Finite Group Databasebeta

← #29779750

Subgroups — #29779750

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454182C211yesmaximal, characteristic, complemented
2C4×#14519765C211yesmaximal, characteristic
3#21447388C212yesmaximal, complemented
4#21447061C212yesmaximal
5#21446516C212yesmaximal
6#21446509C212yesmaximal, complemented
7#21445825C211yesmaximal, characteristic, complemented
8#21445825C211yesmaximal, characteristic, complemented
9#21445820C211yesmaximal, characteristic
10#21445820C211yesmaximal, characteristic
11C61×#1767C211yesmaximal, characteristic, complemented
12C2×#14519769—21no—
13C2×#14519765C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16C2×#14519702C2212yes—
17C2×#14519702C2212yes—
18C2×#14519699C2212yes—
19C2×#14519699C2212yes—
20#16839668C2211yescharacteristic
21#16839664C2211yescharacteristic
22#16839660—21no—
23#16839660—21no—
24#16839660C2212yes—
25#16839660C2212yes—
26#16839653—21no—
27#16839653—21no—
28#16839632C2211yescharacteristic
29#16839632C2211yescharacteristic
30#16839628—21no—
31#16839628C2211yescharacteristic
32C4×#11758198—21no—
33C4×#11758198—21no—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38C2×C4⋊C8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×#97×C61C2212yes—
41C43×C61C2211yescharacteristic
42C61×#336C2212yes—
43C22×Dic122C2311yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46#14519769—44no—
47#14519765—22no—
48#14519765—22no—
49C4×Dic122—22no—
50C4×Dic122—22no—
51C4×D244—22no—
52C4×D244—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—21no—
62C2×#11758196—22no—
63C2×#11758196—22no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×C4×Dic61—21no—
70#14519702—24no—
71#14519702—42no—
72#14519699—24no—
73#14519699—42no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—22no—
79#14519684—22no—
80C22×C8×C61C2312yes—
81C22×C8×C61C2312yes—
82C2×C42×C61D411yescharacteristic
83C2×C42×C61D411yescharacteristic
84C2×C42×C61D411yescharacteristic
85C2×C42×C61D411yescharacteristic
86C2×C42×C61C2311yescharacteristic
87C2×C42×C61C2311yescharacteristic
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—22no—
90C4⋊C8×C61—22no—
91#97×C61—42no—
92C23×D61—21no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—22no—
104C22×Dic61—22no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244#4112yes—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×Dic122—21no—
115C2×Dic122—21no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×Dic122#4112yes—
120C2×C8×C61—22no—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C42×C61—22no—
127C42×C61—22no—
128C42×C61—21no—
129C42×C61—21no—
130C42×C61—21no—
131C42×C61—21no—
132C42×C61—41no—
133C42×C61—41no—
134C42×C61#4112yes—
135C42×C61#4112yes—
136#11758198—24no—
137#11758198—24no—
138#11758198—42no—
139#11758198—42no—
140#11758198—42no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—42no—
150C4⋊Dic61—42no—
151C4⋊Dic61—42no—
152C4⋊Dic61—42no—
153#11758196—44no—
154#11758196—44no—
155#11758196—42no—
156C4×Dic61—42no—
157C23×C61C22×D411yescharacteristic
158C22×D61—22no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61#12412yes—
180C2×C4×C61#12412yes—
181C2×C4×C61AGL(1,ℤ/8)12yes—
182C2×C4×C61#13712yes—
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61H(5,𝔽2)11yescharacteristic
188C2×C4×C612−1+411yescharacteristic
189C2×Dic61—22no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204D244—22no—
205D244—22no—
206D244—41no—
207D244—41no—
208C4×D61—84no—
209Dic122—22no—
210Dic122—22no—
211Dic122—41no—
212Dic122—41no—
213C8×C61—42no—
214C8×C61—42no—
215#29131—611nomaximal, Sylow
216C22×C61#52511yescharacteristic
217C22×C61C2×#12411yescharacteristic
218C22×C61#53711yescharacteristic
219C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
220C22×C61C2×#13711yescharacteristic
221C22×C61#57511yescharacteristic
222C22×C61#57511yescharacteristic
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—42no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—22no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61—41no—
234C4×C61—41no—
235C4×C61#48914yes—
236C4×C61#55211yescharacteristic
237C4×C61#55211yescharacteristic
238C4×C61#55311yescharacteristic
239C4×C61#55311yescharacteristic
240Dic61—42no—
241Dic61—82no—
242Dic61—81no—
243Dic61—81no—
244Dic61—82no—
245#2859—611no—
246C4×#124—611no—
247#2101—612no—
248#2042—612no—
249#1936—612no—
250#1935—612no—
251#1803—611no—
252#1803—611no—
253#1802—611no—
254#1802—611no—
255#1767D6111yescharacteristic, complemented
256C2×C61#256211yescharacteristic
257C2×C61#311511yescharacteristic
258C2×C61#311611yescharacteristic
259C2×C61C2×#48911yescharacteristic
260C2×C61#314211yescharacteristic
261C2×C61#325111yescharacteristic
262C2×C61#325211yescharacteristic
263D61—82no—
264C2×#125—1221no—
265C2×#124—611no—
266C2×C4×Q8—611no—
267C2×C4×D4—611no—
268C2×C4⋊C8C2×D6111yescharacteristic
269C2×C4⋊C8C2×D6111yescharacteristic
270C2×#103—612no—
271C2×#103—612no—
272C2×#102—612no—
273C2×#102—612no—
274C2×#97C2×D6112yes—
275#391—611no—
276#390—611no—
277#389—612no—
278#389—612no—
279#389—1221no—
280#389—1221no—
281#388—1221no—
282#388—1221no—
283#384—611no—
284#384—611no—
285#382—611no—
286#382—1221no—
287C4×C22⋊C4—1221no—
288C4×C22⋊C4—1221no—
289C43C2×D6111yescharacteristic
290#340—612no—
291#340—612no—
292#340—612no—
293#340—612no—
294#336C2×D6112yes—
295C61#2913111yescharacteristic, complemented, Sylow
296C22×Q8—611no—
297C22×D4—611no—
298C23×C4—1221no—
299C22×C8C22×D6112yes—
300C22×C8C22×D6112yes—
301#125—2444no—
302#124—1222no—
303#124—1222no—
304C4×Q8—1222no—
305C4×Q8—1222no—
306C4×D4—1222no—
307C4×D4—1222no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—612no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4—1222no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C42—1221no—
325C2×C42D24411yescharacteristic
326C2×C42D24411yescharacteristic
327C2×C42D24411yescharacteristic
328C2×C42D24411yescharacteristic
329C2×C42C22×D6111yescharacteristic
330C2×C42C22×D6111yescharacteristic
331C2×C42C22×D6111yescharacteristic
332C4⋊C8—22no—
333C4⋊C8—22no—
334#103—1224no—
335#103—2442no—
336#102—1224no—
337#102—2442no—
338#97—42no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342#95—1221no—
343#95—1222no—
344#95—1222no—
345C24—1221no—
346C2×Q8—612no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—1221no—
352C2×D4—612no—
353C2×D4—1221no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—2442no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4C2×D24411yescharacteristic
367C22×C4C2×D24411yescharacteristic
368C22×C4C2×D24411yescharacteristic
369C22×C4C2×D24411yescharacteristic
370C22×C4C2×D24411yescharacteristic
371C22×C4C2×D24411yescharacteristic
372C22×C4C23×D6111yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C4⋊C4—1222no—
380C4⋊C4—1222no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—2444no—
388C4⋊C4—2444no—
389C4⋊C4—2442no—
390C4⋊C4—2442no—
391C4⋊C4—2442no—
392C4⋊C4—2442no—
393C4⋊C4—2442no—
394C22⋊C4—1224no—
395C22⋊C4—1224no—
396C22⋊C4—2442no—
397C22⋊C4—2442no—
398C22⋊C4—2442no—
399C42—22no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—22no—
405C42—41no—
406C42—41no—
407C42—2442no—
408C42#1175822212yes—
409C42#1175822212yes—
410C23—1222no—
411C23—2441no—
412C23—2441no—
413C23—2441no—
414C23—2441no—
415C23—2441no—
416C23—2441no—
417C23C22×D24411yescharacteristic
418Q8—1222no—
419Q8—1222no—
420Q8—2441no—
421Q8—2441no—
422D4—1222no—
423D4—1222no—
424D4—2441no—
425D4—2441no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—1222no—
441C2×C4—4884no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4#1451976512yes—
457C2×C4#1451976512yes—
458C2×C4#1451980512yes—
459C2×C4#1451981012yes—
460C2×C4#41×D6111yescharacteristic
461C2×C4#41×D6111yescharacteristic
462C2×C4C2×#1175822211yescharacteristic
463C2×C4C2×#1175822211yescharacteristic
464C2×C4#1451982211yescharacteristic
465C2×C4#1451982511yescharacteristic
466C8—42no—
467C8—42no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22—2442no—
473C22—2442no—
474C22—2442no—
475C22#1684027411yescharacteristic
476C22C2×#1451976511yescharacteristic
477C22#1684034311yescharacteristic
478C22C2×#1451980511yescharacteristic
479C22C2×#1451981011yescharacteristic
480C22#1684056211yescharacteristic
481C22#1684056211yescharacteristic
482C4—22no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—4881no—
488C4—4882no—
489C4—4881no—
490C4—4882no—
491C4—2442no—
492C4#1684012214yes—
493C4#1684045311yescharacteristic
494C4#1684045311yescharacteristic
495C4#1684046211yescharacteristic
496C4#1684046211yescharacteristic
497C2—4882no—
498C2#2145098011yescharacteristic
499C2#2145597111yescharacteristic
500C2#2145598211yescharacteristic
501C2C2×#1684012211yescharacteristic
502C2#2145622011yescharacteristic
503C2#2145718211yescharacteristic
504C2#2145719311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1767C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1767D61
hypocenterC61#29131
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
1. #21454182C2semidirect
3. #21447388C2semidirect
6. #21446509C2semidirect
7. #21445825C2semidirect
8. #21445825C2semidirect
11. C61×#1767C2semidirect
255. #1767D61semidirect
295. C61#29131semidirect

Maximal subgroups

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

subgroupindex
#214541822
C4×#145197652
#214473882
#214470612
#214465162
#214465092
#214458252
#214458202
C61×#17672
#2913161