Finite Group Databasebeta

← #29709164

Subgroups — #29709164

Order 15616.

924 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758190C212yesmaximal
2C4×#14519777C211yesmaximal, characteristic, complemented
3#21448283C211yesmaximal, characteristic
4#21447297C211yesmaximal, characteristic, complemented
5#21447224C211yesmaximal, characteristic
6#21447199C212yesmaximal, complemented
7#21447137C212yesmaximal
8#21447066C211yesmaximal, characteristic, complemented
9#21446904C211yesmaximal, characteristic
10C4×#14519702C212yesmaximal, complemented
11C4×C4⋊C8×C61C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×C4×Dic122C2212yes—
15C2×C4×D244C2211yescharacteristic
16C2×#14519734—21no—
17C4×#11758190—28no—
18C2×#14519709C2212yes—
19C2×#14519702C2212yes—
20C2×#14519702C2212yes—
21C2×#14519702C2212yes—
22C2×#14519699C2212yes—
23#16839668C2212yes—
24#16839664C2211yescharacteristic
25#16839664C2211yescharacteristic
26#16839660—21no—
27#16839660—21no—
28#16839660C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839653—21no—
31#16839653—21no—
32#16839632C2211yescharacteristic
33C4×C4⋊Dic61—21no—
34C4×C4⋊Dic61—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39C2×C4⋊C8×C61C2212yes—
40C2×C4×C8×C61C2212yes—
41C43×C61C2211yescharacteristic
42C61×#336C2211yescharacteristic
43C61×#336C2211yescharacteristic
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758190—24no—
48C2×#11758190—24no—
49C2×#11758190—24no—
50#14519777—24no—
51C4×Dic122—24no—
52C4×Dic122—24no—
53C4×D244—22no—
54C4×D244D414yes—
55#14519734—44no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—21no—
65C2×#11758196—21no—
66C2×#11758196—22no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61—22no—
74#14519709—24no—
75#14519702—24no—
76#14519702—24no—
77#14519702—24no—
78#14519699—24no—
79#14519684—21no—
80#14519684—21no—
81#14519684—22no—
82#14519684—22no—
83#14519684—21no—
84C22×C8×C61C2312yes—
85C22×C8×C61C2312yes—
86C2×C42×C61D411yescharacteristic
87C2×C42×C61D411yescharacteristic
88C2×C42×C61D411yescharacteristic
89C2×C42×C61D411yescharacteristic
90C2×C42×C61C2311yescharacteristic
91C2×C42×C61C2311yescharacteristic
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61—24no—
94C4×C8×C61—24no—
95C23×D61—21no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2411yescharacteristic
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—21no—
111C2×D244C2×D412yes—
112C2×C4×D61—24no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×Dic122—22no—
116C2×Dic122—22no—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122#4114yes—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61#4114yes—
127C42×C61—22no—
128C42×C61—22no—
129C42×C61—22no—
130C42×C61—22no—
131C42×C61—41no—
132C42×C61SD1614yescomplemented
133C42×C61C2×D412yes—
134C42×C61#4112yes—
135#11758198—24no—
136#11758198—42no—
137#11758198—42no—
138#11758198—42no—
139#11758198—42no—
140C4⋊Dic61—24no—
141C4⋊Dic61—24no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—42no—
148C4⋊Dic61—42no—
149C4⋊Dic61C2×D412yes—
150#11758196—44no—
151#11758196—42no—
152#11758196—42no—
153C4×Dic61—44no—
154C4×Dic61—42no—
155#11758190—216no—
156C23×C61C22×D411yescharacteristic
157C22×D61—22no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C2×C4×C61—21no—
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#12112yes—
176C2×C4×C61#12112yes—
177C2×C4×C61C2×SD1612yes—
178C2×C4×C61C2×SD1612yes—
179C2×C4×C61C2×SD1612yes—
180C2×C4×C61#13512yes—
181C2×C4×C61#13712yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4112yes—
184C2×C4×C61C2×#4112yes—
185C2×C4×C61C2×#4111yescharacteristic
186C2×Dic61—22no—
187C2×Dic61—24no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—42no—
200C2×Dic61—41no—
201D244—24no—
202D244—42no—
203C4×D61—48no—
204Dic122—24no—
205Dic122—24no—
206Dic122—42no—
207Dic122—42no—
208C8×C61—24no—
209C8×C61—42no—
210#23677—611nomaximal, Sylow
211C22×C61C2×#12111yescharacteristic
212C22×C61#52511yescharacteristic
213C22×C61#53211yescharacteristic
214C22×C61C22×SD1611yescharacteristic
215C22×C61C2×#13511yescharacteristic
216C22×C61C2×#13711yescharacteristic
217C22×C61#57511yescharacteristic
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C4×C61—22no—
226C4×C61—22no—
227C4×C61—22no—
228C4×C61—22no—
229C4×C61—41no—
230C4×C61#46114yes—
231C4×C61#52512yes—
232C4×C61#52912yes—
233Dic61—44no—
234Dic61—42no—
235Dic61—81no—
236Dic61—81no—
237Dic61—82no—
238C2×C4×SD16—612no—
239C4×C4⋊Q8—611no—
240#2293—611no—
241#2084—611no—
242#2071—611no—
243#2066—612no—
244#2055—612no—
245#2043—611no—
246#2013—611no—
247C4×C4⋊C8D6111yescharacteristic, complemented
248C4×#103—612no—
249C2×C61#239211yescharacteristic
250C2×C61C2×#46111yescharacteristic
251C2×C61#303511yescharacteristic
252C2×C61#310311yescharacteristic
253C2×C61#310811yescharacteristic
254C2×C61#316711yescharacteristic
255C2×C61#317211yescharacteristic
256D61—82no—
257C22×SD16—612no—
258C2×C4⋊Q8—611no—
259C2×C4×Q8—612no—
260C2×C4×D4—611no—
261C2×#117—1221no—
262C4×SD16—1228no—
263C2×#106—612no—
264C2×C4⋊C8C2×D6112yes—
265C2×#103—612no—
266C2×#103—612no—
267C2×#103—612no—
268C2×#102—612no—
269C2×C4×C8C2×D6112yes—
270#391—612no—
271#390—611no—
272#390—611no—
273#389—611no—
274#389—611no—
275#389—1221no—
276#389—1221no—
277#388—1221no—
278#388—1221no—
279#384—611no—
280C4×C4⋊C4—611no—
281C4×C4⋊C4—1221no—
282C4×C4⋊C4—1221no—
283C4×C22⋊C4—1221no—
284C43C2×D6111yescharacteristic
285#340—612no—
286#340—612no—
287#336C2×D6111yescharacteristic
288#336C2×D6111yescharacteristic
289C61#2367711yescharacteristic, complemented, Sylow
290C22×Q8—612no—
291C22×D4—611no—
292C23×C4—1221no—
293C2×SD16—1224no—
294C2×SD16—1224no—
295C2×SD16—1224no—
296C22×C8C22×D6112yes—
297C22×C8C22×D6112yes—
298C4⋊Q8—1224no—
299C4×Q8—1224no—
300C4×Q8—1224no—
301C4×D4—614no—
302C4×D4—1222no—
303#117—2444no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1222no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C42—1221no—
321C2×C42—1222no—
322C2×C42D24411yescharacteristic
323C2×C42D24411yescharacteristic
324C2×C42D24411yescharacteristic
325C2×C42D24411yescharacteristic
326C2×C42C22×D6111yescharacteristic
327C2×C42C22×D6111yescharacteristic
328C2×C42C22×D6111yescharacteristic
329#106—1224no—
330C4⋊C8—24no—
331#103—1224no—
332#103—1224no—
333#103—1224no—
334#102—1224no—
335C4×C8—24no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339#95—1222no—
340#95—1222no—
341C24—1221no—
342C2×Q8—614no—
343C2×Q8—1222no—
344C2×Q8—1222no—
345C2×Q8—1222no—
346C2×Q8—1222no—
347C2×Q8—1222no—
348C2×D4—612no—
349C2×D4—1222no—
350C2×D4—1222no—
351C2×D4—1221no—
352C22×C4—1222no—
353C22×C4—1221no—
354C22×C4—1224no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1222no—
358C22×C4—2442no—
359C22×C4—2442no—
360C22×C4C2×D24411yescharacteristic
361C22×C4C2×D24411yescharacteristic
362C22×C4C2×D24411yescharacteristic
363C22×C4C2×D24411yescharacteristic
364C22×C4C2×D24411yescharacteristic
365C22×C4C2×D24411yescharacteristic
366C22×C4C23×D6111yescharacteristic
367SD16—12216no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8#1175822214yes—
374C4⋊C4—612no—
375C4⋊C4—1224no—
376C4⋊C4—1224no—
377C4⋊C4—1221no—
378C4⋊C4—1222no—
379C4⋊C4—1222no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—2442no—
383C4⋊C4—2444no—
384C4⋊C4—2442no—
385C4⋊C4—2442no—
386C4⋊C4—2442no—
387C22⋊C4—1224no—
388C22⋊C4—2442no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4—2442no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—41no—
397C42—2444no—
398C42—2442no—
399C42#1175819014yescomplemented
400C42D4×D6112yes—
401C42#1175822612yes—
402C23—1222no—
403C23—2441no—
404C23—2441no—
405C23—2441no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23C22×D24411yescharacteristic
410Q8—1224no—
411Q8—1224no—
412Q8—2442no—
413Q8—2442no—
414D4—1224no—
415D4—2442no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—1222no—
428C2×C4—1224no—
429C2×C4—2442no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2442no—
435C2×C4—2441no—
436C2×C4—2442no—
437C2×C4—2441no—
438C2×C4—2442no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2448no—
443C2×C4#1451975212yes—
444C2×C4#1451975212yes—
445C2×C4C2×#1175819012yes—
446C2×C4C2×#1175819012yes—
447C2×C4C2×#1175819012yes—
448C2×C4#1451979912yes—
449C2×C4#1451981012yes—
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4#41×D6112yes—
452C2×C4C2×#1175822212yes—
453C2×C4C2×#1175822611yescharacteristic
454C8—24no—
455C8—42no—
456C22—4881no—
457C22—4881no—
458C22—4881no—
459C22—4881no—
460C22—2442no—
461C22—2442no—
462C22—2442no—
463C22C2×#1451975211yescharacteristic
464C22#1684027411yescharacteristic
465C22#1684031211yescharacteristic
466C22C22×#1175819011yescharacteristic
467C22C2×#1451979911yescharacteristic
468C22C2×#1451981011yescharacteristic
469C22#1684056211yescharacteristic
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—41no—
475C4—4882no—
476C4—4881no—
477C4—4881no—
478C4—2444no—
479C4—2442no—
480C4#1683996814yes—
481C4#1684027112yes—
482C4#1684029912yes—
483C2—4882no—
484C2#2144905911yescharacteristic
485C2C2×#1683996811yescharacteristic
486C2#2145527611yescharacteristic
487C2#2145585111yescharacteristic
488C2#2145590111yescharacteristic
489C2#2145637811yescharacteristic
490C2#2145642511yescharacteristic
subgroupG/H
centerC22×C4C2×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C4⋊C8×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC4×C4⋊C8D61
hypocenterC61#23677
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×#14519777C2semidirect
4. #21447297C2semidirect
6. #21447199C2semidirect
8. #21447066C2semidirect
10. C4×#14519702C2semidirect
11. C4×C4⋊C8×C61C2semidirect
132. C42×C61SD16semidirect
247. C4×C4⋊C8D61semidirect
289. C61#23677semidirect
399. C42#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117581902
C4×#145197772
#214482832
#214472972
#214472242
#214471992
#214471372
#214470662
#214469042
C4×#145197022
C4×C4⋊C8×C612
#2367761