Finite Group Databasebeta

← #29689719

Subgroups — #29689719

Order 15616.

1386 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3#21448225C211yesmaximal, characteristic, complemented
4#21446778C212yesmaximal, complemented
5#21446623C212yesmaximal, complemented
6#21446140C212yesmaximal, complemented
7#21446138C212yesmaximal, complemented
8#21445830C211yesmaximal, characteristic, complemented
9#21445829C211yesmaximal, characteristic, complemented
10#21445826C211yesmaximal, characteristic, complemented
11C61×#1804C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840309—24no—
14#16840309C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519752C2211yescharacteristic
19C2×#14519750—21no—
20C2×#14519748C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×#14519735C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839814—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519699C2212yes—
28C2×#14519698C2212yes—
29C2×#14519697C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839628C2211yescharacteristic
34#16839621—21no—
35#16839621—21no—
36#16839613—21no—
37#16839613—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—48no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C4×D61—21no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2312yes—
59#14519776—22no—
60#14519776C2×C414yes—
61#14519773—22no—
62#14519773C2×C414yes—
63#14519763—28no—
64#14519752—22no—
65#14519752C2×C414yes—
66#14519750—28no—
67#14519748—22no—
68#14519748C2×C414yes—
69C4×D244—22no—
70C4×D244C2×C414yes—
71C4×C22⋊D61—28no—
72#14519735—42no—
73#14519735—42no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C42×D61C2×C414yes—
88C42×D61D414yes—
89C4⋊#92418—22no—
90#14519699—42no—
91#14519698—42no—
92#14519697—42no—
93#14519689—24no—
94C8×Dic61—24no—
95#14519684—21no—
96#14519684—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C22×C8×C61C2312yes—
100C4×D4×C61—22no—
101C4×D4×C61C2×C414yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—22no—
107#102×C61—42no—
108#95×C61—21no—
109#95×C61—21no—
110C24×C61—21no—
111C23×D61—21no—
112C23×D61C2×D411yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—21no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—44no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137#11758222—432no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—24no—
143C2×D244—21no—
144C2×D244—44no—
145C2×D244C22×C412yes—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—24no—
153C2×C4×D61—44no—
154C2×C4×D61C22⋊C414yes—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61C22×C412yes—
157C2×C4×D61C2×D412yes—
158C2×Dic122—24no—
159C2×Dic122—44no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61C22×C412yes—
167C22⋊C4×C61—24no—
168C42×C61C22×C412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—24no—
171#11758198—24no—
172#11758198—24no—
173#11758198—44no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C22×C412yes—
178#11758196—24no—
179#11758196—44no—
180#11758196—44no—
181C4×Dic61—44no—
182C4×Dic61C22⋊C414yes—
183C4×Dic61C22⋊C414yes—
184C4×Dic61C22×C412yes—
185C4×Dic61C2×D412yes—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C23×C61—22no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61C2×C22⋊C412yes—
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—41no—
208D4×C61—41no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—48no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C22⋊D61—48no—
226C22⋊D61—48no—
227C22⋊D61—48no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61#13514yes—
239C2×Dic61AGL(1,ℤ/8)14yes—
240D244—22no—
241D244—22no—
242D244—48no—
243D244—41no—
244D244—41no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—28no—
250C4×D61—28no—
251C4×D61—48no—
252Dic122—48no—
253C8×C61—42no—
254#92418—42no—
255#21863—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—82no—
260C22×C61—82no—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61#53511yescharacteristic
263C22×C61#53611yescharacteristic
264C22×C61C2×#13512yes—
265C22×C61C2×AGL(1,ℤ/8)12yes—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—82no—
275C2×D61—82no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—82no—
280C4×C61#52011yescharacteristic
281C4×C61#52011yescharacteristic
282C4×C61#53711yescharacteristic
283C4×C61#53711yescharacteristic
284Dic61—22no—
285Dic61—42no—
286Dic61—82no—
287Dic61#41614yes—
288Dic61#41814yes—
289C2×#532—611no—
290C2×#431—611no—
291#2285—611no—
292#1990—612no—
293#1957—612no—
294#1875—612no—
295#1875—612no—
296#1804—611no—
297#1804—611no—
298#1804—611no—
299#1804D6111yescharacteristic, complemented
300C2×C61—82no—
301C2×C61#231111yescharacteristic
302C2×C61C2×#41611yescharacteristic
303C2×C61C2×#41811yescharacteristic
304C2×C61#320712yes—
305C2×C61#322812yes—
306D61—42no—
307D61—82no—
308C22×#41—1221no—
309#532—618no—
310#532—1224no—
311C2×C4⋊Q8—611no—
312C2×#127—611no—
313C2×#124—1221no—
314C2×#121—611no—
315C2×#121—611no—
316C2×#121—1221no—
317C2×C4×D4—611no—
318C2×C4×D4—1221no—
319C2×C4×D4C2×D6111yescharacteristic
320C2×#117—612no—
321C22×C42—611no—
322#431—1224no—
323C2×C4⋊C8—611no—
324C2×C4⋊C8C2×D6111yescharacteristic
325C2×#102—612no—
326C2×#102—612no—
327C2×#102—612no—
328C2×#102C2×D6112yes—
329C2×C22⋊C8—612no—
330C2×C4×C8—612no—
331#384—612no—
332#382—611no—
333#382C2×D6111yescharacteristic
334#380—1221no—
335#380—1221no—
336C4×C4⋊C4—1221no—
337C4×C4⋊C4—1221no—
338#340—612no—
339#340—612no—
340#340—612no—
341#340C2×D6112yes—
342C61#2186311yescharacteristic, complemented, Sylow
343C2×#41—1228no—
344C2×#41—2448no—
345C22×Q8—1221no—
346C22×D4—611no—
347C22×D4—1221no—
348C22×D4—1221no—
349C22×D4—1221no—
350C22×D4—1221no—
351C22×D4C22×D6111yescharacteristic
352C23×C4—21no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—1221no—
357C22×C8—612no—
358C22×C8C22×D6112yes—
359C4⋊Q8—614no—
360C4⋊Q8—1222no—
361#127—614no—
362#127—1222no—
363#124—1228no—
364#121—614no—
365#121—614no—
366#121—1228no—
367#121—1222no—
368#121—1222no—
369C4×D4—22no—
370C4×D4—614no—
371C4×D4—1228no—
372C4×D4—1222no—
373C4×D4C4×D6114yes—
374#117—2442no—
375#117—2442no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4C22×D6112yes—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1222no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—614no—
392C2×C42—614no—
393C2×C42—1222no—
394C2×C42C22×D6111yescharacteristic
395C4⋊C8—22no—
396C4⋊C8—1222no—
397#102—42no—
398#102—2442no—
399#102—2442no—
400#102—2442no—
401C22⋊C8—1224no—
402C4×C8—1224no—
403#95—21no—
404#95—21no—
405#95—1221no—
406#95—1221no—
407C24—21no—
408C24—611no—
409C24—1221no—
410#41—24432no—
411C2×Q8—1224no—
412C2×Q8—2444no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—612no—
419C2×D4—1224no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—1224no—
423C2×D4—1221no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—2444no—
429C2×D4—2444no—
430C2×D4—2444no—
431C2×D4—2444no—
432C2×D4C2×C4×D6112yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—2444no—
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1222no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—612no—
467C4⋊C4—1224no—
468C4⋊C4—1221no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—2444no—
472C4⋊C4—2444no—
473C4⋊C4C2×C4×D6112yes—
474C22⋊C4—24no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C22⋊C4—2444no—
479C42—612no—
480C42—612no—
481C42—614no—
482C42—614no—
483C42—2444no—
484C42C2×C4×D6112yes—
485C42D4×D6112yes—
486C23—22no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—612no—
492C23—1222no—
493C23—1222no—
494C23—1221no—
495C23—1221no—
496C23—1222no—
497C23—2442no—
498C23—2442no—
499C23—2441no—
500C23—2441no—
501C23C2×D4×D6111yescharacteristic
502Q8—2448no—
503D4—22no—
504D4—22no—
505D4—41no—
506D4—41no—
507D4—1222no—
508D4—1222no—
509D4—2448no—
510D4—2448no—
511D4—2448no—
512D4—2448no—
513D4—2441no—
514D4—2441no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—614no—
526C2×C4—614no—
527C2×C4—612no—
528C2×C4—1228no—
529C2×C4—1228no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—2448no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4C22⋊C4×D6112yes—
543C2×C4C22⋊C4×D6112yes—
544C2×C4C22×C4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4#1451982211yescharacteristic
547C2×C4#1451982311yescharacteristic
548C2×C4#1451982411yescharacteristic
549C2×C4#1451982511yescharacteristic
550C8—42no—
551C8—2442no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—82no—
556C22—82no—
557C22—1222no—
558C22—1222no—
559C22—1222no—
560C22—4882no—
561C22—4882no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22C2×C22⋊C4×D6111yescharacteristic
568C22#1684032811yescharacteristic
569C22#1684033511yescharacteristic
570C22#135×D6112yes—
571C22AGL(1,ℤ/8)×D6112yes—
572C4—22no—
573C4—22no—
574C4—42no—
575C4—82no—
576C4—614no—
577C4—614no—
578C4—1222no—
579C4—4882no—
580C4—2442no—
581C4#1684024211yescharacteristic
582C4#1684024311yescharacteristic
583C4#1684034111yescharacteristic
584C4#1684034211yescharacteristic
585C2—82no—
586C2—4882no—
587C2—2442no—
588C2#2144840811yescharacteristic
589C2#416×D6111yescharacteristic
590C2#418×D6111yescharacteristic
591C2#2145675112yes—
592C2#2145695412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1804C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenter#1804D61
hypocenterC61#21863
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#16839814C2semidirect
3. #21448225C2semidirect
4. #21446778C2semidirect
5. #21446623C2semidirect
6. #21446140C2semidirect
7. #21446138C2semidirect
8. #21445830C2semidirect
9. #21445829C2semidirect
10. #21445826C2semidirect
11. C61×#1804C2semidirect
14. #16840309C4semidirect
299. #1804D61semidirect
342. C61#21863semidirect

Maximal subgroups

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

subgroupindex
C2×#168403092
C2×#168398142
#214482252
#214467782
#214466232
#214461402
#214461382
#214458302
#214458292
#214458262
C61×#18042
#2186361