Finite Group Databasebeta

← #29734180

Subgroups — #29734180

Order 15616.

1048 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840481C211yesmaximal, characteristic
2C2×#16840054C211yesmaximal, characteristic, complemented
3#21450808C211yesmaximal, characteristic, complemented
4#21447063C212yesmaximal, complemented
5#21447058C212yesmaximal
6#21446759C211yesmaximal, characteristic, complemented
7#21446752C211yesmaximal, characteristic
8#21446071C212yesmaximal, complemented
9#21446064C212yesmaximal
10#21445650C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic
12#16840481—24no—
13#16840481—24no—
14C2×#14519775—21no—
15C2×#14519775C2211yescharacteristic
16C2×#14519770—21no—
17C2×#14519770—21no—
18C2×#14519764C2211yescharacteristic
19C2×#14519754—21no—
20C2×Q8×Dic61C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24C2×#14519733—21no—
25#16840054—24no—
26C2×#14519710C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519701C2212yes—
30C2×#14519700C2212yes—
31C2×#14519698C2212yes—
32#16839693—21no—
33#16839693—21no—
34#16839669—21no—
35#16839658C2212yes—
36#16839658C2212yes—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#124×C61C2211yescharacteristic
43C61×#489—24no—
44C2×#103×C61C2212yes—
45C2×#102×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122D411yescharacteristic
48C22×Dic122C2311yescharacteristic
49C23×Dic61—21no—
50C22×#92418C2312yes—
51#14519775—22no—
52#14519775—22no—
53#14519775—44no—
54#14519770—28no—
55#14519770—28no—
56#14519764—22no—
57#14519764—22no—
58#14519754—28no—
59Q8×Dic61—22no—
60Q8×Dic61—22no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64C4×Dic122D414yes—
65D4×Dic61—22no—
66D4×Dic61—22no—
67#14519733—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×#11758196D411yescharacteristic
78C2×#11758196D411yescharacteristic
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519710—24no—
88C4⋊#92418—22no—
89#14519702—24no—
90#14519701—24no—
91#14519700—24no—
92#14519698—24no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61—21no—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61—22no—
102#124×C61—22no—
103#124×C61D414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61C2311yescharacteristic
109#103×C61—24no—
110#102×C61—24no—
111#97×C61—22no—
112C24×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—42no—
117C2×Q8×C61—42no—
118C2×Q8×C61—42no—
119C2×Q8×C61C2×D412yes—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—24no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122—22no—
144C2×Dic122—41no—
145C2×Dic122C2×D412yes—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—42no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—24no—
157C22⋊Dic61—24no—
158C22⋊Dic61—24no—
159C4⋊Dic61—24no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61C2×D412yes—
164C4⋊Dic61#4114yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—24no—
168#11758196—22no—
169#11758196—22no—
170#11758196—44no—
171#11758196—44no—
172C4×Dic61—24no—
173C4×Dic61—24no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—42no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—42no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—42no—
187C23×C61C22×D411yescharacteristic
188Q8×C61—24no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61—84no—
192D4×C61—24no—
193D4×C61—41no—
194D4×C61—41no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61#12012yes—
204C2×C4×C61#12012yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4112yes—
208C2×C4×C612−1+412yes—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—22no—
213C2×Dic61—21no—
214C2×Dic61—22no—
215C2×Dic61—21no—
216C2×Dic61—24no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—48no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61AGL(1,ℤ/8)12yes—
228Dic122—22no—
229Dic122—22no—
230Dic122—44no—
231Dic122—42no—
232C8×C61—42no—
233#92418—42no—
234#25661—611nomaximal, Sylow
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61C2×#12011yescharacteristic
241C22×C61#54812yes—
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61#57511yescharacteristic
244C22×C61#57812yes—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—42no—
248C4×C61—82no—
249C4×C61#53211yescharacteristic
250C4×C61#53611yescharacteristic
251C4×C61#54712yes—
252Dic61—22no—
253Dic61—44no—
254Dic61—42no—
255Dic61—41no—
256Dic61—42no—
257Dic61—41no—
258Dic61—41no—
259C2×#556—611no—
260C2×#489D6111yescharacteristic
261C2×#477—611no—
262#2546—611no—
263#2043—612no—
264#2042—612no—
265#1985—611no—
266#1983—611no—
267#1863—612no—
268#1862—612no—
269#1763—611no—
270C2×C61—82no—
271C2×C61#243211yescharacteristic
272C2×C61#300611yescharacteristic
273C2×C61#301111yescharacteristic
274C2×C61#336712yes—
275C2×C61#337712yes—
276#556—1224no—
277#556—1224no—
278C2×C4⋊Q8—611no—
279C2×C4⋊Q8—1221no—
280C2×C4⋊Q8C2×D6111yescharacteristic
281C2×#126—1221no—
282C2×#126—1221no—
283C2×#124—611no—
284C2×#124C2×D6111yescharacteristic
285C2×C22⋊Q8—1221no—
286C2×C4×Q8—611no—
287C2×C4×Q8—611no—
288C2×C4×Q8—611no—
289C2×C4×D4—611no—
290C2×#117—1221no—
291#489—24no—
292#477—1224no—
293C2×#107—612no—
294C2×C4⋊C8—611no—
295C2×#103—612no—
296C2×#103—612no—
297C2×#103—612no—
298C2×#103C2×D6112yes—
299C2×#102—612no—
300C2×#102C2×D6112yes—
301C2×#97C2×D6111yescharacteristic
302#398—1221no—
303#398—1221no—
304#391—1221no—
305#389—612no—
306#389—612no—
307C4×C4⋊C4—611no—
308#340—612no—
309#336—612no—
310#336—612no—
311C61#2566111yescharacteristic, complemented, Sylow
312C22×Q8—21no—
313C22×Q8—611no—
314C22×Q8—611no—
315C22×Q8C22×D6111yescharacteristic
316C22×D4C22×D6111yescharacteristic
317C23×C4—1221no—
318C22×C8—612no—
319C22×C8C22×D6112yes—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324C4⋊Q8—2444no—
325#126—1228no—
326#126—1228no—
327#124—22no—
328#124—1222no—
329#124—1222no—
330#124C22⋊D6114yes—
331C22⋊Q8—1228no—
332C4×Q8—614no—
333C4×Q8—1222no—
334C4×Q8—1222no—
335C4×Q8—1222no—
336C4×Q8—1222no—
337C4×Q8—1222no—
338C4×D4—1222no—
339C4×D4—1222no—
340#117—1228no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4C22×D6112yes—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—1221no—
362C2×C42—1221no—
363C2×C42—1221no—
364C2×C42C22×D6111yescharacteristic
365#107—1224no—
366C4⋊C8—1222no—
367#103—24no—
368#103—1224no—
369#103—1224no—
370#103—1224no—
371#102—24no—
372#102—1224no—
373#97—22no—
374#95—1222no—
375#95—1222no—
376#95—1221no—
377C24—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—612no—
385C2×Q8—1222no—
386C2×Q8—1221no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—2441no—
391C2×Q8C2×C22⋊D6112yes—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4C2×C22⋊D6112yes—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—1224no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4D4×D6111yescharacteristic
409C22×C4D4×D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—612no—
421C4⋊C4—614no—
422C4⋊C4—1224no—
423C4⋊C4—1224no—
424C4⋊C4—1224no—
425C4⋊C4—1221no—
426C4⋊C4—1222no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C4⋊C4—1224no—
431C4⋊C4—2444no—
432C4⋊C4—2444no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C42—1224no—
439C42—1221no—
440C42—1221no—
441C42—1221no—
442C42—1221no—
443C42—1224no—
444C42—2442no—
445C42D4×D6112yes—
446C42C2×C22⋊D6112yes—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23C2×D4×D6111yescharacteristic
453Q8—24no—
454Q8—41no—
455Q8—41no—
456Q8—84no—
457Q8—1222no—
458Q8—1222no—
459Q8—2444no—
460Q8—2442no—
461D4—24no—
462D4—41no—
463D4—41no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—612no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—1222no—
477C2×C4—1221no—
478C2×C4—1221no—
479C2×C4—1222no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1224no—
485C2×C4—2448no—
486C2×C4—2442no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4#1451974512yes—
492C2×C4#1451974512yes—
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4C22×C22⋊D6111yescharacteristic
495C2×C4#41×D6112yes—
496C2×C4#1451982412yes—
497C8—42no—
498C8—2442no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—82no—
503C22—82no—
504C22C2×#1451974511yescharacteristic
505C22#1684042012yes—
506C22AGL(1,ℤ/8)×D6111yescharacteristic
507C22#1684055811yescharacteristic
508C22#1684057812yes—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—82no—
513C4—1222no—
514C4—2441no—
515C4—2441no—
516C4—2444no—
517C4—2441no—
518C4—2442no—
519C4—2442no—
520C4#1684031111yescharacteristic
521C4#1684033211yescharacteristic
522C4#1684041112yes—
523C2—82no—
524C2#2144945011yescharacteristic
525C2#2145506711yescharacteristic
526C2#2145511311yescharacteristic
527C2#2145853412yes—
528C2#2145866012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#489D61
hypocenterC61#25661
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. C2×#16840054C2semidirect
3. #21450808C2semidirect
4. #21447063C2semidirect
6. #21446759C2semidirect
8. #21446071C2semidirect
10. #21445650C2semidirect
311. C61#25661semidirect

Maximal subgroups

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

subgroupindex
C2×#168404812
C2×#168400542
#214508082
#214470632
#214470582
#214467592
#214467522
#214460712
#214460642
#214456502
C2×C61×#4892
#2566161