Finite Group Databasebeta

← #30620155

Subgroups — #30620155

Order 15616.

1410 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#11758217C212yesmaximal, complemented
2C2×#16840749C212yesmaximal, complemented
3#21461962C212yesmaximal, complemented
4#21461846C214yesmaximal, complemented
5#21461683C211yesmaximal, characteristic, complemented
6#21461656C212yesmaximal, complemented
7C4×#14519837C211yesmaximal, characteristic, complemented
8C2×#16840412C211yesmaximal, characteristic
9C22×#11758217C2214yes—
10#16840797—22no—
11#16840793—24no—
12D4×#92420—22no—
13#16840765C2212yes—
14#16840749—28no—
15C2×#14519838C2214yes—
16C2×#14519838C2214yes—
17C2×#14519837C2211yescharacteristic
18C2×#14519837C2212yes—
19C2×#14519837C2211yescharacteristic
20#16840732C2214yes—
21C4×#11758217C2212yes—
22#16840642C2212yes—
23#16840642C2212yes—
24C22×#11758224C2212yes—
25#16840412—22no—
26#16840412—24no—
27#16840412—22no—
28C2×#14519776C2211yescharacteristic
29C2×#14519759C412yes—
30C2×#14519758C412yes—
31C2×#14519754C2212yes—
32C2×#14519748C411yescharacteristic
33C2×D4×Dic61C2211yescharacteristic
34C2×C4×C22⋊D61C2211yescharacteristic
35C22×C4⋊Dic61C411yescharacteristic
36C2×C4⋊C4×D61C411yescharacteristic
37C2×#121×C61C411yescharacteristic
38C2×#11758217—24no—
39C2×#11758217—28no—
40C2×#11758217—44no—
41C2×#11758217C2312yes—
42C2×#11758217C2314yes—
43C2×#11758217C2312yes—
44C22×#92420—24no—
45#14519844—22no—
46C4⋊#92420—22no—
47#14519838—28no—
48#14519838—44no—
49#14519837—24no—
50#14519837—22no—
51#14519837D414yescomplemented
52C22⋊#92420—24no—
53C22⋊#92420—24no—
54#14519832—22no—
55C4×#92420—22no—
56C2×#11758224—24no—
57C2×#11758224—24no—
58C2×#11758224—24no—
59C2×#11758224—24no—
60C2×#11758224—44no—
61C22×Dic122C2312yes—
62C22×C22⋊D61C2×C412yes—
63C22×C22⋊D61C2311yescharacteristic
64C23×Dic61C2×C412yes—
65C23×Dic61C2312yes—
66C22×C4×D61C2×C412yes—
67C22×C4×D61C2×C411yescharacteristic
68#14519776—22no—
69#14519776—22no—
70#14519759—24no—
71#14519759—24no—
72#14519758—24no—
73#14519758—24no—
74#14519754—24no—
75#14519754—24no—
76#14519748—22no—
77#14519748—22no—
78D4×Dic61D412yes—
79D4×Dic61D414yes—
80D4×Dic61D412yes—
81C4×C22⋊D61—22no—
82C4×C22⋊D61—22no—
83C2×C4⋊Dic61—22no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×C4⋊Dic61C2×C412yes—
88C2×C4⋊Dic61C2×C411yescharacteristic
89C2×#11758196C2×C412yes—
90C2×#11758196C2×C412yes—
91C2×#11758196C2311yescharacteristic
92C4⋊C4×D61—22no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C2×C22⋊Dic61C2×C411yescharacteristic
96C2×C22⋊Dic61C2×C412yes—
97C2×C22⋊Dic61C2×C412yes—
98C2×#11758198C2×C411yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C22×D4×C61C2×C412yes—
101C22×D4×C61C2×C411yescharacteristic
102C23×C4×C61C2×C411yescharacteristic
103#121×C61—22no—
104#121×C61—22no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C22⋊C4×C61C2×C412yes—
107C24×C61C22×C412yes—
108C24×C61C22×C411yescharacteristic
109C23×D61C22×C411yescharacteristic
110C2×D4×C61—24no—
111C2×D4×C61—24no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C2×D4×C61C22⋊C412yes—
115C2×D4×C61C22⋊C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—42no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61C2×D412yes—
128C2×C22⋊D61#4112yes—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—24no—
132C22×Dic61—24no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—42no—
136C22×Dic61C22×C412yes—
137C22×Dic61C22×C412yes—
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C22×Dic61C2×D414yes—
141C22×Dic61C2×D414yes—
142C22×Dic61C2411yescharacteristic
143#11758224—48no—
144#11758224—48no—
145#11758224—48no—
146#11758224—48no—
147C2×C4×D61—24no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×Dic122—22no—
155C2×Dic122—24no—
156C2×Dic122—22no—
157C2×Dic122—24no—
158#11758217—22no—
159#11758217—24no—
160#11758217—22no—
161#11758217—22no—
162#11758217—24no—
163#11758217—48no—
164#11758217C2×D414yes—
165C2×#92420—22no—
166C2×#92420—22no—
167C2×#92420—24no—
168C2×#92420—48no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C22⋊C4×C61—24no—
172C22⋊Dic61—24no—
173C22⋊Dic61—22no—
174C22⋊Dic61C22⋊C414yes—
175C22⋊Dic61C22⋊C414yes—
176#11758198—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61C22⋊C412yes—
182C4⋊Dic61C22⋊C411yescharacteristic
183C4⋊Dic61C22⋊C411yescharacteristic
184#11758196—22no—
185#11758196—22no—
186#11758196—24no—
187#11758196—22no—
188C4×Dic61C2×D412yes—
189C4×Dic61C2×D412yes—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61C2×C22⋊C414yes—
199C23×C61C4×D412yes—
200C23×C61C23×C411yescharacteristic
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61C4×D412yes—
206D4×C61—24no—
207D4×C61—42no—
208D4×C61—48no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61C2×C22⋊C411yescharacteristic
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C22⋊D61—24no—
220C22⋊D61—24no—
221C22⋊D61—48no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—24no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—21no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—41no—
233C2×Dic61—44no—
234C2×Dic61—44no—
235C2×Dic61—41no—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C22×D411yescharacteristic
240C2×Dic61C22×D412yes—
241C2×Dic61C2×#4111yescharacteristic
242C2×Dic61H(5,𝔽2)12yes—
243C4×D61—44no—
244C4×D61—44no—
245C4×D61—42no—
246C4×D61—42no—
247Dic122—44no—
248Dic122—44no—
249#92420—24no—
250#92420—44no—
251#92420—42no—
252#22411—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—24no—
256C22×C61—22no—
257C22×C61—24no—
258C22×C61—24no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61C2×C4×D411yescharacteristic
267C22×C61#51811yescharacteristic
268C2×D61—22no—
269C2×D61—42no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—42no—
273C4×C61—21no—
274C4×C61—42no—
275C4×C61—41no—
276C4×C61#41312yescomplemented
277Dic61—22no—
278Dic61—22no—
279Dic61—22no—
280Dic61—42no—
281Dic61—41no—
282Dic61D4212yes—
283Dic61#54612yes—
284C2×#547—611no—
285D4×M16—612no—
286C2×#413—612no—
287#1966—612no—
288#1901—614no—
289#1800—611no—
290#1785—612no—
291C4×#100—611no—
292C2×C61—21no—
293C2×C61—21no—
294C2×C61—44no—
295C2×C61—42no—
296C2×C61D4×C22⋊C411yescharacteristic
297C2×C61C2×#41311yescharacteristic
298C2×C61#288911yescharacteristic
299D61—42no—
300C22×#41—612no—
301C22×M16—614no—
302#547—1222no—
303#547—1224no—
304#547—1222no—
305C2×C4⋊Q8—611no—
306C2×#123—612no—
307C2×#123—612no—
308C2×C22⋊Q8—612no—
309C2×#121—611no—
310C2×#121#692311yescharacteristic
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C22×C4⋊C4—611no—
314C22×C4⋊C4—611no—
315#436—1222no—
316#435—1224no—
317C8×D4—1222no—
318#423—612no—
319#413—1228no—
320C2×#101—614no—
321C2×#101—614no—
322C2×#100—611no—
323C2×#100—612no—
324C2×#100—611no—
325#407—614no—
326C4×M16—612no—
327#343—612no—
328#343—612no—
329C61#2241111yescharacteristic, complemented, Sylow
330C2×#41—1224no—
331C2×#41—1224no—
332C2×#41—1224no—
333C2×#41—1224no—
334C2×#41—2444no—
335C22×Q8—612no—
336C22×D4—611no—
337C22×D4—612no—
338C22×D4C2×#692312yes—
339C22×D4C2×#692311yescharacteristic
340C23×C4—612no—
341C23×C4—612no—
342C23×C4—612no—
343C23×C4—611no—
344C23×C4C2×#692311yescharacteristic
345C2×M16—612no—
346C2×M16—614no—
347C2×M16—612no—
348C2×M16—1224no—
349C2×M16—1228no—
350C2×M16—2444no—
351C22×C8—1224no—
352C4⋊Q8—1222no—
353C4⋊Q8—1222no—
354#123—1224no—
355#123—1224no—
356#123—1224no—
357#123—1224no—
358C22⋊Q8—1224no—
359C22⋊Q8—1224no—
360#121—22no—
361#121—22no—
362#121—1222no—
363#121—1222no—
364C4×D4—612no—
365C4×D4—614no—
366C4×D4—612no—
367C4×D4—1222no—
368C4×D4—1222no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4C2×#692311yescharacteristic
382C2×C22⋊C4—612no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4C2×#692312yes—
387C2×C42—611no—
388C4⋊C8—1222no—
389C4⋊C8—1222no—
390#101—1228no—
391#101—2444no—
392#100—614no—
393#100—1222no—
394#100—1224no—
395C22⋊C8—1224no—
396C22⋊C8—1224no—
397#97—1222no—
398C4×C8—1222no—
399C24—611no—
400C24C22×#692312yes—
401C24C22×#692311yescharacteristic
402#41—2448no—
403#41—2448no—
404#41—2448no—
405#41—2448no—
406C2×Q8—1222no—
407C2×Q8—1224no—
408C2×Q8—1224no—
409C2×Q8—1222no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—612no—
415C2×D4—612no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1224no—
419C2×D4—1224no—
420C2×D4—2442no—
421C2×D4C22⋊#692312yes—
422C2×D4C22⋊#692312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—611no—
427C22×C4—614no—
428C22×C4—614no—
429C22×C4—612no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—611no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1224no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1224no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4C22×#692312yes—
448C22×C4C22×#692311yescharacteristic
449C22×C4C22×#692311yescharacteristic
450M16—614no—
451M16—1222no—
452M16—1224no—
453M16—1222no—
454M16—1222no—
455M16—1224no—
456M16—2448no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1224no—
460C2×C8—2448no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—612no—
464C4⋊C4—611no—
465C4⋊C4—611no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1224no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C22⋊C4—24no—
475C22⋊C4—614no—
476C22⋊C4—614no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C22⋊C4—1224no—
480C42—612no—
481C42—612no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—612no—
491C23—1221no—
492C23—1221no—
493C23—2441no—
494C23—2441no—
495C23C2×C22⋊#692314yes—
496C23D4×#692312yes—
497C23C23×#692311yescharacteristic
498Q8—2444no—
499Q8—2444no—
500D4—24no—
501D4—42no—
502D4—48no—
503D4—1224no—
504D4—1224no—
505D4—2448no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—612no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1221no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—2442no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—2441no—
535C2×C4—2444no—
536C2×C4—2444no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4C2×C22⋊#692311yescharacteristic
540C2×C4C2×C22⋊#692311yescharacteristic
541C8—1224no—
542C8—2444no—
543C8—2442no—
544C22—21no—
545C22—21no—
546C22—24no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—82no—
555C22—82no—
556C22—1222no—
557C22—2442no—
558C22—2442no—
559C22—2441no—
560C22—2441no—
561C22C22×C22⋊#692311yescharacteristic
562C22C2×D4×#692311yescharacteristic
563C22#1684082911yescharacteristic
564C4—21no—
565C4—42no—
566C4—41no—
567C4—612no—
568C4—612no—
569C4—1222no—
570C4—1222no—
571C4—1222no—
572C4—2442no—
573C4—2441no—
574C4#1684074912yescomplemented
575C2—21no—
576C2—21no—
577C2—44no—
578C2—42no—
579C2—2442no—
580C2D4×C22⋊#692311yescharacteristic
581C2C2×#1684074911yescharacteristic
582C2#2146243111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#121×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840412C2
cosocleC22×Dic61C24
hypercenterC2×#121#6923
hypocenterC61#22411
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. D4×#11758217C2semidirect
2. C2×#16840749C2semidirect
3. #21461962C2semidirect
4. #21461846C2semidirect
5. #21461683C2semidirect
6. #21461656C2semidirect
7. C4×#14519837C2semidirect
51. #14519837D4semidirect
276. C4×C61#413semidirect
329. C61#22411semidirect
574. C4#16840749semidirect

Maximal subgroups

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

subgroupindex
D4×#117582172
C2×#168407492
#214619622
#214618462
#214616832
#214616562
C4×#145198372
C2×#168404122
#2241161