Finite Group Databasebeta

← #29722576

Subgroups — #29722576

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×#16840032C211yesmaximal, characteristic, complemented
2#21455440C212yesmaximal, complemented
3C2×#16840015C211yesmaximal, characteristic, complemented
4C2×#16839811C211yesmaximal, characteristic, complemented
5#21447178C212yesmaximal, complemented
6#21447110C212yesmaximal, complemented
7#21446729C211yesmaximal, characteristic, complemented
8#21446599C211yesmaximal, characteristic, complemented
9#21446280C212yesmaximal, complemented
10#21445878C211yesmaximal, characteristic, complemented
11C61×#2930C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14C22×#11758194C2212yes—
15C2×#14519751C2212yes—
16#16840032—24no—
17#16840015—22no—
18#16840015—22no—
19#16840013—22no—
20#16840010—22no—
21#16839996—22no—
22#16839993—22no—
23#16839811—24no—
24C2×#14519713C2212yes—
25C2×#14519713C2211yescharacteristic
26C2×#14519713C2211yescharacteristic
27#16839805C2212yes—
28C2×#14519709C2211yescharacteristic
29#16839766C2212yes—
30#16839763C2212yes—
31C2×#14519699C2212yes—
32C2×#14519693C2212yes—
33C2×#14519693C2212yes—
34#16839530C2212yes—
35#16839530C2212yes—
36C2×#131×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C61×#525C2211yescharacteristic
39C61×#436—22no—
40C61×#435—22no—
41C61×#424C2211yescharacteristic
42C61×#408C2211yescharacteristic
43C61×#407C2211yescharacteristic
44C61×#405C2211yescharacteristic
45C22×D244C2312yes—
46C22×C22⋊D61—22no—
47#14519802—24no—
48#14519802—42no—
49C2×#11758201—22no—
50C2×D488—21no—
51C2×D488—21no—
52C2×D488—22no—
53C2×D488—22no—
54C2×#11758199—22no—
55C2×#11758194—22no—
56C2×#11758194—24no—
57C2×#11758194—42no—
58C2×#11758194C2312yes—
59C2×#11758194C2312yes—
60C22×#92418—22no—
61#14519751—22no—
62#14519751—22no—
63#14519751—22no—
64#14519751—22no—
65C2×C4⋊Dic61C2312yes—
66C2×#11758198—22no—
67#14519713—22no—
68#14519713—24no—
69#14519713D414yescomplemented
70#14519710—22no—
71#14519709—21no—
72#14519709—21no—
73#14519708—22no—
74#14519701—22no—
75#14519699—24no—
76#14519698—22no—
77#14519697—22no—
78#14519697—22no—
79#14519693—24no—
80#14519693—42no—
81C2×#41×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83#131×C61—22no—
84#131×C61—22no—
85C2×M16×C61—22no—
86C2×M16×C61—21no—
87C2×M16×C61—22no—
88C2×M16×C61D412yes—
89C2×M16×C61D412yes—
90C2×M16×C61C2311yescharacteristic
91C2×M16×C61C2311yescharacteristic
92C22×C8×C61—21no—
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2311yescharacteristic
95#123×C61—21no—
96C22⋊Q8×C61D412yes—
97#121×C61D412yes—
98#120×C61—21no—
99C4×D4×C61—22no—
100#117×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102C22⋊C8×C61—22no—
103C22⋊C8×C61—22no—
104#97×C61—21no—
105C4×C8×C61—21no—
106C24×C61#4111yescharacteristic
107C23×D61—22no—
108#41×C61—21no—
109#41×C61—21no—
110#41×C61—21no—
111#41×C61—21no—
112C2×Q8×C61C2×D411yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61C2×D411yescharacteristic
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61#4111yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—84no—
128C22×Dic61—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135M16×C61—24no—
136M16×C61—24no—
137M16×C61—21no—
138M16×C61—21no—
139M16×C61—21no—
140M16×C61—21no—
141M16×C61—42no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—42no—
147C2×C8×C61C2×D412yes—
148C2×C8×C61C2×D412yes—
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61#4112yes—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—21no—
156C22⋊C4×C61—22no—
157C42×C61—21no—
158#11758201—44no—
159#11758199—44no—
160#11758198—42no—
161#11758198—42no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164#11758194—22no—
165#11758194—22no—
166#11758194—22no—
167#11758194—44no—
168#11758194C2×D414yes—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—44no—
172D488—42no—
173D488—42no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—42no—
178C23×C61#12112yes—
179C23×C61C2×#4111yescharacteristic
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—21no—
186Q8×C61—21no—
187D4×C61—21no—
188D4×C61—21no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—82no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—21no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61#12112yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)11yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C22⋊D61—84no—
210C22⋊D61—84no—
211C2×Dic61—42no—
212C2×Dic61—82no—
213D244—42no—
214D244—42no—
215D244—42no—
216D244—42no—
217C8×C61—22no—
218C8×C61—41no—
219C8×C61—41no—
220C8×C61#12112yes—
221C8×C61#12112yes—
222#92418—24no—
223#92418—42no—
224#24806—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—22no—
229C22×C61—41no—
230C22×C61—42no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61C2×#12111yescharacteristic
234C22×C61#54811yescharacteristic
235C22×C61#55311yescharacteristic
236C2×D61—82no—
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—162no—
240C2×D61—162no—
241C4×C61—22no—
242C4×C61—41no—
243C4×C61—42no—
244C4×C61C2×#12111yescharacteristic
245C4×C61#53711yescharacteristic
246C4×C61D4211yescharacteristic
247C4×C61#54711yescharacteristic
248Dic61—82no—
249C2×#472—611no—
250#3055—612no—
251C2×#469—611no—
252#2930D6111yescharacteristic, complemented
253C2×#430—611no—
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—81no—
264C2×C61#241611yescharacteristic
265C2×C61#306812yes—
266D61—162no—
267C2×AGL(1,ℤ/8)—612no—
268C22×D8—611no—
269C2×#131C2×D6111yescharacteristic
270C22×M16—612no—
271C22×M16C2×D6111yescharacteristic
272#525C2×D6111yescharacteristic
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—22no—
282#435—22no—
283#430—1224no—
284C2×#108—611no—
285C2×#108—612no—
286C2×#108—611no—
287#428—612no—
288C2×#106—611no—
289#424C2×D6111yescharacteristic
290#418—612no—
291#417—612no—
292C2×#102—612no—
293C2×#100—612no—
294C2×#100—612no—
295#408C2×D6111yescharacteristic
296#407C2×D6111yescharacteristic
297#405C2×D6111yescharacteristic
298#343—612no—
299#343—612no—
300C61#2480611yescharacteristic, complemented, Sylow
301C2×#41C22×D6111yescharacteristic
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—1222no—
310C2×D8—1222no—
311C2×D8—1221no—
312C2×D8—1221no—
313#131—22no—
314#131—22no—
315C2×M16—22no—
316C2×M16—22no—
317C2×M16—21no—
318C2×M16—612no—
319C2×M16—612no—
320C2×M16—1222no—
321C2×M16—1224no—
322C2×M16—2442no—
323C2×M16C22⋊D6112yes—
324C2×M16C22⋊D6112yes—
325C2×M16C22×D6111yescharacteristic
326C2×M16C22×D6111yescharacteristic
327C22×C8—21no—
328C22×C8—1222no—
329C22×C8C22×D6111yescharacteristic
330C22×C8C22×D6111yescharacteristic
331#123—21no—
332C22⋊Q8D24412yes—
333#121—1222no—
334#121—1222no—
335#121—1222no—
336#121—1222no—
337#121D24412yes—
338#120—21no—
339C4×D4—22no—
340#117C22×D6111yescharacteristic
341C2×C4⋊C4—612no—
342C2×C22⋊C4—1222no—
343#108—614no—
344#108—1224no—
345#108—1222no—
346#107—1222no—
347#106—1221no—
348#106—1221no—
349#106—1222no—
350C4⋊C8—22no—
351#103—1222no—
352#102—1222no—
353#102—1224no—
354#102—1222no—
355#102—1222no—
356#100—1224no—
357#100—2442no—
358C22⋊C8—22no—
359C22⋊C8—22no—
360#97—21no—
361C4×C8—21no—
362C24—1222no—
363C24#1175822211yescharacteristic
364#41—21no—
365#41—21no—
366#41—21no—
367#41—21no—
368C2×Q8C2×D24411yescharacteristic
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4—4884no—
378C2×D4—2442no—
379C2×D4—2442no—
380C2×D4C2×D24411yescharacteristic
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—1222no—
387C22×C4C2×D24412yes—
388C22×C4#1175822211yescharacteristic
389C22×C4C2×C22⋊D6112yes—
390C22×C4C23×D6111yescharacteristic
391SD16—2444no—
392D8—2444no—
393D8—2442no—
394D8—2442no—
395M16—24no—
396M16—24no—
397M16—21no—
398M16—21no—
399M16—21no—
400M16—21no—
401M16—42no—
402M16—614no—
403M16—1222no—
404M16—1222no—
405M16—1222no—
406M16—2444no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—42no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—2444no—
415C2×C8D4×D6112yes—
416C2×C8#1175822412yes—
417C2×C8C2×C22⋊D6112yes—
418C2×C8C2×C22⋊D6111yescharacteristic
419C2×C8C2×C22⋊D6111yescharacteristic
420C4⋊C4—22no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4C2×D24412yes—
424C22⋊C4—21no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—2442no—
428C22⋊C4—2442no—
429C42—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—42no—
434C23—4882no—
435C23—4882no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23#1451975112yes—
440C23C2×#1175822211yescharacteristic
441Q8—21no—
442Q8—21no—
443D4—21no—
444D4—21no—
445D4—41no—
446D4—41no—
447D4—82no—
448D4—4884no—
449D4—4884no—
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—22no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—4882no—
465C2×C4—2442no—
466C2×C4#1451975112yes—
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C22×C22⋊D6111yescharacteristic
469C2×C4C22×D24411yescharacteristic
470C2×C4C2×#1175822411yescharacteristic
471C2×C4#1451982311yescharacteristic
472C2×C4#1451982411yescharacteristic
473C8—22no—
474C8—41no—
475C8—41no—
476C8—1224no—
477C8—2442no—
478C8#1451974912yes—
479C8#1451974912yes—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—42no—
486C22—81no—
487C22—81no—
488C22—9762no—
489C22—9762no—
490C22—4882no—
491C22—4882no—
492C22—4882no—
493C22C2×#1451975111yescharacteristic
494C22#1684042111yescharacteristic
495C22#1684045811yescharacteristic
496C4—22no—
497C4—41no—
498C4—42no—
499C4—4882no—
500C4C2×#1451974911yescharacteristic
501C4#1684034011yescharacteristic
502C4D4×D24411yescharacteristic
503C4#1684041611yescharacteristic
504C2—21no—
505C2—21no—
506C2—42no—
507C2—81no—
508C2—9762no—
509C2#2144925911yescharacteristic
510C2#2145554712yes—
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2930C2
socleC22×C61C2×#121
hypercenter#2930D61
hypocenterC61#24806
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2930

Complements

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

NH ≅ G/Ntype
1. C2×#16840032C2semidirect
2. #21455440C2semidirect
3. C2×#16840015C2semidirect
4. C2×#16839811C2semidirect
5. #21447178C2semidirect
6. #21447110C2semidirect
7. #21446729C2semidirect
8. #21446599C2semidirect
9. #21446280C2semidirect
10. #21445878C2semidirect
11. C61×#2930C2semidirect
69. #14519713D4semidirect
252. #2930D61semidirect
300. C61#24806semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214554402
C2×#168400152
C2×#168398112
#214471782
#214471102
#214467292
#214465992
#214462802
#214458782
C61×#29302
#2480661