Finite Group Databasebeta

← #29734171

Subgroups — #29734171

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×#16840121C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3#21450804C211yesmaximal, characteristic, complemented
4#21447062C212yesmaximal, complemented
5#21447055C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446752C211yesmaximal, characteristic
8#21446073C212yesmaximal, complemented
9#21446066C212yesmaximal
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519775—21no—
14#16840121—24no—
15#16840057—24no—
16C2×#14519710C2212yes—
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519688C2211yescharacteristic
25#16839692—21no—
26#16839692—21no—
27#16839667—21no—
28#16839660C2212yes—
29#16839660C2212yes—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2212yes—
34C61×#556—24no—
35C61×#556—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#126×C61—21no—
38C2×#126×C61—21no—
39C2×#124×C61C2211yescharacteristic
40C2×#124×C61C2211yescharacteristic
41C2×C22⋊Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61C2211yescharacteristic
46C2×#117×C61—21no—
47C22×Dic122—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52#14519775—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×C4⋊Dic61C2312yes—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×C4×Dic61—21no—
59#14519710—24no—
60C4⋊#92418—22no—
61#14519701—24no—
62#14519701—24no—
63#14519701—24no—
64#14519701—24no—
65#14519698—24no—
66#14519698—24no—
67#14519688—22no—
68#14519684—21no—
69#14519684—22no—
70#14519684—22no—
71C22×Q8×C61D411yescharacteristic
72C22×Q8×C61C2311yescharacteristic
73C22×Q8×C61C2311yescharacteristic
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61—21no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61—22no—
78#126×C61—28no—
79#126×C61—28no—
80#124×C61—22no—
81#124×C61—22no—
82#124×C61—22no—
83#124×C61D414yes—
84C22⋊Q8×C61—28no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61D414yes—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93#117×C61—28no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112C24×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—41no—
122C2×Q8×C61C2×D412yes—
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—22no—
142C22×Dic61—22no—
143C2×Dic122—42no—
144C2×Dic122—42no—
145C2×Dic122—42no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D412yes—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C42×C61—24no—
161C42×C61—24no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61#4114yes—
173#11758196—44no—
174#11758196—44no—
175C4×Dic61—42no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
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—22no—
189Q8×C61—22no—
190Q8×C61—24no—
191Q8×C61—44no—
192Q8×C61—41no—
193Q8×C61—41no—
194Q8×C61—42no—
195D4×C61—24no—
196D4×C61—41no—
197D4×C61—41no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—48no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61#12012yes—
219C2×C4×C61#12012yes—
220C2×C4×C61AGL(1,ℤ/8)12yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4112yes—
224C2×C4×C612−1+412yes—
225C2×Dic61—24no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231Dic122—84no—
232#92418—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—22no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61—41no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61#53211yescharacteristic
255C4×C61#53611yescharacteristic
256C4×C61#54712yes—
257Dic61—44no—
258Dic61—82no—
259C2×#556D6111yescharacteristic
260C2×#489—611no—
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—24no—
277#556—24no—
278C2×C4⋊Q8—611no—
279C2×C4⋊Q8—1221no—
280C2×C4⋊Q8C2×D6111yescharacteristic
281C2×#126—21no—
282C2×#126—21no—
283C2×#124C2×D6111yescharacteristic
284C2×#124C2×D6111yescharacteristic
285C2×C22⋊Q8—21no—
286C2×C4×Q8C2×D6111yescharacteristic
287C2×C4×Q8C2×D6111yescharacteristic
288C2×C4×Q8C2×D6111yescharacteristic
289C2×C4×D4C2×D6111yescharacteristic
290C2×#117—21no—
291#489—1224no—
292#477—1224no—
293C2×#107—612no—
294C2×C4⋊C8—611no—
295C2×#103—612no—
296C2×#103—612no—
297C2×#103—612no—
298C2×#103—612no—
299C2×#102—612no—
300C2×#102—612no—
301C2×#97—611no—
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—1221no—
313C22×Q8C22⋊D6111yescharacteristic
314C22×Q8C22×D6111yescharacteristic
315C22×Q8C22×D6111yescharacteristic
316C22×D4C22×D6111yescharacteristic
317C23×C4—21no—
318C22×C8—612no—
319C22×C8—612no—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324C4⋊Q8—2444no—
325#126—28no—
326#126—28no—
327#124—22no—
328#124—22no—
329#124—22no—
330#124C22⋊D6114yes—
331C22⋊Q8—28no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8C22⋊D6114yes—
338C4×D4—22no—
339C4×D4—22no—
340#117—28no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4C22⋊D6111yescharacteristic
351C2×C4⋊C4C22⋊D6111yescharacteristic
352C2×C4⋊C4C22⋊D6111yescharacteristic
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—1221no—
362C2×C42C22×D6111yescharacteristic
363C2×C42C22×D6111yescharacteristic
364C2×C42C22×D6111yescharacteristic
365#107—1224no—
366C4⋊C8—1222no—
367#103—1224no—
368#103—1224no—
369#103—1224no—
370#103—1224no—
371#102—1224no—
372#102—1224no—
373#97—1222no—
374#95—1221no—
375#95—1222no—
376#95—1222no—
377C24—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—41no—
387C2×Q8—2442no—
388C2×Q8—2442no—
389C2×Q8—2442no—
390C2×Q8C2×C22⋊D6112yes—
391C2×Q8C2×C22⋊D6112yes—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4C2×C22⋊D6112yes—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C23×D6111yescharacteristic
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—614no—
426C4⋊C4—1222no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—2444no—
430C4⋊C4—2442no—
431C4⋊C4—2444no—
432C4⋊C4C2×C22⋊D6112yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C42—24no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—24no—
444C42—2442no—
445C42C2×C22⋊D6112yes—
446C42C2×C22⋊D6112yes—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23C22×C22⋊D6111yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—24no—
456Q8—44no—
457Q8—41no—
458Q8—41no—
459Q8—42no—
460Q8—4884no—
461D4—24no—
462D4—41no—
463D4—41no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—48no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—1224no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4#1451974612yes—
491C2×C4#1451974612yes—
492C2×C4#1451980212yes—
493C2×C4C22×C22⋊D6111yescharacteristic
494C2×C4C22×C22⋊D6111yescharacteristic
495C2×C4#41×D6112yes—
496C2×C4#1451982412yes—
497C8—2442no—
498C8—2442no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—82no—
503C22—82no—
504C22C2×#1451974611yescharacteristic
505C22#1684041912yes—
506C22C2×#1451980211yescharacteristic
507C22#1684056011yescharacteristic
508C22#1684057712yes—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—4882no—
519C4—2444no—
520C4#1684030811yescharacteristic
521C4#1684033311yescharacteristic
522C4#1684041112yes—
523C2—82no—
524C2#2144944311yescharacteristic
525C2#2145507111yescharacteristic
526C2#2145511411yescharacteristic
527C2#2145853312yes—
528C2#2145865512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#556D61
hypocenterC61#25661
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
3. #21450804C2semidirect
4. #21447062C2semidirect
6. #21446760C2semidirect
8. #21446073C2semidirect
10. #21445649C2semidirect
311. C61#25661semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168400572
#214508042
#214470622
#214470552
#214467602
#214467522
#214460732
#214460662
#214456492
C2×C61×#5562
#2566161