Finite Group Databasebeta

← #29721584

Subgroups — #29721584

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839964C211yesmaximal, characteristic
2C2×#16839964C212yesmaximal
3C2×C4×#11758201C211yesmaximal, characteristic
4C4⋊C4×D244C211yesmaximal, characteristic
5#21447200C212yesmaximal, complemented
6#21447062C212yesmaximal, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446538C211yesmaximal, characteristic, complemented
9#21446156C212yesmaximal
10#21445824C211yesmaximal, characteristic, complemented
11C61×#2546C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×#11758201C2211yescharacteristic
14C2×C4×D244C2212yes—
15C2×C4×D244C2211yescharacteristic
16C2×C4⋊C4×D61—21no—
17#16839964—28no—
18#16839964—24no—
19C4×#11758201—24no—
20C2×#14519708C2211yescharacteristic
21C2×C4⋊#92418C2212yes—
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519701C2212yes—
24C2×#14519701C2212yes—
25C2×#14519701C2212yes—
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2212yes—
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839660C2212yes—
31#16839636—22no—
32#16839632C2211yescharacteristic
33#16839619—21no—
34#16839524C2212yes—
35#16839517C2212yes—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×C4⋊Q8×C61C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#398—22no—
40C61×#391C2212yes—
41C61×#389—21no—
42C61×#389—21no—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49C2×#11758201—24no—
50C2×#11758201—24no—
51C2×#11758201—24no—
52C2×#11758201D414yes—
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55C22×#92418C2311yescharacteristic
56C4×D244—22no—
57C4×D244—24no—
58C4×D244—42no—
59C4×D244D414yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C4⋊C4×D61—44no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68#14519708—22no—
69C4⋊#92418—22no—
70C4⋊#92418—24no—
71#14519701—24no—
72#14519701—24no—
73#14519701—22no—
74#14519701—24no—
75#14519701—42no—
76#14519697—22no—
77C4×#92418—22no—
78#14519684—22no—
79C22×Q8×C61C2312yes—
80C22×Q8×C61C2311yescharacteristic
81C4⋊Q8×C61—24no—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—42no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C42×C61—21no—
96C2×C42×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#95×C61—22no—
99#95×C61—22no—
100#95×C61D412yes—
101C23×D61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—21no—
107C2×Q8×C61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244C2×D412yes—
128C2×D244C2×D412yes—
129C2×D244#4112yes—
130C2×C4×D61—24no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—44no—
134C2×C4×D61—44no—
135C2×C4×D61—82no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—44no—
140C4⋊C4×C61—44no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61SD1614yescomplemented
145C42×C61—22no—
146C42×C61—22no—
147C42×C61—42no—
148C42×C61C2×D412yes—
149C42×C61#4112yes—
150#11758201—28no—
151#11758201—48no—
152#11758198—24no—
153#11758198—44no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—42no—
158C4⋊Dic61—81no—
159C4⋊Dic61C2×D412yes—
160#11758196—44no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418C2×D412yes—
170C23×C61C22×D411yescharacteristic
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—22no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178Q8×C61—22no—
179Q8×C61—22no—
180Q8×C61—41no—
181Q8×C61—41no—
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61—42no—
185Q8×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61#12112yes—
203C2×C4×C61#12112yes—
204C2×C4×C61C2×SD1614yes—
205C2×C4×C61C2×SD1612yes—
206C2×C4×C61#13712yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—22no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219D244—22no—
220D244—22no—
221D244—22no—
222D244#12114yes—
223C4×D61—48no—
224C4×D61—88no—
225#92418—22no—
226#92418—42no—
227#92418—41no—
228#24730—611nomaximal, Sylow
229C22×C61C2×#12111yescharacteristic
230C22×C61D4211yescharacteristic
231C22×C61#54811yescharacteristic
232C22×C61C22×SD1611yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C22×C61#57711yescharacteristic
235C22×C61#57811yescharacteristic
236C2×D61—24no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C4×C61—24no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—82no—
248C4×C61—81no—
249C4×C61#46114yescomplemented
250C4×C61C2×#12111yescharacteristic
251C4×C61#53711yescharacteristic
252C4×C61#54711yescharacteristic
253C4×C61#55311yescharacteristic
254Dic61—42no—
255Dic61—82no—
256C2×#461—611no—
257C2×#461—612no—
258C2×C4×SD16—611no—
259#2546D6111yescharacteristic, complemented
260D4×C4⋊C4—611no—
261#2066—612no—
262#2043—612no—
263#1956—611no—
264#1939—611no—
265#1877—612no—
266#1803—611no—
267C2×C61#241611yescharacteristic
268C2×C61C2×#46111yescharacteristic
269C2×C61#304611yescharacteristic
270C2×C61#327611yescharacteristic
271C2×C61D4×SD1611yescharacteristic
272C2×C61#335911yescharacteristic
273C2×C61#337211yescharacteristic
274D61—44no—
275C22×SD16—611no—
276C22×SD16—612no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C2×C4⋊Q8C2×D6112yes—
279C2×C4×Q8C2×D6111yescharacteristic
280C2×C4×D4—611no—
281C2×C4×D4—612no—
282C22×C4⋊C4—1221no—
283#461—1224no—
284#461—1228no—
285C4×SD16—1224no—
286C2×#106—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—612no—
289C2×#103—612no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#103—611no—
293C2×#103—612no—
294C2×#102—611no—
295C2×C4×C8—611no—
296#398—22no—
297#391C2×D6112yes—
298#389—21no—
299#389—21no—
300#389—612no—
301#385—1222no—
302#384—611no—
303#380—1221no—
304C4×C4⋊C4C2×D6111yescharacteristic
305#340—612no—
306#336—612no—
307C61#2473011yescharacteristic, complemented, Sylow
308C22×Q8C22×D6112yes—
309C22×Q8C22×D6111yescharacteristic
310C22×D4—611no—
311C23×C4—1221no—
312C23×C4—1222no—
313C2×SD16—614no—
314C2×SD16—1222no—
315C2×SD16—1222no—
316C2×SD16—1224no—
317C2×SD16—1224no—
318C2×SD16—1224no—
319C22×C8—611no—
320C22×C8—612no—
321C22×C8—611no—
322C4⋊Q8—24no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—42no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×D4—614no—
329C4×D4—1224no—
330C4×D4—1222no—
331C4×D4—2442no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—2444no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1222no—
348C2×C42—21no—
349C2×C42C22×D6112yes—
350C2×C42C22×D6111yescharacteristic
351#106—1222no—
352C4⋊C8—1224no—
353C4⋊C8—1222no—
354#103—1222no—
355#103—1224no—
356#103—1224no—
357#103—1224no—
358#103—2442no—
359#102—1222no—
360C4×C8—1222no—
361#95—22no—
362#95—22no—
363#95—1222no—
364#95C22⋊D6112yes—
365C24—1221no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8D4×D6112yes—
378C2×D4—612no—
379C2×D4—612no—
380C2×D4—612no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—1221no—
388C22×C4—1224no—
389C22×C4—1222no—
390C22×C4—4882no—
391C22×C4—2444no—
392C22×C4—2444no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4C2×C22⋊D6112yes—
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6112yes—
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400SD16—1228no—
401SD16—2448no—
402C2×C8—612no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1221no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1221no—
410C2×C8—1221no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—612no—
420C4⋊C4—1221no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—4881no—
424C4⋊C4—2444no—
425C4⋊C4—2442no—
426C4⋊C4#1175820114yescomplemented
427C22⋊C4—1224no—
428C22⋊C4—2444no—
429C42—22no—
430C42—22no—
431C42—42no—
432C42D4×D6112yes—
433C42#1175822412yes—
434C23—1222no—
435C23—1222no—
436C23—1222no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C22×C22⋊D6111yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—41no—
445Q8—41no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450D4—614no—
451D4—1222no—
452D4—1222no—
453D4—1222no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—1222no—
471C2×C4—4888no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2448no—
478C2×C4#1451974812yes—
479C2×C4#1451974812yes—
480C2×C4C2×#1175820114yes—
481C2×C4C2×#1175820112yes—
482C2×C4#1451980812yes—
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4#41×D6111yescharacteristic
486C2×C4C2×#1175822411yescharacteristic
487C2×C4#1451982211yescharacteristic
488C2×C4#1451982411yescharacteristic
489C8—1222no—
490C8—2442no—
491C8—2441no—
492C22—1224no—
493C22—2442no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22C2×#1451974811yescharacteristic
500C22D4×C22⋊D6111yescharacteristic
501C22#1684041911yescharacteristic
502C22C22×#1175820111yescharacteristic
503C22C2×#1451980811yescharacteristic
504C22#1684057011yescharacteristic
505C22#1684057711yescharacteristic
506C4—22no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—81no—
511C4—82no—
512C4—4882no—
513C4—2442no—
514C4#1683996414yescomplemented
515C4#121×D6111yescharacteristic
516C4#1684034111yescharacteristic
517C4#1684041211yescharacteristic
518C4#1684045911yescharacteristic
519C2—2444no—
520C2#2144925511yescharacteristic
521C2C2×#1683996411yescharacteristic
522C2#2145536511yescharacteristic
523C2#2145746511yescharacteristic
524C2D4×#1175820111yescharacteristic
525C2#2145845711yescharacteristic
526C2#2145859611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2546C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2546D61
hypocenterC61#24730
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2546

Complements

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

NH ≅ G/Ntype
5. #21447200C2semidirect
6. #21447062C2semidirect
7. #21446622C2semidirect
8. #21446538C2semidirect
10. #21445824C2semidirect
11. C61×#2546C2semidirect
144. C4⋊C4×C61SD16semidirect
249. C4×C61#461semidirect
259. #2546D61semidirect
307. C61#24730semidirect
426. C4⋊C4#11758201semidirect
514. C4#16839964semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
C2×C4×#117582012
C4⋊C4×D2442
#214472002
#214470622
#214466222
#214465382
#214461562
#214458242
C61×#25462
#2473061