298C22×D4—1221no—
299C23×C4—1222no—
300C2×SD16—1224no—
301C2×SD16—1224no—
302C2×SD16—2442no—
303C22×C8C22×D6112yes—
304C22×C8C22×D6111yescharacteristic
305C22×C8C22×D6111yescharacteristic
306C4⋊Q8—1222no—
307C4⋊Q8—1222no—
308#127—614no—
309#127—1222no—
310C4×Q8—1222no—
311C4×Q8—1222no—
312C4×D4—1224no—
313C4×D4—2442no—
314C4×D4—2444no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—612no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4D24411yescharacteristic
322C2×C4⋊C4D24411yescharacteristic
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1222no—
326C2×C42—1221no—
327C2×C42—1221no—
328C2×C42C22×D6112yes—
329C2×C42C22×D6111yescharacteristic
330C4⋊C8—22no—
331C4⋊C8—42no—
332#103—1222no—
333#103—1222no—
334#102—1224no—
335#102—1224no—
336#102—1224no—
337C4×C8—22no—
338#95—1221no—
339#95—1222no—
340#95—1222no—
341#95—1222no—
342#95—1222no—
343#95D24412yes—
344C24—1222no—
345C2×Q8—612no—
346C2×Q8—1221no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—2442no—
352C2×Q8—2442no—
353C2×Q8—2442no—
354C2×D4—614no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—1222no—
359C2×D4—1222no—
360C2×D4—1224no—
361C2×D4—1224no—
362C2×D4—4881no—
363C2×D4—4881no—
364C2×D4—2442no—
365C2×D4—2442no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—1221no—
371C22×C4—4882no—
372C22×C4—2444no—
373C22×C4—2444no—
374C22×C4C2×D24412yes—
375C22×C4C2×D24412yes—
376C22×C4C2×D24411yescharacteristic
377C22×C4C2×D24411yescharacteristic
378C22×C4C23×D6111yescharacteristic
379SD16—2448no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8—22no—
388C2×C8—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—612no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—1222no—
398C4⋊C4—1221no—
399C4⋊C4—2442no—
400C4⋊C4—2442no—
401C22⋊C4—4882no—
402C22⋊C4—2444no—
403C22⋊C4—2444no—
404C42—22no—
405C42—22no—
406C42—2442no—
407C42—2442no—
408C42D4×D6112yes—
409C42#1175822612yes—
410C23—1224no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23—2442no—
417C23C22×D24411yescharacteristic
418Q8—1222no—
419Q8—1222no—
420Q8—4884no—
421Q8—2441no—
422Q8—2441no—
423D4—1224no—
424D4—1224no—
425D4—4884no—
426D4—2448no—
427D4—2442no—
428D4—2442no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—4888no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2442no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4#1451975212yes—
454C2×C4#1451975212yes—
455C2×C4#1451979912yes—
456C2×C4#1451980512yes—
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4#41×D6111yescharacteristic
459C2×C4#41×D6111yescharacteristic
460C2×C4C2×#1175822611yescharacteristic
461C2×C4#1451982211yescharacteristic
462C2×C4#1451982511yescharacteristic
463C8—41no—
464C8—42no—
465C8—41no—
466C22—4882no—
467C22—4882no—
468C22—4882no—
469C22—4882no—
470C22—2444no—
471C22—2444no—
472C22—2444no—
473C22C2×#1451975211yescharacteristic
474C22#1684040711yescharacteristic
475C22#1684042411yescharacteristic
476C22C2×#1451979911yescharacteristic
477C22C2×#1451980511yescharacteristic
478C22#1684057411yescharacteristic
479C22#1684057411yescharacteristic
480C4—22no—
481C4—42no—
482C4—42no—
483C4—4881no—
484C4—4882no—
485C4—4881no—
486C4—2442no—
487C4—2442no—
488C4#1683997814yes—
489C4#1684029711yescharacteristic
490C4#1684038611yescharacteristic
491C4#1684040911yescharacteristic
492C4#1684041411yescharacteristic
493C2—4884no—
494C2#2144939411yescharacteristic
495C2C2×#1683997811yescharacteristic
496C2#2145533211yescharacteristic
497C2#2145775911yescharacteristic
498C2#2145805111yescharacteristic
499C2#2145851711yescharacteristic
500C2#2145855011yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1939C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#1939D61
hypocenterC61#25325
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
3. #21451034C2semidirect
5. #21447234C2semidirect
6. #21447137C2semidirect
7. #21446770C2semidirect
8. #21446152C2semidirect
9. #21445830C2semidirect
10. C4×#14519702C2semidirect
11. C61×#1939C2semidirect
250. #1939D61semidirect
293. C61#25325semidirect

Maximal subgroups

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

subgroupindex
C2×#168401092
C2×#168399632
#214510342
#214488832
#214472342
#214471372
#214467702
#214461522
#214458302
C4×#145197022
C61×#19392
#2532561