Finite Group Databasebeta

← #29734467

Subgroups — #29734467

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456372C211yesmaximal, characteristic, complemented
2#21455641C211yesmaximal, characteristic, complemented
3#21454189C211yesmaximal, characteristic, complemented
4#21447033C212yesmaximal, complemented
5#21447026C212yesmaximal
6#21446797C211yesmaximal, characteristic, complemented
7#21446669C211yesmaximal, characteristic, complemented
8#21446034C212yesmaximal, complemented
9#21446029C212yesmaximal, complemented
10C4×#14519704C211yesmaximal, characteristic, complemented
11C61×#3510C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840168—22no—
14#16840161—22no—
15#16840069—22no—
16#16840062—22no—
17#16839805C2212yes—
18#16839805C2212yes—
19#16839805C2212yes—
20#16839786C2211yescharacteristic
21C2×#14519704C2212yes—
22#16839763C2212yes—
23C2×#14519696C2212yes—
24C2×#14519694C2212yes—
25C4×#11758194C2211yescharacteristic
26#16839669C2212yes—
27#16839665C2212yes—
28#16839656—24no—
29C42×Dic61C2211yescharacteristic
30#16839521C2212yes—
31#16839521C2212yes—
32#16839521C2212yes—
33C61×#559—22no—
34C61×#556—22no—
35C61×#551—24no—
36C61×#542—22no—
37C61×#536C2211yescharacteristic
38C61×#535C2211yescharacteristic
39C61×#533C2211yescharacteristic
40C61×#529—22no—
41C2×#124×C61C2211yescharacteristic
42C61×#519C2211yescharacteristic
43C61×#518C2211yescharacteristic
44C4×#41×C61C2211yescharacteristic
45C2×#11758194C2312yes—
46C2×#11758194C2312yes—
47#14519769—22no—
48#14519769—21no—
49#14519769—21no—
50C2×C4⋊Dic61C2312yes—
51C2×#11758196—22no—
52C2×C22⋊Dic61—24no—
53C2×C4×Dic61—22no—
54C2×C4×Dic61C2312yes—
55#14519710—22no—
56#14519710—22no—
57#14519710—22no—
58#14519708—22no—
59#14519708—22no—
60#14519708—22no—
61C4⋊#92418—22no—
62#14519704—24no—
63#14519701—22no—
64#14519698—22no—
65#14519696—24no—
66#14519694—24no—
67#14519688—21no—
68C4×#92418—21no—
69#14519684—24no—
70C2×#41×C61C2311yescharacteristic
71C22×Q8×C61C2311yescharacteristic
72C22×D4×C61C2311yescharacteristic
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61D412yes—
75#127×C61—21no—
76#126×C61—24no—
77#126×C61—24no—
78#125×C61D412yes—
79#124×C61—24no—
80#124×C61—21no—
81#124×C61—21no—
82#124×C61—21no—
83#124×C61—22no—
84#124×C61—21no—
85#123×C61—24no—
86#123×C61—24no—
87#123×C61—22no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91#121×C61—22no—
92#120×C61—22no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×Q8×C61D412yes—
97C4×D4×C61—24no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61D412yes—
102#117×C61—24no—
103#117×C61—22no—
104#117×C61—21no—
105#117×C61C2311yescharacteristic
106#117×C61C2311yescharacteristic
107#117×C61C2311yescharacteristic
108C2×C22⋊C4×C61—22no—
109C2×C22⋊C4×C61D414yes—
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C24×C61C2×D412yes—
113#41×C61—22no—
114#41×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61C2×D411yescharacteristic
121C2×D4×C61—21no—
122C2×D4×C61—24no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61C2×D411yescharacteristic
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61C2×D412yes—
143C4⋊C4×C61C2×D412yes—
144C4⋊C4×C61C2×D412yes—
145C22⋊C4×C61—24no—
146C22⋊C4×C61—24no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—21no—
149C22⋊C4×C61—21no—
150C22⋊C4×C61—21no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—22no—
154C42×C61—24no—
155C42×C61—22no—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D412yes—
162C22⋊Dic61—48no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165#11758196—42no—
166#11758196—42no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C4×Dic61—44no—
170#11758194—22no—
171#11758194—22no—
172#11758194—22no—
173#11758194#4114yes—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—22no—
177C23×C61—21no—
178C23×C61—24no—
179C23×C61—44no—
180C23×C61C22×D411yescharacteristic
181Q8×C61—21no—
182Q8×C61—21no—
183Q8×C61—41no—
184Q8×C61—41no—
185Q8×C61—42no—
186Q8×C61—82no—
187D4×C61—21no—
188D4×C61—21no—
189D4×C61—42no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—42no—
193D4×C61—84no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—82no—
203C2×C4×C61—84no—
204C2×C4×C61#12012yes—
205C2×C4×C61#12012yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4112yes—
209C2×C4×C612−1+412yes—
210C2×Dic61—24no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—44no—
214C2×Dic61—42no—
215#92418—24no—
216#92418—42no—
217#25689—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—41no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—44no—
223C22×C61—84no—
224C22×C61C2×#12011yescharacteristic
225C22×C61#54712yes—
226C4×C61—22no—
227C4×C61—22no—
228C4×C61—42no—
229C4×C61—41no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61#53211yescharacteristic
234C4×C61#53611yescharacteristic
235C4×C61#54812yes—
236Dic61—44no—
237Dic61—82no—
238#3510D6111yescharacteristic
239#3166—611no—
240#3080—611no—
241#2861—611no—
242#2038—612no—
243#2037—612no—
244#1994—611no—
245#1967—611no—
246#1857—612no—
247#1856—612no—
248C4×#104—611no—
249C2×C61—22no—
250C2×C61—81no—
251C2×C61—82no—
252C2×C61#243211yescharacteristic
253C2×C61#301812yes—
254#559—22no—
255#556—22no—
256#551—24no—
257#542—22no—
258#536C2×D6111yescharacteristic
259#535C2×D6111yescharacteristic
260#533C2×D6111yescharacteristic
261#529—22no—
262C2×#125—611no—
263C2×#124C2×D6111yescharacteristic
264#519C2×D6111yescharacteristic
265#518C2×D6111yescharacteristic
266C4×#41C2×D6111yescharacteristic
267#501—1222no—
268#499—1222no—
269#479—1222no—
270#478—1222no—
271#428—612no—
272#428—612no—
273#428—612no—
274#423—611no—
275C2×#104—612no—
276#417—612no—
277C2×#101—612no—
278C2×#100—612no—
279C4×M16—611no—
280#391—612no—
281#390—612no—
282#388—1224no—
283C43—611no—
284#339—612no—
285#339—612no—
286#339—612no—
287C61#2568911yescharacteristic, complemented, Sylow
288C2×#41C22×D6111yescharacteristic
289C22×Q8C22×D6111yescharacteristic
290C22×D4C22×D6111yescharacteristic
291C2×M16—612no—
292C2×M16—612no—
293C4⋊Q8—21no—
294C4⋊Q8C22⋊D6112yes—
295#127—21no—
296#126—24no—
297#126—24no—
298#125—1222no—
299#125—1221no—
300#125—1221no—
301#125C22⋊D6112yes—
302#124—24no—
303#124—21no—
304#124—21no—
305#124—21no—
306#124—21no—
307#124—22no—
308#123—24no—
309#123—24no—
310#123—22no—
311C22⋊Q8—24no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314#121—22no—
315#120—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8C22⋊D6112yes—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4C22⋊D6112yes—
325#117—24no—
326#117—22no—
327#117—21no—
328#117C22×D6111yescharacteristic
329#117C22×D6111yescharacteristic
330#117C22×D6111yescharacteristic
331C2×C4⋊C4—612no—
332C2×C4⋊C4—1222no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—1224no—
335C2×C22⋊C4C22⋊D6114yes—
336C2×C42—21no—
337C2×C42—612no—
338C2×C42—1222no—
339C2×C42C22×D6111yescharacteristic
340#107—1222no—
341#107—1222no—
342#107—1222no—
343#106—1222no—
344#106—1222no—
345#106—1222no—
346C4⋊C8—1222no—
347#104—1224no—
348#103—1222no—
349#102—1222no—
350#101—1224no—
351#100—1224no—
352#97—1221no—
353C4×C8—1221no—
354#95—1224no—
355C24C2×C22⋊D6112yes—
356#41—22no—
357#41—22no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8C2×C22⋊D6111yescharacteristic
364C2×D4—21no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4C2×C22⋊D6111yescharacteristic
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4C2×C22⋊D6112yes—
377C22×C4C2×C22⋊D6112yes—
378C22×C4C23×D6111yescharacteristic
379M16—614no—
380M16—1222no—
381M16—1222no—
382M16—1222no—
383C2×C8—1222no—
384C2×C8—1222no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C4⋊C4—2442no—
394C4⋊C4—2442no—
395C4⋊C4C2×C22⋊D6112yes—
396C4⋊C4C2×C22⋊D6112yes—
397C4⋊C4C2×C22⋊D6112yes—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—22no—
407C22⋊C4—2448no—
408C42—24no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—22no—
413C42—1222no—
414C42—1222no—
415C42—2444no—
416C42C2×C22⋊D6111yescharacteristic
417C42C2×C22⋊D6111yescharacteristic
418C42C2×C22⋊D6112yes—
419C23—22no—
420C23—21no—
421C23—24no—
422C23—44no—
423C23C22×C22⋊D6111yescharacteristic
424Q8—21no—
425Q8—21no—
426Q8—41no—
427Q8—41no—
428Q8—42no—
429Q8—82no—
430D4—21no—
431D4—21no—
432D4—42no—
433D4—41no—
434D4—41no—
435D4—42no—
436D4—84no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—1224no—
448C2×C4—2444no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4#1451974612yes—
453C2×C4#1451974612yes—
454C2×C4C22×C22⋊D6111yescharacteristic
455C2×C4C22×C22⋊D6111yescharacteristic
456C2×C4#41×D6112yes—
457C2×C4#1451982412yes—
458C8—1224no—
459C8—2442no—
460C22—22no—
461C22—42no—
462C22—44no—
463C22—41no—
464C22—42no—
465C22—84no—
466C22C2×#1451974611yescharacteristic
467C22#1684041112yes—
468C4—22no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—41no—
475C4—4882no—
476C4—2444no—
477C4#1684030811yescharacteristic
478C4#1684033311yescharacteristic
479C4#1684041912yes—
480C2—22no—
481C2—82no—
482C2—81no—
483C2#2144944311yescharacteristic
484C2#2145516812yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3510C2
socleC22×C61C2×#120
prime radicalC61×#535C22
hypercenter#3510D61
hypocenterC61#25689
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3510

Complements

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

NH ≅ G/Ntype
1. #21456372C2semidirect
2. #21455641C2semidirect
3. #21454189C2semidirect
4. #21447033C2semidirect
6. #21446797C2semidirect
7. #21446669C2semidirect
8. #21446034C2semidirect
9. #21446029C2semidirect
10. C4×#14519704C2semidirect
287. C61#25689semidirect

Maximal subgroups

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

subgroupindex
#214563722
#214556412
#214541892
#214470332
#214470262
#214467972
#214466692
#214460342
#214460292
C4×#145197042
C61×#35102
#2568961