Finite Group Databasebeta

← #29729934

Subgroups — #29729934

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×#16840110C211yesmaximal, characteristic
2C2×#16839963C211yesmaximal, characteristic
3#21451035C211yesmaximal, characteristic, complemented
4#21448883C211yesmaximal, characteristic
5#21447236C212yesmaximal, complemented
6#21447138C212yesmaximal, complemented
7#21446535C211yesmaximal, characteristic, complemented
8#21446154C212yesmaximal, complemented
9#21445828C212yesmaximal, complemented
10C4×#14519700C211yesmaximal, characteristic, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15C2×C4×D244—21no—
16C2×#14519739C2212yes—
17#16840110—24no—
18#16839963—24no—
19C2×#14519706C2212yes—
20C2×#14519700C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519699C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839700—22no—
27#16839669—21no—
28#16839669—21no—
29#16839667C2212yes—
30#16839664—21no—
31#16839663C2212yes—
32#16839640—22no—
33#16839627C2212yes—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839516C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C61×#340C2212yes—
42C61×#340C2212yes—
43C22×Dic122C2311yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×C4×D61—22no—
48C2×#11758190—24no—
49C2×#11758190—24no—
50C2×#11758190—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519774—22no—
54#14519774D414yes—
55C4×Dic122—22no—
56C4×Dic122—22no—
57C4×D244—44no—
58#14519739—24no—
59#14519739—42no—
60C2×C4⋊Dic61D411yescharacteristic
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×#11758196—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2312yes—
69#14519706—42no—
70#14519700—22no—
71#14519700—22no—
72#14519699—24no—
73#14519697—24no—
74#14519697—24no—
75C4×#92418—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79#14519684—22no—
80#14519684D412yes—
81C22×Q8×C61—21no—
82C22×C8×C61C2312yes—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C42×C61—21no—
90C2×C42×C61C2311yescharacteristic
91C4⋊C8×C61—22no—
92#95×C61—21no—
93C23×D61—22no—
94C2×Q8×C61—42no—
95C2×Q8×C61—42no—
96C2×Q8×C61—42no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2411yescharacteristic
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61C2×D412yes—
106C22×Dic61C2×D412yes—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—24no—
112C2×D244—22no—
113C2×D244—24no—
114C2×D244—42no—
115C2×D244—42no—
116C2×D244—81no—
117C2×D244—81no—
118C2×D244C2×D414yes—
119C2×C4×D61—44no—
120C2×C4×D61—44no—
121C2×C4×D61—82no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122#4112yes—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—42no—
137C4⋊C4×C61#4112yes—
138C42×C61—42no—
139C42×C61C2×D412yes—
140C42×C61#4112yes—
141#11758198—44no—
142#11758198—44no—
143#11758198—82no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148#11758196—42no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C4×Dic61—42no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—21no—
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—
167Q8×C61—84no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61#12112yes—
180C2×C4×C61#12112yes—
181C2×C4×C61#13512yes—
182C2×C4×C61AGL(1,ℤ/8)12yes—
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61H(5,𝔽2)11yescharacteristic
188C2×C4×C612−1+411yescharacteristic
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201D244—24no—
202D244—24no—
203D244—42no—
204D244—42no—
205D244—48no—
206D244—84no—
207C4×D61—88no—
208Dic122—22no—
209Dic122—22no—
210Dic122—41no—
211Dic122—41no—
212C8×C61—42no—
213#92418—41no—
214#92418—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—81no—
233C4×C61—82no—
234C4×C61#46314yes—
235C4×C61#52911yescharacteristic
236C4×C61#54211yescharacteristic
237C4×C61#54611yescharacteristic
238C4×C61#54711yescharacteristic
239Dic61—42no—
240Dic61—42no—
241Dic61—42no—
242Dic61—81no—
243C2×#486—611no—
244C2×#460—611no—
245#2568—611no—
246#2366—611no—
247#2073—612no—
248#2055—612no—
249#1988D6111yescharacteristic, complemented
250#1939—611no—
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⋊Q8C2×D6111yescharacteristic
264C2×#127—611no—
265C2×C4×Q8—611no—
266C2×C4×D4—612no—
267C2×C4×D4—1221no—
268#486—1224no—
269#460—1224no—
270C2×C4⋊C8—612no—
271C2×C4⋊C8C2×D6111yescharacteristic
272C2×#103—611no—
273C2×#103—611no—
274C2×#102—612no—
275C2×#102—612no—
276C2×#102—612no—
277C2×C4×C8—611no—
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—611no—
289#340—611no—
290#340C2×D6112yes—
291#340C2×D6112yes—
292#336—612no—
293C61#2532511yescharacteristic, complemented, Sylow
294C22×Q8—21no—
295C22×Q8—611no—
296C22×D4—612no—
297C22×D4—1221no—
298C22×D4—1221no—
299C23×C4—1222no—
300C2×SD16—1224no—
301C2×SD16—1224no—
302C2×SD16—2442no—
303C22×C8—611no—
304C22×C8—611no—
305C22×C8C22×D6112yes—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308#127—614no—
309#127—1222no—
310C4×Q8—1222no—
311C4×Q8—1222no—
312C4×D4—1224no—
313C4×D4—2444no—
314C4×D4—2442no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4C22×D6112yes—
321C2×C4⋊C4C22×D6111yescharacteristic
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1222no—
326C2×C42—21no—
327C2×C42—612no—
328C2×C42—1221no—
329C2×C42C22×D6111yescharacteristic
330C4⋊C8—22no—
331C4⋊C8—2442no—
332#103—1222no—
333#103—1222no—
334#102—1224no—
335#102—1224no—
336#102—1224no—
337C4×C8—1222no—
338#95—21no—
339#95—612no—
340#95—1222no—
341#95—1222no—
342#95—1222no—
343#95—1222no—
344C24—1222no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—612no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—1221no—
352C2×Q8—1221no—
353C2×Q8—1221no—
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—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—4882no—
374C22×C4—2444no—
375C22×C4—2444no—
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4C23×D6111yescharacteristic
379SD16—2448no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—1221no—
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—1221no—
387C2×C8—1221no—
388C2×C8—1221no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—42no—
395C4⋊C4—1222no—
396C4⋊C4—1222no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—2442no—
400C4⋊C4#1175822612yes—
401C22⋊C4—4882no—
402C22⋊C4—2444no—
403C22⋊C4—2444no—
404C42—42no—
405C42—1222no—
406C42—1222no—
407C42—2442no—
408C42#1175822212yes—
409C42D4×D6112yes—
410C23—1224no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23—2442no—
417C23C2×D4×D6111yescharacteristic
418Q8—84no—
419Q8—1222no—
420Q8—1222no—
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—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—4888no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4#1451975012yes—
454C2×C4#1451975012yes—
455C2×C4#1451979812yes—
456C2×C4#1451980112yes—
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4#41×D6111yescharacteristic
459C2×C4C2×#1175822211yescharacteristic
460C2×C4C2×#1175822611yescharacteristic
461C2×C4#1451982311yescharacteristic
462C2×C4#1451982511yescharacteristic
463C8—42no—
464C8—2441no—
465C8—2441no—
466C22—4882no—
467C22—4882no—
468C22—4882no—
469C22—4882no—
470C22—2444no—
471C22—2444no—
472C22—2444no—
473C22C2×#1451975011yescharacteristic
474C22#1684040311yescharacteristic
475C22#1684042211yescharacteristic
476C22C2×#1451979811yescharacteristic
477C22C2×#1451980111yescharacteristic
478C22#1684057111yescharacteristic
479C22#1684057111yescharacteristic
480C4—22no—
481C4—42no—
482C4—81no—
483C4—82no—
484C4—4881no—
485C4—2442no—
486C4—2442no—
487C4—2442no—
488C4#1683997614yes—
489C4#1684029911yescharacteristic
490C4#1684038411yescharacteristic
491C4#1684040511yescharacteristic
492C4#1684041511yescharacteristic
493C2—4884no—
494C2#2144939711yescharacteristic
495C2C2×#1683997611yescharacteristic
496C2#2145533011yescharacteristic
497C2#2145775511yescharacteristic
498C2#2145804711yescharacteristic
499C2#2145851411yescharacteristic
500C2#2145854711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#1988D61
hypocenterC61#25325
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. #21451035C2semidirect
5. #21447236C2semidirect
6. #21447138C2semidirect
7. #21446535C2semidirect
8. #21446154C2semidirect
9. #21445828C2semidirect
10. C4×#14519700C2semidirect
11. C61×#1988C2semidirect
249. #1988D61semidirect
293. C61#25325semidirect

Maximal subgroups

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

subgroupindex
C2×#168401102
C2×#168399632
#214510352
#214488832
#214472362
#214471382
#214465352
#214461542
#214458282
C4×#145197002
C61×#19882
#2532561