Finite Group Databasebeta

← #29576575

Subgroups — #29576575

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456572C211yesmaximal, characteristic
2#21448869C211yesmaximal, characteristic
3#21444386C211yesmaximal, characteristic
4C2×#16839499C211yesmaximal, characteristic, complemented
5#21443697C211yesmaximal, characteristic, complemented
6#21443667C211yesmaximal, characteristic, complemented
7C61×#1996C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519737—21no—
10#16840090—21no—
11#16840080C412yescomplemented
12#16839993—22no—
13#16839912—21no—
14#16839901C412yescomplemented
15C2×C22⋊#92418C2211yescharacteristic
16C2×C22⋊#92418C2211yescharacteristic
17#16839653—21no—
18#16839647—21no—
19#16839640—21no—
20#16839639—21no—
21C4×#11758198—21no—
22C2×#14519684C2211yescharacteristic
23#16839499—21no—
24#16839499—21no—
25#16839495—22no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#380C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C61×#342—21no—
30C61×#342—21no—
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C23×Dic61—21no—
34C22×C4×D61—21no—
35C2×#11758201—22no—
36C22×#92418—21no—
37C22×#92418—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751C2×C412yes—
41#14519751D412yes—
42#14519737—42no—
43#14519737—42no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×C4×Dic61—21no—
51#14519708—22no—
52#14519701—22no—
53#14519697—22no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418C2×C412yescomplemented
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684D411yescharacteristic
62#14519684D411yescharacteristic
63C22×Q8×C61C2×C411yescharacteristic, complemented
64C23×C4×C61C2311yescharacteristic
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—21no—
68C22⋊Q8×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61D411yescharacteristic
76C2×C42×C61—21no—
77#95×C61—21no—
78#95×C61—21no—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—81no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—41no—
89C22×C4×C61—81no—
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C2×C22⋊D61—22no—
97C2×C22⋊D61—22no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—41no—
101C2×C22⋊D61—41no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—42no—
106C22×Dic61—41no—
107C22×Dic61—41no—
108C22×Dic61—81no—
109C2×D244—21no—
110C2×D244—81no—
111C2×D244C22⋊C412yes—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×C4×D61—81no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C4⋊C4×C61—81no—
120C22⋊C4×C61—21no—
121C22⋊C4×C61—21no—
122C22⋊C4×C61—21no—
123C22⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C22⋊C4×C61SD1612yescomplemented
128C42×C61—81no—
129#11758201—82no—
130#11758198—22no—
131#11758198—22no—
132#11758198—42no—
133#11758198—42no—
134#11758198—81no—
135C4⋊Dic61—21no—
136C4⋊Dic61C22⋊C412yes—
137#11758196—42no—
138C4×Dic61—42no—
139C2×#92418—22no—
140C2×#92418—41no—
141C2×#92418—81no—
142C2×#92418—81no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61C4×D411yescharacteristic
146C23×C61#12011yescharacteristic
147C23×C61#12111yescharacteristic
148C23×C61C2×SD1611yescharacteristic
149C23×C61#13511yescharacteristic
150C23×C61AGL(1,ℤ/8)11yescharacteristic
151C23×C61#13711yescharacteristic
152C22×D61—22no—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159Q8×C61—42no—
160Q8×C61—81no—
161C2×C4×C61—22no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—81no—
176C2×C4×C61—81no—
177C2×C4×C61—81no—
178C2×C4×C61—81no—
179C2×C4×C61—81no—
180C2×C4×C61C2×C22⋊C411yescharacteristic
181C2×C4×C61C4×D411yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12311yescharacteristic
184C22⋊D61—42no—
185C22⋊D61—42no—
186C22⋊D61—42no—
187C22⋊D61—42no—
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—82no—
198C2×Dic61—82no—
199D244—42no—
200D244—81no—
201C4×D61—84no—
202#92418—42no—
203#92418—81no—
204#9275—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61#38611yescharacteristic
216C22×C61C4×SD1611yescharacteristic
217C22×C61#44011yescharacteristic
218C22×C61#45011yescharacteristic
219C22×C61#45211yescharacteristic
220C22×C61#46111yescharacteristic
221C22×C61#46311yescharacteristic
222C2×D61—42no—
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61—81no—
237C4×C61—161no—
238Dic61—42no—
239Dic61—42no—
240Dic61—81no—
241#3187—611no—
242#2364—611no—
243#1996D6111yescharacteristic, complemented
244#1519—611no—
245C2×#328—611no—
246#1294—611no—
247#1282—611no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61—21no—
254C2×C61#187711yescharacteristic
255C2×C61#190611yescharacteristic
256C2×C61#191311yescharacteristic
257D61—82no—
258C2×C22⋊Q8C2×D6111yescharacteristic
259C2×#121—611no—
260C2×C4×D4—1221no—
261#483—1221no—
262#481—612no—
263#465—1222no—
264#450—1221no—
265#448—612no—
266C2×C22⋊C8—611no—
267C2×C22⋊C8—611no—
268#388—1221no—
269#387—1221no—
270#386—1221no—
271#386—1221no—
272#380C2×D6111yescharacteristic
273C4×C22⋊C4—1221no—
274C4×C22⋊C4C2×D6111yescharacteristic
275C2×#95—611no—
276#342—21no—
277#342—21no—
278#328—1221no—
279#328—1221no—
280#327—1222no—
281C61#927511yescharacteristic, complemented, Sylow
282C22×Q8C4×D6111yescharacteristic, complemented
283C22×D4—611no—
284C22×D4—1221no—
285C23×C4—1221no—
286C23×C4—1221no—
287C23×C4C22×D6111yescharacteristic
288C2×SD16—1222no—
289C22×C8—1221no—
290C22×C8—1221no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295#121—612no—
296#121—612no—
297#121—1221no—
298#121—1221no—
299C4×D4—2442no—
300C4×D4—2442no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4C4×D6111yescharacteristic, complemented
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4C22⋊D6111yescharacteristic
313C2×C22⋊C4C22⋊D6111yescharacteristic
314C2×C42—21no—
315C2×C42—1221no—
316#106—1222no—
317#103—1222no—
318#102—1222no—
319C22⋊C8—612no—
320C22⋊C8—1221no—
321C22⋊C8—1221no—
322C22⋊C8—1221no—
323#95—21no—
324#95—21no—
325#95—611no—
326#95—611no—
327#95—1221no—
328#95—1221no—
329#95—1221no—
330C24—1221no—
331C24C2×C22⋊D6111yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—81no—
335C2×D4—612no—
336C2×D4—1222no—
337C2×D4—1222no—
338C2×D4—1221no—
339C2×D4—4881no—
340C2×D4—2441no—
341C2×D4—2441no—
342C2×D4—2441no—
343C2×D4—2441no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—81no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—4881no—
354C22×C4—4881no—
355C22×C4—2442no—
356C22×C4—2442no—
357C22×C4—2442no—
358C22×C4—2441no—
359C22×C4—2441no—
360C22×C4C2×C4×D6111yescharacteristic
361C22×C4#1175822211yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4D4×D6111yescharacteristic
364C22×C4D4×D6111yescharacteristic
365C22×C4#1175822411yescharacteristic
366SD16—4882no—
367C2×C8—1222no—
368C2×C8—4881no—
369C2×C8—4881no—
370C2×C8—2441no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—81no—
376C4⋊C4—612no—
377C4⋊C4—1221no—
378C4⋊C4—2442no—
379C22⋊C4—21no—
380C22⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—1222no—
387C22⋊C4—1222no—
388C22⋊C4—4881no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4#1175820112yescomplemented
392C42—81no—
393C42—2442no—
394C23—41no—
395C23—41no—
396C23—1222no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23—2441no—
403C23C4×C22⋊D6111yescharacteristic
404C23#1451974511yescharacteristic
405C23#1451974811yescharacteristic
406C23C2×#1175820111yescharacteristic
407C23#1451979711yescharacteristic
408C23#1451980211yescharacteristic
409C23#1451980811yescharacteristic
410Q8—42no—
411Q8—81no—
412D4—4881no—
413D4—2442no—
414D4—2442no—
415D4—2442no—
416D4—2442no—
417D4—2442no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—4884no—
440C2×C4—4882no—
441C2×C4—4882no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4C22⋊C4×D6111yescharacteristic
450C2×C4#1451973711yescharacteristic
451C2×C4#1451975011yescharacteristic
452C2×C4#1451975911yescharacteristic
453C8—4881no—
454C8—2442no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22—2442no—
470C22—2442no—
471C22—2442no—
472C22#1683963811yescharacteristic
473C22C4×#1175820111yescharacteristic
474C22#1683986011yescharacteristic
475C22#1683991211yescharacteristic
476C22#1683992211yescharacteristic
477C22#1683996411yescharacteristic
478C22#1683997411yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—161no—
488C4—4881no—
489C4—2442no—
490C4—2442no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—4882no—
498C2#2144615611yescharacteristic
499C2#2144631711yescharacteristic
500C2#2144635811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1996C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1996D61
hypocenterC61#9275
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
4. C2×#16839499C2semidirect
5. #21443697C2semidirect
6. #21443667C2semidirect
7. C61×#1996C2semidirect
11. #16840080C4semidirect
14. #16839901C4semidirect
57. C22⋊#92418C2×C4semidirect
63. C22×Q8×C61C2×C4semidirect
71. C2×C4⋊C4×C61C2×C4semidirect
127. C22⋊C4×C61SD16semidirect
243. #1996D61semidirect
281. C61#9275semidirect
282. C22×Q8C4×D61semidirect
305. C2×C4⋊C4C4×D61semidirect
391. C22⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214565722
#214488692
#214443862
C2×#168394992
#214436972
#214436672
C61×#19962
#927561