Finite Group Databasebeta

← #29713966

Subgroups — #29713966

Order 15616.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840054C211yesmaximal, characteristic, complemented
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3#21450741C211yesmaximal, characteristic
4#21447019C212yesmaximal, complemented
5#21446827C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic
8#21446126C212yesmaximal, complemented
9#21446007C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C61×#554C211yesmaximal, characteristic
12C22×#11758202C2211yescharacteristic
13C2×#14519756C2212yes—
14C2×C4×Dic122C2211yescharacteristic
15#16840054—24no—
16C4×#11758202—24no—
17C2×#14519710C2212yes—
18C2×#14519710C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2212yes—
22C2×#14519700C2212yes—
23C2×#14519700C2211yescharacteristic
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×C4×#92418C2211yescharacteristic
27#16839685—22no—
28#16839665—21no—
29#16839658—21no—
30#16839632C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#554—24no—
36C61×#554—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#125×C61—21no—
39C2×C22⋊Q8×C61—22no—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C2×#11758202—22no—
47C2×#11758202—22no—
48C2×#11758202—22no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2312yes—
51C22×#92418C2311yescharacteristic
52#14519756—24no—
53#14519756—42no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—21no—
61C2×#11758196—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C4×Dic61—21no—
65#14519710—24no—
66#14519710—22no—
67C4⋊#92418—22no—
68#14519701—22no—
69#14519701—42no—
70#14519700—24no—
71#14519700—22no—
72C22⋊#92418—24no—
73C22⋊#92418—42no—
74C4×#92418—22no—
75#14519684—22no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61D411yescharacteristic
79C23×C4×C61—22no—
80C23×C4×C61C2312yes—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61D414yes—
83#125×C61—28no—
84C22⋊Q8×C61—216no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yes—
89C4×D4×C61—28no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—24no—
97C2×C4⋊C4×C61—24no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61D412yes—
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D412yes—
106C2×Q8×C61—24no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—81no—
113C2×Q8×C61—81no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61Q1614yescomplemented
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—28no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—82no—
129C22×C4×C61—82no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122#4112yes—
143C4⋊C4×C61—28no—
144C4⋊C4×C61—28no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D414yes—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—24no—
154C42×C61—24no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—44no—
158C22⋊Dic61—82no—
159#11758202—28no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—42no—
165C4⋊Dic61—81no—
166#11758196—44no—
167#11758196—42no—
168C4×Dic61—42no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418#4112yes—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61C2×Q1614yes—
185C23×C61C22×D411yescharacteristic
186Q8×C61—22no—
187Q8×C61—22no—
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—84no—
191D4×C61—28no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—88no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12012yes—
213C2×C4×C61C2×Q1612yes—
214C2×C4×C61#13712yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)11yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230Dic122—22no—
231Dic122—22no—
232Dic122—41no—
233Dic122—41no—
234#92418—22no—
235#92418—41no—
236#92418—42no—
237#24122—611nomaximal, Sylow
238C22×C61—24no—
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61C2×#12011yescharacteristic
246C22×C61#54611yescharacteristic
247C22×C61#54711yescharacteristic
248C22×C61C22×Q1611yescharacteristic
249C22×C61C2×#13711yescharacteristic
250C22×C61#57611yescharacteristic
251C22×C61#57711yescharacteristic
252C4×C61—22no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—42no—
256C4×C61—44no—
257C4×C61#45114yes—
258C4×C61#52511yescharacteristic
259C4×C61#53711yescharacteristic
260C4×C61#54811yescharacteristic
261C4×C61#55011yescharacteristic
262Dic61—42no—
263Dic61—82no—
264Dic61—81no—
265C2×#554D6111yescharacteristic
266C2×#477—611no—
267C2×C4×Q16—611no—
268#2538—611no—
269#2036—612no—
270#2001—612no—
271#1990—611no—
272#1954—611no—
273#1872—612no—
274#1852—612no—
275#1803—611no—
276C2×C61—44no—
277C2×C61#240711yescharacteristic
278C2×C61C2×#45111yescharacteristic
279C2×C61#301211yescharacteristic
280C2×C61#329911yescharacteristic
281C2×C61#333311yescharacteristic
282C2×C61#333811yescharacteristic
283C2×C61#335611yescharacteristic
284C22×Q16—611no—
285#554—24no—
286#554—24no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×#125—21no—
289C2×C22⋊Q8—22no—
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8C2×D6112yes—
292C2×C4×Q8—611no—
293C2×C4×Q8C2×D6111yescharacteristic
294C2×C4×D4—21no—
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4C2×D6112yes—
297#477—1224no—
298C4×Q16—1224no—
299C2×#107—612no—
300C2×#107—611no—
301C2×C4⋊C8—611no—
302C2×#103—611no—
303C2×#103—612no—
304C2×#103—611no—
305C2×#103—612no—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×C4×C8—611no—
309#396—1222no—
310#390—1221no—
311#389—1221no—
312#384—611no—
313#380—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317C61#2412211yescharacteristic, complemented, Sylow
318C22×Q8—21no—
319C22×Q8—611no—
320C22×Q8C22×D6111yescharacteristic
321C22×D4C22⋊D6111yescharacteristic
322C23×C4—22no—
323C23×C4C22×D6112yes—
324C2×Q16—1222no—
325C2×Q16—1222no—
326C2×Q16—1222no—
327C22×C8—611no—
328C22×C8—611no—
329C22×C8—612no—
330C4⋊Q8—22no—
331C4⋊Q8C22⋊D6114yes—
332#125—28no—
333C22⋊Q8—216no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—1224no—
337C22⋊Q8—2442no—
338C4×Q8—22no—
339C4×Q8—1222no—
340C4×Q8—1222no—
341C4×Q8C22⋊D6114yes—
342C4×D4—28no—
343C4×D4—22no—
344C4×D4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4C22⋊D6111yescharacteristic
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C4⋊C4C22×D6112yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4C22⋊D6112yes—
364C2×C42—21no—
365C2×C42—1221no—
366C2×C42C22×D6111yescharacteristic
367#107—1222no—
368#107—1224no—
369C4⋊C8—1222no—
370#103—1222no—
371#103—1222no—
372#103—1224no—
373#103—2442no—
374C22⋊C8—1224no—
375C22⋊C8—2442no—
376C4×C8—1222no—
377#95—1222no—
378C24C2×C22⋊D6112yes—
379C2×Q8—21no—
380C2×Q8—24no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—612no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8C2×C22⋊D6112yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4#1175820214yescomplemented
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1222no—
412C22×C4C2×C22⋊D6112yes—
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C23×D6111yescharacteristic
416Q16—1228no—
417C2×C8—612no—
418C2×C8—1221no—
419C2×C8—1221no—
420C2×C8—1221no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8—1221no—
424C2×C8—1222no—
425C2×C8—1221no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—4881no—
438C4⋊C4—2444no—
439C4⋊C4—2442no—
440C4⋊C4—2442no—
441C4⋊C4C2×C22⋊D6112yes—
442C4⋊C4C2×C22⋊D6114yes—
443C22⋊C4—28no—
444C22⋊C4—24no—
445C22⋊C4—4882no—
446C22⋊C4—2444no—
447C42—24no—
448C42—2442no—
449C42C2×C22⋊D6112yes—
450C42C2×C22⋊D6112yes—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23C2×#1175820214yes—
458C23C22×C22⋊D6111yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—41no—
462Q8—41no—
463Q8—84no—
464Q8—1222no—
465Q8—1222no—
466Q8—2441no—
467Q8—2441no—
468D4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—88no—
488C2×C4—1222no—
489C2×C4—2441no—
490C2×C4—2442no—
491C2×C4—2441no—
492C2×C4—2442no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4#1451974612yes—
498C2×C4#1451974612yes—
499C2×C4C2×#1175820212yes—
500C2×C4#1451980812yes—
501C2×C4C22×C22⋊D6111yescharacteristic
502C2×C4C22×C22⋊D6111yescharacteristic
503C2×C4#41×D6111yescharacteristic
504C2×C4#41×D6111yescharacteristic
505C2×C4#1451982311yescharacteristic
506C2×C4#1451982411yescharacteristic
507C8—1222no—
508C8—2441no—
509C8—2442no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#1451974611yescharacteristic
518C22#1684040011yescharacteristic
519C22#1684041111yescharacteristic
520C22C22×#1175820211yescharacteristic
521C22C2×#1451980811yescharacteristic
522C22#1684056511yescharacteristic
523C22#1684057311yescharacteristic
524C4—22no—
525C4—42no—
526C4—44no—
527C4—42no—
528C4—42no—
529C4—4881no—
530C4—4882no—
531C4—2442no—
532C4#1683991814yes—
533C4#1684026911yescharacteristic
534C4#1684034011yescharacteristic
535C4#1684041911yescharacteristic
536C4#1684043111yescharacteristic
537C2—44no—
538C2#2144915811yescharacteristic
539C2C2×#1683991811yescharacteristic
540C2#2145512311yescharacteristic
541C2#2145768911yescharacteristic
542C2#2145815111yescharacteristic
543C2#2145820711yescharacteristic
544C2#2145843411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×#554D61
hypocenterC61#24122
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16840054C2semidirect
2. C2×C4×#11758202C2semidirect
4. #21447019C2semidirect
5. #21446827C2semidirect
6. #21446781C2semidirect
8. #21446126C2semidirect
9. #21446007C2semidirect
10. #21445824C2semidirect
117. C2×D4×C61Q16semidirect
317. C61#24122semidirect
396. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
C2×#168400542
C2×C4×#117582022
#214507412
#214470192
#214468272
#214467812
#214466142
#214461262
#214460072
#214458242
C2×C61×#5542
#2412261