Finite Group Databasebeta

← #29734392

Subgroups — #29734392

Order 15616.

1088 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840368C211yesmaximal, characteristic
2C2×#16840128C211yesmaximal, characteristic, complemented
3#21451650C211yesmaximal, characteristic, complemented
4#21447010C212yesmaximal, complemented
5#21447003C212yesmaximal, complemented
6#21446789C211yesmaximal, characteristic, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446077C212yesmaximal, complemented
9#21446063C212yesmaximal, complemented
10#21445586C211yesmaximal, characteristic, complemented
11C2×C61×#499C211yesmaximal, characteristic, complemented
12#16840368—24no—
13#16840368—24no—
14C2×#14519774C2211yescharacteristic
15C2×#14519762—22no—
16C2×#14519752—22no—
17C2×#14519752C2212yes—
18C2×#14519739C2212yes—
19C2×#14519734C2211yescharacteristic
20#16840128—24no—
21C2×#14519699C2212yes—
22C2×#14519699C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×#14519689C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839695—22no—
29#16839684—22no—
30#16839658C2212yes—
31#16839632C2211yescharacteristic
32#16839619C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839516C2211yescharacteristic
36#16839516C2211yescharacteristic
37C2×#125×C61C2211yescharacteristic
38C2×#125×C61C2211yescharacteristic
39C61×#499—24no—
40C2×#107×C61C2212yes—
41C2×#106×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×D244—22no—
44C22×D244—22no—
45C22×D244C2312yes—
46C22×C4×D61—22no—
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2311yescharacteristic
49C22×#92418C2311yescharacteristic
50#14519774—22no—
51#14519774D414yes—
52#14519762—216no—
53#14519752—216no—
54#14519752—24no—
55#14519752—24no—
56#14519739—24no—
57#14519739—24no—
58#14519734—22no—
59#14519734—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×#11758196D412yes—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198D412yes—
68C2×C4×Dic61C2311yescharacteristic
69#14519699—24no—
70#14519699—24no—
71#14519697—24no—
72#14519697—24no—
73#14519689—22no—
74#14519689—22no—
75#14519688—22no—
76#14519684—22no—
77#14519684—22no—
78C22×C8×C61C2311yescharacteristic
79C22×C8×C61C2311yescharacteristic
80#125×C61—22no—
81#125×C61—22no—
82#125×C61—22no—
83#125×C61D414yes—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89#107×C61—24no—
90#106×C61—24no—
91C4×C8×C61—22no—
92C23×D61—22no—
93C23×D61C2×D411yescharacteristic
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—22no—
99C22×Dic61C2×D412yes—
100C22×Dic61C2×D411yescharacteristic
101C2×D244—22no—
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—28no—
107C2×D244—28no—
108C2×D244—82no—
109C2×D244—82no—
110C2×D244—82no—
111C2×D244—82no—
112C2×D244C2×D414yes—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×C4×D61—28no—
120C2×C4×D61—82no—
121C2×C4×D61—82no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61#4112yes—
127C2×C8×C61#4112yes—
128C4⋊C4×C61—28no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—44no—
133C4⋊C4×C61C2×D414yes—
134C42×C61C2×D412yes—
135C42×C61C2×D412yes—
136#11758198—28no—
137#11758198—28no—
138#11758198—24no—
139#11758198—82no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—82no—
143#11758196—24no—
144#11758196—82no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—21no—
153C23×C61C22×D411yescharacteristic
154C22×D61—21no—
155C22×D61—24no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—22no—
159C22×D61—22no—
160C22×D61—42no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—24no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61#12014yes—
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×C4×C612−1+411yescharacteristic
180C2×C4×C612−1+411yescharacteristic
181C2×Dic61—22no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190D244—24no—
191D244—24no—
192D244—42no—
193D244—42no—
194D244—88no—
195D244—88no—
196C4×D61—41no—
197C4×D61—41no—
198C4×D61—41no—
199C4×D61—41no—
200C4×D61—88no—
201C8×C61—22no—
202C8×C61—22no—
203#92418—41no—
204#92418—41no—
205#25682—611nomaximal, Sylow
206C22×C61C2×#12011yescharacteristic
207C22×C61#53211yescharacteristic
208C22×C61#53611yescharacteristic
209C22×C61#57612yes—
210C22×C61#57712yes—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—41no—
214C2×D61—44no—
215C2×D61—42no—
216C2×D61—44no—
217C2×D61—42no—
218C2×D61—44no—
219C2×D61—82no—
220C2×D61—82no—
221C2×D61—82no—
222C2×D61—82no—
223C4×C61—24no—
224C4×C61—44no—
225C4×C61—82no—
226C4×C61#54711yescharacteristic
227C4×C61#54711yescharacteristic
228C4×C61#54811yescharacteristic
229C4×C61#54811yescharacteristic
230Dic61—41no—
231Dic61—42no—
232Dic61—82no—
233C2×#540—611no—
234C2×#499D6111yescharacteristic, complemented
235C2×#490—611no—
236#2629—611no—
237#2034—612no—
238#2033—612no—
239#1992—611no—
240#1957—611no—
241#1864—612no—
242#1862—612no—
243#1750—611no—
244C2×C61#243211yescharacteristic
245C2×C61#301212yes—
246C2×C61#317411yescharacteristic
247C2×C61#317611yescharacteristic
248C2×C61#324112yes—
249D61—81no—
250D61—84no—
251#540—1224no—
252#540—1224no—
253C2×#127—611no—
254C2×#125C2×D6111yescharacteristic
255C2×#125C2×D6111yescharacteristic
256C2×#123—1222no—
257C2×#121—612no—
258C2×#121—1222no—
259C2×C4×D4—612no—
260C2×#117—611no—
261#499—24no—
262#490—1224no—
263C2×#107C2×D6112yes—
264C2×#106C2×D6112yes—
265C2×#102—612no—
266C2×#102—612no—
267C2×#102—612no—
268C2×#102—612no—
269C2×C22⋊C8—611no—
270C2×C22⋊C8—611no—
271C2×#97—611no—
272C2×C4×C8C2×D6111yescharacteristic
273#398—1222no—
274#396—1222no—
275#389—612no—
276#384—611no—
277#380—612no—
278#340—612no—
279#340—612no—
280#336—611no—
281#336—611no—
282C61#2568211yescharacteristic, complemented, Sylow
283C22×D4—612no—
284C22×D4—1222no—
285C22×D4—1222no—
286C23×C4—611no—
287C23×C4—1222no—
288C22×C8—611no—
289C22×C8—611no—
290C22×C8C22×D6111yescharacteristic
291C22×C8C22×D6111yescharacteristic
292#127—614no—
293#127—1222no—
294#125—22no—
295#125—22no—
296#125—22no—
297#125C22⋊D6114yes—
298#123—12216no—
299#121—1224no—
300#121—12216no—
301#121—1224no—
302C4×D4—1224no—
303C4×D4—1224no—
304#117—1222no—
305#117—1222no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—612no—
310C2×C4⋊C4—1222no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4C22×D6112yes—
313C2×C4⋊C4C22×D6112yes—
314C2×C22⋊C4—612no—
315C2×C22⋊C4—1222no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1222no—
318C2×C42—611no—
319C2×C42C22×D6111yescharacteristic
320#107—24no—
321#106—24no—
322#102—1224no—
323#102—1224no—
324#102—1224no—
325#102—1224no—
326C22⋊C8—1222no—
327C22⋊C8—1222no—
328#97—1222no—
329C4×C8—22no—
330#95—1222no—
331#95—1222no—
332C24—611no—
333C24—1222no—
334C2×D4—614no—
335C2×D4—1228no—
336C2×D4—1222no—
337C2×D4—1222no—
338C2×D4—1222no—
339C2×D4—1222no—
340C2×D4—1222no—
341C2×D4—1228no—
342C2×D4—4882no—
343C2×D4—4882no—
344C2×D4—4882no—
345C2×D4—4882no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—611no—
349C22×C4—612no—
350C22×C4—1228no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1222no—
358C22×C4—4882no—
359C22×C4—4882no—
360C22×C4D4×D6112yes—
361C22×C4C23×D6111yescharacteristic
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—1221no—
367C2×C8—1221no—
368C2×C8—1221no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8#1175822612yes—
373C2×C8#1175822612yes—
374C4⋊C4—22no—
375C4⋊C4—28no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—44no—
379C4⋊C4—1224no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—4882no—
383C4⋊C4—4882no—
384C4⋊C4C2×C22⋊D6114yes—
385C22⋊C4—1228no—
386C22⋊C4—1228no—
387C22⋊C4—1224no—
388C22⋊C4—4882no—
389C42—1221no—
390C42—1221no—
391C42D4×D6112yes—
392C42C2×C22⋊D6112yes—
393C23—1224no—
394C23—1221no—
395C23—1222no—
396C23—1221no—
397C23—1222no—
398C23—1221no—
399C23—2442no—
400C23—2442no—
401C23—2442no—
402C23—2442no—
403C23—2442no—
404C23—2442no—
405C23C2×D4×D6111yescharacteristic
406D4—1224no—
407D4—1224no—
408D4—4888no—
409D4—4888no—
410D4—2442no—
411D4—2442no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—1222no—
421C2×C4—1221no—
422C2×C4—1222no—
423C2×C4—1221no—
424C2×C4—1222no—
425C2×C4—1221no—
426C2×C4—4888no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4#1451974514yes—
435C2×C4C2×D4×D6111yescharacteristic
436C2×C4C22×C22⋊D6111yescharacteristic
437C2×C4C2×#1175822611yescharacteristic
438C2×C4C2×#1175822611yescharacteristic
439C2×C4#1451982511yescharacteristic
440C2×C4#1451982511yescharacteristic
441C8—22no—
442C8—22no—
443C8—2441no—
444C8—2441no—
445C22—4882no—
446C22—4882no—
447C22—4882no—
448C22—4882no—
449C22—2444no—
450C22—2442no—
451C22—2444no—
452C22—2441no—
453C22—2441no—
454C22—2442no—
455C22—2441no—
456C22—2444no—
457C22C2×#1451974511yescharacteristic
458C22#1684031311yescharacteristic
459C22#1684033711yescharacteristic
460C22#1684056412yes—
461C22#1684056912yes—
462C4—24no—
463C4—44no—
464C4—82no—
465C4—4882no—
466C4—2442no—
467C4—2441no—
468C4#1684041511yescharacteristic
469C4#1684041511yescharacteristic
470C4#1684042311yescharacteristic
471C4#1684042311yescharacteristic
472C2—4884no—
473C2—4881no—
474C2#2144944411yescharacteristic
475C2#2145511912yes—
476C2#2145644911yescharacteristic
477C2#2145647011yescharacteristic
478C2#2145707512yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#499C2
socleC23×C61C22×D4
prime radicalC2×#16840368C2
hypercenterC2×#499D61
hypocenterC61#25682
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
2. C2×#16840128C2semidirect
3. #21451650C2semidirect
4. #21447010C2semidirect
5. #21447003C2semidirect
6. #21446789C2semidirect
7. #21446625C2semidirect
8. #21446077C2semidirect
9. #21446063C2semidirect
10. #21445586C2semidirect
11. C2×C61×#499C2semidirect
234. C2×#499D61semidirect
282. C61#25682semidirect

Maximal subgroups

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

subgroupindex
C2×#168403682
C2×#168401282
#214516502
#214470102
#214470032
#214467892
#214466252
#214460772
#214460632
#214455862
C2×C61×#4992
#2568261