Finite Group Databasebeta

← #29685215

Subgroups — #29685215

Order 15616.

1142 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×M16×Dic61C211yesmaximal, characteristic
3#21448214C211yesmaximal, characteristic
4#21446790C212yesmaximal, complemented
5#21446637C212yesmaximal, complemented
6#21446253C211yesmaximal, characteristic
7#21446252C211yesmaximal, characteristic
8#21446250C211yesmaximal, characteristic
9#103×Dic61C212yesmaximal, complemented
10#21445638C212yesmaximal, complemented
11C61×#1894C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13#16840314—24no—
14#16840314C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519775C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519754—21no—
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519742—21no—
22C2×#14519742C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4×Dic61C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28M16×Dic61—24no—
29C2×#14519687C2212yes—
30C2×C8×Dic61C2212yes—
31#16839629—21no—
32#16839627C2212yes—
33#16839623—21no—
34#16839622C2211yescharacteristic
35C4⋊C4×Dic61C2212yes—
36C22⋊C4×Dic61—21no—
37#16839607—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C2×#11758194—22no—
51C2×#11758194—24no—
52C22×#92418C2312yes—
53#14519776—22no—
54#14519776C2×C414yes—
55#14519775—22no—
56#14519775C2×C414yes—
57#14519767—28no—
58#14519756—22no—
59#14519756C2×C414yes—
60#14519754—28no—
61Q8×Dic61—22no—
62Q8×Dic61C2×C414yes—
63#14519742—28no—
64#14519742—22no—
65#14519742C2×C414yes—
66#14519733—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61C2×C414yes—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D414yes—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519702—24no—
87#14519701—24no—
88#14519700—24no—
89#14519687—24no—
90C8×Dic61—24no—
91#14519684—21no—
92#14519684—22no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—24no—
99C22×C8×C61C2312yes—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61C2×C414yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107#103×C61—24no—
108#95×C61—21no—
109C24×C61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C22×C412yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—24no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—41no—
128C22×Dic61—41no—
129C22×Dic61C22⋊C414yes—
130C22×Dic61C22⋊C414yes—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—24no—
139C2×Dic122—42no—
140C2×Dic122—42no—
141C2×Dic122C22×C412yes—
142M16×C61—28no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61#4114yes—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C22×C412yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—42no—
153C22⋊Dic61—24no—
154C22⋊Dic61—42no—
155C4⋊Dic61—21no—
156C4⋊Dic61—24no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61C22×C412yes—
160#11758196—24no—
161#11758196—24no—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165C4×Dic61—24no—
166C4×Dic61—24no—
167C4×Dic61—44no—
168C4×Dic61C22⋊C414yes—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22×C412yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C2×D412yes—
173C4×Dic61C2×D412yes—
174#11758194—28no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418#4114yes—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61C2×C22⋊C412yes—
183C23×C61C22×D411yescharacteristic
184Q8×C61—24no—
185Q8×C61—41no—
186Q8×C61—41no—
187C2×C4×C61—22no—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61C2×C22⋊C412yes—
200C2×C4×C61C23×C411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4112yes—
203C2×C4×C61C2×#4112yes—
204C2×Dic61—22no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—28no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—44no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61C2×C22⋊C412yes—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61#13714yes—
224Dic122—24no—
225Dic122—48no—
226Dic122—41no—
227Dic122—41no—
228C8×C61—24no—
229#92418—24no—
230#21404—611nomaximal, Sylow
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61C22×C22⋊C411yescharacteristic
237C22×C61#53212yes—
238C22×C61C2×#13712yes—
239C22×C61#57511yescharacteristic
240C22×C61#57511yescharacteristic
241C4×C61—22no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—82no—
245C4×C61C4×#4112yes—
246C4×C61#52511yescharacteristic
247C4×C61#52511yescharacteristic
248Dic61—22no—
249Dic61—22no—
250Dic61—22no—
251Dic61—42no—
252Dic61—42no—
253Dic61—82no—
254Dic61#41914yes—
255C2×#533—611no—
256C2×C4×M16—611no—
257#2283—611no—
258#1993—612no—
259#1960—612no—
260#1894—611no—
261#1894—611no—
262#1894—611no—
263#1894D6111yescharacteristic
264C4×#103—612no—
265C4×#103—612no—
266C2×C61—42no—
267C2×C61#229711yescharacteristic
268C2×C61C2×#41911yescharacteristic
269C2×C61#289611yescharacteristic
270C2×C61#317212yes—
271C2×C61#317212yes—
272C22×M16—611no—
273C22×M16C2×D6111yescharacteristic
274#533—618no—
275#533—1224no—
276C2×C4⋊Q8—611no—
277C2×C4⋊Q8—611no—
278C2×#125—1221no—
279C2×C22⋊Q8—611no—
280C2×C22⋊Q8—1221no—
281C2×C22⋊Q8C2×D6111yescharacteristic
282C2×C4×Q8—611no—
283C2×C4×Q8—611no—
284C2×C4×Q8—1221no—
285C2×#117—1221no—
286C22×C42—611no—
287C2×#103—612no—
288C2×#103—612no—
289C2×#103—612no—
290C2×#103C2×D6112yes—
291C4×M16—1224no—
292C2×#97—612no—
293C2×C4×C8—612no—
294#383—1221no—
295#382—612no—
296#381—1221no—
297#380—611no—
298#380C2×D6111yescharacteristic
299C4×C4⋊C4—612no—
300C4×C22⋊C4—1221no—
301#376—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306C61#2140411yescharacteristic, complemented, Sylow
307C22×Q8—611no—
308C22×Q8—1221no—
309C22×Q8C22×D6111yescharacteristic
310C23×C4—611no—
311C23×C4—611no—
312C23×C4C22×D6111yescharacteristic
313C2×M16—22no—
314C2×M16—24no—
315C2×M16—1224no—
316C2×M16—1222no—
317C22×C8—612no—
318C22×C8C22×D6112yes—
319C4⋊Q8—614no—
320C4⋊Q8—614no—
321C4⋊Q8—1222no—
322C4⋊Q8—1222no—
323#125—1228no—
324C22⋊Q8—22no—
325C22⋊Q8—614no—
326C22⋊Q8—1228no—
327C22⋊Q8—1222no—
328C22⋊Q8C4×D6114yes—
329C4×Q8—614no—
330C4×Q8—614no—
331C4×Q8—1222no—
332C4×Q8—1228no—
333C4×Q8—1222no—
334#117—1228no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4C22×D6112yes—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C42—614no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—614no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—1221no—
357C2×C42—1221no—
358C2×C42—1222no—
359#103—24no—
360#103—1224no—
361#103—1224no—
362#103—1224no—
363#97—1224no—
364C4×C8—1224no—
365#95—21no—
366#95—1222no—
367#95—1222no—
368#95—1222no—
369#95—1221no—
370C24D4×D6111yescharacteristic
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—612no—
375C2×Q8—1224no—
376C2×Q8—1221no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—2442no—
380C2×Q8—2442no—
381C2×Q8C2×C4×D6112yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—614no—
392C22×C4—614no—
393C22×C4—1224no—
394C22×C4—1221no—
395C22×C4—1221no—
396C22×C4—1222no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4C2×C4×D6112yes—
400C22×C4D4×D6112yes—
401C22×C4D4×D6111yescharacteristic
402C22×C4C23×D6111yescharacteristic
403M16—28no—
404M16—1228no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—614no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8#1175822414yes—
411C4⋊C4—21no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—612no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1221no—
422C4⋊C4—1224no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4C2×C4×D6112yes—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—1224no—
429C22⋊C4—2442no—
430C42—612no—
431C42—612no—
432C42—612no—
433C42—612no—
434C42—614no—
435C42—614no—
436C42—1224no—
437C42—1224no—
438C42—2444no—
439C23—22no—
440C23—21no—
441C23—21no—
442C23—22no—
443C23C22⋊C4×D6112yes—
444C23C2×D4×D6111yescharacteristic
445Q8—24no—
446Q8—41no—
447Q8—41no—
448Q8—1224no—
449Q8—2448no—
450Q8—2441no—
451Q8—2441no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—612no—
465C2×C4—614no—
466C2×C4—612no—
467C2×C4—612no—
468C2×C4—612no—
469C2×C4—1228no—
470C2×C4—1222no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—2444no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4C22⋊C4×D6112yes—
485C2×C4C22×C4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4#41×D6112yes—
488C2×C4C2×#1175822412yes—
489C8—24no—
490C8—1224no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22C2×C22⋊C4×D6111yescharacteristic
497C22#1684030712yes—
498C22#137×D6112yes—
499C22#1684055811yescharacteristic
500C22#1684056111yescharacteristic
501C4—22no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—614no—
506C4—1222no—
507C4—1222no—
508C4—1222no—
509C4—4882no—
510C4—2442no—
511C4—2442no—
512C4C4×#1175822412yes—
513C4#1684026811yescharacteristic
514C4#1684027011yescharacteristic
515C2—42no—
516C2#2144830411yescharacteristic
517C2#419×D6111yescharacteristic
518C2#2145433511yescharacteristic
519C2#2145641812yes—
520C2#2145642012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1894C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1894D61
hypocenterC61#21404
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
4. #21446790C2semidirect
5. #21446637C2semidirect
9. #103×Dic61C2semidirect
10. #21445638C2semidirect
14. #16840314C4semidirect
306. C61#21404semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×M16×Dic612
#214482142
#214467902
#214466372
#214462532
#214462522
#214462502
#103×Dic612
#214456382
C61×#18942
#2140461