Finite Group Databasebeta

← #29732233

Subgroups — #29732233

Order 15616.

1132 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840004C212yesmaximal
3#21454157C211yesmaximal, characteristic
4#21447065C211yesmaximal, characteristic, complemented
5#21446957C211yesmaximal, characteristic, complemented
6#21446921C211yesmaximal, characteristic, complemented
7#21446241C212yesmaximal
8#21446161C212yesmaximal, complemented
9#21446070C212yesmaximal, complemented
10#336×D61C211yesmaximal, characteristic
11C61×#2057C211yesmaximal, characteristic, complemented
12C22×Dic244C2212yes—
13C22×C8×D61C2212yes—
14#16840317—28no—
15C2×#14519777C2211yescharacteristic
16C2×#14519768—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519757C2212yes—
19C2×#14519753—21no—
20C2×Q8×Dic61—21no—
21C2×#14519742C2212yes—
22C2×#14519735—21no—
23C2×C42×D61C2211yescharacteristic
24#16840004—28no—
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29#16839668C2211yescharacteristic
30#16839667C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839649—21no—
33#16839648—21no—
34#16839633—21no—
35#16839624—21no—
36#16839619C2211yescharacteristic
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#107×C61C2212yes—
42C61×#389C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×Dic122C2312yes—
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C2×Dic244—24no—
51C2×Dic244—24no—
52C2×Dic244—42no—
53C22×#92418C2312yes—
54C2×C8×D61—24no—
55C2×C8×D61—24no—
56C2×C8×D61—42no—
57#14519777—24no—
58#14519768—28no—
59#14519757—24no—
60#14519757—24no—
61#14519757D418yes—
62#14519753—28no—
63Q8×Dic61—28no—
64#14519742—24no—
65#14519742—24no—
66#14519735—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758198—22no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C42×D61—24no—
82#14519702—24no—
83#14519700—24no—
84#14519700—24no—
85#14519700—42no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61—21no—
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101#107×C61—24no—
102#95×C61—21no—
103C23×D61C2×D411yescharacteristic
104C2×Q8×C61—24no—
105C2×Q8×C61—42no—
106C2×Q8×C61—42no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—22no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—41no—
127C2×C4×D61Q1614yescomplemented
128C2×C4×D61C2×D414yes—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122C2×D414yes—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61#4114yes—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61C2×D414yes—
149C42×C61—22no—
150#11758198—28no—
151#11758198—42no—
152#11758198—42no—
153C4⋊Dic61—24no—
154C4⋊Dic61—24no—
155C4⋊Dic61—28no—
156C4⋊Dic61—22no—
157C4⋊Dic61—42no—
158#11758196—28no—
159#11758196—42no—
160C4×Dic61—28no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163C4×Dic61—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167Dic244—48no—
168C8×D61—48no—
169C23×C61C22×D411yescharacteristic
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—21no—
174C22×D61#12012yes—
175C22×D61C2×Q1612yes—
176C22×D61#13712yes—
177Q8×C61—48no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—24no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61C2×Q1612yes—
192C2×C4×C61#13512yes—
193C2×C4×C61C22×D412yes—
194C2×C4×C61C2×#4112yes—
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—24no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61#12012yes—
214C2×Dic61C2×Q1612yes—
215C4×D61—28no—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—44no—
219Dic122—24no—
220Dic122—24no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—42no—
225C8×C61—24no—
226#92418—42no—
227#25498—611nomaximal, Sylow
228C22×C61C2×#12011yescharacteristic
229C22×C61#53211yescharacteristic
230C22×C61#53711yescharacteristic
231C22×C61C22×Q1611yescharacteristic
232C22×C61C2×#13511yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C22×C61#57511yescharacteristic
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61#45114yes—
242C4×C61—22no—
243C4×C61—22no—
244C4×C61—44no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61#54612yes—
248C4×C61#54712yes—
249Dic61—22no—
250Dic61—22no—
251Dic61—44no—
252Dic61—41no—
253Dic61—81no—
254Dic61—81no—
255C2×#533—611no—
256C2×#467—612no—
257#2855—611no—
258#2057D6111yescharacteristic, complemented
259#2043—611no—
260#2024—611no—
261#2016—611no—
262#1892—612no—
263#1878—612no—
264#1863—612no—
265C2×#336—611no—
266C2×C61#243011yescharacteristic
267C2×C61C2×#45111yescharacteristic
268C2×C61#300211yescharacteristic
269C2×C61#317011yescharacteristic
270C2×C61#318111yescharacteristic
271C2×C61#324811yescharacteristic
272C2×C61#324911yescharacteristic
273D61—24no—
274C22×Q16—612no—
275C23×C8—612no—
276#533—1228no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×#125—1221no—
280C2×C22⋊Q8—612no—
281C2×C22⋊Q8—611no—
282C2×C22⋊Q8—1221no—
283C2×C4×Q8—612no—
284C2×C4×Q8—1221no—
285C2×#117—1221no—
286C22×C42—611no—
287#467—1228no—
288C2×#107C2×D6112yes—
289C2×#103—612no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#103—612no—
293#391—611no—
294#391—611no—
295#389—611no—
296#389C2×D6111yescharacteristic
297#387—1221no—
298#387—1221no—
299#385—1221no—
300#381—1221no—
301#380—611no—
302#340—612no—
303#340C2×D6112yes—
304#336—612no—
305#336—611no—
306#336C2×D6111yescharacteristic
307C61#2549811yescharacteristic, complemented, Sylow
308C22×Q8—21no—
309C22×Q8—612no—
310C22×Q8—611no—
311C23×C4—611no—
312C23×C4—611no—
313C23×C4—611no—
314C2×Q16—1224no—
315C2×Q16—1224no—
316C2×Q16—2442no—
317C22×C8—612no—
318C22×C8—1224no—
319C22×C8—1224no—
320C22×C8—2442no—
321C22×C8C22×D6112yes—
322C4⋊Q8—24no—
323C4⋊Q8—1224no—
324#125—1228no—
325C22⋊Q8—618no—
326C22⋊Q8—1224no—
327C22⋊Q8—1228no—
328C22⋊Q8—1224no—
329C4×Q8—1224no—
330C4×Q8—1228no—
331C4×Q8—1224no—
332#117—1228no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C4⋊C4C22×D6111yescharacteristic
344C2×C4⋊C4C22×D6112yes—
345C2×C22⋊C4—1222no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C42—611no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—1224no—
352C2×C42—1222no—
353C2×C42C22×D6111yescharacteristic
354#107—24no—
355#103—1224no—
356#103—1224no—
357#103—1224no—
358#103—2442no—
359#95—21no—
360#95—1221no—
361#95—1221no—
362#95—1221no—
363#95—1221no—
364#95—1221no—
365#95—1221no—
366C24—611no—
367C2×Q8—24no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—614no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8—1221no—
378C2×Q8—1222no—
379C2×Q8—1221no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—614no—
386C22×C4—614no—
387C22×C4—611no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—2441no—
399C22×C4D4×D6111yescharacteristic
400C22×C4D4×D6111yescharacteristic
401C22×C4C23×D6111yescharacteristic
402Q16—2448no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—2448no—
409C2×C8#1175822614yes—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—42no—
416C4⋊C4—1228no—
417C4⋊C4—1228no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1222no—
421C4⋊C4—2442no—
422C4⋊C4—2442no—
423C4⋊C4D4×D6114yes—
424C22⋊C4—1228no—
425C22⋊C4—2442no—
426C22⋊C4—2442no—
427C42—22no—
428C42—1222no—
429C42—1222no—
430C42—1228no—
431C42—1222no—
432C23—612no—
433C23—612no—
434C23—612no—
435C23—1221no—
436C23—1221no—
437C23—1221no—
438C23—1221no—
439C23C2×D4×D6111yescharacteristic
440Q8—48no—
441Q8—1224no—
442Q8—1224no—
443Q8—2442no—
444Q8—2442no—
445Q8—2442no—
446Q8—2442no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—612no—
461C2×C4—612no—
462C2×C4—1228no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1224no—
468C2×C4—1224no—
469C2×C4—1221no—
470C2×C4—1224no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—2444no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4Q16×D6112yes—
483C2×C4#1451979612yes—
484C2×C4C2×D4×D6112yes—
485C2×C4C2×#1175822612yes—
486C2×C4#1451982311yescharacteristic
487C2×C4#1451982511yescharacteristic
488C8—24no—
489C8—2442no—
490C22—614no—
491C22—1222no—
492C22—1222no—
493C22—1222no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22#120×D6111yescharacteristic
498C22#1684031311yescharacteristic
499C22#1684034211yescharacteristic
500C22C2×Q16×D6111yescharacteristic
501C22C2×#1451979611yescharacteristic
502C22#137×D6111yescharacteristic
503C22#1684055811yescharacteristic
504C4—22no—
505C4—22no—
506C4—44no—
507C4—81no—
508C4—81no—
509C4—1222no—
510C4—1222no—
511C4—4881no—
512C4—4881no—
513C4—2441no—
514C4—2444no—
515C4#1684040312yes—
516C4#1684041512yes—
517C2—1224no—
518C2#2144943011yescharacteristic
519C2#451×D6111yescharacteristic
520C2#2145503211yescharacteristic
521C2#2145640611yescharacteristic
522C2#2145653011yescharacteristic
523C2#2145715811yescharacteristic
524C2#2145716311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2057C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2057D61
hypocenterC61#25498
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
4. #21447065C2semidirect
5. #21446957C2semidirect
6. #21446921C2semidirect
8. #21446161C2semidirect
9. #21446070C2semidirect
11. C61×#2057C2semidirect
127. C2×C4×D61Q16semidirect
258. #2057D61semidirect
307. C61#25498semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168400042
#214541572
#214470652
#214469572
#214469212
#214462412
#214461612
#214460702
#336×D612
C61×#20572
#2549861