Finite Group Databasebeta

← #29718429

Subgroups — #29718429

Order 15616.

984 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840081C211yesmaximal, characteristic
2C22×#14519702C211yesmaximal, characteristic, complemented
3#21449477C211yesmaximal, characteristic
4#21448694C211yesmaximal, characteristic, complemented
5#21447199C212yesmaximal, complemented
6#21447061C212yesmaximal, complemented
7#21446749C212yesmaximal, complemented
8#21446454C212yesmaximal, complemented
9#21446247C211yesmaximal, characteristic
10#21445755C211yesmaximal, characteristic, complemented
11C61×#1850C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C2×#14519765C2212yes—
14C2×#14519751C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16840081—24no—
17C2×#14519709C2211yescharacteristic
18C2×#14519709C2211yescharacteristic
19C2×#14519702—24no—
20C2×#14519702C2212yes—
21C2×#14519702C2212yes—
22C2×#14519702C2212yes—
23C2×#14519702C2212yes—
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2212yes—
27#16839675—22no—
28#16839668C2212yes—
29#16839660C2212yes—
30#16839645—22no—
31#16839639—21no—
32#16839639—21no—
33#16839622C2211yescharacteristic
34#16839620—21no—
35C4×C4⋊Dic61C2212yes—
36C22⋊C4×Dic61—21no—
37#16839525C2212yes—
38C23×C8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2212yes—
40C2×C22⋊C8×C61C2211yescharacteristic
41C4×C22⋊C4×C61C2211yescharacteristic
42C61×#336C2212yes—
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—22no—
46C22×Dic122—21no—
47C22×Dic122—42no—
48C22×Dic122C2312yes—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C23×Dic61—21no—
52C23×Dic61—21no—
53#14519765—24no—
54#14519765—42no—
55#14519751—22no—
56#14519751—22no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61D414yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C2×C22⋊Dic61—21no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×C4×Dic61—22no—
70#14519709—22no—
71#14519709—22no—
72#14519702—24no—
73#14519702—24no—
74#14519702—24no—
75#14519702—42no—
76#14519699—22no—
77#14519699—22no—
78#14519699—42no—
79#14519684—22no—
80#14519684—22no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61—22no—
86C22×C8×C61—42no—
87C22×C8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C42×C61C2312yes—
93C4⋊C8×C61—42no—
94C22⋊C8×C61—22no—
95#95×C61D412yes—
96C24×C61C2×D411yescharacteristic
97C23×D61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D412yes—
104C22×C4×C61#4112yes—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—42no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—42no—
109C22×Dic61—22no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—24no—
113C22×Dic61—21no—
114C22×Dic61—22no—
115C22×Dic61—44no—
116C22×Dic61—42no—
117C22×Dic61—82no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—22no—
121C2×D244—22no—
122C2×Dic122—28no—
123C2×Dic122—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—48no—
129C2×Dic122—44no—
130C2×Dic122—44no—
131C2×Dic122—42no—
132C2×Dic122#4114yes—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—44no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C42×C61—22no—
144C42×C61—22no—
145C22⋊Dic61—42no—
146#11758198—44no—
147#11758198—44no—
148#11758198—42no—
149C4⋊Dic61—24no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61C2×D412yes—
157C4⋊Dic61C2×D412yes—
158#11758196—82no—
159C4×Dic61—82no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61#12112yes—
163C23×C61#13512yes—
164C23×C61#13712yes—
165C23×C61C22×D411yescharacteristic
166C22×D61—41no—
167C22×D61—42no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—42no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61#12112yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4112yes—
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61H(5,𝔽2)12yes—
186C22⋊D61—84no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—24no—
190C2×Dic61—48no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—41no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—84no—
202D244—41no—
203D244—41no—
204D244—42no—
205Dic122—24no—
206Dic122—24no—
207Dic122—48no—
208Dic122—42no—
209Dic122—42no—
210C8×C61—41no—
211C8×C61—41no—
212C8×C61—42no—
213#24483—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—24no—
217C22×C61—24no—
218C22×C61#46414yes—
219C22×C61C2×#12111yescharacteristic
220C22×C61#54612yes—
221C22×C61C2×#13511yescharacteristic
222C22×C61C2×#13711yescharacteristic
223C22×C61#57712yes—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—82no—
228C2×D61—82no—
229C4×C61—22no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61#52511yescharacteristic
233C4×C61#53511yescharacteristic
234C4×C61#55112yes—
235Dic61—44no—
236Dic61—42no—
237Dic61—42no—
238Dic61—81no—
239Dic61—82no—
240C2×#481—611no—
241C22×#103—611no—
242#2436—611no—
243#2345—611no—
244#2066—612no—
245#2042—612no—
246#1983—612no—
247#1927—612no—
248#1893—611no—
249#1850D6111yescharacteristic, complemented
250#1787—611no—
251C2×C61—24no—
252C2×C61#241111yescharacteristic
253C2×C61C2×#46411yescharacteristic
254C2×C61#304211yescharacteristic
255C2×C61#330112yes—
256C2×C61#331612yes—
257D61—161no—
258C23×Q8—611no—
259C23×C8C2×D6111yescharacteristic
260C2×#124—612no—
261C2×#121—611no—
262C22×C4⋊C4—611no—
263#481—1224no—
264C2×#106—611no—
265C2×#106—611no—
266C2×C4⋊C8C2×D6112yes—
267C2×#103—612no—
268C2×#103—612no—
269C2×#103—612no—
270C2×#103—612no—
271C2×#103—1224no—
272C2×#102—611no—
273C2×#102—611no—
274C2×#102—612no—
275C2×C22⋊C8C2×D6111yescharacteristic
276#393—1222no—
277#391—612no—
278#389—612no—
279#387—1222no—
280#386—1221no—
281#386—1221no—
282#380—611no—
283#380—1221no—
284C4×C4⋊C4—612no—
285C4×C22⋊C4—1221no—
286C4×C22⋊C4C2×D6111yescharacteristic
287#340—612no—
288#336C2×D6112yes—
289C61#2448311yescharacteristic, complemented, Sylow
290C22×Q8—612no—
291C22×Q8—1224no—
292C22×Q8—1221no—
293C22×Q8—1222no—
294C22×Q8—1222no—
295C22×Q8—2442no—
296C22×D4—611no—
297C22×D4—1221no—
298C23×C4—1221no—
299C23×C4—1221no—
300C23×C4C22×D6111yescharacteristic
301C22×C8—22no—
302C22×C8—42no—
303C22×C8C22×D6112yes—
304C22×C8C22×D6111yescharacteristic
305C22×C8C22×D6111yescharacteristic
306#124—1224no—
307#124—2442no—
308#121—1222no—
309#121—1222no—
310C2×C4⋊C4—614no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—612no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1222no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4D24411yescharacteristic
323C2×C22⋊C4D24411yescharacteristic
324C2×C42—1222no—
325C2×C42C22×D6112yes—
326#106—1222no—
327#106—1222no—
328C4⋊C8—42no—
329#103—1224no—
330#103—1224no—
331#103—1224no—
332#103—2442no—
333#102—1222no—
334#102—1222no—
335#102—2442no—
336C22⋊C8—22no—
337#95—1222no—
338#95—1222no—
339#95—1221no—
340#95—1222no—
341#95—1222no—
342#95D24412yes—
343C24—1221no—
344C24C2×D24411yescharacteristic
345C2×Q8—614no—
346C2×Q8—1228no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×Q8—1222no—
350C2×Q8—1222no—
351C2×Q8—1222no—
352C2×Q8—2444no—
353C2×Q8—2448no—
354C2×Q8—2444no—
355C2×Q8—2442no—
356C2×D4—1221no—
357C2×D4—1222no—
358C2×D4—1221no—
359C2×D4—1222no—
360C2×D4—2442no—
361C2×D4—2442no—
362C2×D4—2442no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—1224no—
366C22×C4—1221no—
367C22×C4—1222no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1222no—
371C22×C4—4882no—
372C22×C4—2444no—
373C22×C4—2442no—
374C22×C4C2×D24412yes—
375C22×C4C2×D24412yes—
376C22×C4C2×D24411yescharacteristic
377C22×C4D4×D6112yes—
378C22×C4#1175822612yes—
379C22×C4C23×D6111yescharacteristic
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—44no—
388C4⋊C4—612no—
389C4⋊C4—612no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1224no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—1221no—
396C4⋊C4—1222no—
397C4⋊C4—4882no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—2444no—
401C22⋊C4—2444no—
402C22⋊C4—2442no—
403C22⋊C4—2442no—
404C42—22no—
405C42—22no—
406C42—4882no—
407C23—22no—
408C23—22no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23—2442no—
413C23—2442no—
414C23#1451975212yes—
415C23#1451979912yes—
416C23#1451981012yes—
417C23C22×D24411yescharacteristic
418Q8—1224no—
419Q8—1224no—
420Q8—2448no—
421Q8—2442no—
422Q8—2442no—
423D4—4884no—
424D4—2441no—
425D4—2441no—
426D4—2442no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—1224no—
440C2×C4—4884no—
441C2×C4—2441no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2448no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4#1451975212yes—
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4#41×D6112yes—
455C2×C4C2×#1175822611yescharacteristic
456C2×C4#1451982212yes—
457C8—41no—
458C8—41no—
459C8—42no—
460C22—24no—
461C22—24no—
462C22—22no—
463C22—22no—
464C22—4882no—
465C22—4882no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22#1683998514yes—
470C22C2×#1451975211yescharacteristic
471C22#1684040712yes—
472C22C2×#1451979911yescharacteristic
473C22C2×#1451981011yescharacteristic
474C22#1684057412yes—
475C4—22no—
476C4—42no—
477C4—42no—
478C4—4882no—
479C4—4881no—
480C4—2442no—
481C4—2442no—
482C4—2444no—
483C4#1684027011yescharacteristic
484C4#1684032711yescharacteristic
485C4#1684043912yes—
486C2—24no—
487C2—9761no—
488C2#2144920211yescharacteristic
489C2C2×#1683998511yescharacteristic
490C2#2145533211yescharacteristic
491C2#2145772312yes—
492C2#2145793212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1850D61
hypocenterC61#24483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
2. C22×#14519702C2semidirect
4. #21448694C2semidirect
5. #21447199C2semidirect
6. #21447061C2semidirect
7. #21446749C2semidirect
8. #21446454C2semidirect
10. #21445755C2semidirect
11. C61×#1850C2semidirect
249. #1850D61semidirect
289. C61#24483semidirect

Maximal subgroups

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

subgroupindex
C2×#168400812
C22×#145197022
#214494772
#214486942
#214471992
#214470612
#214467492
#214464542
#214462472
#214457552
C61×#18502
#2448361