Finite Group Databasebeta

← #29685388

Subgroups — #29685388

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic
3C4×M16×D61C211yesmaximal, characteristic
4#21446808C211yesmaximal, characteristic, complemented
5#21446807C211yesmaximal, characteristic, complemented
6#21446664C211yesmaximal, characteristic, complemented
7#21446284C214yesmaximal
8#101×Dic61C214yesmaximal, complemented
9C61×#1966C211yesmaximal, characteristic, complemented
10C2×M16×D61C2212yes—
11#16840317—21no—
12#16840317—21no—
13#16840317—21no—
14#16840317—21no—
15#16840317—21no—
16#16840317—21no—
17#16840317—21no—
18#16840317—21no—
19C2×#14519777C2211yescharacteristic
20C2×#14519768C412yes—
21C2×#14519757C412yes—
22C2×#14519753C2212yes—
23C2×Q8×Dic61C2212yes—
24C2×#14519742C412yes—
25C2×#14519735C412yes—
26C2×C42×D61C2211yescharacteristic
27#16839818—21no—
28#16839818—21no—
29#16839814—21no—
30#16839814—21no—
31#16839786C2211yescharacteristic
32C2×#14519696C2214yes—
33#16839725C2212yes—
34C4×#11758194C2211yescharacteristic
35#16839714—21no—
36M16×Dic61C2212yes—
37M16×Dic61C2212yes—
38C4×#11758189—21no—
39#97×D61—21no—
40C4×C8×D61—21no—
41#16839530C2214yes—
42#16839530C2214yes—
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#423C2211yescharacteristic
45C2×#101×C61C2214yes—
46C4×M16×C61C2211yescharacteristic
47C22×Dic122C2×C412yes—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758194C2312yes—
52C2×#11758194C2312yes—
53C2×#11758189—22no—
54M16×D61—24no—
55M16×D61—24no—
56C2×C8×D61—22no—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777—21no—
61#14519768—22no—
62#14519768—22no—
63#14519768—22no—
64#14519768—22no—
65#14519757—22no—
66#14519757—22no—
67#14519757—22no—
68#14519757—22no—
69#14519753—22no—
70#14519753—22no—
71#14519753—22no—
72#14519753—22no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76Q8×Dic61—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519742—22no—
81#14519735—22no—
82#14519735—22no—
83#14519735—22no—
84#14519735—22no—
85C2×C4⋊Dic61C2×C412yes—
86C2×C4⋊Dic61C2×C414yes—
87C2×C4⋊Dic61C2312yes—
88C2×#11758196C2×C414yes—
89C2×#11758198C2×C414yes—
90C2×C4×Dic61C2×C414yes—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C42×D61D411yescharacteristic
102C4⋊#92418—21no—
103C4⋊#92418—21no—
104#14519696—28no—
105#14519689—22no—
106#14519689—22no—
107#14519688—21no—
108#14519687—22no—
109#14519687—22no—
110C4×#92418—21no—
111C8×Dic61—22no—
112C8×Dic61—22no—
113C22×Q8×C61C2312yes—
114C2×M16×C61C2312yes—
115C2×M16×C61C2312yes—
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C2×C4⋊C4×C61C2×C412yes—
121C2×C4⋊C4×C61C2×C412yes—
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—21no—
124C4⋊C8×C61—21no—
125#101×C61—28no—
126#97×C61—21no—
127C4×C8×C61—21no—
128C23×D61C22×C411yescharacteristic
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—24no—
132C2×Q8×C61—24no—
133C22×C4×C61C22×C414yes—
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C414yes—
140C22×Dic61C22×C411yescharacteristic
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—24no—
144C2×C4×D61—42no—
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C22⋊C411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—44no—
160M16×C61—22no—
161M16×C61—24no—
162M16×C61#4114yes—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C42×C61C2×D411yescharacteristic
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173#11758198—24no—
174#11758198—24no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—24no—
180C4⋊Dic61—24no—
181#11758196—24no—
182#11758196—24no—
183C4×Dic61—24no—
184C4×Dic61—24no—
185C4×Dic61C22⋊C411yescharacteristic
186C4×Dic61C22⋊C411yescharacteristic
187C4×Dic61C22⋊C411yescharacteristic
188C4×Dic61C22⋊C411yescharacteristic
189C4×Dic61C22⋊C411yescharacteristic
190C4×Dic61C22⋊C411yescharacteristic
191C4×Dic61C22⋊C411yescharacteristic
192C4×Dic61C22⋊C411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197#11758194—22no—
198#11758194—24no—
199#11758194#4114yes—
200C2×#92418—22no—
201C2×#92418—22no—
202#11758189—28no—
203C8×D61—28no—
204C23×C61C23×C411yescharacteristic
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—41no—
208C22×D61C2×C22⋊C411yescharacteristic
209C22×D61C2×C22⋊C411yescharacteristic
210Q8×C61—44no—
211Q8×C61—42no—
212Q8×C61—42no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—28no—
217C2×C4×C61C2×C22⋊C411yescharacteristic
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4112yes—
224C2×C4×C61C2×#4112yes—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—41no—
229C2×Dic61—44no—
230C2×Dic61C2×C22⋊C411yescharacteristic
231C2×Dic61C2×C22⋊C411yescharacteristic
232C2×Dic61C2×C22⋊C411yescharacteristic
233C2×Dic61C2×C22⋊C411yescharacteristic
234C2×Dic61C2×C22⋊C411yescharacteristic
235C2×Dic61C2×C22⋊C411yescharacteristic
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61#10114yescomplemented
250Dic122—44no—
251Dic122—44no—
252C8×C61—24no—
253C8×C61—42no—
254#92418—24no—
255#92418—42no—
256#21425—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61C22×C22⋊C411yescharacteristic
260C22×C61C4×#4112yes—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—44no—
267C2×D61C2×#10112yes—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—44no—
271C4×C61C2×#10112yes—
272C4×C61#41312yes—
273C4×C61#52512yes—
274C4×C61#53211yescharacteristic
275C4×C61#53211yescharacteristic
276Dic61—21no—
277Dic61—21no—
278Dic61—21no—
279Dic61—21no—
280Dic61—21no—
281Dic61—44no—
282Dic61C2×#10112yes—
283C2×#533—611no—
284#2920—611no—
285C2×C4×M16—611no—
286#1997—611no—
287#1997—611no—
288#1966—611no—
289#1966D6111yescharacteristic, complemented
290#1901—614no—
291C4×#101—614no—
292C2×C61—22no—
293C2×C61#229711yescharacteristic
294C2×C61C22×#10111yescharacteristic
295C2×C61C2×#41311yescharacteristic
296D61—22no—
297D61—41no—
298C22×M16—612no—
299#533—1221no—
300#533—1221no—
301#533—1221no—
302#533—1221no—
303#533—1221no—
304#533—1221no—
305#533—1221no—
306#533—1221no—
307C2×C4⋊Q8—611no—
308C2×C4⋊Q8C2×D6111yescharacteristic
309C2×#125—612no—
310C2×C22⋊Q8—612no—
311C2×C22⋊Q8—612no—
312C2×C4×Q8—612no—
313C2×C4×Q8—612no—
314C2×#117—612no—
315C22×C42—611no—
316#432—1221no—
317#432—1221no—
318#431—1221no—
319#431—1221no—
320#423—611no—
321#423C2×D6111yescharacteristic
322C2×#101—614no—
323C2×#101C2×D6114yes—
324#407—612no—
325C4×M16—612no—
326C4×M16—612no—
327C4×M16—611no—
328C4×M16—1221no—
329C4×M16—1221no—
330C4×M16C2×D6111yescharacteristic
331C2×#97—1221no—
332C2×C4×C8—1221no—
333#343—614no—
334#343—614no—
335C61#2142511yescharacteristic, complemented, Sylow
336C22×Q8—612no—
337C22×Q8C22×D6112yes—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C2×M16—612no—
342C2×M16—612no—
343C2×M16—1222no—
344C2×M16—1224no—
345C2×M16—1224no—
346C2×M16C22×D6112yes—
347C2×M16C22×D6112yes—
348C22×C8—1222no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357#125—1222no—
358#125—1222no—
359#125—1222no—
360#125—1222no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C22⋊Q8—1222no—
365C22⋊Q8—1222no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C22⋊Q8—1222no—
369C4×Q8—1222no—
370C4×Q8—1222no—
371C4×Q8—1222no—
372C4×Q8—1222no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375C4×Q8—1222no—
376C4×Q8—1222no—
377#117—1222no—
378#117—1222no—
379#117—1222no—
380#117—1222no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—614no—
383C2×C4⋊C4—614no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4C4×D6112yes—
386C2×C4⋊C4C4×D6112yes—
387C2×C22⋊C4—614no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—614no—
400C2×C42C22×D6111yescharacteristic
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405#101—28no—
406#101—1228no—
407C22⋊C8—1222no—
408C22⋊C8—1222no—
409#97—21no—
410#97—1222no—
411#97—1222no—
412#97—1221no—
413C4×C8—21no—
414C4×C8—1222no—
415C4×C8—1222no—
416C4×C8—1221no—
417C24—611no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8—2444no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—614no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—2442no—
445C22×C4C2×C4×D6114yes—
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4C2×C4×D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449M16—22no—
450M16—24no—
451M16—614no—
452M16—1228no—
453M16—1222no—
454M16—1224no—
455M16#1175822414yes—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—1228no—
459C2×C8—1222no—
460C2×C8—1222no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C42—611no—
476C42—611no—
477C42—611no—
478C42—611no—
479C42—611no—
480C42—611no—
481C42—611no—
482C42—611no—
483C42—611no—
484C42—611no—
485C42—611no—
486C42—611no—
487C42—1224no—
488C42—1224no—
489C42D4×D6111yescharacteristic
490C42D4×D6111yescharacteristic
491C42D4×D6111yescharacteristic
492C42D4×D6111yescharacteristic
493C23—611no—
494C23—611no—
495C23—1222no—
496C23—1222no—
497C23—2441no—
498C23C22×C4×D6111yescharacteristic
499Q8—44no—
500Q8—42no—
501Q8—42no—
502Q8—2444no—
503Q8—2444no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—611no—
509C2×C4—614no—
510C2×C4—611no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—611no—
514C2×C4—611no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1224no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2444no—
533C2×C4C22⋊C4×D6111yescharacteristic
534C2×C4C22⋊C4×D6111yescharacteristic
535C2×C4C22⋊C4×D6111yescharacteristic
536C2×C4C22⋊C4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4#41×D6112yes—
540C2×C4C2×#1175822412yes—
541C8—24no—
542C8—42no—
543C8—1224no—
544C8—2442no—
545C22—21no—
546C22—21no—
547C22—612no—
548C22—1221no—
549C22—1221no—
550C22—1221no—
551C22—1221no—
552C22—1221no—
553C22—2444no—
554C22C2×C22⋊C4×D6111yescharacteristic
555C22C4×#1175822412yes—
556C4—21no—
557C4—21no—
558C4—44no—
559C4—612no—
560C4—1221no—
561C4—1221no—
562C4—1221no—
563C4—1221no—
564C4—1221no—
565C4—2444no—
566C4#101×D6112yes—
567C4#1683974512yes—
568C4#1684027212yes—
569C4#1684031011yescharacteristic
570C4#1684031111yescharacteristic
571C2—22no—
572C2—1222no—
573C2—2441no—
574C2#2144830611yescharacteristic
575C2C2×#101×D6111yescharacteristic
576C2C2×#1683974511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1966C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22×C4
cosocleC22×C4×C61C24
hypercenter#1966D61
hypocenterC61#21425
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
4. #21446808C2semidirect
5. #21446807C2semidirect
6. #21446664C2semidirect
8. #101×Dic61C2semidirect
9. C61×#1966C2semidirect
249. C4×D61#101semidirect
289. #1966D61semidirect
335. C61#21425semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214544512
C4×M16×D612
#214468082
#214468072
#214466642
#214462842
#101×Dic612
C61×#19662
#2142561