Finite Group Databasebeta

← #29716400

Subgroups — #29716400

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840039C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic
3#21450737C211yesmaximal, characteristic
4#21447154C212yesmaximal, complemented
5#21446962C212yesmaximal
6#21446781C211yesmaximal, characteristic, complemented
7#21446756C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal
9#21446073C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C61×#557C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C2×#14519753—21no—
14C2×C4×D244C2211yescharacteristic
15#16840039—24no—
16C4×#11758201—24no—
17C2×#14519710C2212yes—
18C2×#14519708C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2212yes—
21C2×#14519701C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×C4×#92418C2211yescharacteristic
25#16839684—22no—
26#16839664C2212yes—
27#16839660C2212yes—
28#16839632C2211yescharacteristic
29#16839619—21no—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#557—24no—
36C61×#557—24no—
37C2×C4⋊Q8×C61—21no—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#125×C61—22no—
40C2×#125×C61C2212yes—
41C2×C4×Q8×C61—21no—
42C2×C4×Q8×C61C2212yes—
43C2×C4×Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49C2×#11758201Q814yescomplemented
50C22×#92418C2311yescharacteristic
51C22×#92418C2312yes—
52C22×#92418C2311yescharacteristic
53#14519753—44no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64#14519710—24no—
65#14519708—22no—
66C4⋊#92418—22no—
67#14519701—22no—
68#14519701—24no—
69#14519697—24no—
70#14519697—22no—
71C4×#92418—22no—
72#14519684—22no—
73C22×Q8×C61—21no—
74C22×Q8×C61D411yescharacteristic
75C22×Q8×C61C2311yescharacteristic
76C4⋊Q8×C61—28no—
77C4⋊Q8×C61—22no—
78C4⋊Q8×C61D414yes—
79#125×C61—216no—
80#125×C61—24no—
81#125×C61—24no—
82C4×Q8×C61—28no—
83C4×Q8×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yes—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61D412yes—
95C2×C4⋊C4×C61D411yescharacteristic
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61—22no—
100C2×C42×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102C23×D61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—24no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—42no—
111C2×Q8×C61—42no—
112C2×Q8×C61C2×D412yes—
113C2×Q8×C61C2×Q812yes—
114C2×Q8×C61#4112yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—22no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244C2×Q812yes—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C4⋊C4×C61—28no—
135C4⋊C4×C61—28no—
136C4⋊C4×C61—28no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61C2×D414yes—
144C4⋊C4×C61C2×D412yes—
145C42×C61—28no—
146C42×C61—24no—
147C42×C61—22no—
148C42×C61—22no—
149C42×C61C2×D412yes—
150C42×C61C2×D412yes—
151#11758201—28no—
152#11758198—44no—
153#11758198—42no—
154#11758198—82no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—42no—
160C4⋊Dic61—81no—
161#11758196—44no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418C2×Q812yes—
171C23×C61C22×D411yescharacteristic
172C22×D61—22no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179Q8×C61—22no—
180Q8×C61—22no—
181Q8×C61—22no—
182Q8×C61—44no—
183Q8×C61—48no—
184Q8×C61C22⋊Q814yescomplemented
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—24no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61#12012yes—
203C2×C4×C61#12012yes—
204C2×C4×C61#13512yes—
205C2×C4×C61AGL(1,ℤ/8)12yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×Q811yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C612−1+411yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218D244—22no—
219D244—22no—
220D244—41no—
221D244—41no—
222C4×D61—84no—
223#92418—22no—
224#92418—41no—
225#92418—42no—
226#24328—611nomaximal, Sylow
227C22×C61C2×#12011yescharacteristic
228C22×C61#54811yescharacteristic
229C22×C61D4×Q811yescharacteristic
230C22×C61C2×#13511yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61#57711yescharacteristic
233C22×C61#57811yescharacteristic
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C4×C61—24no—
242C4×C61—22no—
243C4×C61—42no—
244C4×C61—44no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61#44914yes—
248C4×C61C2×C22⋊Q811yescharacteristic
249C4×C61#53611yescharacteristic
250C4×C61#54811yescharacteristic
251C4×C61D4×Q811yescharacteristic
252Dic61—82no—
253Dic61—81no—
254C2×#557D6111yescharacteristic, complemented
255C2×#474—611no—
256C2×C4×SD16—611no—
257#2538—611no—
258#2059—612no—
259#2025—612no—
260#1990—611no—
261#1984—611no—
262#1880—612no—
263#1863—612no—
264#1803—611no—
265C2×C61#240911yescharacteristic
266C2×C61C2×#44911yescharacteristic
267C2×C61#301311yescharacteristic
268C2×C61#336411yescharacteristic
269C2×C61#336811yescharacteristic
270C2×C61#338111yescharacteristic
271C2×C61#338611yescharacteristic
272D61—82no—
273C22×SD16—611no—
274#557—24no—
275#557—24no—
276C2×C4⋊Q8—21no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C2×#125—22no—
279C2×#125C2×D6112yes—
280C2×C22⋊Q8—1221no—
281C2×C4×Q8—21no—
282C2×C4×Q8C2×D6112yes—
283C2×C4×Q8C2×D6111yescharacteristic
284C2×C4×Q8C2×D6111yescharacteristic
285C2×C4×D4—611no—
286#474—1224no—
287C4×SD16—1224no—
288C2×#107—612no—
289C2×#106—611no—
290C2×C4⋊C8—611no—
291C2×#103—612no—
292C2×#103—611no—
293C2×#102—612no—
294C2×#102—611no—
295C2×C4×C8—611no—
296#396—1222no—
297#390—612no—
298#389—612no—
299#384—611no—
300#380—1221no—
301#340—612no—
302#340—612no—
303#340—612no—
304#336—612no—
305#336—612no—
306C61#2432811yescharacteristic, complemented, Sylow
307C22×Q8—21no—
308C22×Q8C22⋊D6111yescharacteristic
309C22×Q8C22×D6111yescharacteristic
310C22×D4—611no—
311C23×C4—1221no—
312C2×SD16—614no—
313C2×SD16—1222no—
314C2×SD16—1222no—
315C22×C8—611no—
316C22×C8—611no—
317C22×C8—612no—
318C4⋊Q8—28no—
319C4⋊Q8—22no—
320C4⋊Q8C22⋊D6114yes—
321#125—216no—
322#125—24no—
323#125—24no—
324C22⋊Q8—2444no—
325C4×Q8—28no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8C22⋊D6114yes—
332C4×D4—1222no—
333C4×D4—1222no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22⋊D6112yes—
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C4⋊C4C22×D6112yes—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1222no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42C22×D6112yes—
354C2×C42C22×D6111yescharacteristic
355#107—1224no—
356#106—1222no—
357C4⋊C8—1222no—
358#103—1224no—
359#103—1222no—
360#102—1222no—
361#102—1224no—
362C4×C8—1222no—
363#95—1222no—
364C24—1221no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8Q8×D6112yes—
375C2×Q8#1175822612yes—
376C2×Q8C2×C22⋊D6112yes—
377C2×D4—612no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1221no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—1221no—
387C22×C4—1222no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4C2×C22⋊D6112yes—
392C22×C4C2×C22⋊D6112yes—
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396SD16—1228no—
397C2×C8—612no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1222no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—1221no—
416C4⋊C4—1221no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—4881no—
420C4⋊C4—2444no—
421C4⋊C4—2442no—
422C4⋊C4C2×C22⋊D6112yes—
423C4⋊C4C2×C22⋊D6114yes—
424C22⋊C4—4882no—
425C22⋊C4—2444no—
426C22⋊C4—2442no—
427C42—28no—
428C42—22no—
429C42—22no—
430C42—24no—
431C42C2×C22⋊D6112yes—
432C42C2×C22⋊D6112yes—
433C23—1222no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23C22×C22⋊D6111yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—22no—
444Q8—44no—
445Q8—48no—
446Q8#1451975314yescomplemented
447D4—1222no—
448D4—1222no—
449D4—2441no—
450D4—2441no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—4884no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4#1451974612yes—
476C2×C4#1451974612yes—
477C2×C4#1451979712yes—
478C2×C4#1451980212yes—
479C2×C4C22×C22⋊D6111yescharacteristic
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4C2×Q8×D6111yescharacteristic
482C2×C4C2×#1175822611yescharacteristic
483C2×C4#1451982511yescharacteristic
484C2×C4#1451982511yescharacteristic
485C8—1222no—
486C8—2441no—
487C8—2442no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22—4881no—
492C22—2442no—
493C22—2442no—
494C22—2442no—
495C22C2×#1451974611yescharacteristic
496C22#1684042311yescharacteristic
497C22Q8×C22⋊D6111yescharacteristic
498C22C2×#1451979711yescharacteristic
499C22C2×#1451980211yescharacteristic
500C22#1684057011yescharacteristic
501C22#1684057711yescharacteristic
502C4—22no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—44no—
508C4—4882no—
509C4—4881no—
510C4#1683990714yes—
511C4C2×#1451975311yescharacteristic
512C4#1684033611yescharacteristic
513C4#1684042311yescharacteristic
514C4Q8×C22⋊D6111yescharacteristic
515C2—4882no—
516C2#2144917811yescharacteristic
517C2C2×#1683990711yescharacteristic
518C2#2145513411yescharacteristic
519C2#2145850411yescharacteristic
520C2#2145854811yescharacteristic
521C2#2145869011yescharacteristic
522C2#2145873611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#557C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#557D61
hypocenterC61#24328
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #21447154C2semidirect
6. #21446781C2semidirect
7. #21446756C2semidirect
9. #21446073C2semidirect
10. #21445824C2semidirect
11. C2×C61×#557C2semidirect
49. C2×#11758201Q8semidirect
184. Q8×C61C22⋊Q8semidirect
254. C2×#557D61semidirect
306. C61#24328semidirect
446. Q8#14519753semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C2×C4×#117582012
#214507372
#214471542
#214469622
#214467812
#214467562
#214461742
#214460732
#214458242
C2×C61×#5572
#2432861