Finite Group Databasebeta

← #29722570

Subgroups — #29722570

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840033C211yesmaximal, characteristic, complemented
2#21455438C212yesmaximal, complemented
3C2×#16840015C211yesmaximal, characteristic, complemented
4#21454522C211yesmaximal, characteristic, complemented
5#21447174C212yesmaximal, complemented
6#21447113C212yesmaximal, complemented
7#21446727C211yesmaximal, characteristic, complemented
8#21446600C211yesmaximal, characteristic, complemented
9#21446282C212yesmaximal, complemented
10#21445875C211yesmaximal, characteristic, complemented
11C2×C61×#430C211yesmaximal, characteristic, complemented
12C2×#14519805C2212yes—
13C22×D488C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840273C2211yescharacteristic
16C2×#14519751C2212yes—
17#16840033—24no—
18#16840015—22no—
19#16840015—22no—
20#16840015—22no—
21#16840008—22no—
22#16839998—22no—
23#16839991—22no—
24#16839838—22no—
25#16839831—22no—
26#16839804C2212yes—
27C2×#14519709C2211yescharacteristic
28#16839788C2211yescharacteristic
29#16839768C2212yes—
30#16839764C2212yes—
31C2×#14519699C2212yes—
32C2×#14519693C2212yes—
33C2×#14519693C2212yes—
34#16839728C2211yescharacteristic
35#16839726C2211yescharacteristic
36#16839717C2211yescharacteristic
37#16839530C2212yes—
38#16839530C2212yes—
39C22×M16×C61C2212yes—
40C22×M16×C61C2211yescharacteristic
41C61×#430—24no—
42C2×#108×C61C2212yes—
43C2×#108×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C2×#11758222C2311yescharacteristic
46C22×D244C2312yes—
47C22×C22⋊D61—22no—
48#14519805—24no—
49#14519805—42no—
50C2×#11758190—22no—
51C2×D488—22no—
52C2×D488—21no—
53C2×D488—21no—
54C2×D488—22no—
55C2×D488—22no—
56#14519783—22no—
57#14519783—22no—
58C2×#11758194C2311yescharacteristic
59C2×#11758194C2311yescharacteristic
60C2×#11758189—21no—
61C2×C8×D61—21no—
62#14519760—21no—
63#14519756D412yes—
64#14519751—22no—
65#14519751—22no—
66#14519751—22no—
67#14519751—22no—
68#14519751D412yes—
69#14519746—21no—
70C4×C22⋊D61—22no—
71#14519736C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758198—22no—
74#14519711—22no—
75#14519709—21no—
76#14519709—21no—
77#14519709—22no—
78#14519706—22no—
79#14519702—22no—
80#14519699—22no—
81#14519699—24no—
82#14519699—22no—
83#14519699—22no—
84#14519693—24no—
85#14519693—42no—
86C22⋊#92418—22no—
87#14519689—22no—
88#14519687—21no—
89C8×Dic61—21no—
90C23×C4×C61C2311yescharacteristic
91C2×M16×C61—22no—
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61—24no—
95C2×M16×C61—42no—
96C2×M16×C61D412yes—
97C2×M16×C61D412yes—
98C2×M16×C61C2312yes—
99C2×M16×C61C2312yes—
100C22×C8×C61—22no—
101C22×C8×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103#108×C61—24no—
104#108×C61—22no—
105#108×C61D414yescomplemented
106C24×C61#4111yescharacteristic
107C23×D61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—21no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—84no—
121C22×Dic61—22no—
122#11758222—21no—
123#11758222—21no—
124#11758222—21no—
125#11758222—21no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244C2×D411yescharacteristic
133C2×C4×D61—21no—
134C2×Dic122C2×D411yescharacteristic
135M16×C61—22no—
136M16×C61—24no—
137M16×C61—24no—
138M16×C61—22no—
139M16×C61—22no—
140M16×C61—44no—
141M16×C61C2×D414yes—
142C2×C8×C61—21no—
143C2×C8×C61—22no—
144C2×C8×C61—21no—
145C2×C8×C61—22no—
146C2×C8×C61—44no—
147C2×C8×C61C2×D412yes—
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D412yes—
151C2×C8×C61#4112yes—
152C22⋊Dic61—22no—
153C22⋊Dic61—21no—
154#11758198—22no—
155#11758198—42no—
156#11758198—42no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61C2×D412yes—
160#11758196—22no—
161C4×Dic61—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—21no—
165#11758194—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168D488—44no—
169D488—42no—
170D488—42no—
171#11758190—44no—
172#11758189—42no—
173C8×D61—42no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—42no—
177C23×C61#12112yes—
178C23×C61C2×#4111yescharacteristic
179C22×D61—21no—
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—82no—
184C22×D61—82no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61#12112yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61H(5,𝔽2)11yescharacteristic
197C2×C4×C612−1+411yescharacteristic
198C22⋊D61—41no—
199C22⋊D61—41no—
200C22⋊D61—82no—
201C22⋊D61—84no—
202C22⋊D61—84no—
203C2×Dic61—22no—
204C2×Dic61—21no—
205C2×Dic61—42no—
206C2×Dic61—82no—
207D244—21no—
208D244—21no—
209D244—42no—
210D244—42no—
211D244—42no—
212D244—42no—
213C4×D61—41no—
214C4×D61—41no—
215Dic122—21no—
216Dic122—21no—
217C8×C61—22no—
218C8×C61—24no—
219C8×C61—42no—
220C8×C61#12112yes—
221C8×C61#12112yes—
222#92418—41no—
223#92418—41no—
224#24806—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—22no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61C2×#12111yescharacteristic
232C22×C61#54811yescharacteristic
233C22×C61#55311yescharacteristic
234C2×D61—41no—
235C2×D61—42no—
236C2×D61—82no—
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—162no—
240C2×D61—162no—
241C4×C61—22no—
242C4×C61C2×#12111yescharacteristic
243C4×C61#53711yescharacteristic
244C4×C61D4211yescharacteristic
245C4×C61#54711yescharacteristic
246Dic61—41no—
247Dic61—42no—
248Dic61—82no—
249C2×#472—611no—
250#3055—612no—
251C2×#469—611no—
252#2930—611no—
253C2×#430D6111yescharacteristic, complemented
254#2063—612no—
255#2051—612no—
256#1978—611no—
257#1951—611no—
258#1900—612no—
259#1815—611no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61—42no—
263C2×C61#241611yescharacteristic
264C2×C61#306812yes—
265D61—81no—
266D61—162no—
267C2×AGL(1,ℤ/8)—612no—
268C22×D8—611no—
269C2×#131—611no—
270C22×M16C2×D6112yes—
271C22×M16C2×D6111yescharacteristic
272#525—611no—
273C2×#121—612no—
274#472—1224no—
275#469—1222no—
276#469—1222no—
277#469—1222no—
278#468—1222no—
279#466—1222no—
280#465—1222no—
281#436—1222no—
282#435—1222no—
283#430—24no—
284C2×#108C2×D6112yes—
285C2×#108C2×D6111yescharacteristic
286C2×#108C2×D6111yescharacteristic
287#428—612no—
288C2×#106—611no—
289#424—611no—
290#418—612no—
291#417—612no—
292C2×#102—612no—
293C2×#100—612no—
294C2×#100—612no—
295#408—611no—
296#407—611no—
297#405—611no—
298#343—612no—
299#343—612no—
300C61#2480611yescharacteristic, complemented, Sylow
301C2×#41—611no—
302C22×D4—612no—
303C22×D4—1222no—
304C23×C4C22×D6111yescharacteristic
305AGL(1,ℤ/8)—1224no—
306AGL(1,ℤ/8)—2442no—
307C2×SD16—1222no—
308C2×D8—1222no—
309C2×D8—1221no—
310C2×D8—1221no—
311C2×D8—1222no—
312C2×D8—1222no—
313#131—1222no—
314#131—1222no—
315C2×M16—22no—
316C2×M16—22no—
317C2×M16—22no—
318C2×M16—24no—
319C2×M16—42no—
320C2×M16—611no—
321C2×M16—611no—
322C2×M16—1221no—
323C2×M16D24412yes—
324C2×M16D24412yes—
325C2×M16C22×D6112yes—
326C2×M16C22×D6112yes—
327C22×C8—22no—
328C22×C8—1221no—
329C22×C8C22×D6111yescharacteristic
330C22×C8C22×D6111yescharacteristic
331#123—1221no—
332C22⋊Q8—612no—
333#121—612no—
334#121—1222no—
335#121—1222no—
336#121—1222no—
337#121—1222no—
338#120—1221no—
339C4×D4—1222no—
340#117—611no—
341C2×C4⋊C4—612no—
342C2×C22⋊C4—1222no—
343#108—24no—
344#108—22no—
345#108C22⋊D6114yescomplemented
346#107—1222no—
347#106—1222no—
348#106—1221no—
349#106—1221no—
350C4⋊C8—1222no—
351#103—1222no—
352#102—1222no—
353#102—1222no—
354#102—1222no—
355#102—1224no—
356#100—1224no—
357#100—2442no—
358C22⋊C8—1222no—
359C22⋊C8—1222no—
360#97—1221no—
361C4×C8—1221no—
362C24—1222no—
363C24#1175822611yescharacteristic
364#41—1221no—
365#41—1221no—
366#41—1221no—
367#41—1221no—
368C2×Q8—611no—
369C2×D4—611no—
370C2×D4—1222no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4—1222no—
378C2×D4—4884no—
379C2×D4—2442no—
380C2×D4—2442no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—1221no—
386C22×C4—1222no—
387C22×C4C2×D24412yes—
388C22×C4D4×D6112yes—
389C22×C4#1175822611yescharacteristic
390C22×C4C23×D6111yescharacteristic
391SD16—2444no—
392D8—2444no—
393D8—2442no—
394D8—2442no—
395M16—22no—
396M16—24no—
397M16—24no—
398M16—22no—
399M16—22no—
400M16—44no—
401M16—1221no—
402M16—1221no—
403M16—1221no—
404M16—1221no—
405M16—2442no—
406M16C2×C22⋊D6114yes—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—44no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—2442no—
415C2×C8C2×D24412yes—
416C2×C8C2×D24411yescharacteristic
417C2×C8C2×D24411yescharacteristic
418C2×C8#1175822212yes—
419C2×C8C2×C22⋊D6112yes—
420C4⋊C4—612no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C22⋊C4—1221no—
425C22⋊C4—1222no—
426C22⋊C4—1222no—
427C22⋊C4—2442no—
428C22⋊C4—2442no—
429C42—1221no—
430C23—21no—
431C23—21no—
432C23—42no—
433C23—1221no—
434C23—4882no—
435C23—4882no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23#1451975212yes—
440C23C2×#1175822611yescharacteristic
441Q8—1221no—
442Q8—1221no—
443D4—1221no—
444D4—1221no—
445D4—4884no—
446D4—4884no—
447D4—4882no—
448D4—2441no—
449D4—2441no—
450D4—2442no—
451D4—2442no—
452D4—2442no—
453D4—2442no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—1222no—
461C2×C4—1221no—
462C2×C4—4882no—
463C2×C4—2442no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4#1451975212yes—
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C22×C22⋊D6111yescharacteristic
469C2×C4C22×D24411yescharacteristic
470C2×C4C2×#1175822211yescharacteristic
471C2×C4#1451982211yescharacteristic
472C2×C4#1451982511yescharacteristic
473C8—22no—
474C8—24no—
475C8—42no—
476C8—2441no—
477C8—2441no—
478C8#1451975112yes—
479C8#1451975112yes—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—81no—
485C22—81no—
486C22—9762no—
487C22—9762no—
488C22—4882no—
489C22—4882no—
490C22—4882no—
491C22—2441no—
492C22—2442no—
493C22C2×#1451975211yescharacteristic
494C22#1684042311yescharacteristic
495C22#1684046011yescharacteristic
496C4—22no—
497C4—4882no—
498C4—2441no—
499C4—2442no—
500C4C2×#1451975111yescharacteristic
501C4#1684034311yescharacteristic
502C4D4×C22⋊D6111yescharacteristic
503C4#1684041311yescharacteristic
504C2—21no—
505C2—21no—
506C2—42no—
507C2—9762no—
508C2—4881no—
509C2#2144925311yescharacteristic
510C2#2145554812yes—
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#430C2
socleC22×C61C2×#121
hypercenterC2×#430D61
hypocenterC61#24806
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
1. C2×#16840033C2semidirect
2. #21455438C2semidirect
3. C2×#16840015C2semidirect
4. #21454522C2semidirect
5. #21447174C2semidirect
6. #21447113C2semidirect
7. #21446727C2semidirect
8. #21446600C2semidirect
9. #21446282C2semidirect
10. #21445875C2semidirect
11. C2×C61×#430C2semidirect
105. #108×C61D4semidirect
253. C2×#430D61semidirect
300. C61#24806semidirect
345. #108C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400332
#214554382
C2×#168400152
#214545222
#214471742
#214471132
#214467272
#214466002
#214462822
#214458752
C2×C61×#4302
#2480661