Finite Group Databasebeta

← #29702251

Subgroups — #29702251

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839957C211yesmaximal, characteristic
2C2×C4×#11758199C212yesmaximal, complemented
3#21448883C211yesmaximal, characteristic, complemented
4#21446624C211yesmaximal, characteristic, complemented
5#21446575C211yesmaximal, characteristic, complemented
6#21446501C212yesmaximal, complemented
7#21446167C212yesmaximal, complemented
8#21446065C212yesmaximal, complemented
9C4⋊C4×#92418C211yesmaximal, characteristic, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244C2212yes—
15C2×#14519739—21no—
16#16839957C418yescomplemented
17C4×#11758199—28no—
18C2×#14519710C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26C2×C4×#92418C2212yes—
27#16839664C2212yes—
28#16839663—21no—
29#16839640—22no—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2211yescharacteristic
32#16839524C2211yescharacteristic
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4×D4×C61C2212yes—
36C2×C4×D4×C61C2211yescharacteristic
37C22×C4⋊C4×C61—21no—
38C61×#389C2212yes—
39C61×#385—22no—
40C61×#384C2211yescharacteristic
41C61×#380—21no—
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×C4×D61—22no—
47C2×#11758199—24no—
48C2×#11758199—24no—
49C2×#11758199C2×C418yescomplemented
50C22×#92418C2311yescharacteristic
51C22×#92418C2312yes—
52C22×#92418C2311yescharacteristic
53#14519774—22no—
54#14519774C2×C414yescomplemented
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—
64#14519710—24no—
65C4⋊#92418C2×C414yescomplemented
66#14519698—22no—
67#14519698D414yescomplemented
68#14519697—24no—
69#14519697—42no—
70#14519697C2×C418yescomplemented
71C4×#92418—22no—
72C4×#92418—24no—
73#14519684—21no—
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61—21no—
76C23×C4×C61—22no—
77C4×D4×C61—24no—
78C4×D4×C61—22no—
79C4×D4×C61—42no—
80C4×D4×C61C2×C414yescomplemented
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—44no—
85C2×C4⋊C4×C61D411yescharacteristic
86C2×C4⋊C4×C61D411yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—22no—
91C2×C42×C61C2312yes—
92C2×C42×C61C2311yescharacteristic
93#95×C61—22no—
94#95×C61D412yes—
95C24×C61—21no—
96C23×D61—22no—
97C2×D4×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61C22×C412yes—
101C2×D4×C61C2×D412yes—
102C2×D4×C61#4112yes—
103C22×C4×C61—21no—
104C22×C4×C61—24no—
105C22×C4×C61—22no—
106C22×C4×C61—44no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—44no—
110C22×C4×C61—82no—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—21no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—24no—
123C2×D244—22no—
124C2×D244—24no—
125C2×D244—42no—
126C2×D244—42no—
127C2×D244—81no—
128C2×D244—81no—
129C2×D244C22×C414yes—
130C2×C4×D61—44no—
131C2×C4×D61—44no—
132C2×C4×D61—82no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—44no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—81no—
140C4⋊C4×C61D814yescomplemented
141C4⋊C4×C61C22×C412yes—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—44no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61C22×C412yes—
147C42×C61#4112yes—
148#11758199—216no—
149#11758198—44no—
150#11758198—44no—
151#11758198—82no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61C2×D412yes—
156C4×Dic61—42no—
157C2×#92418—22no—
158C2×#92418—21no—
159C2×#92418—22no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418C22×C414yes—
165C2×#92418C2×D412yes—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61C22×D411yescharacteristic
174C22×D61—24no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181D4×C61—22no—
182D4×C61—22no—
183D4×C61—22no—
184D4×C61C4×D414yescomplemented
185C2×C4×C61—22no—
186C2×C4×C61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—48no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—88no—
200C2×C4×C61C4×D412yes—
201C2×C4×C61C4×D412yes—
202C2×C4×C61C2×D814yes—
203C2×C4×C61C2×D812yes—
204C2×C4×C61#13512yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61H(5,𝔽2)11yescharacteristic
210C2×C4×C612−1+411yescharacteristic
211C2×Dic61—22no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217D244—24no—
218D244—24no—
219D244—42no—
220D244—42no—
221D244—48no—
222D244—84no—
223C4×D61—88no—
224#92418—24no—
225#92418—22no—
226#92418—41no—
227#22995—611nomaximal, Sylow
228C22×C61—24no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61C2×C4×D411yescharacteristic
236C22×C61D4211yescharacteristic
237C22×C61#54811yescharacteristic
238C22×C61C22×D811yescharacteristic
239C22×C61C2×#13511yescharacteristic
240C22×C61#57611yescharacteristic
241C22×C61#57711yescharacteristic
242C2×D61—44no—
243C2×D61—44no—
244C2×D61—44no—
245C2×D61—82no—
246C2×D61—82no—
247C2×D61—82no—
248C2×D61—82no—
249C4×C61—22no—
250C4×C61—24no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—82no—
254C4×C61C4×D814yescomplemented
255C4×C61C2×C4×D411yescharacteristic
256C4×C61#52011yescharacteristic
257C4×C61#54711yescharacteristic
258C4×C61#55311yescharacteristic
259Dic61—42no—
260Dic61—81no—
261C2×#459—611no—
262C2×C4×D8—612no—
263#2366—611no—
264D4×C4⋊C4D6111yescharacteristic, complemented
265#1957—611no—
266#1947—611no—
267#1934—612no—
268#1879—612no—
269#1862—612no—
270C8×C4⋊C4—611no—
271C4×#102—611no—
272C2×C61—44no—
273C2×C61#236111yescharacteristic
274C2×C61C2×C4×D811yescharacteristic
275C2×C61#294611yescharacteristic
276C2×C61D4×D811yescharacteristic
277C2×C61#328411yescharacteristic
278C2×C61#335811yescharacteristic
279C2×C61#336111yescharacteristic
280D61—84no—
281C22×D8—612no—
282C2×#127—611no—
283C2×C4×D4—612no—
284C2×C4×D4—1221no—
285C2×C4×D4C2×D6112yes—
286C2×C4×D4C2×D6111yescharacteristic
287C22×C4⋊C4—21no—
288#459—618no—
289C4×D8—1228no—
290C2×#107—612no—
291C2×C4⋊C8—611no—
292C2×#102—611no—
293C2×#102—612no—
294C2×#102—611no—
295C2×#102—612no—
296C2×#102—612no—
297C2×C4×C8—612no—
298C2×C4×C8—611no—
299#390—612no—
300#390—1221no—
301#389C2×D6112yes—
302#386—1222no—
303#385—22no—
304#384C2×D6111yescharacteristic
305#380—21no—
306C4×C4⋊C4—611no—
307C4×C4⋊C4C2×D6111yescharacteristic
308#340—611no—
309#340—612no—
310#340—611no—
311#336—612no—
312C61#2299511yescharacteristic, complemented, Sylow
313C22×D4—612no—
314C22×D4—1221no—
315C22×D4—1221no—
316C22×D4C22×D6111yescharacteristic
317C23×C4—21no—
318C23×C4—22no—
319C23×C4—1222no—
320C2×D8—618no—
321C2×D8—1224no—
322C2×D8—1224no—
323C22×C8—612no—
324C22×C8—611no—
325C22×C8—611no—
326#127—614no—
327#127—1222no—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—42no—
331C4×D4—1224no—
332C4×D4—2442no—
333C4×D4—2444no—
334C4×D4C4×D6114yescomplemented
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4C22⋊D6111yescharacteristic
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C4⋊C4C22×D6112yes—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C42—1221no—
351C2×C42C22×D6112yes—
352C2×C42C22×D6111yescharacteristic
353#107—1224no—
354C4⋊C8—614no—
355#102—614no—
356#102—618no—
357#102—1224no—
358#102—1222no—
359#102—2442no—
360C4×C8—1224no—
361C4×C8—1222no—
362#95—22no—
363#95—1221no—
364#95C22⋊D6112yes—
365C24—21no—
366C24—1222no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—614no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4—1224no—
377C2×D4—1224no—
378C2×D4—4881no—
379C2×D4—4881no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4C2×C4×D6112yes—
383C2×D4D4×D6112yes—
384C2×D4#1175822612yes—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—82no—
393C22×C4—1221no—
394C22×C4—1221no—
395C22×C4—4882no—
396C22×C4—2444no—
397C22×C4—2444no—
398C22×C4C2×C22⋊D6112yes—
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403D8—12216no—
404C2×C8—614no—
405C2×C8—612no—
406C2×C8—1222no—
407C2×C8—1221no—
408C2×C8—1222no—
409C2×C8—1221no—
410C2×C8—1221no—
411C2×C8—1221no—
412C2×C8—1221no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—81no—
420C4⋊C4—612no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4#1175819914yescomplemented
425C4⋊C4C2×C4×D6112yes—
426C22⋊C4—24no—
427C22⋊C4—44no—
428C22⋊C4—4882no—
429C22⋊C4—2444no—
430C22⋊C4—2444no—
431C42—22no—
432C42—22no—
433C42—2442no—
434C42C2×C4×D6112yes—
435C42#1175822212yes—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—1224no—
444C23—2442no—
445C23—2442no—
446C23—2442no—
447C23—2442no—
448C23—2442no—
449C23—2442no—
450C23C22×C22⋊D6111yescharacteristic
451D4—22no—
452D4—22no—
453D4—22no—
454D4—1224no—
455D4—1224no—
456D4—4884no—
457D4—2442no—
458D4—2448no—
459D4—2442no—
460D4#1451973914yescomplemented
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—48no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—88no—
476C2×C4—1222no—
477C2×C4—4888no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4C4×C22⋊D6112yes—
484C2×C4C4×C22⋊D6112yes—
485C2×C4C2×#1175819914yes—
486C2×C4C2×#1175819912yes—
487C2×C4#1451979712yes—
488C2×C4C22×C4×D6111yescharacteristic
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4C2×#1175822211yescharacteristic
491C2×C4C2×#1175822611yescharacteristic
492C2×C4#1451982211yescharacteristic
493C2×C4#1451982511yescharacteristic
494C8—1224no—
495C8—1222no—
496C8—2441no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—4882no—
505C22—4882no—
506C22—4882no—
507C22—4882no—
508C22—2444no—
509C22—2444no—
510C22—2444no—
511C22C2×C4×C22⋊D6111yescharacteristic
512C22D4×C22⋊D6111yescharacteristic
513C22#1684042311yescharacteristic
514C22C22×#1175819911yescharacteristic
515C22C2×#1451979711yescharacteristic
516C22#1684056511yescharacteristic
517C22#1684057011yescharacteristic
518C4—22no—
519C4—24no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—4881no—
524C4—2442no—
525C4C4×#1175819914yescomplemented
526C4C2×#1451973911yescharacteristic
527C4#1684024311yescharacteristic
528C4#1684041311yescharacteristic
529C4#1684046011yescharacteristic
530C2—44no—
531C2—4884no—
532C2#2144883511yescharacteristic
533C2C2×C4×#1175819911yescharacteristic
534C2#2145464711yescharacteristic
535C2D4×#1175819911yescharacteristic
536C2#2145754511yescharacteristic
537C2#2145845411yescharacteristic
538C2#2145847911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingD4×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
cosocleC22×C4×C61C24
hypercenterD4×C4⋊C4D61
hypocenterC61#22995
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758199C2semidirect
3. #21448883C2semidirect
4. #21446624C2semidirect
5. #21446575C2semidirect
6. #21446501C2semidirect
7. #21446167C2semidirect
8. #21446065C2semidirect
9. C4⋊C4×#92418C2semidirect
10. C4×#14519698C2semidirect
11. D4×C4⋊C4×C61C2semidirect
16. #16839957C4semidirect
49. C2×#11758199C2×C4semidirect
54. #14519774C2×C4semidirect
65. C4⋊#92418C2×C4semidirect
67. #14519698D4semidirect
70. #14519697C2×C4semidirect
80. C4×D4×C61C2×C4semidirect
140. C4⋊C4×C61D8semidirect
184. D4×C61C4×D4semidirect
254. C4×C61C4×D8semidirect
264. D4×C4⋊C4D61semidirect
312. C61#22995semidirect
334. C4×D4C4×D61semidirect
424. C4⋊C4#11758199semidirect
460. D4#14519739semidirect
525. C4C4×#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168399572
C2×C4×#117581992
#214488832
#214466242
#214465752
#214465012
#214461672
#214460652
C4⋊C4×#924182
C4×#145196982
D4×C4⋊C4×C612
#2299561