Finite Group Databasebeta

← #29689802

Subgroups — #29689802

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840368C211yesmaximal, characteristic
2C2×#16839824C211yesmaximal, characteristic, complemented
3#21448440C211yesmaximal, characteristic, complemented
4#21446785C211yesmaximal, characteristic, complemented
5#21446643C211yesmaximal, characteristic, complemented
6#21446624C211yesmaximal, characteristic, complemented
7#21446140C212yesmaximal, complemented
8#21446135C212yesmaximal, complemented
9#21445828C212yesmaximal, complemented
10#21445818C212yesmaximal, complemented
11C61×#1992C211yesmaximal, characteristic, complemented
12#16840368—24no—
13#16840368C418yescomplemented
14C2×#14519774C2211yescharacteristic
15C2×#14519762—22no—
16C2×#14519752—22no—
17C2×#14519752C2212yes—
18C2×#14519739C2212yes—
19C2×#14519735C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22#16839824—24no—
23C2×#14519706C2212yes—
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28C2×#14519689C2212yes—
29C2×C4×#92418C2211yescharacteristic
30#16839657—22no—
31#16839633—22no—
32#16839631C2211yescharacteristic
33#16839628C2211yescharacteristic
34#16839627C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#125×C61C2211yescharacteristic
38C2×#97×C61C2211yescharacteristic
39C61×#384C2211yescharacteristic
40C61×#340C2212yes—
41C61×#340C2212yes—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244C2312yes—
45C22×C4×D61—22no—
46C22×C4×D61—21no—
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2312yes—
49#14519774—22no—
50#14519774C2×C414yescomplemented
51#14519762—216no—
52#14519752—24no—
53#14519752—216no—
54#14519752C2×C418yes—
55#14519739—24no—
56#14519739C2×C418yescomplemented
57#14519735—42no—
58#14519734C2×C414yes—
59#14519734D414yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758196—21no—
63C2×#11758196—22no—
64C2×#11758196D412yes—
65C4⋊C4×D61—42no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—21no—
69C2×#11758198D412yes—
70C2×C4×Dic61—21no—
71C2×C4×Dic61C2311yescharacteristic
72#14519706—24no—
73#14519699—42no—
74#14519699—42no—
75#14519697—42no—
76#14519697—42no—
77#14519689—24no—
78C4×#92418—22no—
79#14519684—22no—
80#14519684—22no—
81C22×C8×C61C2312yes—
82#125×C61—22no—
83#125×C61C2×C414yescomplemented
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90#97×C61—22no—
91C23×D61—22no—
92C23×D61C2×D411yescharacteristic
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61C2×D412yes—
101C22×Dic61C2×D411yescharacteristic
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—28no—
108C2×D244—28no—
109C2×D244—82no—
110C2×D244—82no—
111C2×D244—82no—
112C2×D244—82no—
113C2×D244C22×C414yes—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—28no—
117C2×C4×D61—42no—
118C2×C4×D61—82no—
119C2×C4×D61—82no—
120C2×C4×D61—82no—
121C2×C4×D61C22×C412yes—
122C2×C4×D61C2×D412yes—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C4⋊C4×C61—28no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—82no—
131C4⋊C4×C61C22×C414yes—
132C42×C61C22×C412yes—
133C42×C61C2×D412yes—
134#11758198—28no—
135#11758198—28no—
136#11758198—42no—
137#11758198C22⋊C418yes—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140#11758196—42no—
141#11758196—82no—
142#11758196C22⋊C418yes—
143C4×Dic61—42no—
144C4×Dic61C22×C412yes—
145C4×Dic61C2×D412yes—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—22no—
149C23×C61C22×D411yescharacteristic
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—22no—
153C22×D61—22no—
154C22×D61—24no—
155C22×D61—42no—
156C22×D61—42no—
157C22×D61—42no—
158C22×D61—42no—
159C22×D61—42no—
160C22×D61—42no—
161C22×D61C2×C22⋊C412yes—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—24no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61C2×C22⋊C414yes—
172C2×C4×C61C23×C411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61H(5,𝔽2)12yes—
175C2×C4×C612−1+412yes—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61C2×C22⋊C414yes—
185C2×Dic61C2×C22⋊C412yes—
186D244—24no—
187D244—24no—
188D244—42no—
189D244—42no—
190D244—88no—
191D244—88no—
192C4×D61—24no—
193C4×D61—24no—
194C4×D61—88no—
195C4×D61—84no—
196C8×C61—42no—
197#92418—42no—
198#21872—611nomaximal, Sylow
199C22×C61C22×C22⋊C411yescharacteristic
200C22×C61#53511yescharacteristic
201C22×C61#53611yescharacteristic
202C22×C61#57612yes—
203C22×C61#57712yes—
204C2×D61—22no—
205C2×D61—24no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—42no—
209C2×D61—44no—
210C2×D61—44no—
211C2×D61—44no—
212C2×D61—82no—
213C2×D61—82no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—24no—
217C4×C61—44no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61#52012yes—
221C4×C61#53712yes—
222Dic61—22no—
223Dic61—24no—
224Dic61—81no—
225Dic61—81no—
226C2×#540—611no—
227C2×#433—611no—
228#2317—611no—
229#1992D6111yescharacteristic, complemented
230#1991—611no—
231#1961—611no—
232#1957—611no—
233#1875—612no—
234#1874—612no—
235#1804—612no—
236#1802—612no—
237C2×C61#231111yescharacteristic
238C2×C61#289712yes—
239C2×C61#322011yescharacteristic
240C2×C61#322111yescharacteristic
241C2×C61#324112yes—
242D61—42no—
243D61—84no—
244#540—618no—
245#540—1224no—
246C2×#127—611no—
247C2×#125C2×D6111yescharacteristic
248C2×#123—1222no—
249C2×#121—612no—
250C2×#121—1222no—
251C2×C4×D4—612no—
252C2×#117—611no—
253C2×#117—611no—
254C22×C4⋊C4—611no—
255#433—1224no—
256C2×C4⋊C8—612no—
257C2×#102—612no—
258C2×#102—612no—
259C2×#102—612no—
260C2×#102—612no—
261C2×C22⋊C8—612no—
262C2×#97C2×D6111yescharacteristic
263C2×C4×C8—611no—
264#389—1222no—
265#385—1222no—
266#384—611no—
267#384C2×D6111yescharacteristic
268#382—611no—
269#382—611no—
270#340—612no—
271#340—612no—
272#340C2×D6112yes—
273#340C2×D6112yes—
274C61#2187211yescharacteristic, complemented, Sylow
275C22×D4—612no—
276C22×D4—1222no—
277C22×D4—1222no—
278C23×C4—611no—
279C23×C4—1222no—
280C23×C4—1221no—
281C22×C8—612no—
282C22×C8C22×D6112yes—
283#127—614no—
284#127—1222no—
285#125—22no—
286#125C4×D6114yescomplemented
287#123—12216no—
288#121—618no—
289#121—12216no—
290#121—1224no—
291C4×D4—618no—
292C4×D4—1224no—
293#117—614no—
294#117—614no—
295#117—2442no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—612no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—1222no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—2442no—
304C2×C4⋊C4C22×D6111yescharacteristic
305C2×C4⋊C4C22×D6111yescharacteristic
306C2×C4⋊C4C22×D6112yes—
307C2×C22⋊C4—612no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1222no—
310C2×C22⋊C4—1221no—
311C2×C42—611no—
312C2×C42—1221no—
313C2×C42C22×D6111yescharacteristic
314C4⋊C8—1224no—
315#102—2442no—
316#102—2442no—
317#102—2442no—
318#102—2442no—
319C22⋊C8—1224no—
320#97—22no—
321C4×C8—1222no—
322#95—1222no—
323#95—1222no—
324C24—611no—
325C24—1222no—
326C2×D4—614no—
327C2×D4—1228no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—1222no—
333C2×D4—1228no—
334C2×D4—4882no—
335C2×D4—4882no—
336C2×D4—4882no—
337C2×D4—4882no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—612no—
342C22×C4—611no—
343C22×C4—612no—
344C22×C4—612no—
345C22×C4—1228no—
346C22×C4—1222no—
347C22×C4—1222no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—4882no—
351C22×C4—4882no—
352C22×C4—4882no—
353C22×C4—2442no—
354C22×C4D4×D6112yes—
355C22×C4C23×D6111yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—1222no—
360C2×C8—1222no—
361C2×C8—1222no—
362C4⋊C4—22no—
363C4⋊C4—28no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—82no—
367C4⋊C4—618no—
368C4⋊C4—1222no—
369C4⋊C4—1222no—
370C4⋊C4—4882no—
371C4⋊C4—2442no—
372C4⋊C4C2×C4×D6114yes—
373C22⋊C4—618no—
374C22⋊C4—1228no—
375C22⋊C4—1228no—
376C22⋊C4—2442no—
377C42—612no—
378C42—612no—
379C42—2442no—
380C42C2×C4×D6112yes—
381C42D4×D6112yes—
382C23—612no—
383C23—1222no—
384C23—1222no—
385C23—1221no—
386C23—1221no—
387C23—1224no—
388C23—2442no—
389C23—2442no—
390C23—2442no—
391C23—2442no—
392C23—2442no—
393C23—2442no—
394C23C2×D4×D6111yescharacteristic
395D4—1224no—
396D4—1224no—
397D4—4888no—
398D4—4888no—
399D4—2442no—
400D4—2442no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—614no—
411C2×C4—612no—
412C2×C4—1222no—
413C2×C4—1222no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1224no—
417C2×C4—1224no—
418C2×C4—4888no—
419C2×C4—4884no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4C22⋊C4×D6114yes—
425C2×C4C22×C4×D6111yescharacteristic
426C2×C4C2×D4×D6111yescharacteristic
427C2×C4#1451982312yes—
428C2×C4#1451982512yes—
429C8—42no—
430C8—2442no—
431C22—1224no—
432C22—1222no—
433C22—4882no—
434C22—4882no—
435C22—4882no—
436C22—4882no—
437C22—2444no—
438C22—2444no—
439C22—2444no—
440C22—2442no—
441C22—2441no—
442C22—2441no—
443C22C2×C22⋊C4×D6111yescharacteristic
444C22#1684033011yescharacteristic
445C22#1684033711yescharacteristic
446C22#1684056612yes—
447C22#1684057112yes—
448C4—24no—
449C4—44no—
450C4—81no—
451C4—81no—
452C4—1224no—
453C4—1222no—
454C4—4881no—
455C4—4881no—
456C4#1684024412yes—
457C4#1684034212yes—
458C2—4884no—
459C2—2442no—
460C2#2144841011yescharacteristic
461C2#2145434312yes—
462C2#2145687011yescharacteristic
463C2#2145688111yescharacteristic
464C2#2145707312yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1992C2
socleC23×C61C22×D4
prime radicalC2×#16840368C2
cosocleC22×C4×C61C24
hypercenter#1992D61
hypocenterC61#21872
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
2. C2×#16839824C2semidirect
3. #21448440C2semidirect
4. #21446785C2semidirect
5. #21446643C2semidirect
6. #21446624C2semidirect
7. #21446140C2semidirect
8. #21446135C2semidirect
9. #21445828C2semidirect
10. #21445818C2semidirect
11. C61×#1992C2semidirect
13. #16840368C4semidirect
50. #14519774C2×C4semidirect
56. #14519739C2×C4semidirect
83. #125×C61C2×C4semidirect
229. #1992D61semidirect
274. C61#21872semidirect
286. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403682
C2×#168398242
#214484402
#214467852
#214466432
#214466242
#214461402
#214461352
#214458282
#214458182
C61×#19922
#2187261