Finite Group Databasebeta

← #29729541

Subgroups — #29729541

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839963C211yesmaximal, characteristic
2#21451189C211yesmaximal, characteristic, complemented
3#21450818C211yesmaximal, characteristic
4#21447192C212yesmaximal, complemented
5#21447137C212yesmaximal, complemented
6#21446673C212yesmaximal, complemented
7#21446625C211yesmaximal, characteristic
8#21446583C211yesmaximal, characteristic, complemented
9#21446502C212yesmaximal, complemented
10C4×#14519702C211yesmaximal, characteristic, complemented
11C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519766—21no—
15C2×C4×Dic122C2211yescharacteristic
16#16839963—24no—
17C2×#14519709C2212yes—
18C2×#14519702C2211yescharacteristic
19C2×#14519702C2211yescharacteristic
20C2×#14519699C2212yes—
21C2×#14519699C2212yes—
22C2×#14519699C2212yes—
23#16839695—22no—
24#16839679—22no—
25#16839669—21no—
26#16839668C2212yes—
27#16839664C2212yes—
28#16839664C2212yes—
29#16839663—21no—
30#16839660C2212yes—
31#16839658—21no—
32#16839632C2211yescharacteristic
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×Dic122C2311yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C2×#11758190—24no—
48C2×#11758190—24no—
49C2×#11758190—42no—
50#14519774—22no—
51#14519774—22no—
52#14519766—44no—
53C4×Dic122—22no—
54C4×Dic122—22no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2312yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—21no—
62C2×#11758196—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69#14519709—42no—
70#14519702—22no—
71#14519702—22no—
72#14519699—24no—
73#14519699—24no—
74#14519699—42no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684—21no—
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—22no—
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—21no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—42no—
114C2×D244—42no—
115C2×D244—42no—
116C2×D244—42no—
117C2×D244—81no—
118C2×D244—81no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122#4112yes—
125C2×C8×C61—21no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—21no—
132C2×C8×C61C2×D414yes—
133C2×C8×C61#4112yes—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—22no—
136C42×C61—22no—
137C42×C61—22no—
138C42×C61C2×D412yes—
139C42×C61#4112yes—
140#11758198—44no—
141#11758198—44no—
142#11758198—82no—
143#11758198—82no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—42no—
153C4⋊Dic61—81no—
154#11758196—42no—
155#11758196—82no—
156C4×Dic61—42no—
157C4×Dic61—42no—
158#11758190—48no—
159C23×C61C22×D411yescharacteristic
160C22×D61—42no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—42no—
167Q8×C61—28no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61#12112yes—
176C2×C4×C61#12112yes—
177C2×C4×C61C2×SD1614yes—
178C2×C4×C61C2×SD1612yes—
179C2×C4×C61#13512yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61H(5,𝔽2)11yescharacteristic
185C2×C4×C612−1+411yescharacteristic
186C2×Dic61—22no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201D244—42no—
202D244—42no—
203D244—42no—
204D244—42no—
205D244—84no—
206D244—84no—
207Dic122—22no—
208Dic122—22no—
209Dic122—41no—
210Dic122—41no—
211C8×C61—24no—
212C8×C61—22no—
213C8×C61—41no—
214#25295—611nomaximal, Sylow
215C22×C61C2×#12111yescharacteristic
216C22×C61#54811yescharacteristic
217C22×C61#55011yescharacteristic
218C22×C61C22×SD1611yescharacteristic
219C22×C61C2×#13511yescharacteristic
220C22×C61#57611yescharacteristic
221C22×C61#57711yescharacteristic
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—82no—
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—82no—
229C4×C61—24no—
230C4×C61—22no—
231C4×C61—42no—
232C4×C61#46514yes—
233C4×C61#52911yescharacteristic
234C4×C61#54211yescharacteristic
235C4×C61#54611yescharacteristic
236C4×C61#54811yescharacteristic
237Dic61—42no—
238Dic61—81no—
239Dic61—81no—
240Dic61—82no—
241Dic61—81no—
242C2×#460—611no—
243C2×C8×Q8D6111yescharacteristic, complemented
244#2582—611no—
245#2548—611no—
246#2065—612no—
247#2055—612no—
248#1968—612no—
249#1957—611no—
250#1948—611no—
251#1934—612no—
252C4×#103—611no—
253C2×C61#242611yescharacteristic
254C2×C61C2×#46511yescharacteristic
255C2×C61#306211yescharacteristic
256C2×C61#336211yescharacteristic
257C2×C61#336511yescharacteristic
258C2×C61#339211yescharacteristic
259C2×C61#339611yescharacteristic
260D61—162no—
261C22×SD16—612no—
262C2×#127—611no—
263C2×#124—1221no—
264C2×C4×Q8—611no—
265C2×C4×Q8C2×D6111yescharacteristic
266#460—1224no—
267C8×Q8—24no—
268C2×#106—612no—
269C2×C4⋊C8C2×D6112yes—
270C2×C4⋊C8C2×D6111yescharacteristic
271C2×#103—611no—
272C2×#103—611no—
273C2×#102—612no—
274C2×#102—612no—
275C2×#102—612no—
276C2×C4×C8C2×D6112yes—
277C2×C4×C8C2×D6111yescharacteristic
278#398—1222no—
279#394—1222no—
280#391—612no—
281#391—1221no—
282#390—612no—
283#390—612no—
284#390—1221no—
285#389—612no—
286#389—1221no—
287#384—611no—
288C4×C4⋊C4—611no—
289#340—612no—
290#340—611no—
291#340—611no—
292C61#2529511yescharacteristic, complemented, Sylow
293C22×Q8—611no—
294C22×Q8D24411yescharacteristic
295C22×D4—612no—
296C22×D4—1221no—
297C22×D4—1221no—
298C2×SD16—1224no—
299C2×SD16—1224no—
300C2×SD16—2442no—
301C22×C8C22×D6111yescharacteristic
302C22×C8C22×D6111yescharacteristic
303C22×C8C22×D6112yes—
304#127—1222no—
305#127—1222no—
306#124—2444no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—1222no—
310C4×Q8—1222no—
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—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4D24411yescharacteristic
320C2×C4⋊C4D24412yes—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C42—1221no—
326C2×C42—1221no—
327C2×C42C22×D6112yes—
328C2×C42C22×D6111yescharacteristic
329#106—2442no—
330C4⋊C8—24no—
331C4⋊C8C22⋊D6114yes—
332#103—1222no—
333#103—1222no—
334#102—1224no—
335#102—1224no—
336#102—2442no—
337C4×C8—24no—
338C4×C8—22no—
339#95—1221no—
340#95—1222no—
341#95—1222no—
342#95—1222no—
343C24—1222no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—612no—
347C2×Q8—1221no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—1221no—
352C2×Q8#1175819014yescomplemented
353C2×D4—1222no—
354C2×D4—1222no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—1222no—
359C2×D4—4881no—
360C2×D4—4881no—
361C2×D4—2442no—
362C2×D4—2442no—
363C2×D4—2442no—
364C2×D4—2442no—
365C22×C4—1221no—
366C22×C4—1222no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4C2×D24412yes—
371C22×C4C2×D24411yescharacteristic
372C22×C4C2×D24412yes—
373C22×C4C2×D24411yescharacteristic
374C22×C4C23×D6111yescharacteristic
375SD16—2448no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8#1175822612yes—
384C2×C8C2×C22⋊D6114yes—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1222no—
391C4⋊C4—1221no—
392C4⋊C4—1222no—
393C4⋊C4—1221no—
394C4⋊C4—1221no—
395C4⋊C4—4882no—
396C4⋊C4—4881no—
397C4⋊C4—2442no—
398C4⋊C4—2442no—
399C22⋊C4—4882no—
400C22⋊C4—4882no—
401C22⋊C4—2444no—
402C22⋊C4—2444no—
403C42—22no—
404C42—22no—
405C42—2442no—
406C42—2442no—
407C42#1175822212yes—
408C42C2×C22⋊D6112yes—
409C23—2442no—
410C23—2442no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23C22×D24411yescharacteristic
417Q8—28no—
418Q8—1222no—
419Q8—1222no—
420Q8—2441no—
421Q8—2441no—
422D4—4884no—
423D4—4884no—
424D4—2442no—
425D4—2442no—
426D4—2442no—
427D4—2442no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—1222no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2442no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4#1451975112yes—
451C2×C4#1451975112yes—
452C2×C4C2×#1175819014yes—
453C2×C4C2×#1175819012yes—
454C2×C4#1451979912yes—
455C2×C4C22×C22⋊D6111yescharacteristic
456C2×C4#41×D6111yescharacteristic
457C2×C4C2×#1175822211yescharacteristic
458C2×C4C2×#1175822611yescharacteristic
459C2×C4#1451982311yescharacteristic
460C2×C4#1451982511yescharacteristic
461C8—24no—
462C8—22no—
463C8—41no—
464C22—4882no—
465C22—4882no—
466C22—4882no—
467C22—4882no—
468C22—4882no—
469C22—4882no—
470C22—4882no—
471C22C2×#1451975111yescharacteristic
472C22#1684042411yescharacteristic
473C22#1684043411yescharacteristic
474C22C22×#1175819011yescharacteristic
475C22C2×#1451979911yescharacteristic
476C22#1684056711yescharacteristic
477C22#1684057411yescharacteristic
478C4—24no—
479C4—22no—
480C4—42no—
481C4—4881no—
482C4—4881no—
483C4—4881no—
484C4—4882no—
485C4—2442no—
486C4#1683999114yes—
487C4#1684029911yescharacteristic
488C4#1684038411yescharacteristic
489C4#1684040011yescharacteristic
490C4#1684042311yescharacteristic
491C2—9762no—
492C2#2144937911yescharacteristic
493C2C2×#1683999111yescharacteristic
494C2#2145549911yescharacteristic
495C2#2145849111yescharacteristic
496C2#2145851711yescharacteristic
497C2#2145880111yescharacteristic
498C2#2145883211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×Q8×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenterC2×C8×Q8D61
hypocenterC61#25295
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. #21451189C2semidirect
4. #21447192C2semidirect
5. #21447137C2semidirect
6. #21446673C2semidirect
8. #21446583C2semidirect
9. #21446502C2semidirect
10. C4×#14519702C2semidirect
11. C2×C8×Q8×C61C2semidirect
96. C2×Q8×C61SD16semidirect
243. C2×C8×Q8D61semidirect
292. C61#25295semidirect
352. C2×Q8#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168399632
#214511892
#214508182
#214471922
#214471372
#214466732
#214466252
#214465832
#214465022
C4×#145197022
C2×C8×Q8×C612
#2529561