Finite Group Databasebeta

← #29729528

Subgroups — #29729528

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839964C211yesmaximal, characteristic
2C2×Q8×#92418C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic
4#21447200C212yesmaximal, complemented
5#21447134C212yesmaximal
6#21446676C212yesmaximal
7#21446622C211yesmaximal, characteristic, complemented
8#21446582C211yesmaximal, characteristic, complemented
9#21446498C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic
11C61×#2568C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519753—21no—
14C2×C4×D244C2211yescharacteristic
15#16839964—24no—
16Q8×#92418—24no—
17C2×#14519708C2212yes—
18C2×C4⋊#92418C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×C4×#92418C2211yescharacteristic
26C2×C4×#92418C2212yes—
27#16839684—22no—
28#16839664C2212yes—
29#16839660C2212yes—
30#16839632C2211yescharacteristic
31#16839619—21no—
32#16839524C2211yescharacteristic
33#16839524C2211yescharacteristic
34#16839524C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#400—22no—
38C61×#391—21no—
39C61×#391C2212yes—
40C61×#391C2212yes—
41C61×#382—21no—
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C2×#11758201—24no—
46C2×#11758201—24no—
47C2×#11758201—42no—
48C22×#92418C2311yescharacteristic
49C22×#92418C2312yes—
50C22×#92418C2311yescharacteristic
51#14519753—44no—
52C4×D244—22no—
53C4×D244—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2312yes—
58C2×#11758196—22no—
59C2×#11758198—21no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62#14519708—42no—
63C4⋊#92418—24no—
64C4⋊#92418D414yes—
65#14519701—24no—
66#14519701—24no—
67#14519701—42no—
68#14519697—22no—
69#14519697—22no—
70C4×#92418—22no—
71C4×#92418—24no—
72#14519684—22no—
73C22×Q8×C61D411yescharacteristic
74C22×Q8×C61C2312yes—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61—22no—
77C4×Q8×C61—22no—
78C4×Q8×C61—22no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61D412yes—
82C2×C4⋊C4×C61D411yescharacteristic
83C2×C4⋊C4×C61C2311yescharacteristic
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C42×C61—21no—
86C2×C42×C61C2312yes—
87C2×C42×C61C2311yescharacteristic
88#95×C61—21no—
89#95×C61—22no—
90#95×C61—22no—
91#95×C61—22no—
92#95×C61—22no—
93C23×D61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61SD1614yescomplemented
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—22no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244#4112yes—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C42×C61—22no—
131C42×C61—22no—
132C42×C61—42no—
133C42×C61C2×D412yes—
134C42×C61#4112yes—
135#11758201—48no—
136#11758198—42no—
137#11758198—44no—
138#11758198—82no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—22no—
143C4⋊Dic61—42no—
144C4⋊Dic61—81no—
145#11758196—44no—
146C2×#92418—21no—
147C2×#92418—22no—
148C2×#92418—22no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418C2×D414yes—
154C2×#92418#4112yes—
155C23×C61C22×D411yescharacteristic
156C22×D61—22no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163Q8×C61—28no—
164Q8×C61—42no—
165Q8×C61—42no—
166Q8×C61—42no—
167Q8×C61—42no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61#12112yes—
185C2×C4×C61#12112yes—
186C2×C4×C61C2×SD1614yes—
187C2×C4×C61C2×SD1612yes—
188C2×C4×C61#13512yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61H(5,𝔽2)11yescharacteristic
194C2×C4×C612−1+411yescharacteristic
195C2×Dic61—41no—
196C2×Dic61—42no—
197C2×Dic61—41no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201D244—22no—
202D244—22no—
203D244—41no—
204D244—41no—
205C4×D61—84no—
206#92418—24no—
207#92418—22no—
208#92418—41no—
209#25293—611nomaximal, Sylow
210C22×C61C2×#12111yescharacteristic
211C22×C61#54811yescharacteristic
212C22×C61#55011yescharacteristic
213C22×C61C22×SD1611yescharacteristic
214C22×C61C2×#13511yescharacteristic
215C22×C61#57711yescharacteristic
216C22×C61#57811yescharacteristic
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C4×C61—24no—
225C4×C61—22no—
226C4×C61—42no—
227C4×C61—81no—
228C4×C61—81no—
229C4×C61—82no—
230C4×C61#46514yes—
231C4×C61#52911yescharacteristic
232C4×C61#54211yescharacteristic
233C4×C61#54611yescharacteristic
234C4×C61#54811yescharacteristic
235Dic61—82no—
236Dic61—81no—
237C2×#461—611no—
238C2×C8×Q8—611no—
239#2568D6111yescharacteristic, complemented
240#2538—611no—
241#2066—612no—
242#2055—612no—
243#1968—612no—
244#1956—611no—
245#1948—611no—
246#1933—612no—
247C4×#102—611no—
248C2×C61#242611yescharacteristic
249C2×C61C2×#46511yescharacteristic
250C2×C61#306311yescharacteristic
251C2×C61#335911yescharacteristic
252C2×C61#336011yescharacteristic
253C2×C61#339211yescharacteristic
254C2×C61#339311yescharacteristic
255D61—82no—
256C22×SD16—612no—
257C2×C4⋊Q8C2×D6111yescharacteristic
258C2×C22⋊Q8—1221no—
259C2×C4×Q8C2×D6111yescharacteristic
260C2×C4×D4—611no—
261#461—1224no—
262C8×Q8—1224no—
263C2×#106—612no—
264C2×C4⋊C8—611no—
265C2×C4⋊C8—612no—
266C2×#103—612no—
267C2×#103—612no—
268C2×#103—612no—
269C2×#102—611no—
270C2×#102—611no—
271C2×C4×C8—612no—
272C2×C4×C8—611no—
273#400—22no—
274#396—1222no—
275#391—21no—
276#391C2×D6112yes—
277#391C2×D6112yes—
278#390—612no—
279#389—612no—
280#384—611no—
281#382—21no—
282#380—1221no—
283C4×C4⋊C4C2×D6111yescharacteristic
284#340—612no—
285#340—611no—
286#340—611no—
287C61#2529311yescharacteristic, complemented, Sylow
288C22×Q8C22⋊D6111yescharacteristic
289C22×Q8C22×D6112yes—
290C22×D4—611no—
291C23×C4—1221no—
292C2×SD16—1224no—
293C2×SD16—1224no—
294C2×SD16—2442no—
295C22×C8—612no—
296C22×C8—611no—
297C22×C8—611no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C22⋊Q8—2444no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×D4—1222no—
304C4×D4—1222no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4C22⋊D6111yescharacteristic
313C2×C4⋊C4C22⋊D6112yes—
314C2×C4⋊C4C22×D6111yescharacteristic
315C2×C4⋊C4C22×D6111yescharacteristic
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1222no—
319C2×C42—21no—
320C2×C42C22×D6112yes—
321C2×C42C22×D6111yescharacteristic
322#106—2442no—
323C4⋊C8—614no—
324C4⋊C8—1224no—
325#103—1224no—
326#103—1224no—
327#103—2442no—
328#102—1222no—
329#102—1222no—
330C4×C8—1224no—
331C4×C8—1222no—
332#95—21no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—1222no—
338C24—1221no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8#1175820114yescomplemented
348C2×D4—612no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—1221no—
352C2×D4—1221no—
353C2×D4—1221no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—1221no—
358C22×C4—1222no—
359C22×C4—2442no—
360C22×C4—2442no—
361C22×C4—2442no—
362C22×C4C2×C22⋊D6112yes—
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C2×C22⋊D6112yes—
365C22×C4C2×C22⋊D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367SD16—2448no—
368C2×C8—614no—
369C2×C8—612no—
370C2×C8—1221no—
371C2×C8—1222no—
372C2×C8—1222no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8—1221no—
376C2×C8—1221no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C4⋊C4—4881no—
390C4⋊C4—2444no—
391C4⋊C4—2442no—
392C22⋊C4—4882no—
393C22⋊C4—2442no—
394C22⋊C4—2444no—
395C42—22no—
396C42—22no—
397C42—42no—
398C42D4×D6112yes—
399C42#1175822412yes—
400C23—1222no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23—2441no—
406C23—2441no—
407C23C22×C22⋊D6111yescharacteristic
408Q8—28no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413D4—1222no—
414D4—1222no—
415D4—2441no—
416D4—2441no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—4884no—
434C2×C4—2441no—
435C2×C4—2442no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4#1451974912yes—
441C2×C4#1451974912yes—
442C2×C4C2×#1175820114yes—
443C2×C4C2×#1175820112yes—
444C2×C4#1451979712yes—
445C2×C4C2×D4×D6111yescharacteristic
446C2×C4#41×D6111yescharacteristic
447C2×C4#41×D6111yescharacteristic
448C2×C4C2×#1175822411yescharacteristic
449C2×C4#1451982311yescharacteristic
450C2×C4#1451982411yescharacteristic
451C8—1224no—
452C8—1222no—
453C8—2441no—
454C22—4881no—
455C22—4881no—
456C22—4881no—
457C22—4881no—
458C22—2442no—
459C22—2442no—
460C22—2442no—
461C22C2×#1451974911yescharacteristic
462C22#1684041911yescharacteristic
463C22#1684043111yescharacteristic
464C22C22×#1175820111yescharacteristic
465C22C2×#1451979711yescharacteristic
466C22#1684057011yescharacteristic
467C22#1684057711yescharacteristic
468C4—24no—
469C4—22no—
470C4—42no—
471C4—81no—
472C4—81no—
473C4—82no—
474C4—4881no—
475C4—4882no—
476C4#1683999314yes—
477C4#1684029111yescharacteristic
478C4#1684038111yescharacteristic
479C4#1684040511yescharacteristic
480C4#1684042011yescharacteristic
481C2—4882no—
482C2#2144938811yescharacteristic
483C2C2×#1683999311yescharacteristic
484C2#2145550611yescharacteristic
485C2#2145845711yescharacteristic
486C2#2145846411yescharacteristic
487C2#2145879611yescharacteristic
488C2#2145880311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2568C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2568D61
hypocenterC61#25293
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#92418C2semidirect
4. #21447200C2semidirect
7. #21446622C2semidirect
8. #21446582C2semidirect
9. #21446498C2semidirect
11. C61×#2568C2semidirect
102. C2×Q8×C61SD16semidirect
239. #2568D61semidirect
287. C61#25293semidirect
347. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
C2×Q8×#924182
#214507372
#214472002
#214471342
#214466762
#214466222
#214465822
#214464982
C4×#145196972
C61×#25682
#2529361