Finite Group Databasebeta

← #29689793

Subgroups — #29689793

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×#16840367C211yesmaximal, characteristic
2#21448442C211yesmaximal, characteristic, complemented
3#21446788C211yesmaximal, characteristic, complemented
4#21446784C211yesmaximal, characteristic, complemented
5#21446643C211yesmaximal, characteristic, complemented
6#21446625C211yesmaximal, characteristic, complemented
7#21446139C212yesmaximal, complemented
8#21446134C212yesmaximal, complemented
9#21445830C212yesmaximal, complemented
10#21445820C212yesmaximal, complemented
11C2×C61×#433C211yesmaximal, characteristic, complemented
12#16840367—24no—
13#16840367C418yescomplemented
14C2×#14519774C2211yescharacteristic
15C2×#14519769C2211yescharacteristic
16C2×#14519762—22no—
17C2×#14519751C2212yes—
18C2×#14519750—22no—
19C2×C4×D244C2212yes—
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22C2×#14519699C2212yes—
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26#16839661—22no—
27#16839637—22no—
28#16839632C2211yescharacteristic
29#16839632C2211yescharacteristic
30#16839628C2211yescharacteristic
31#16839628C2211yescharacteristic
32#16839525C2212yes—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36C2×#117×C61C2211yescharacteristic
37C61×#433—24no—
38C2×C4⋊C8×C61C2212yes—
39C2×C22⋊C8×C61C2212yes—
40C2×#97×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C22×D244—22no—
43C22×D244C2312yes—
44C22×C22⋊D61—22no—
45C23×Dic61—21no—
46C22×C4×D61—22no—
47#14519774—22no—
48#14519774C2×C414yescomplemented
49#14519769—22no—
50#14519769C2×C414yescomplemented
51#14519762—216no—
52#14519751—24no—
53#14519751C2×C418yes—
54#14519750—216no—
55C4×D244—24no—
56C4×D244C2×C418yescomplemented
57#14519736—42no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—42no—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×#11758196—22no—
68C2×C22⋊Dic61—21no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—21no—
72#14519699—42no—
73#14519699—42no—
74#14519699—42no—
75#14519699—42no—
76#14519684—22no—
77#14519684—22no—
78C23×C4×C61C2311yescharacteristic
79C22×C8×C61C2312yes—
80C22×C8×C61C2312yes—
81#117×C61C2×C414yes—
82#117×C61D414yes—
83C2×C4⋊C4×C61D412yes—
84C2×C22⋊C4×C61D412yes—
85C2×C42×C61C2311yescharacteristic
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—24no—
88C22⋊C8×C61—24no—
89#97×C61—22no—
90C4×C8×C61—22no—
91C24×C61C2×D411yescharacteristic
92C23×D61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61C22×C412yes—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2411yescharacteristic
101C2×C22⋊D61—28no—
102C2×C22⋊D61—82no—
103C2×C22⋊D61—82no—
104C22×Dic61—22no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—42no—
110C22×Dic61—82no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—28no—
117C2×D244—82no—
118C2×D244—82no—
119C2×D244C22×C414yes—
120C2×C4×D61—28no—
121C2×C4×D61—82no—
122C2×C4×D61—82no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C4⋊C4×C61C22⋊C418yes—
130C22⋊C4×C61C22⋊C418yes—
131C42×C61C22×C412yes—
132C42×C61C22×C412yes—
133C42×C61C2×D412yes—
134C42×C61C2×D412yes—
135C22⋊Dic61—42no—
136#11758198—28no—
137#11758198—28no—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—42no—
144C4⋊Dic61—82no—
145C4⋊Dic61—82no—
146C4⋊Dic61C22×C414yes—
147#11758196—28no—
148C4×Dic61—42no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—22no—
152C23×C61—22no—
153C23×C61C2×C22⋊C412yes—
154C23×C61C22×D411yescharacteristic
155C22×D61—24no—
156C22×D61—42no—
157C22×D61—42no—
158C22×D61—42no—
159C22×D61—42no—
160C22×D61—42no—
161C22×D61—42no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—24no—
169C2×C4×C61—24no—
170C2×C4×C61C2×C22⋊C414yes—
171C2×C4×C61C2×C22⋊C412yes—
172C2×C4×C61C2×C22⋊C414yes—
173C2×C4×C61C23×C411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61H(5,𝔽2)12yes—
176C2×C4×C612−1+412yes—
177C22⋊D61—88no—
178C2×Dic61—24no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—84no—
190D244—24no—
191D244—24no—
192D244—42no—
193D244—42no—
194D244—88no—
195C4×D61—88no—
196C8×C61—42no—
197C8×C61—42no—
198#21872—611nomaximal, Sylow
199C22×C61—22no—
200C22×C61—24no—
201C22×C61—41no—
202C22×C61—41no—
203C22×C61—42no—
204C22×C61C22×C22⋊C411yescharacteristic
205C22×C61#53511yescharacteristic
206C22×C61#53611yescharacteristic
207C22×C61#57612yes—
208C22×C61#57712yes—
209C2×D61—44no—
210C2×D61—44no—
211C2×D61—44no—
212C2×D61—82no—
213C2×D61—82no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—22no—
217C4×C61—24no—
218C4×C61—24no—
219C4×C61#52012yes—
220C4×C61#53712yes—
221Dic61—44no—
222Dic61—81no—
223Dic61—81no—
224Dic61—81no—
225Dic61—81no—
226C2×#540—611no—
227C2×#433D6111yescharacteristic, complemented
228#2317—611no—
229#1992—611no—
230#1991—611no—
231#1961—611no—
232#1957—611no—
233#1875—612no—
234#1874—612no—
235#1804—612no—
236#1802—612no—
237C2×C61—42no—
238C2×C61#231111yescharacteristic
239C2×C61#289712yes—
240C2×C61#322011yescharacteristic
241C2×C61#322111yescharacteristic
242C2×C61#324112yes—
243D61—84no—
244#540—618no—
245#540—1224no—
246C2×#127—611no—
247C2×#125—611no—
248C2×#123—1222no—
249C2×#121—612no—
250C2×#121—1222no—
251C2×C4×D4—612no—
252C2×#117—611no—
253C2×#117C2×D6111yescharacteristic
254C22×C4⋊C4—611no—
255#433—24no—
256C2×C4⋊C8C2×D6112yes—
257C2×#102—612no—
258C2×#102—612no—
259C2×#102—612no—
260C2×#102—612no—
261C2×C22⋊C8C2×D6112yes—
262C2×#97C2×D6111yescharacteristic
263C2×C4×C8C2×D6111yescharacteristic
264#389—1222no—
265#385—1222no—
266#384—611no—
267#384—611no—
268#382—611no—
269#382—611no—
270#340—612no—
271#340—612no—
272#340—612no—
273#340—612no—
274C61#2187211yescharacteristic, complemented, Sylow
275C22×D4—612no—
276C22×D4—1222no—
277C22×D4—1222no—
278C23×C4—1222no—
279C23×C4—1221no—
280C23×C4C22×D6111yescharacteristic
281C22×C8C22×D6112yes—
282C22×C8C22×D6112yes—
283#127—614no—
284#127—1222no—
285#125—614no—
286#125—1222no—
287#123—12216no—
288#121—618no—
289#121—1224no—
290#121—12216no—
291C4×D4—618no—
292C4×D4—1224no—
293#117—2442no—
294#117C4×D6114yes—
295#117C22⋊D6114yes—
296C2×C4⋊C4—612no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—1222no—
302C2×C4⋊C4—1222no—
303C2×C4⋊C4—1222no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—2442no—
306C2×C4⋊C4D24412yes—
307C2×C22⋊C4—1222no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4D24412yes—
311C2×C42—1221no—
312C2×C42C22×D6111yescharacteristic
313C2×C42C22×D6111yescharacteristic
314C4⋊C8—24no—
315#102—2442no—
316#102—2442no—
317#102—2442no—
318#102—2442no—
319C22⋊C8—24no—
320#97—22no—
321C4×C8—22no—
322#95—1222no—
323#95—1222no—
324C24—1222no—
325C24C2×D24411yescharacteristic
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—22no—
340C22×C4—1222no—
341C22×C4—1228no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—4882no—
347C22×C4—4882no—
348C22×C4—4882no—
349C22×C4—2442no—
350C22×C4C2×C4×D6112yes—
351C22×C4C2×D24412yes—
352C22×C4C2×D24412yes—
353C22×C4C2×D24411yescharacteristic
354C22×C4C2×C22⋊D6112yes—
355C22×C4C23×D6111yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C4⋊C4—614no—
363C4⋊C4—1222no—
364C4⋊C4—1228no—
365C4⋊C4—1222no—
366C4⋊C4—1222no—
367C4⋊C4—1222no—
368C4⋊C4—1222no—
369C4⋊C4—4882no—
370C4⋊C4—4882no—
371C4⋊C4—2442no—
372C4⋊C4#1175819818yes—
373C22⋊C4—1228no—
374C22⋊C4—1228no—
375C22⋊C4—2442no—
376C22⋊C4#1175819818yes—
377C42—2442no—
378C42C2×C4×D6112yes—
379C42C2×C4×D6112yes—
380C42C2×C22⋊D6112yes—
381C42C2×C22⋊D6112yes—
382C23—22no—
383C23—22no—
384C23—21no—
385C23—21no—
386C23—1224no—
387C23—2442no—
388C23—2442no—
389C23—2442no—
390C23—2442no—
391C23—2442no—
392C23—2442no—
393C23C2×#1175819812yes—
394C23C22×D24411yescharacteristic
395D4—1224no—
396D4—1224no—
397D4—4888no—
398D4—4888no—
399D4—2442no—
400D4—2442no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—1224no—
410C2×C4—4888no—
411C2×C4—4884no—
412C2×C4—2442no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4C2×#1175819814yes—
423C2×C4C2×#1175819814yes—
424C2×C4C2×#1175819812yes—
425C2×C4C22×C4×D6111yescharacteristic
426C2×C4C22×C22⋊D6111yescharacteristic
427C2×C4#1451982312yes—
428C2×C4#1451982412yes—
429C8—42no—
430C8—42no—
431C22—24no—
432C22—22no—
433C22—42no—
434C22—41no—
435C22—41no—
436C22—4882no—
437C22—4882no—
438C22—4882no—
439C22—4882no—
440C22—2444no—
441C22—2444no—
442C22—2444no—
443C22C22×#1175819811yescharacteristic
444C22#1684033111yescharacteristic
445C22#1684033811yescharacteristic
446C22#1684056712yes—
447C22#1684057412yes—
448C4—24no—
449C4—24no—
450C4—22no—
451C4—4881no—
452C4—4881no—
453C4—4881no—
454C4—4881no—
455C4—2444no—
456C4#1684024212yes—
457C4#1684034012yes—
458C2—42no—
459C2—4884no—
460C2#2144840711yescharacteristic
461C2#2145434612yes—
462C2#2145687111yescharacteristic
463C2#2145688211yescharacteristic
464C2#2145707412yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#433C2
socleC23×C61C22×D4
prime radicalC2×#16840367C2
cosocleC22×C4×C61C24
hypercenterC2×#433D61
hypocenterC61#21872
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. #21448442C2semidirect
3. #21446788C2semidirect
4. #21446784C2semidirect
5. #21446643C2semidirect
6. #21446625C2semidirect
7. #21446139C2semidirect
8. #21446134C2semidirect
9. #21445830C2semidirect
10. #21445820C2semidirect
11. C2×C61×#433C2semidirect
13. #16840367C4semidirect
48. #14519774C2×C4semidirect
50. #14519769C2×C4semidirect
56. C4×D244C2×C4semidirect
227. C2×#433D61semidirect
274. C61#21872semidirect

Maximal subgroups

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

subgroupindex
C2×#168403672
#214484422
#214467882
#214467842
#214466432
#214466252
#214461392
#214461342
#214458302
#214458202
C2×C61×#4332
#2187261