Finite Group Databasebeta

← #29779558

Subgroups — #29779558

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454166C211yesmaximal, characteristic
2#391×D61C211yesmaximal, characteristic
3#21447229C211yesmaximal, characteristic, complemented
4#21447227C211yesmaximal, characteristic, complemented
5#21446957C211yesmaximal, characteristic, complemented
6#21446507C212yesmaximal, complemented
7#21446506C212yesmaximal, complemented
8#21445904C212yesmaximal
9#21445752C212yesmaximal
10#21445580C211yesmaximal, characteristic
11C61×#2024C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C2×#14519753—21no—
14C2×C4⋊C4×D61C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16C2×#14519702C2212yes—
17C2×#14519701C2212yes—
18C2×#14519700C2212yes—
19C2×#14519689C2212yes—
20C2×#14519687C2212yes—
21#16839669—21no—
22#16839669C2211yescharacteristic
23#16839668C2211yescharacteristic
24#16839667C2211yescharacteristic
25#16839666—21no—
26#16839666—21no—
27#16839658—21no—
28#16839658C2212yes—
29#16839632C2211yescharacteristic
30#16839619—21no—
31#16839619C2212yes—
32#95×D61—21no—
33#95×D61—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#103×C61C2212yes—
40C61×#391C2211yescharacteristic
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C22×Dic122—21no—
44C22×Dic122C2311yescharacteristic
45C2×Q8×D61—22no—
46C2×Q8×D61—22no—
47C2×Q8×D61—24no—
48C2×Q8×D61—42no—
49C22×C4×D61—21no—
50C22×C4×D61—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55#14519753—44no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—21no—
61C2×#11758196—22no—
62C4⋊C4×D61—24no—
63C2×#11758198—22no—
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C42×D61—42no—
68#14519702—24no—
69#14519701—24no—
70#14519700—42no—
71#14519689—24no—
72#14519687—42no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684—22no—
80#14519684—22no—
81C22×Q8×C61—21no—
82C22×Q8×C61C2311yescharacteristic
83C22×C8×C61C2312yes—
84C4⋊Q8×C61—42no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C42×C61C2311yescharacteristic
89#103×C61—42no—
90#95×C61—21no—
91#95×C61—21no—
92C23×D61C2×D411yescharacteristic
93C2×Q8×C61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C2×Q8×C61#4112yes—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61C2×D411yescharacteristic
110C22×Dic61C2×D411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112Q8×D61—48no—
113Q8×D61—48no—
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61—22no—
117C2×C4×D61—24no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61#4114yes—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—24no—
133C2×Dic122—21no—
134C2×Dic122—42no—
135C2×Dic122—42no—
136C2×Dic122#4112yes—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—42no—
144C42×C61—22no—
145#11758198—44no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—42no—
150C4⋊Dic61—42no—
151#11758196—44no—
152#11758196—42no—
153C4×Dic61—22no—
154C4×Dic61—41no—
155C4×Dic61—41no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C23×C61C22×D411yescharacteristic
160C22×D61—21no—
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61#12412yes—
165C22×D61AGL(1,ℤ/8)12yes—
166C22×D61#13712yes—
167Q8×C61—24no—
168Q8×C61—42no—
169Q8×C61—84no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—22no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61#13712yes—
184C2×C4×C61#13712yes—
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4112yes—
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61H(5,𝔽2)12yes—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—22no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61#12412yes—
207C4×D61—24no—
208C4×D61—24no—
209C4×D61—44no—
210C4×D61—44no—
211C4×D61—48no—
212C4×D61—84no—
213Dic122—24no—
214Dic122—48no—
215Dic122—42no—
216C8×C61—42no—
217#92418—42no—
218#29111—611nomaximal, Sylow
219C22×C61#52511yescharacteristic
220C22×C61C2×#12411yescharacteristic
221C22×C61#53411yescharacteristic
222C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C61C2×#13711yescharacteristic
224C22×C61C2×#13711yescharacteristic
225C22×C61C2×#13711yescharacteristic
226C2×D61—22no—
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—22no—
232C2×D61#48914yes—
233C4×C61—22no—
234C4×C61—22no—
235C4×C61—42no—
236C4×C61—81no—
237C4×C61—82no—
238C4×C61#55112yes—
239C4×C61#55112yes—
240Dic61—22no—
241Dic61—42no—
242Dic61—41no—
243Dic61—41no—
244Dic61—82no—
245Dic61—81no—
246#2857—611no—
247C2×#391—611no—
248#2072—611no—
249#2072—611no—
250#2024—611no—
251#2024D6111yescharacteristic, complemented
252#1935—612no—
253#1935—612no—
254#1823—612no—
255#1787—612no—
256#1749—611no—
257C2×C61#256111yescharacteristic
258C2×C61#311211yescharacteristic
259C2×C61#311311yescharacteristic
260C2×C61C2×#48911yescharacteristic
261C2×C61#313911yescharacteristic
262C2×C61#319311yescharacteristic
263C2×C61#319511yescharacteristic
264D61—24no—
265C23×Q8—611no—
266C2×C4⋊Q8C2×D6111yescharacteristic
267C2×C22⋊Q8—1221no—
268C22×C4⋊C4—611no—
269C22×C42—611no—
270C2×#103—612no—
271C2×#103—612no—
272C2×#103—612no—
273C2×#103C2×D6112yes—
274C2×C22⋊C8—612no—
275C2×#97—612no—
276#391—611no—
277#391—611no—
278#391—611no—
279#391—1221no—
280#391—1221no—
281#391—1221no—
282#391C2×D6111yescharacteristic
283#389—612no—
284#389—1221no—
285#384—611no—
286#380—612no—
287#380—1221no—
288C2×#95—1221no—
289C2×#95—1221no—
290#340—612no—
291#340—612no—
292#340—612no—
293#340C2×D6112yes—
294#336—611no—
295#336C2×D6111yescharacteristic
296C61#2911111yescharacteristic, complemented, Sylow
297C22×Q8—21no—
298C22×Q8—611no—
299C22×Q8—1221no—
300C22×Q8—1224no—
301C22×Q8—1222no—
302C22×Q8—1222no—
303C22×Q8—2442no—
304C22×Q8C22×D6111yescharacteristic
305C23×C4—611no—
306C23×C4—611no—
307C23×C4—611no—
308C23×C4—1221no—
309C23×C4—1221no—
310C22×C8—612no—
311C22×C8C22×D6112yes—
312C4⋊Q8—42no—
313C22⋊Q8—2444no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—1224no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C4⋊C4C22×D6112yes—
324C2×C22⋊C4—1222no—
325C2×C42—611no—
326C2×C42—611no—
327C2×C42—611no—
328C2×C42—2442no—
329C2×C42C22×D6111yescharacteristic
330#103—42no—
331#103—1224no—
332#103—1224no—
333#103—2442no—
334C22⋊C8—1224no—
335#97—2442no—
336#95—21no—
337#95—21no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342#95—1222no—
343#95—1222no—
344#95—1221no—
345#95—1221no—
346C24—611no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—612no—
354C2×Q8—1224no—
355C2×Q8—1221no—
356C2×Q8—1222no—
357C2×Q8—1222no—
358C2×Q8—2442no—
359C2×Q8—2442no—
360C2×Q8—2448no—
361C2×Q8—2448no—
362C2×Q8#1175822612yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—611no—
367C22×C4—614no—
368C22×C4—611no—
369C22×C4—611no—
370C22×C4—1222no—
371C22×C4—1222no—
372C22×C4—1224no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1222no—
376C22×C4—1221no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1221no—
380C22×C4—1222no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4D4×D6111yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4C23×D6111yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—1222no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4—2442no—
405C4⋊C4—2442no—
406C4⋊C4—2444no—
407C4⋊C4—2442no—
408C22⋊C4—2444no—
409C42—22no—
410C42—1222no—
411C42—2441no—
412C42—2441no—
413C23—612no—
414C23—612no—
415C23—612no—
416C23—1221no—
417C23—1221no—
418C23—1221no—
419C23—1221no—
420C23C2×D4×D6111yescharacteristic
421Q8—24no—
422Q8—42no—
423Q8—84no—
424Q8—1224no—
425Q8—2448no—
426Q8—2442no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—612no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—1224no—
444C2×C4—1224no—
445C2×C4—1222no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—4884no—
453C2×C4—2444no—
454C2×C4—2444no—
455C2×C4—2448no—
456C2×C4—2442no—
457C2×C4—2441no—
458C2×C4—2442no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4#1451980612yes—
465C2×C4#1451980912yes—
466C2×C4#41×D6112yes—
467C2×C4#41×D6111yescharacteristic
468C2×C4C2×#1175822611yescharacteristic
469C2×C4#1451982212yes—
470C8—42no—
471C8—2442no—
472C22—614no—
473C22—1222no—
474C22—1222no—
475C22—1222no—
476C22—1222no—
477C22—1222no—
478C22—1222no—
479C22#1684027511yescharacteristic
480C22#124×D6111yescharacteristic
481C22#1684032211yescharacteristic
482C22AGL(1,ℤ/8)×D6111yescharacteristic
483C22#137×D6111yescharacteristic
484C22C2×#1451980611yescharacteristic
485C22C2×#1451980911yescharacteristic
486C4—22no—
487C4—22no—
488C4—42no—
489C4—81no—
490C4—82no—
491C4—1222no—
492C4—4881no—
493C4—4882no—
494C4—2441no—
495C4—2442no—
496C4—2441no—
497C4#1684043912yes—
498C4#1684044212yes—
499C2—1224no—
500C2#2145097311yescharacteristic
501C2#2145594311yescharacteristic
502C2#2145595111yescharacteristic
503C2#489×D6111yescharacteristic
504C2#2145619611yescharacteristic
505C2#2145662211yescharacteristic
506C2#2145664111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2024D61
hypocenterC61#29111
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
3. #21447229C2semidirect
4. #21447227C2semidirect
5. #21446957C2semidirect
6. #21446507C2semidirect
7. #21446506C2semidirect
11. C61×#2024C2semidirect
251. #2024D61semidirect
296. C61#29111semidirect

Maximal subgroups

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

subgroupindex
#214541662
#391×D612
#214472292
#214472272
#214469572
#214465072
#214465062
#214459042
#214457522
#214455802
C61×#20242
#2911161