Finite Group Databasebeta

← #29689391

Subgroups — #29689391

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×#16840337C211yesmaximal, characteristic
2C2×#16839821C211yesmaximal, characteristic, complemented
3#21448427C211yesmaximal, characteristic, complemented
4#21446748C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446640C211yesmaximal, characteristic, complemented
7#21446633C211yesmaximal, characteristic, complemented
8#21446630C211yesmaximal, characteristic, complemented
9#21445758C212yesmaximal
10#21445752C212yesmaximal, complemented
11C61×#1956C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic
14#16840337—24no—
15#16840337C418yescomplemented
16C2×#14519775C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519762—22no—
20C2×#14519755—22no—
21C2×#14519734C2211yescharacteristic
22#16839821—24no—
23C2×#14519706C2212yes—
24C2×#14519702C2212yes—
25C2×#14519701C2212yes—
26C2×#14519701C2212yes—
27C2×#14519700C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30C2×#14519689C2212yes—
31C2×#14519688C2211yescharacteristic
32#16839652—22no—
33#16839643—22no—
34#16839628C2211yescharacteristic
35#16839619C2212yes—
36C4⋊C4×Dic61C2212yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#103×C61C2212yes—
39C2×#103×C61C2212yes—
40C2×C4×C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C2×#11758226—22no—
43C2×#11758226—42no—
44C2×#11758226—42no—
45C2×#11758226C2×C414yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C2×Q8×D61—22no—
49C2×Q8×D61—42no—
50C2×Q8×D61—42no—
51C2×Q8×D61C2×C414yes—
52C22×D244—21no—
53C22×D244C2311yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58#14519775—22no—
59#14519775C2×C414yes—
60#14519766—22no—
61#14519766C2×C414yes—
62#14519765—22no—
63#14519765C2×C414yes—
64#14519762—216no—
65#14519755—216no—
66#14519734C2×C414yes—
67#14519734D414yes—
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—22no—
70C2×#11758196D412yes—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198D412yes—
75C2×C4×Dic61—22no—
76C2×C4×Dic61C2311yescharacteristic
77#14519706—24no—
78#14519702—42no—
79#14519701—42no—
80#14519701—42no—
81#14519700—42no—
82#14519699—42no—
83#14519697—42no—
84#14519689—24no—
85#14519688—22no—
86#14519684—22no—
87#14519684—22no—
88#14519684—22no—
89C22×Q8×C61C2311yescharacteristic
90C22×Q8×C61C2311yescharacteristic
91C22×C8×C61C2312yes—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61C2×C414yes—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#103×C61—42no—
99#103×C61—42no—
100C4×C8×C61—22no—
101C23×D61—21no—
102C23×D61C2×D411yescharacteristic
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61C22×C412yes—
110C2×Q8×C61C22×C412yes—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—21no—
117C22×Dic61—22no—
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D412yes—
120#11758226—48no—
121#11758226—48no—
122Q8×D61—48no—
123Q8×D61—48no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—24no—
127C2×D244—21no—
128C2×D244—42no—
129C2×D244—42no—
130C2×D244C22×C412yes—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—24no—
134C2×C4×D61—24no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61C22×C412yes—
140C2×C4×D61C2×D412yes—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—21no—
144C2×Dic122—24no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C2×Dic122C22×C412yes—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—28no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—82no—
155C42×C61C22×C412yes—
156C42×C61C2×D412yes—
157#11758198—28no—
158#11758198—28no—
159#11758198—82no—
160#11758198C22⋊C418yes—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163#11758196—28no—
164#11758196C22⋊C418yes—
165C4×Dic61—82no—
166C4×Dic61C22×C412yes—
167C4×Dic61C2×D412yes—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61C22×D411yescharacteristic
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61C2×C22⋊C412yes—
182Q8×C61—24no—
183Q8×C61—24no—
184Q8×C61—42no—
185Q8×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61C2×C22⋊C414yes—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61H(5,𝔽2)12yes—
201C2×C4×C61H(5,𝔽2)12yes—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61C2×C22⋊C414yes—
213C2×Dic61C2×C22⋊C412yes—
214D244—24no—
215D244—48no—
216D244—42no—
217C4×D61—24no—
218C4×D61—24no—
219C4×D61—48no—
220C4×D61—48no—
221Dic122—24no—
222Dic122—48no—
223Dic122—42no—
224C8×C61—42no—
225#92418—42no—
226#21814—611nomaximal, Sylow
227C22×C61C22×C22⋊C411yescharacteristic
228C22×C61#53511yescharacteristic
229C22×C61#53511yescharacteristic
230C22×C61#57712yes—
231C22×C61#57812yes—
232C2×D61—22no—
233C2×D61—24no—
234C2×D61—41no—
235C2×D61—41no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—44no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—82no—
242C4×C61—24no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—82no—
246C4×C61#51812yes—
247C4×C61#53512yes—
248Dic61—22no—
249Dic61—24no—
250Dic61—42no—
251Dic61—82no—
252C2×#536—611no—
253C2×#433—611no—
254#2314—611no—
255#1983—612no—
256#1981—612no—
257#1960—611no—
258#1959—611no—
259#1958—611no—
260#1956D6111yescharacteristic, complemented
261#1788—612no—
262#1787—612no—
263C2×C61#230911yescharacteristic
264C2×C61#289812yes—
265C2×C61#322111yescharacteristic
266C2×C61#322311yescharacteristic
267C2×C61#322512yes—
268D61—42no—
269D61—82no—
270C22×#41—611no—
271C23×Q8—611no—
272#536—618no—
273#536—1224no—
274C2×C4⋊Q8—611no—
275C2×C4⋊Q8C2×D6111yescharacteristic
276C2×#124—611no—
277C2×#124—611no—
278C2×#123—1222no—
279C2×C22⋊Q8—1222no—
280C2×#117—611no—
281#433—1224no—
282C2×C4⋊C8—612no—
283C2×#103—612no—
284C2×#103—612no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#103C2×D6112yes—
288C2×#103C2×D6112yes—
289C2×#102—612no—
290C2×#102—612no—
291C2×C22⋊C8—612no—
292C2×#97—611no—
293C2×C4×C8C2×D6111yescharacteristic
294#388—1222no—
295#387—1222no—
296#384C2×D6111yescharacteristic
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—1221no—
308C22×Q8—1222no—
309C22×Q8—2442no—
310C22×Q8—2442no—
311C22×Q8C22×D6111yescharacteristic
312C22×Q8C22×D6111yescharacteristic
313C22×D4—611no—
314C22×D4—1221no—
315C23×C4—611no—
316C23×C4—1221no—
317C23×C4—1221no—
318C22×C8—612no—
319C22×C8C22×D6112yes—
320C4⋊Q8—22no—
321C4⋊Q8—614no—
322C4⋊Q8—1222no—
323C4⋊Q8C4×D6114yes—
324#124—614no—
325#124—614no—
326#124—1222no—
327#124—1222no—
328#123—12216no—
329C22⋊Q8—12216no—
330#117—614no—
331#117—614no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4C22×D6112yes—
338C2×C22⋊C4—612no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C42—611no—
343C2×C42—1222no—
344C2×C42C22×D6111yescharacteristic
345C4⋊C8—1224no—
346#103—42no—
347#103—42no—
348#103—2442no—
349#103—2442no—
350#103—2442no—
351#103—2442no—
352#102—2442no—
353#102—2442no—
354C22⋊C8—1224no—
355#97—1222no—
356C4×C8—22no—
357#95—1222no—
358#95—1222no—
359#95—1222no—
360C24—611no—
361C24—1221no—
362#41—2448no—
363#41—2448no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—612no—
371C2×Q8—1224no—
372C2×Q8—1221no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—2442no—
376C2×Q8—2442no—
377C2×Q8—2448no—
378C2×Q8—2448no—
379C2×Q8C2×C4×D6112yes—
380C2×Q8C2×C4×D6112yes—
381C2×D4—612no—
382C2×D4—1224no—
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—1222no—
386C2×D4—2442no—
387C2×D4—2442no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—611no—
392C22×C4—612no—
393C22×C4—612no—
394C22×C4—612no—
395C22×C4—1224no—
396C22×C4—1224no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1221no—
400C22×C4—1222no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4—2442no—
405C22×C4D4×D6112yes—
406C22×C4C23×D6111yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—82no—
417C4⋊C4—618no—
418C4⋊C4—1228no—
419C4⋊C4—1222no—
420C4⋊C4—1222no—
421C22⋊C4—618no—
422C22⋊C4—1228no—
423C22⋊C4—1228no—
424C22⋊C4—4882no—
425C42—612no—
426C42—612no—
427C42—4882no—
428C42C2×C4×D6112yes—
429C42D4×D6112yes—
430C23—612no—
431C23—1222no—
432C23—1222no—
433C23—1221no—
434C23—1221no—
435C23—1222no—
436C23—2442no—
437C23—2442no—
438C23—2441no—
439C23—2441no—
440C23C2×D4×D6111yescharacteristic
441Q8—24no—
442Q8—24no—
443Q8—42no—
444Q8—42no—
445Q8—1224no—
446Q8—2448no—
447Q8—2442no—
448D4—1224no—
449D4—2448no—
450D4—2442no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—614no—
463C2×C4—612no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1224no—
469C2×C4—1224no—
470C2×C4—1222no—
471C2×C4—2448no—
472C2×C4—2448no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4C22⋊C4×D6114yes—
479C2×C4C22×C4×D6111yescharacteristic
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4#1451982212yes—
482C2×C4#1451982312yes—
483C8—42no—
484C8—2442no—
485C22—1224no—
486C22—1222no—
487C22—4882no—
488C22—4881no—
489C22—4881no—
490C22—2444no—
491C22—2442no—
492C22—2442no—
493C22—2441no—
494C22—2441no—
495C22C2×C22⋊C4×D6111yescharacteristic
496C22#1684032511yescharacteristic
497C22#1684033011yescharacteristic
498C22#1684056812yes—
499C22#1684057512yes—
500C4—24no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—1224no—
505C4—1222no—
506C4—4882no—
507C4—2442no—
508C4#1684023412yes—
509C4#1684032812yes—
510C2—4882no—
511C2—2442no—
512C2#2144839811yescharacteristic
513C2#2145434712yes—
514C2#2145687311yescharacteristic
515C2#2145689511yescharacteristic
516C2#2145693112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1956D61
hypocenterC61#21814
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#16839821C2semidirect
3. #21448427C2semidirect
4. #21446748C2semidirect
5. #21446740C2semidirect
6. #21446640C2semidirect
7. #21446633C2semidirect
8. #21446630C2semidirect
10. #21445752C2semidirect
11. C61×#1956C2semidirect
15. #16840337C4semidirect
260. #1956D61semidirect
300. C61#21814semidirect

Maximal subgroups

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

subgroupindex
C2×#168403372
C2×#168398212
#214484272
#214467482
#214467402
#214466402
#214466332
#214466302
#214457582
#214457522
C61×#19562
#2181461