Finite Group Databasebeta

← #29689400

Subgroups — #29689400

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840338C211yesmaximal, characteristic
2#21448428C211yesmaximal, characteristic, complemented
3#21446749C212yesmaximal, complemented
4#21446741C212yesmaximal, complemented
5#21446639C211yesmaximal, characteristic, complemented
6#21446635C211yesmaximal, characteristic, complemented
7#21446630C211yesmaximal, characteristic, complemented
8#21446621C211yesmaximal, characteristic, complemented
9#21445760C212yesmaximal
10#21445755C212yesmaximal, complemented
11C2×C61×#433C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14#16840338—24no—
15#16840338C418yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519765C2211yescharacteristic
20C2×#14519761—22no—
21C2×#14519757—22no—
22C2×#14519702C2212yes—
23C2×#14519702C2212yes—
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30#16839655—22no—
31#16839648—22no—
32#16839632C2211yescharacteristic
33#16839628C2211yescharacteristic
34#16839622C2212yes—
35C4×C4⋊Dic61C2212yes—
36C2×#117×C61C2211yescharacteristic
37C61×#433—24no—
38C2×C4⋊C8×C61C2212yes—
39C2×C22⋊C8×C61C2212yes—
40C2×#97×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C2×#11758222—22no—
43C2×#11758222—42no—
44C2×#11758222—42no—
45C2×#11758222C2×C414yes—
46C22×Dic122—21no—
47C22×Dic122—22no—
48C22×Dic122—42no—
49C22×Dic122—42no—
50C22×Dic122C2×C414yes—
51C22×Dic122C2311yescharacteristic
52C22×Dic122C2311yescharacteristic
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C22×C4×D61—21no—
58#14519777—22no—
59#14519777—22no—
60#14519777C2×C414yes—
61#14519777C2×C414yes—
62#14519765—22no—
63#14519765—22no—
64#14519765C2×C414yes—
65#14519765C2×C414yes—
66#14519761—216no—
67#14519757—216no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2312yes—
73C2×C22⋊Dic61—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×C4×Dic61—22no—
77#14519702—42no—
78#14519702—42no—
79#14519702—42no—
80#14519702—42no—
81#14519702—42no—
82#14519702—42no—
83#14519699—42no—
84#14519699—42no—
85#14519684—22no—
86#14519684—22no—
87#14519684—22no—
88C23×C4×C61C2311yescharacteristic
89C22×C8×C61C2312yes—
90C22×C8×C61C2312yes—
91#117×C61C2×C414yes—
92#117×C61D414yes—
93C2×C4⋊C4×C61D412yes—
94C2×C22⋊C4×C61D412yes—
95C2×C42×C61C2311yescharacteristic
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—24no—
98C22⋊C8×C61—24no—
99#97×C61—22no—
100C4×C8×C61—22no—
101C24×C61C2×D411yescharacteristic
102C23×D61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61C22×C412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2411yescharacteristic
111C2×C22⋊D61—24no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C22×Dic61—24no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—42no—
121C22×Dic61—42no—
122#11758222—48no—
123#11758222—48no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—21no—
127C2×D244C22×C412yes—
128C2×C4×D61—24no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—24no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—48no—
142C2×Dic122—42no—
143C2×Dic122—42no—
144C2×Dic122—48no—
145C2×Dic122C22×C412yes—
146C2×Dic122C22×C412yes—
147C2×Dic122C22×C412yes—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61C22⋊C418yes—
155C22⋊C4×C61C22⋊C418yes—
156C42×C61C22×C412yes—
157C42×C61C22×C412yes—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160C22⋊Dic61—82no—
161#11758198—28no—
162#11758198—28no—
163C4⋊Dic61—28no—
164C4⋊Dic61—28no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—82no—
170C4×Dic61—82no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61C2×C22⋊C412yes—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—42no—
181C22×D61—42no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61C2×C22⋊C414yes—
191C2×C4×C61C2×C22⋊C414yes—
192C2×C4×C61C2×C22⋊C412yes—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61H(5,𝔽2)12yes—
196C2×C4×C61H(5,𝔽2)12yes—
197C22⋊D61—48no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—48no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214D244—24no—
215D244—42no—
216C4×D61—48no—
217Dic122—24no—
218Dic122—24no—
219Dic122—24no—
220Dic122—48no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224C8×C61—42no—
225C8×C61—42no—
226#21814—611nomaximal, Sylow
227C22×C61—22no—
228C22×C61—24no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—42no—
232C22×C61C22×C22⋊C411yescharacteristic
233C22×C61#53511yescharacteristic
234C22×C61#53511yescharacteristic
235C22×C61#57712yes—
236C22×C61#57812yes—
237C2×D61—42no—
238C2×D61—44no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—82no—
242C4×C61—22no—
243C4×C61—24no—
244C4×C61—24no—
245C4×C61#51812yes—
246C4×C61#53512yes—
247Dic61—42no—
248Dic61—42no—
249Dic61—42no—
250Dic61—82no—
251Dic61—82no—
252C2×#536—611no—
253C2×#433D6111yescharacteristic, complemented
254#2314—611no—
255#1983—612no—
256#1981—612no—
257#1960—611no—
258#1959—611no—
259#1958—611no—
260#1956—611no—
261#1788—612no—
262#1787—612no—
263C2×C61—42no—
264C2×C61#230911yescharacteristic
265C2×C61#289812yes—
266C2×C61#322111yescharacteristic
267C2×C61#322311yescharacteristic
268C2×C61#322512yes—
269D61—82no—
270C22×#41—611no—
271C23×Q8—611no—
272#536—618no—
273#536—1224no—
274C2×C4⋊Q8—611no—
275C2×C4⋊Q8—611no—
276C2×#124—611no—
277C2×#124—611no—
278C2×#123—1222no—
279C2×C22⋊Q8—1222no—
280C2×#117C2×D6111yescharacteristic
281#433—24no—
282C2×C4⋊C8C2×D6112yes—
283C2×#103—612no—
284C2×#103—612no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#103—612no—
288C2×#103—612no—
289C2×#102—612no—
290C2×#102—612no—
291C2×C22⋊C8C2×D6112yes—
292C2×#97C2×D6111yescharacteristic
293C2×C4×C8C2×D6111yescharacteristic
294#388—1222no—
295#387—1222no—
296#384—611no—
297#382—611no—
298#380—612no—
299C4×C4⋊C4—612no—
300C61#2181411yescharacteristic, complemented, Sylow
301C2×#41—614no—
302C2×#41—1222no—
303C2×#41—2442no—
304C2×#41—2442no—
305C22×Q8—614no—
306C22×Q8—611no—
307C22×Q8—611no—
308C22×Q8—611no—
309C22×Q8—1221no—
310C22×Q8—1222no—
311C22×Q8—2442no—
312C22×Q8—2442no—
313C22×D4—611no—
314C22×D4—1221no—
315C23×C4—1221no—
316C23×C4—1221no—
317C23×C4C22×D6111yescharacteristic
318C22×C8C22×D6112yes—
319C22×C8C22×D6112yes—
320C4⋊Q8—614no—
321C4⋊Q8—614no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324#124—614no—
325#124—614no—
326#124—1222no—
327#124—1222no—
328#123—12216no—
329C22⋊Q8—12216no—
330#117C4×D6114yes—
331#117C22⋊D6114yes—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4D24412yes—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4D24412yes—
342C2×C42—1222no—
343C2×C42C22×D6111yescharacteristic
344C2×C42C22×D6111yescharacteristic
345C4⋊C8—24no—
346#103—2442no—
347#103—2442no—
348#103—2442no—
349#103—2442no—
350#103—2442no—
351#103—2442no—
352#102—2442no—
353#102—2442no—
354C22⋊C8—24no—
355#97—22no—
356C4×C8—22no—
357#95—1222no—
358#95—1222no—
359#95—1222no—
360C24—1221no—
361C24C2×D24411yescharacteristic
362#41—2448no—
363#41—2448no—
364C2×Q8—612no—
365C2×Q8—612no—
366C2×Q8—612no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8—1221no—
370C2×Q8—1224no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8—2448no—
378C2×Q8—2442no—
379C2×Q8—2442no—
380C2×Q8—2448no—
381C2×D4—612no—
382C2×D4—1224no—
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—1222no—
386C2×D4—2442no—
387C2×D4—2442no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—1221no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1224no—
394C22×C4—1224no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4C2×C4×D6112yes—
402C22×C4C2×D24411yescharacteristic
403C22×C4C2×D24412yes—
404C22×C4C2×D24412yes—
405C22×C4C2×C22⋊D6112yes—
406C22×C4C23×D6111yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—1228no—
414C4⋊C4—1228no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—4882no—
420C4⋊C4#1175819818yes—
421C22⋊C4—1228no—
422C22⋊C4—1228no—
423C22⋊C4—4882no—
424C22⋊C4#1175819818yes—
425C42—4882no—
426C42C2×C4×D6112yes—
427C42C2×C4×D6112yes—
428C42C2×C22⋊D6112yes—
429C42C2×C22⋊D6112yes—
430C23—22no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—1222no—
435C23—2442no—
436C23—2442no—
437C23—2441no—
438C23—2441no—
439C23C2×#1175819812yes—
440C23C22×D24411yescharacteristic
441Q8—1224no—
442Q8—1224no—
443Q8—1224no—
444Q8—2448no—
445Q8—2442no—
446Q8—2442no—
447Q8—2442no—
448D4—1224no—
449D4—2448no—
450D4—2442no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—2448no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2448no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4C2×#1175819814yes—
477C2×C4C2×#1175819814yes—
478C2×C4C2×#1175819812yes—
479C2×C4C22×C4×D6111yescharacteristic
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4#1451982312yes—
482C2×C4#1451982312yes—
483C8—42no—
484C8—42no—
485C22—24no—
486C22—22no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—4881no—
491C22—4882no—
492C22—4881no—
493C22—2444no—
494C22—2442no—
495C22C22×#1175819811yescharacteristic
496C22#1684033111yescharacteristic
497C22#1684033111yescharacteristic
498C22#1684057412yes—
499C22#1684057912yes—
500C4—24no—
501C4—24no—
502C4—22no—
503C4—4882no—
504C4—4882no—
505C4—2442no—
506C4—2442no—
507C4—2442no—
508C4#1684023612yes—
509C4#1684032612yes—
510C2—42no—
511C2—4882no—
512C2#2144839911yescharacteristic
513C2#2145435312yes—
514C2#2145688211yescharacteristic
515C2#2145690411yescharacteristic
516C2#2145693012yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#433C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#433D61
hypocenterC61#21814
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
2. #21448428C2semidirect
3. #21446749C2semidirect
4. #21446741C2semidirect
5. #21446639C2semidirect
6. #21446635C2semidirect
7. #21446630C2semidirect
8. #21446621C2semidirect
10. #21445755C2semidirect
11. C2×C61×#433C2semidirect
15. #16840338C4semidirect
253. C2×#433D61semidirect
300. C61#21814semidirect

Maximal subgroups

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

subgroupindex
C2×#168403382
#214484282
#214467492
#214467412
#214466392
#214466352
#214466302
#214466212
#214457602
#214457552
C2×C61×#4332
#2181461