Finite Group Databasebeta

← #29689763

Subgroups — #29689763

Order 15616.

1366 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic, complemented
3#21448222C211yesmaximal, characteristic, complemented
4#21446790C212yesmaximal, complemented
5#21446641C212yesmaximal, complemented
6#21446142C211yesmaximal, characteristic, complemented
7#21446141C211yesmaximal, characteristic, complemented
8#21446139C211yesmaximal, characteristic, complemented
9#21445818C212yesmaximal, complemented
10#21445817C212yesmaximal, complemented
11C61×#1875C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13#16840307—24no—
14#16840307C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519751C2211yescharacteristic
19C2×#14519749—21no—
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519737—21no—
23C2×#14519736C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25#16839816—24no—
26C2×#14519706C2212yes—
27C2×#14519699C2212yes—
28C2×#14519698C2212yes—
29C2×#14519697C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yes—
32#16839631C2212yes—
33#16839627C2212yes—
34#16839620—21no—
35#16839620—21no—
36C4×#11758196—21no—
37C4×#11758196—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#121×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—28no—
47C2×#11758224—48no—
48C22×Dic122—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56C22×C4×D61—21no—
57C22×#92418C2312yes—
58#14519776—22no—
59#14519776C2×C414yes—
60#14519773—22no—
61#14519773C2×C414yes—
62#14519763—28no—
63#14519751—22no—
64#14519751C2×C414yes—
65#14519749—28no—
66D4×Dic61—22no—
67D4×Dic61C2×C414yes—
68#14519739—22no—
69#14519739C2×C414yes—
70#14519737—28no—
71#14519736—42no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2×C414yes—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D414yes—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519706—24no—
89#14519699—42no—
90#14519698—42no—
91#14519697—42no—
92C22⋊#92418—22no—
93#14519687—24no—
94#14519684—22no—
95#14519684—22no—
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100#121×C61—22no—
101#121×C61C2×C414yes—
102#117×C61—42no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108#102×C61—42no—
109C22⋊C8×C61—22no—
110C24×C61—21no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C2×D4×C61—24no—
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—44no—
118C2×D4×C61C22×C412yes—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—44no—
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—24no—
136C22×Dic61—22no—
137C22×Dic61—42no—
138C22×Dic61—44no—
139C22×Dic61C22⋊C414yes—
140C22×Dic61C22⋊C414yes—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145#11758224—432no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—21no—
149C2×D244C22×C412yes—
150C2×C4×D61—24no—
151C2×C4×D61—44no—
152C2×Dic122—24no—
153C2×Dic122—44no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C22×C412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—42no—
163C42×C61—42no—
164C22⋊Dic61—24no—
165C22⋊Dic61—42no—
166#11758198—24no—
167#11758198—24no—
168C4⋊Dic61—21no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61C22×C412yes—
172#11758196—24no—
173#11758196—44no—
174#11758196—44no—
175C4×Dic61—42no—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22×C412yes—
179C4×Dic61C22×C412yes—
180C4×Dic61C2×D412yes—
181C4×Dic61C2×D412yes—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61C2×C22⋊C412yes—
195C23×C61C22×D411yescharacteristic
196C22×D61—22no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—42no—
200C22×D61—42no—
201D4×C61—24no—
202D4×C61—48no—
203D4×C61—42no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61H(5,𝔽2)12yes—
217C2×C4×C612−1+412yes—
218C22⋊D61—48no—
219C22⋊D61—48no—
220C22⋊D61—48no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—28no—
225C2×Dic61—22no—
226C2×Dic61—44no—
227C2×Dic61—42no—
228C2×Dic61—48no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61AGL(1,ℤ/8)14yes—
237D244—24no—
238D244—42no—
239C4×D61—48no—
240Dic122—48no—
241C8×C61—42no—
242#92418—42no—
243#21869—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61C22×C22⋊C411yescharacteristic
255C22×C61#53511yescharacteristic
256C22×C61#53611yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)12yes—
258C22×C61#57512yes—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—82no—
263C2×D61—82no—
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—82no—
267C4×C61#52012yes—
268C4×C61#53712yes—
269Dic61—22no—
270Dic61—22no—
271Dic61—22no—
272Dic61—42no—
273Dic61—82no—
274Dic61#41814yes—
275C2×#532—611no—
276C2×#432—611no—
277#2285—611no—
278#1993—612no—
279#1961—612no—
280#1875—611no—
281#1875—611no—
282#1875—611no—
283#1875D6111yescharacteristic, complemented
284#1802—612no—
285#1802—612no—
286C2×C61—42no—
287C2×C61—82no—
288C2×C61#231111yescharacteristic
289C2×C61C2×#41811yescharacteristic
290C2×C61#289611yescharacteristic
291C2×C61#321412yes—
292C2×C61#323512yes—
293D61—82no—
294C22×#41—1221no—
295#532—618no—
296#532—1224no—
297C2×C4⋊Q8—611no—
298C2×#127—611no—
299C2×#124—1221no—
300C2×#121—611no—
301C2×#121—1221no—
302C2×#121C2×D6111yescharacteristic
303C2×C4×D4—611no—
304C2×C4×D4—611no—
305C2×C4×D4—1221no—
306C2×#117—611no—
307C2×#117C2×D6111yescharacteristic
308C22×C42—611no—
309#432—1224no—
310C2×C4⋊C8—612no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#102—612no—
314C2×#102C2×D6112yes—
315C2×C22⋊C8—611no—
316C2×C22⋊C8C2×D6111yescharacteristic
317C2×#97—612no—
318#384—612no—
319#382—612no—
320#380—1221no—
321#380—1221no—
322C4×C4⋊C4—1221no—
323C4×C4⋊C4—1221no—
324#340—612no—
325#340—612no—
326#340—612no—
327#340C2×D6112yes—
328C61#2186911yescharacteristic, complemented, Sylow
329C2×#41—1228no—
330C2×#41—2448no—
331C22×Q8—1221no—
332C22×D4—21no—
333C22×D4—611no—
334C22×D4—1221no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4C22×D6111yescharacteristic
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1221no—
341C23×C4—1221no—
342C23×C4C22×D6111yescharacteristic
343C22×C8—612no—
344C22×C8C22×D6112yes—
345C4⋊Q8—614no—
346C4⋊Q8—1222no—
347#127—614no—
348#127—1222no—
349#124—1228no—
350#121—22no—
351#121—614no—
352#121—1228no—
353#121—1222no—
354#121C4×D6114yes—
355C4×D4—614no—
356C4×D4—614no—
357C4×D4—1228no—
358C4×D4—1222no—
359C4×D4—1222no—
360#117—42no—
361#117—2442no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4C22×D6112yes—
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C42—21no—
376C2×C42—614no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—614no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—1221no—
383C4⋊C8—1224no—
384#102—42no—
385#102—2442no—
386#102—2442no—
387#102—2442no—
388C22⋊C8—22no—
389C22⋊C8—1222no—
390#97—1224no—
391#95—1222no—
392#95—1222no—
393C24—21no—
394C24—1221no—
395C24D4×D6111yescharacteristic
396#41—24432no—
397C2×Q8—1224no—
398C2×Q8—2444no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—44no—
404C2×D4—612no—
405C2×D4—1224no—
406C2×D4—1224no—
407C2×D4—1224no—
408C2×D4—1221no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—2444no—
412C2×D4—2444no—
413C2×D4—2444no—
414C2×D4C2×C4×D6112yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—614no—
424C22×C4—614no—
425C22×C4—1224no—
426C22×C4—1224no—
427C22×C4—1224no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—2444no—
431C22×C4—2444no—
432C22×C4—2442no—
433C22×C4C2×C4×D6112yes—
434C22×C4D4×D6112yes—
435C22×C4D4×D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1222no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—612no—
447C4⋊C4—1224no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—2444no—
452C4⋊C4—2444no—
453C4⋊C4C2×C4×D6112yes—
454C22⋊C4—24no—
455C22⋊C4—42no—
456C22⋊C4—1224no—
457C22⋊C4—1224no—
458C22⋊C4—1224no—
459C22⋊C4—2442no—
460C42—42no—
461C42—612no—
462C42—612no—
463C42—612no—
464C42—612no—
465C42—614no—
466C42—614no—
467C42—2442no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—1222no—
478C23—2442no—
479C23—2442no—
480C23—2441no—
481C23—2441no—
482C23C22⋊C4×D6112yes—
483C23C2×D4×D6111yescharacteristic
484Q8—2448no—
485D4—24no—
486D4—48no—
487D4—42no—
488D4—1224no—
489D4—2448no—
490D4—2448no—
491D4—2448no—
492D4—2442no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—612no—
503C2×C4—614no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—1228no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—2448no—
513C2×C4—2448no—
514C2×C4—2444no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4C22⋊C4×D6112yes—
520C2×C4C22×C4×D6111yescharacteristic
521C2×C4C2×D4×D6111yescharacteristic
522C2×C4#1451982212yes—
523C2×C4#1451982412yes—
524C8—42no—
525C8—2442no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—4882no—
537C22—4882no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22C2×C22⋊C4×D6111yescharacteristic
542C22#1684032711yescharacteristic
543C22#1684033411yescharacteristic
544C22AGL(1,ℤ/8)×D6112yes—
545C22#1684055912yes—
546C4—22no—
547C4—42no—
548C4—82no—
549C4—614no—
550C4—1222no—
551C4—1222no—
552C4—1222no—
553C4—4882no—
554C4—2442no—
555C4#1684024112yes—
556C4#1684033912yes—
557C2—42no—
558C2—82no—
559C2—4882no—
560C2#2144840611yescharacteristic
561C2#418×D6111yescharacteristic
562C2#2145433411yescharacteristic
563C2#2145681112yes—
564C2#2145701412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1875C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
cosocleC22×C4×C61C24
hypercenter#1875D61
hypocenterC61#21869
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. C2×#16839816C2semidirect
3. #21448222C2semidirect
4. #21446790C2semidirect
5. #21446641C2semidirect
6. #21446142C2semidirect
7. #21446141C2semidirect
8. #21446139C2semidirect
9. #21445818C2semidirect
10. #21445817C2semidirect
11. C61×#1875C2semidirect
14. #16840307C4semidirect
283. #1875D61semidirect
328. C61#21869semidirect

Maximal subgroups

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

subgroupindex
C2×#168403072
C2×#168398162
#214482222
#214467902
#214466412
#214461422
#214461412
#214461392
#214458182
#214458172
C61×#18752
#2186961