Finite Group Databasebeta

← #29690413

Subgroups — #29690413

Order 15616.

874 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×#16839784C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic
3#21448223C211yesmaximal, characteristic
4#21446583C211yesmaximal, characteristic, complemented
5#21446582C211yesmaximal, characteristic, complemented
6#21446581C212yesmaximal, complemented
7#21446575C211yesmaximal, characteristic, complemented
8#21446574C212yesmaximal, complemented
9#21445853C212yesmaximal
10#21445851C212yesmaximal
11C61×#1947C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13C2×#14519735—21no—
14C2×#14519734—21no—
15C2×C42×D61C2211yescharacteristic
16C2×#14519710C2212yes—
17C2×#14519709C2212yes—
18C2×#14519708C2212yes—
19#16839784C418yescomplemented
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519706C2212yes—
22#16839632C2211yescharacteristic
23#16839631C2211yescharacteristic
24#16839631C2211yescharacteristic
25#16839627—21no—
26#16839626—21no—
27#16839621—21no—
28#16839619—21no—
29#16839619C2211yescharacteristic
30#16839619C2211yescharacteristic
31#16839619C2212yes—
32C4⋊C4×Dic61—21no—
33C4⋊C4×Dic61C2212yes—
34C4×#11758196—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#107×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C2×#11758189—42no—
48C2×#11758189—42no—
49C2×#11758189C2×C418yes—
50C22×#92418C2312yes—
51#14519735—44no—
52#14519734—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C42×D61C2×C414yes—
71C42×D61Q814yes—
72#14519710—42no—
73#14519709—42no—
74#14519708—42no—
75C4⋊#92418C2×C414yes—
76#14519706C2×C418yes—
77#14519684—21no—
78#14519684—21no—
79#14519684—21no—
80#14519684—22no—
81#14519684—22no—
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90#107×C61—42no—
91C4⋊C8×C61C2×C414yes—
92#95×C61—21no—
93C23×D61C2×D411yescharacteristic
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—22no—
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—22no—
113C2×C4×D61—22no—
114C2×C4×D61C4⋊C414yes—
115C2×C4×D61C4⋊C414yes—
116C2×C4×D61C22×C412yes—
117C2×C4×D61C2×Q812yes—
118C2×C8×C61—22no—
119C2×C8×C61—22no—
120C2×C8×C61C22×C414yes—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—42no—
128C4⋊C4×C61—42no—
129C42×C61—42no—
130C42×C61C22×C412yes—
131C42×C61C2×Q812yes—
132#11758198—42no—
133#11758198—42no—
134#11758198—44no—
135C4⋊Dic61—22no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—42no—
142C4⋊Dic61—42no—
143#11758196—42no—
144#11758196—42no—
145#11758196—42no—
146#11758196—42no—
147C4×Dic61—42no—
148C4×Dic61—44no—
149C4×Dic61C4⋊C414yes—
150C4×Dic61C4⋊C414yes—
151C4×Dic61C22×C412yes—
152C4×Dic61C2×Q812yes—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418C22×C414yes—
156#11758189—48no—
157C23×C61C22×D411yescharacteristic
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—21no—
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61C2×C4⋊C412yes—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61C2×C4⋊C412yes—
177C2×C4×C61C2×C4⋊C412yes—
178C2×C4×C61AGL(1,ℤ/8)12yes—
179C2×C4×C61#13712yes—
180C2×C4×C61C23×C411yescharacteristic
181C2×C4×C61C22×Q811yescharacteristic
182C2×C4×C61H(5,𝔽2)11yescharacteristic
183C2×C4×C61H(5,𝔽2)11yescharacteristic
184C2×C4×C612−1+411yescharacteristic
185C2×C4×C612−1+411yescharacteristic
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—41no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—42no—
199C2×Dic61C2×C4⋊C412yes—
200C2×Dic61C2×C4⋊C412yes—
201C2×Dic61C2×C4⋊C412yes—
202C2×Dic61AGL(1,ℤ/8)12yes—
203C2×Dic61#13712yes—
204C4×D61—22no—
205C4×D61—22no—
206C4×D61—22no—
207C4×D61—22no—
208C4×D61—28no—
209C4×D61—28no—
210C8×C61—24no—
211#92418—24no—
212#21964—611nomaximal, Sylow
213C22×C61C22×C4⋊C411yescharacteristic
214C22×C61#53711yescharacteristic
215C22×C61#53711yescharacteristic
216C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
217C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
218C22×C61C2×#13711yescharacteristic
219C22×C61C2×#13711yescharacteristic
220C2×D61—22no—
221C2×D61—22no—
222C2×D61—22no—
223C2×D61—41no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—41no—
227C4×C61—22no—
228C4×C61—82no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61#42814yes—
232C4×C61#51811yescharacteristic
233C4×C61#51911yescharacteristic
234C4×C61#54411yescharacteristic
235C4×C61#54411yescharacteristic
236Dic61—22no—
237Dic61—22no—
238Dic61—81no—
239Dic61—82no—
240Dic61—81no—
241Dic61#42814yes—
242C2×#423—611no—
243#2285—611no—
244#2285—611no—
245#1948—611no—
246#1948—611no—
247#1948—612no—
248#1947—612no—
249#1947—611no—
250#1947D6111yescharacteristic, complemented
251#1809—612no—
252#1809—612no—
253C2×C61#231611yescharacteristic
254C2×C61C2×#42811yescharacteristic
255C2×C61C2×#42811yescharacteristic
256C2×C61#325311yescharacteristic
257C2×C61#325411yescharacteristic
258C2×C61#325411yescharacteristic
259C2×C61#325511yescharacteristic
260D61—42no—
261C22×M16—612no—
262C2×#117—1221no—
263C2×#117—1221no—
264C22×C42—611no—
265C2×#107—612no—
266C2×#107C2×D6112yes—
267C2×#106—612no—
268C2×#106—612no—
269#423—618no—
270C2×C4⋊C8—612no—
271C2×C4⋊C8—611no—
272C2×C4⋊C8C2×D6111yescharacteristic
273#384—611no—
274#384—611no—
275#384—611no—
276#384C2×D6111yescharacteristic
277#382—1221no—
278#382—1221no—
279#380—611no—
280#380—612no—
281#380—611no—
282#380—1221no—
283#380—1221no—
284C4×C4⋊C4—612no—
285C4×C4⋊C4—611no—
286C4×C4⋊C4—1221no—
287C4×C4⋊C4—1221no—
288C4×C4⋊C4C2×D6111yescharacteristic
289#340—612no—
290#340—612no—
291#340—612no—
292#340C2×D6112yes—
293C61#2196411yescharacteristic, complemented, Sylow
294C23×C4—611no—
295C23×C4—611no—
296C23×C4—611no—
297C2×M16—618no—
298C2×M16—2442no—
299C2×M16—2442no—
300C22×C8—612no—
301C22×C8C22×D6112yes—
302#117—2444no—
303#117—2444no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4C22×D6111yescharacteristic
316C2×C4⋊C4C22×D6112yes—
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1222no—
321C2×C42—21no—
322C2×C42—611no—
323C2×C42—611no—
324C2×C42—611no—
325C2×C42—614no—
326C2×C42—614no—
327C2×C42—1222no—
328C2×C42—1221no—
329C2×C42C22×D6111yescharacteristic
330#107—42no—
331#107—2442no—
332#106—2442no—
333#106—2442no—
334C4⋊C8—618no—
335C4⋊C8—614no—
336C4⋊C8C4×D6114yes—
337#95—21no—
338#95—1222no—
339#95—1221no—
340#95—1221no—
341#95—1222no—
342#95—1221no—
343C24—611no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—611no—
350C22×C4—612no—
351C22×C4—612no—
352C22×C4—614no—
353C22×C4—614no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4D4×D6111yescharacteristic
366C22×C4D4×D6111yescharacteristic
367C22×C4C23×D6111yescharacteristic
368M16—2448no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—614no—
372C2×C8—1222no—
373C2×C8—1222no—
374C2×C8C2×C4×D6114yes—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1222no—
388C4⋊C4—1221no—
389C4⋊C4—2442no—
390C4⋊C4—2442no—
391C4⋊C4—2442no—
392C4⋊C4—2442no—
393C4⋊C4—2442no—
394C4⋊C4—2442no—
395C22⋊C4—2444no—
396C22⋊C4—2442no—
397C22⋊C4—2442no—
398C42—42no—
399C42—612no—
400C42—612no—
401C42—614no—
402C42—614no—
403C42—2444no—
404C42—2442no—
405C42C2×C4×D6112yes—
406C42Q8×D6112yes—
407C23—612no—
408C23—1221no—
409C23—1221no—
410C23—1221no—
411C23—1221no—
412C23—1221no—
413C23—1221no—
414C23C2×D4×D6111yescharacteristic
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—612no—
427C2×C4—612no—
428C2×C4—612no—
429C2×C4—612no—
430C2×C4—612no—
431C2×C4—1228no—
432C2×C4—1221no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—1228no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—2442no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2442no—
447C2×C4—2441no—
448C2×C4—2442no—
449C2×C4—2441no—
450C2×C4C4⋊C4×D6112yes—
451C2×C4C4⋊C4×D6112yes—
452C2×C4#1451980112yes—
453C2×C4#1451980912yes—
454C2×C4C22×C4×D6111yescharacteristic
455C2×C4C2×Q8×D6111yescharacteristic
456C2×C4#1451982211yescharacteristic
457C2×C4#1451982311yescharacteristic
458C2×C4#1451982411yescharacteristic
459C2×C4#1451982511yescharacteristic
460C8—24no—
461C8—1224no—
462C22—1222no—
463C22—1222no—
464C22—1222no—
465C22—2441no—
466C22—2441no—
467C22—2441no—
468C22—2441no—
469C22C2×C4⋊C4×D6111yescharacteristic
470C22#1684034111yescharacteristic
471C22#1684034211yescharacteristic
472C22AGL(1,ℤ/8)×D6111yescharacteristic
473C22C2×#1451980111yescharacteristic
474C22#137×D6111yescharacteristic
475C22C2×#1451980911yescharacteristic
476C4—22no—
477C4—82no—
478C4—81no—
479C4—81no—
480C4—614no—
481C4—1222no—
482C4—1222no—
483C4—4881no—
484C4—4882no—
485C4—4881no—
486C4#1683980314yes—
487C4#1684023411yescharacteristic
488C4#1684023811yescharacteristic
489C4#1684038911yescharacteristic
490C4#1684039211yescharacteristic
491C2—2442no—
492C2#2144843411yescharacteristic
493C2#428×D6111yescharacteristic
494C2C2×#1683980311yescharacteristic
495C2#2145719811yescharacteristic
496C2#2145720511yescharacteristic
497C2#2145720811yescharacteristic
498C2#2145721111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1947D61
hypocenterC61#21964
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. #21446583C2semidirect
5. #21446582C2semidirect
6. #21446581C2semidirect
7. #21446575C2semidirect
8. #21446574C2semidirect
11. C61×#1947C2semidirect
19. #16839784C4semidirect
250. #1947D61semidirect
293. C61#21964semidirect

Maximal subgroups

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

subgroupindex
C2×#168397842
#214482242
#214482232
#214465832
#214465822
#214465812
#214465752
#214465742
#214458532
#214458512
C61×#19472
#2196461