Finite Group Databasebeta

← #29729519

Subgroups — #29729519

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839968C211yesmaximal, characteristic
2#21451031C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic
4#21447199C212yesmaximal, complemented
5#21447137C212yesmaximal
6#21446673C212yesmaximal
7#21446621C211yesmaximal, characteristic, complemented
8#21446583C211yesmaximal, characteristic, complemented
9#21446495C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic
11C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519753—21no—
15C2×C4×D244C2211yescharacteristic
16#16839968—24no—
17C2×#14519709C2212yes—
18C2×#14519702C2212yes—
19C2×#14519702C2212yes—
20C2×#14519702C2212yes—
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23#16839701—22no—
24#16839684—22no—
25#16839669—21no—
26#16839668C2212yes—
27#16839668C2212yes—
28#16839664C2212yes—
29#16839660C2212yes—
30#16839632C2211yescharacteristic
31#16839627—21no—
32#16839619—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C8×Q8×C61—24no—
39C2×C4⋊C8×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C2×C4×C8×C61C2212yes—
43C22×Dic122C2312yes—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C2×#11758190—24no—
47C2×#11758190—24no—
48C2×#11758190—42no—
49#14519777—22no—
50#14519777—22no—
51#14519753—44no—
52C4×D244—22no—
53C4×D244—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×C4×Dic61—21no—
67#14519709—42no—
68#14519702—24no—
69#14519702—24no—
70#14519702—42no—
71#14519699—22no—
72#14519699—22no—
73#14519684—22no—
74#14519684—21no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79C22×Q8×C61D411yescharacteristic
80C22×C8×C61C2311yescharacteristic
81C22×C8×C61C2312yes—
82C22×C8×C61C2311yescharacteristic
83C4×Q8×C61—22no—
84C4×Q8×C61—22no—
85C2×C4⋊C4×C61D412yes—
86C2×C4⋊C4×C61D411yescharacteristic
87C2×C42×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—24no—
90C4⋊C8×C61D414yes—
91C4×C8×C61—24no—
92C4×C8×C61—22no—
93C23×D61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61SD1614yescomplemented
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—22no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—22no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244#4112yes—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×Dic122—22no—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122—22no—
121C2×Dic122—22no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—21no—
126C2×C8×C61—21no—
127C2×C8×C61—21no—
128C2×C8×C61—21no—
129C2×C8×C61C2×D414yes—
130C2×C8×C61#4112yes—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—22no—
133C42×C61—22no—
134C42×C61—22no—
135C42×C61C2×D412yes—
136C42×C61#4112yes—
137#11758198—42no—
138#11758198—44no—
139#11758198—82no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—42no—
149C4⋊Dic61—42no—
150C4⋊Dic61—42no—
151C4⋊Dic61—81no—
152#11758196—44no—
153C4×Dic61—42no—
154#11758190—48no—
155C23×C61C22×D411yescharacteristic
156C22×D61—22no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163Q8×C61—28no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61#12112yes—
172C2×C4×C61#12112yes—
173C2×C4×C61C2×SD1614yes—
174C2×C4×C61C2×SD1612yes—
175C2×C4×C61#13512yes—
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C61H(5,𝔽2)11yescharacteristic
181C2×C4×C612−1+411yescharacteristic
182C2×Dic61—41no—
183C2×Dic61—42no—
184C2×Dic61—41no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197D244—22no—
198D244—22no—
199D244—41no—
200D244—41no—
201C4×D61—84no—
202Dic122—42no—
203Dic122—42no—
204Dic122—42no—
205Dic122—42no—
206C8×C61—24no—
207C8×C61—22no—
208C8×C61—41no—
209#25293—611nomaximal, Sylow
210C22×C61C2×#12111yescharacteristic
211C22×C61#54811yescharacteristic
212C22×C61#55011yescharacteristic
213C22×C61C22×SD1611yescharacteristic
214C22×C61C2×#13511yescharacteristic
215C22×C61#57711yescharacteristic
216C22×C61#57811yescharacteristic
217C2×D61—42no—
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C4×C61—24no—
225C4×C61—22no—
226C4×C61—42no—
227C4×C61#46514yes—
228C4×C61#52911yescharacteristic
229C4×C61#54211yescharacteristic
230C4×C61#54611yescharacteristic
231C4×C61#54811yescharacteristic
232Dic61—81no—
233Dic61—82no—
234Dic61—81no—
235Dic61—82no—
236Dic61—81no—
237C2×#461—611no—
238C2×C8×Q8D6111yescharacteristic, complemented
239#2568—611no—
240#2538—611no—
241#2066—612no—
242#2055—612no—
243#1968—612no—
244#1956—611no—
245#1948—611no—
246#1933—612no—
247C4×#102—611no—
248C2×C61#242611yescharacteristic
249C2×C61C2×#46511yescharacteristic
250C2×C61#306311yescharacteristic
251C2×C61#335911yescharacteristic
252C2×C61#336011yescharacteristic
253C2×C61#339211yescharacteristic
254C2×C61#339311yescharacteristic
255D61—82no—
256C22×SD16—612no—
257C2×C4⋊Q8—611no—
258C2×C22⋊Q8—1221no—
259C2×C4×Q8C2×D6111yescharacteristic
260C2×C4×D4—611no—
261#461—1224no—
262C8×Q8—24no—
263C2×#106—612no—
264C2×C4⋊C8C2×D6112yes—
265C2×C4⋊C8C2×D6111yescharacteristic
266C2×#103—612no—
267C2×#103—612no—
268C2×#103—612no—
269C2×#102—611no—
270C2×#102—611no—
271C2×C4×C8C2×D6112yes—
272C2×C4×C8C2×D6111yescharacteristic
273#400—1222no—
274#396—1222no—
275#391—612no—
276#391—612no—
277#391—1221no—
278#390—612no—
279#389—612no—
280#384—611no—
281#382—1221no—
282#380—1221no—
283C4×C4⋊C4—611no—
284#340—612no—
285#340—611no—
286#340—611no—
287C61#2529311yescharacteristic, complemented, Sylow
288C22×Q8—612no—
289C22×Q8D24411yescharacteristic
290C22×D4—611no—
291C23×C4—1221no—
292C2×SD16—1224no—
293C2×SD16—1224no—
294C2×SD16—2442no—
295C22×C8C22×D6111yescharacteristic
296C22×C8C22×D6111yescharacteristic
297C22×C8C22×D6112yes—
298C4⋊Q8—1222no—
299C4⋊Q8—1222no—
300C22⋊Q8—2444no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×D4—1222no—
304C4×D4—1222no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4D24411yescharacteristic
315C2×C4⋊C4D24412yes—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1222no—
319C2×C42—1221no—
320C2×C42C22×D6112yes—
321C2×C42C22×D6111yescharacteristic
322#106—2442no—
323C4⋊C8—24no—
324C4⋊C8C22⋊D6114yes—
325#103—1224no—
326#103—1224no—
327#103—2442no—
328#102—1222no—
329#102—1222no—
330C4×C8—24no—
331C4×C8—22no—
332#95—1221no—
333#95—1222no—
334#95—1222no—
335#95—1222no—
336#95—1222no—
337#95—1222no—
338C24—1221no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—1222no—
342C2×Q8—1222no—
343C2×Q8—1222no—
344C2×Q8—1222no—
345C2×Q8—1222no—
346C2×Q8—1222no—
347C2×Q8#1175819014yescomplemented
348C2×D4—612no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—1221no—
352C2×D4—1221no—
353C2×D4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—1221no—
358C22×C4—1222no—
359C22×C4—2442no—
360C22×C4—2442no—
361C22×C4—2442no—
362C22×C4C2×D24412yes—
363C22×C4C2×D24411yescharacteristic
364C22×C4C2×D24412yes—
365C22×C4C2×D24411yescharacteristic
366C22×C4C23×D6111yescharacteristic
367SD16—2448no—
368C2×C8—21no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8#1175822612yes—
376C2×C8C2×C22⋊D6114yes—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1222no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—4881no—
388C4⋊C4—2442no—
389C4⋊C4—2442no—
390C4⋊C4—2444no—
391C4⋊C4—2442no—
392C22⋊C4—4882no—
393C22⋊C4—2442no—
394C22⋊C4—2444no—
395C42—22no—
396C42—22no—
397C42—2442no—
398C42#1175822212yes—
399C42C2×C22⋊D6112yes—
400C23—1222no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23—2441no—
406C23—2441no—
407C23C22×D24411yescharacteristic
408Q8—28no—
409Q8—2442no—
410Q8—2442no—
411Q8—2442no—
412Q8—2442no—
413D4—1222no—
414D4—1222no—
415D4—2441no—
416D4—2441no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—4884no—
425C2×C4—2441no—
426C2×C4—2442no—
427C2×C4—2442no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2442no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4—2441no—
439C2×C4—2442no—
440C2×C4#1451975112yes—
441C2×C4#1451975112yes—
442C2×C4C2×#1175819014yes—
443C2×C4C2×#1175819012yes—
444C2×C4#1451979912yes—
445C2×C4C22×C22⋊D6111yescharacteristic
446C2×C4#41×D6111yescharacteristic
447C2×C4C2×#1175822211yescharacteristic
448C2×C4C2×#1175822611yescharacteristic
449C2×C4#1451982311yescharacteristic
450C2×C4#1451982511yescharacteristic
451C8—24no—
452C8—22no—
453C8—41no—
454C22—4881no—
455C22—4881no—
456C22—4881no—
457C22—4881no—
458C22—2442no—
459C22—2442no—
460C22—2442no—
461C22C2×#1451975111yescharacteristic
462C22#1684042411yescharacteristic
463C22#1684043411yescharacteristic
464C22C22×#1175819011yescharacteristic
465C22C2×#1451979911yescharacteristic
466C22#1684057411yescharacteristic
467C22#1684057911yescharacteristic
468C4—24no—
469C4—22no—
470C4—42no—
471C4—4881no—
472C4—4881no—
473C4—4882no—
474C4—4881no—
475C4—4882no—
476C4#1683999114yes—
477C4#1684029911yescharacteristic
478C4#1684038411yescharacteristic
479C4#1684040011yescharacteristic
480C4#1684042311yescharacteristic
481C2—4882no—
482C2#2144937911yescharacteristic
483C2C2×#1683999111yescharacteristic
484C2#2145551011yescharacteristic
485C2#2145846211yescharacteristic
486C2#2145847011yescharacteristic
487C2#2145880111yescharacteristic
488C2#2145880811yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×C8×Q8D61
hypocenterC61#25293
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #21451031C2semidirect
4. #21447199C2semidirect
7. #21446621C2semidirect
8. #21446583C2semidirect
9. #21446495C2semidirect
11. C2×C8×Q8×C61C2semidirect
96. C2×Q8×C61SD16semidirect
238. C2×C8×Q8D61semidirect
287. C61#25293semidirect
347. C2×Q8#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168399682
#214510312
#214507372
#214471992
#214471372
#214466732
#214466212
#214465832
#214464952
C4×#145196992
C2×C8×Q8×C612
#2529361