Finite Group Databasebeta

← #29779579

Subgroups — #29779579

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454171C211yesmaximal, characteristic, complemented
2#21447227C211yesmaximal, characteristic, complemented
3#21447222C211yesmaximal, characteristic, complemented
4#21446960C211yesmaximal, characteristic, complemented
5#21446955C211yesmaximal, characteristic, complemented
6#21446519C212yesmaximal, complemented
7#21446512C212yesmaximal, complemented
8#21445914C212yesmaximal, complemented
9#21445759C212yesmaximal
10#21445581C211yesmaximal, characteristic, complemented
11C61×#2297C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519756—21no—
14C2×#14519701C2212yes—
15C2×#14519701C2212yes—
16C2×#14519698C2212yes—
17C2×#14519698C2212yes—
18C2×C22⋊#92418C2212yes—
19C2×#14519688C2212yes—
20#16839668—21no—
21#16839660—21no—
22#16839660C2212yes—
23#16839632C2211yescharacteristic
24#16839622—21no—
25#16839622C2212yes—
26#16839524C2212yes—
27#16839524C2212yes—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839517C2211yescharacteristic
31#16839517C2211yescharacteristic
32C22×#41×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C61×#391C2211yescharacteristic
36C61×#391C2211yescharacteristic
37C61×#390C2211yescharacteristic
38C61×#390C2211yescharacteristic
39C61×#386—21no—
40C61×#386—21no—
41C4×C22⋊C4×C61—21no—
42C4×C22⋊C4×C61—21no—
43C22×Dic122—21no—
44C22×#92418C2312yes—
45C22×#92418C2312yes—
46#14519777—42no—
47#14519756—44no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2312yes—
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—22no—
54C2×C22⋊Dic61—22no—
55#14519701—24no—
56#14519701—42no—
57#14519698—24no—
58#14519698—42no—
59C22⋊#92418—24no—
60#14519688—42no—
61#14519684—22no—
62#14519684—22no—
63C2×#41×C61—22no—
64C2×#41×C61—22no—
65C2×#41×C61—24no—
66C2×#41×C61—42no—
67C22×Q8×C61C2311yescharacteristic
68C22×D4×C61—21no—
69C22×D4×C61C2311yescharacteristic
70C23×C4×C61—21no—
71C23×C4×C61D411yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74#117×C61—24no—
75C2×C4⋊C4×C61C2311yescharacteristic
76C2×C4⋊C4×C61C2311yescharacteristic
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—21no—
81C2×C22⋊C4×C61—21no—
82C2×C42×C61—21no—
83C2×C42×C61—42no—
84C2×C42×C61D411yescharacteristic
85C2×C42×C61D411yescharacteristic
86C2×C42×C61C2311yescharacteristic
87C2×C42×C61C2311yescharacteristic
88#95×C61—21no—
89#95×C61—21no—
90#95×C61—21no—
91#95×C61—21no—
92C24×C61—21no—
93C24×C61C2×D411yescharacteristic
94#41×C61—48no—
95#41×C61—48no—
96C2×Q8×C61—21no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61#4112yes—
100C2×D4×C61—24no—
101C2×D4×C61—22no—
102C2×D4×C61—21no—
103C2×D4×C61—22no—
104C2×D4×C61—42no—
105C2×D4×C61—42no—
106C2×D4×C61#4112yes—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—24no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61#4114yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—22no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×Dic122—42no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C22⋊C4×C61—42no—
137C22⋊C4×C61—42no—
138C22⋊C4×C61—42no—
139C22⋊C4×C61—42no—
140C22⋊C4×C61—42no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61—41no—
144C42×C61—41no—
145C42×C61—42no—
146C22⋊Dic61—44no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61—42no—
153C4⋊Dic61—42no—
154#11758196—44no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—22no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61#12412yes—
175C23×C61C22×D411yescharacteristic
176Q8×C61—24no—
177Q8×C61—42no—
178D4×C61—24no—
179D4×C61—48no—
180D4×C61—42no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—44no—
197C2×C4×C61—44no—
198C2×C4×C61—48no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61#12412yes—
208C2×C4×C61AGL(1,ℤ/8)12yes—
209C2×C4×C61#13712yes—
210C2×C4×C61C2×#4112yes—
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61H(5,𝔽2)12yes—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220Dic122—84no—
221#92418—42no—
222#92418—42no—
223#29113—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61#52511yescharacteristic
239C22×C61C2×#12411yescharacteristic
240C22×C61#53411yescharacteristic
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61C2×#13711yescharacteristic
243C22×C61#57511yescharacteristic
244C22×C61#57511yescharacteristic
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—22no—
248C4×C61—42no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61—81no—
252C4×C61—81no—
253C4×C61#55112yes—
254C4×C61#55112yes—
255Dic61—82no—
256Dic61—82no—
257#2857—611no—
258#2297D6111yescharacteristic
259#2072—611no—
260#2071—611no—
261#2024—611no—
262#2023—611no—
263#1936—612no—
264#1935—612no—
265#1825—612no—
266#1788—612no—
267#1749—611no—
268C2×C61—42no—
269C2×C61—82no—
270C2×C61#256111yescharacteristic
271C2×C61#311511yescharacteristic
272C2×C61#311611yescharacteristic
273C2×C61#313511yescharacteristic
274C2×C61#314111yescharacteristic
275C2×C61#319411yescharacteristic
276C2×C61#319611yescharacteristic
277C22×#41C2×D6111yescharacteristic
278C2×C4⋊Q8—611no—
279C2×C22⋊Q8—1221no—
280C2×#117C2×D6111yescharacteristic
281C22×C42C2×D6111yescharacteristic
282C2×#103—612no—
283C2×#103—612no—
284C2×#102—612no—
285C2×#102—612no—
286C2×C22⋊C8—612no—
287C2×#97—612no—
288#391—1221no—
289#391C2×D6111yescharacteristic
290#391C2×D6111yescharacteristic
291#390C2×D6111yescharacteristic
292#390C2×D6111yescharacteristic
293#389—612no—
294#389—1221no—
295#386—21no—
296#386—21no—
297#384—611no—
298#380—612no—
299#380—1221no—
300C4×C22⋊C4—21no—
301C4×C22⋊C4—21no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340—612no—
306#336—611no—
307#336—611no—
308C61#2911311yescharacteristic, complemented, Sylow
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—24no—
312C2×#41—42no—
313C22×Q8—1221no—
314C22×Q8C22×D6111yescharacteristic
315C22×D4—21no—
316C22×D4C22×D6111yescharacteristic
317C23×C4—21no—
318C23×C4C22⋊D6111yescharacteristic
319C23×C4C22⋊D6111yescharacteristic
320C23×C4C22×D6111yescharacteristic
321C22×C8—612no—
322C22×C8—612no—
323C4⋊Q8—2442no—
324C22⋊Q8—2444no—
325#117—24no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C4⋊C4C22×D6111yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—1222no—
340C2×C42—21no—
341C2×C42—42no—
342C2×C42C22⋊D6111yescharacteristic
343C2×C42C22⋊D6111yescharacteristic
344C2×C42C22×D6111yescharacteristic
345C2×C42C22×D6111yescharacteristic
346#103—1224no—
347#103—2442no—
348#102—1224no—
349#102—2442no—
350C22⋊C8—1224no—
351#97—2442no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—1222no—
357#95—1222no—
358C24—21no—
359C24C2×C22⋊D6111yescharacteristic
360#41—48no—
361#41—48no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—2442no—
366C2×Q8—2442no—
367C2×Q8—2442no—
368C2×Q8#1175822212yes—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4#1175822212yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4#1175822414yes—
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—1222no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—2444no—
413C4⋊C4—2442no—
414C4⋊C4—2442no—
415C4⋊C4—2442no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—2444no—
422C42—22no—
423C42—22no—
424C42—41no—
425C42—41no—
426C42—42no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23#1451976412yes—
441C23C22×C22⋊D6111yescharacteristic
442Q8—24no—
443Q8—42no—
444Q8—4884no—
445D4—24no—
446D4—48no—
447D4—42no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4#1451976412yes—
481C2×C4#1451980212yes—
482C2×C4#1451980712yes—
483C2×C4C2×#1175822412yes—
484C2×C4C2×#1175822211yescharacteristic
485C2×C4C2×#1175822211yescharacteristic
486C2×C4#1451982212yes—
487C8—2442no—
488C8—2442no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22#1684027311yescharacteristic
504C22C2×#1451976411yescharacteristic
505C22#1684032111yescharacteristic
506C22C2×#1451980211yescharacteristic
507C22C2×#1451980711yescharacteristic
508C22#1684056011yescharacteristic
509C22#1684056011yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—42no—
515C4—41no—
516C4—81no—
517C4—81no—
518C4—4882no—
519C4—4882no—
520C4#1684043812yes—
521C4#1684043812yes—
522C2—42no—
523C2—82no—
524C2#2145097711yescharacteristic
525C2#2145597011yescharacteristic
526C2#2145598111yescharacteristic
527C2#2145616411yescharacteristic
528C2#2145621211yescharacteristic
529C2#2145662911yescharacteristic
530C2#2145664511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2297C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenter#2297D61
hypocenterC61#29113
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. #21454171C2semidirect
2. #21447227C2semidirect
3. #21447222C2semidirect
4. #21446960C2semidirect
5. #21446955C2semidirect
6. #21446519C2semidirect
7. #21446512C2semidirect
8. #21445914C2semidirect
10. #21445581C2semidirect
308. C61#29113semidirect

Maximal subgroups

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

subgroupindex
#214541712
#214472272
#214472222
#214469602
#214469552
#214465192
#214465122
#214459142
#214457592
#214455812
C61×#22972
#2911361