Finite Group Databasebeta

← #29729929

Subgroups — #29729929

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×#16839962C211yesmaximal, characteristic
3#21448883C211yesmaximal, characteristic
4#21447233C212yesmaximal, complemented
5#21447134C212yesmaximal, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446538C211yesmaximal, characteristic, complemented
8#21446156C212yesmaximal, complemented
9#21445826C212yesmaximal, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C61×#2568C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244C2212yes—
15C2×#14519739—21no—
16#16840110—24no—
17#16839962—24no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×C4⋊#92418C2212yes—
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839664C2212yes—
27#16839663—21no—
28#16839640—22no—
29C4×C4⋊Dic61C2211yescharacteristic
30#16839524C2211yescharacteristic
31#16839524C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#400—22no—
38C61×#391—21no—
39C61×#391—21no—
40C61×#391C2212yes—
41C61×#382C2212yes—
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×C4×D61—22no—
47C2×#11758201—24no—
48C2×#11758201—24no—
49C2×#11758201—42no—
50C22×#92418C2311yescharacteristic
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53#14519774—22no—
54#14519774D414yes—
55C4×D244—24no—
56C4×D244—42no—
57#14519739—44no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×C4×Dic61—21no—
64C4⋊#92418—22no—
65C4⋊#92418—42no—
66#14519701—22no—
67#14519701—22no—
68#14519697—24no—
69#14519697—24no—
70#14519697—24no—
71C4×#92418—22no—
72#14519684—21no—
73C22×Q8×C61—21no—
74C22×Q8×C61C2311yescharacteristic
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×C61D411yescharacteristic
82C2×C4⋊C4×C61D411yescharacteristic
83C2×C4⋊C4×C61C2312yes—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C42×C61—21no—
86C2×C42×C61C2312yes—
87C2×C42×C61C2311yescharacteristic
88#95×C61—22no—
89#95×C61—22no—
90#95×C61—22no—
91#95×C61—22no—
92#95×C61D412yes—
93C23×D61—22no—
94C2×Q8×C61—21no—
95C2×Q8×C61—21no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61—42no—
100C2×Q8×C61—42no—
101C2×Q8×C61—42no—
102C2×Q8×C61#4112yes—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—21no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—24no—
118C2×D244—22no—
119C2×D244—24no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—81no—
123C2×D244—81no—
124C2×D244C2×D414yes—
125C2×C4×D61—44no—
126C2×C4×D61—44no—
127C2×C4×D61—82no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—42no—
135C4⋊C4×C61—42no—
136C42×C61—22no—
137C42×C61—22no—
138C42×C61—42no—
139C42×C61C2×D412yes—
140C42×C61#4112yes—
141#11758201—48no—
142#11758198—44no—
143#11758198—44no—
144#11758198—82no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61#4112yes—
149C4×Dic61—42no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—22no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—22no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—21no—
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—22no—
168Q8×C61—22no—
169Q8×C61—41no—
170Q8×C61—41no—
171Q8×C61—84no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61#12112yes—
190C2×C4×C61#12112yes—
191C2×C4×C61#13512yes—
192C2×C4×C61AGL(1,ℤ/8)12yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)11yescharacteristic
198C2×C4×C612−1+411yescharacteristic
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205D244—24no—
206D244—24no—
207D244—42no—
208D244—42no—
209D244—48no—
210D244—84no—
211C4×D61—88no—
212#92418—41no—
213#92418—42no—
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—42no—
233C4×C61—42no—
234C4×C61—82no—
235C4×C61—81no—
236C4×C61#46314yes—
237C4×C61#52911yescharacteristic
238C4×C61#54211yescharacteristic
239C4×C61#54611yescharacteristic
240C4×C61#54711yescharacteristic
241Dic61—42no—
242Dic61—81no—
243C2×#486—611no—
244C2×#460—611no—
245#2568D6111yescharacteristic, complemented
246#2366—611no—
247#2073—612no—
248#2055—612no—
249#1988—611no—
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×Q8C2×D6111yescharacteristic
266C2×C4×D4—612no—
267C2×C4×D4—1221no—
268#486—1224no—
269#460—1224no—
270C2×C4⋊C8—611no—
271C2×C4⋊C8—612no—
272C2×#103—611no—
273C2×#103—611no—
274C2×#102—612no—
275C2×#102—612no—
276C2×#102—612no—
277C2×C4×C8—611no—
278#400—22no—
279#391—21no—
280#391—21no—
281#391C2×D6112yes—
282#390—612no—
283#390—1221no—
284#386—1222no—
285#382C2×D6112yes—
286C4×C4⋊C4—611no—
287C4×C4⋊C4C2×D6111yescharacteristic
288#340—612no—
289#340—611no—
290#340—612no—
291#340—611no—
292#336—612no—
293C61#2532511yescharacteristic, complemented, Sylow
294C22×Q8—21no—
295C22×Q8C22×D6111yescharacteristic
296C22×D4—612no—
297C22×D4—1221no—
298C22×D4—1221no—
299C23×C4—1222no—
300C2×SD16—1224no—
301C2×SD16—1224no—
302C2×SD16—2442no—
303C22×C8—612no—
304C22×C8—611no—
305C22×C8—611no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308#127—614no—
309#127—1222no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×D4—1224no—
313C4×D4—2442no—
314C4×D4—2444no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4C22⋊D6111yescharacteristic
320C2×C4⋊C4C22⋊D6111yescharacteristic
321C2×C4⋊C4C22×D6111yescharacteristic
322C2×C4⋊C4C22×D6112yes—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1222no—
326C2×C42—21no—
327C2×C42—1221no—
328C2×C42C22×D6112yes—
329C2×C42C22×D6111yescharacteristic
330C4⋊C8—1222no—
331C4⋊C8—2442no—
332#103—1222no—
333#103—1222no—
334#102—1224no—
335#102—1224no—
336#102—1224no—
337C4×C8—1222no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—1221no—
343#95C22⋊D6112yes—
344C24—1222no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8#1175822612yes—
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—1221no—
370C22×C4—1221no—
371C22×C4—4882no—
372C22×C4—2444no—
373C22×C4—2444no—
374C22×C4C2×C22⋊D6112yes—
375C22×C4C2×C22⋊D6112yes—
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C2×C22⋊D6111yescharacteristic
378C22×C4C23×D6111yescharacteristic
379SD16—2448no—
380C2×C8—1221no—
381C2×C8—1221no—
382C2×C8—1222no—
383C2×C8—1221no—
384C2×C8—1221no—
385C2×C8—1222no—
386C2×C8—1221no—
387C2×C8—1222no—
388C2×C8—1221no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—612no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C22⋊C4—4882no—
402C22⋊C4—2444no—
403C22⋊C4—2444no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—2442no—
408C42D4×D6112yes—
409C42#1175822412yes—
410C23—1224no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23—2442no—
417C23C22×C22⋊D6111yescharacteristic
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422Q8—84no—
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—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—1222no—
447C2×C4—4888no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4#1451974812yes—
454C2×C4#1451974812yes—
455C2×C4#1451979712yes—
456C2×C4#1451980212yes—
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4#41×D6111yescharacteristic
459C2×C4C2×#1175822411yescharacteristic
460C2×C4C2×#1175822611yescharacteristic
461C2×C4#1451982311yescharacteristic
462C2×C4#1451982511yescharacteristic
463C8—2441no—
464C8—2442no—
465C8—2441no—
466C22—4882no—
467C22—4882no—
468C22—4882no—
469C22—4882no—
470C22—2444no—
471C22—2444no—
472C22—2444no—
473C22C2×#1451974811yescharacteristic
474C22#1684040011yescharacteristic
475C22#1684042311yescharacteristic
476C22C2×#1451979711yescharacteristic
477C22C2×#1451980211yescharacteristic
478C22#1684057011yescharacteristic
479C22#1684057011yescharacteristic
480C4—22no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—81no—
486C4—4881no—
487C4—2442no—
488C4#1683997414yes—
489C4#1684029811yescharacteristic
490C4#1684038511yescharacteristic
491C4#1684040311yescharacteristic
492C4#1684041411yescharacteristic
493C2—4884no—
494C2#2144939211yescharacteristic
495C2C2×#1683997411yescharacteristic
496C2#2145532711yescharacteristic
497C2#2145775211yescharacteristic
498C2#2145804411yescharacteristic
499C2#2145851511yescharacteristic
500C2#2145854811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2568C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2568D61
hypocenterC61#25325
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2568

Complements

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

NH ≅ G/Ntype
4. #21447233C2semidirect
5. #21447134C2semidirect
6. #21446772C2semidirect
7. #21446538C2semidirect
8. #21446156C2semidirect
9. #21445826C2semidirect
10. C4×#14519701C2semidirect
11. C61×#2568C2semidirect
245. #2568D61semidirect
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×#168399622
#214488832
#214472332
#214471342
#214467722
#214465382
#214461562
#214458262
C4×#145197012
C61×#25682
#2532561