Finite Group Databasebeta

← #29688913

Subgroups — #29688913

Order 15616.

1842 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D61C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic, complemented
3C2×C22⋊C8×D61C211yesmaximal, characteristic, complemented
4#21446113C214yesmaximal, complemented
5#21446108C214yesmaximal, complemented
6#21445885C211yesmaximal, characteristic, complemented
7#21445885C211yesmaximal, characteristic, complemented
8#21445729C211yesmaximal, characteristic, complemented
9C61×#1779C211yesmaximal, characteristic, complemented
10C23×C22⋊D61C411yescharacteristic
11C23×C4×D61C2211yescharacteristic
12C22×#11758189C2212yescomplemented
13C22×C8×D61C2212yescomplemented
14C2×C4×C22⋊D61—24no—
15C2×C4×C22⋊D61—24no—
16C2×C4×C22⋊D61C412yes—
17C2×C4×C22⋊D61C412yes—
18C2×C4×C22⋊D61C2212yes—
19C2×C4×C22⋊D61C2212yes—
20C22×#11758196C411yescharacteristic
21C22×C22⋊Dic61C411yescharacteristic
22C22×#11758198C411yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839724—24no—
25C2×C22⋊#92418C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C2×C22⋊#92418C2214yescomplemented
28C2×#14519689C2212yescomplemented
29C2×#14519689C2212yescomplemented
30C22⋊C8×D61—24no—
31#16839516C2214yescomplemented
32#16839516C2214yescomplemented
33C24×C4×C61C2211yescharacteristic
34C2×C22⋊C8×C61C2211yescharacteristic
35C2×C22⋊C8×C61C2211yescharacteristic
36C2×C22⋊C8×C61C2214yescomplemented
37C24×D61C2×C411yescharacteristic
38C22×C22⋊D61—24no—
39C22×C22⋊D61—28no—
40C22×C22⋊D61C2×C412yes—
41C22×C22⋊D61C2×C412yes—
42C23×Dic61C2×C411yescharacteristic
43C23×Dic61C2×C411yescharacteristic
44C23×Dic61C2×C411yescharacteristic
45C22×C4×D61—44no—
46C22×C4×D61C2×C411yescharacteristic
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61D414yes—
49C22×C4×D61D414yes—
50C22×C4×D61C2311yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758189—24no—
53C2×#11758189—44no—
54C22×#92418C2312yes—
55C22×#92418C2312yes—
56C2×C8×D61—24no—
57C2×C8×D61—44no—
58C4×C22⋊D61—28no—
59C4×C22⋊D61—28no—
60C4×C22⋊D61—24no—
61C4×C22⋊D61—24no—
62C4×C22⋊D61—24no—
63C4×C22⋊D61—24no—
64C2×#11758196—24no—
65C2×#11758196C2×C412yes—
66C2×#11758196C2×C412yes—
67C2×C22⋊Dic61—24no—
68C2×C22⋊Dic61C2×C412yes—
69C2×C22⋊Dic61C2×C412yes—
70C2×#11758198—24no—
71C2×#11758198C2×C412yes—
72C2×#11758198C2×C412yes—
73C2×C4×Dic61—22no—
74C2×C4×Dic61—22no—
75C2×C4×Dic61C2×C412yes—
76C2×C4×Dic61C2312yes—
77C22⋊#92418—22no—
78C22⋊#92418—22no—
79C22⋊#92418—44no—
80#14519689—24no—
81#14519689—24no—
82C25×C61C2×C411yescharacteristic
83C23×C4×C61—22no—
84C23×C4×C61—22no—
85C23×C4×C61—44no—
86C23×C4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C412yes—
88C23×C4×C61C2312yes—
89C23×C4×C61C2311yescharacteristic
90C22×C8×C61C2312yes—
91C22×C8×C61C2312yes—
92C22⋊C8×C61—22no—
93C22⋊C8×C61—22no—
94C22⋊C8×C61—44no—
95C24×C61—22no—
96C24×C61—24no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—42no—
100C24×C61C22×C412yes—
101C24×C61C22×C411yescharacteristic
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—24no—
105C23×D61—42no—
106C23×D61C22⋊C414yes—
107C23×D61C22×C411yescharacteristic
108C23×D61C22×C411yescharacteristic
109C22×C4×C61—24no—
110C22×C4×C61—22no—
111C22×C4×C61—24no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—24no—
115C22×C4×C61—24no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—48no—
120C22×C4×C61—48no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—232no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61M1618yescomplemented
133C22×Dic61—22no—
134C22×Dic61—24no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—42no—
139C22×Dic61C22⋊C414yes—
140C22×Dic61C22×C412yes—
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C412yes—
144C2×C4×D61—24no—
145C2×C4×D61—24no—
146C2×C4×D61—48no—
147C2×C4×D61—48no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61C22⋊C414yes—
151C2×C4×D61C22⋊C414yes—
152C2×C4×D61C2×D412yes—
153C2×C4×D61C2×D412yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61C2×D412yes—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—24no—
159C2×C8×C61—24no—
160C22⋊Dic61—24no—
161C22⋊Dic61—24no—
162#11758198—24no—
163#11758198—24no—
164#11758196—24no—
165#11758196—24no—
166C4×Dic61—24no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—24no—
172C2×#92418—24no—
173#11758189—48no—
174C8×D61—48no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—28no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—48no—
185C23×C61—48no—
186C23×C61—44no—
187C23×C61—44no—
188C23×C61C2×C22⋊C412yes—
189C23×C61C2×M1614yes—
190C23×C61#13112yes—
191C23×C61#13112yes—
192C23×C61C23×C411yescharacteristic
193C22×D61—22no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—28no—
201C22×D61—28no—
202C22×D61—44no—
203C22×D61—48no—
204C22×D61—48no—
205C22×D61C2×C22⋊C412yes—
206C22×D61C2×C22⋊C412yes—
207C22×D61C2×M1614yes—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—48no—
215C2×C4×C61—44no—
216C2×C4×C61—44no—
217C2×C4×C61—44no—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61H(5,𝔽2)14yes—
222C22⋊D61—216no—
223C22⋊D61—216no—
224C2×Dic61—28no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—44no—
233C2×Dic61—44no—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×M1614yes—
237C4×D61—24no—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—44no—
242C4×D61—44no—
243C8×C61—42no—
244C8×C61—42no—
245#92418—42no—
246#92418—42no—
247#21740—611nomaximal, Sylow
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—24no—
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61—24no—
255C22×C61—416no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61#40814yes—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61#51812yes—
263C22×C61C22×M1612yes—
264C22×C61C2×#13111yescharacteristic
265C22×C61C2×#13111yescharacteristic
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—24no—
269C2×D61—24no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—416no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61#40714yes—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61#53412yes—
280C4×C61#53412yes—
281Dic61—22no—
282Dic61—22no—
283Dic61—24no—
284Dic61—42no—
285C22×C4×D4—611no—
286C2×#407—611no—
287C22×C22⋊C8—611no—
288#1870—614no—
289#1869—614no—
290#1817—611no—
291#1817—611no—
292#1779—611no—
293#1779D6111yescharacteristic, complemented
294C2×C61—24no—
295C2×C61—44no—
296C2×C61#230811yescharacteristic
297C2×C61C2×#40711yescharacteristic
298C2×C61C2×#40811yescharacteristic
299C2×C61#297412yes—
300C2×C61#297412yes—
301D61—24no—
302D61—42no—
303C23×D4—611no—
304C24×C4—611no—
305C24×C4C2×D6111yescharacteristic
306C22×M16—612no—
307C23×C8—612no—
308C2×C4×D4—612no—
309C2×C4×D4—612no—
310C2×C4×D4—612no—
311C2×C4×D4—612no—
312C2×C4×D4—1224no—
313C2×C4×D4—1224no—
314C22×C4⋊C4—611no—
315C22×C22⋊C4—611no—
316C22×C22⋊C4—611no—
317C22×C42—611no—
318#407—1224no—
319C2×C22⋊C8—611no—
320C2×C22⋊C8—611no—
321C2×C22⋊C8—612no—
322C2×C22⋊C8—612no—
323C2×C22⋊C8—614no—
324C2×C22⋊C8—1224no—
325C2×C22⋊C8C2×D6111yescharacteristic
326C2×C22⋊C8C2×D6111yescharacteristic
327C2×C22⋊C8C2×D6114yescomplemented
328#336—614no—
329#336—614no—
330C61#2174011yescharacteristic, complemented, Sylow
331C25—611no—
332C25C4×D6111yescharacteristic
333C22×D4—612no—
334C22×D4—612no—
335C22×D4—1228no—
336C22×D4—1224no—
337C23×C4—22no—
338C23×C4—22no—
339C23×C4—44no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—614no—
346C23×C4—614no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—2444no—
350C23×C4C4×D6111yescharacteristic
351C23×C4C4×D6112yes—
352C23×C4C22×D6112yes—
353C23×C4C22×D6111yescharacteristic
354C2×M16—1224no—
355C2×M16—2444no—
356C22×C8—612no—
357C22×C8—612no—
358C22×C8—1224no—
359C22×C8—2444no—
360C22×C8C22×D6112yes—
361C22×C8C22×D6112yes—
362C4×D4—1224no—
363C4×D4—1224no—
364C4×D4—1224no—
365C4×D4—1224no—
366C4×D4—1228no—
367C4×D4—1228no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1224no—
371C2×C22⋊C4—612no—
372C2×C22⋊C4—612no—
373C2×C22⋊C4—612no—
374C2×C22⋊C4—612no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—1224no—
377C2×C42—612no—
378C2×C42—612no—
379C2×C42—1222no—
380C2×C42—1222no—
381C22⋊C8—22no—
382C22⋊C8—22no—
383C22⋊C8—44no—
384C22⋊C8—1224no—
385C22⋊C8—1224no—
386C22⋊C8—1222no—
387C22⋊C8—1222no—
388C22⋊C8—2444no—
389C24—22no—
390C24—24no—
391C24—22no—
392C24—22no—
393C24—42no—
394C24—611no—
395C24—611no—
396C24—614no—
397C24—1224no—
398C24—1222no—
399C24—1222no—
400C24—2442no—
401C24C2×C4×D6112yes—
402C24C2×C4×D6111yescharacteristic
403C2×D4—618no—
404C2×D4—1224no—
405C2×D4—1224no—
406C2×D4—1224no—
407C2×D4—1224no—
408C2×D4—12232no—
409C2×D4—1224no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—612no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—614no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—614no—
431C22×C4—614no—
432C22×C4—612no—
433C22×C4—1224no—
434C22×C4—1222no—
435C22×C4—1224no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—2448no—
441C22×C4—2448no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4C2×C4×D6112yes—
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4D4×D6112yes—
448C22×C4D4×D6112yes—
449C22×C4C23×D6111yescharacteristic
450M16—2448no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1224no—
458C2×C8—1224no—
459C2×C8—2448no—
460C4⋊C4—1224no—
461C4⋊C4—1224no—
462C22⋊C4—1224no—
463C22⋊C4—1224no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C42—1222no—
467C42—1222no—
468C42—1224no—
469C23—22no—
470C23—22no—
471C23—28no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—48no—
479C23—48no—
480C23—44no—
481C23—44no—
482C23—612no—
483C23—614no—
484C23—612no—
485C23—1228no—
486C23—1222no—
487C23—1221no—
488C23—1221no—
489C23—1222no—
490C23—1228no—
491C23—1222no—
492C23—1221no—
493C23—1221no—
494C23—2444no—
495C23—2448no—
496C23—2448no—
497C23C22⋊C4×D6112yes—
498C23M16×D6114yes—
499C23#1451978312yes—
500C23#1451978512yes—
501C23C22×C4×D6111yescharacteristic
502D4—12216no—
503D4—12216no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—614no—
517C2×C4—1228no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—2444no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2444no—
533C2×C4C22⋊C4×D6112yes—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4#1451982214yes—
537C8—42no—
538C8—42no—
539C8—2442no—
540C8—2442no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—416no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—614no—
554C22—1224no—
555C22—1224no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—1222no—
560C22—1222no—
561C22—24416no—
562C22—2442no—
563C22—2442no—
564C22#1683972714yes—
565C22C2×C22⋊C4×D6111yescharacteristic
566C22#1684023512yes—
567C22C2×M16×D6112yes—
568C22C2×#1451978311yescharacteristic
569C22C2×#1451978511yescharacteristic
570C4—22no—
571C4—42no—
572C4—1224no—
573C4—1222no—
574C4—1222no—
575C4—2442no—
576C4#1684032212yes—
577C4#1684032212yes—
578C2—24no—
579C2—44no—
580C2—1224no—
581C2—2442no—
582C2#2144839511yescharacteristic
583C2#407×D6111yescharacteristic
584C2C2×#1683972711yescharacteristic
585C2#2145485012yes—
586C2#2145485512yes—
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1779C2
prime radicalC23×C22⋊D61C4
cosocleC22×C4×C61C24
hypercenter#1779D61
hypocenterC61#21740
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#16839724C2semidirect
3. C2×C22⋊C8×D61C2semidirect
4. #21446113C2semidirect
5. #21446108C2semidirect
6. #21445885C2semidirect
7. #21445885C2semidirect
8. #21445729C2semidirect
9. C61×#1779C2semidirect
12. C22×#11758189C22semidirect
13. C22×C8×D61C22semidirect
27. C2×C22⋊#92418C22semidirect
28. C2×#14519689C22semidirect
29. C2×#14519689C22semidirect
31. #16839516C22semidirect
32. #16839516C22semidirect
36. C2×C22⋊C8×C61C22semidirect
132. C2×C22⋊D61M16semidirect
293. #1779D61semidirect
327. C2×C22⋊C8C2×D61semidirect
330. C61#21740semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D612
C2×#168397242
C2×C22⋊C8×D612
#214461132
#214461082
#214458852
#214457292
C61×#17792
#2174061