Finite Group Databasebeta

← #29597619

Subgroups — #29597619

Order 15616.

646 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455105C211yesmaximal, characteristic
2#21454283C211yesmaximal, characteristic, complemented
3#21447490C211yesmaximal, characteristic, complemented
4#21447470C211yesmaximal, characteristic, complemented
5#21443949C211yesmaximal, characteristic, complemented
6#21443939C211yesmaximal, characteristic, complemented
7C61×#1796C211yesmaximal, characteristic, complemented
8C2×#14519802C411yescharacteristic, complemented
9C22×#11758201C411yescharacteristic, complemented
10#16840326C2211yescharacteristic
11#16840240—21no—
12#16839907—21no—
13#16839907—21no—
14#16839901—21no—
15#16839901—21no—
16#16839771C411yescharacteristic, complemented
17#16839758C411yescharacteristic, complemented
18#16839736—22no—
19#16839736—22no—
20#16839736—21no—
21#16839736C2211yescharacteristic
22#16839736C2211yescharacteristic
23#16839729C2211yescharacteristic
24#16839554—22no—
25#16839550—22no—
26C2×2−1+4×C61C2211yescharacteristic
27C61×#421—21no—
28C61×#419—21no—
29C61×#413C2211yescharacteristic
30C61×#410C2211yescharacteristic
31C22×D244C2×C411yescharacteristic
32#14519802—42no—
33C2×#11758201—21no—
34C2×#11758201—21no—
35C2×#11758201—21no—
36C2×#11758201—42no—
37C2×#11758199—21no—
38C2×#11758194C2×C411yescharacteristic
39C22×#92418C2×C411yescharacteristic
40#14519773—21no—
41#14519766—21no—
42#14519751—21no—
43#14519751—21no—
44#14519745—21no—
45#14519745—21no—
46Q8×Dic61—42no—
47Q8×Dic61—42no—
48#14519736—24no—
49#14519736—21no—
50#14519736—21no—
51#14519736C2×C411yescharacteristic
52#14519701—21no—
53#14519701—21no—
54#14519701—21no—
55#14519698—21no—
56#14519692—24no—
57#14519692—21no—
58#14519692—21no—
59#14519692D412yes—
60#14519692D412yes—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
632−1+4×C61—42no—
642−1+4×C61—42no—
65C2×#41×C61—21no—
66C2×#41×C61—21no—
67C2×#41×C61—21no—
68C2×#41×C61C2×C411yescharacteristic
69C2×#41×C61D411yescharacteristic
70C2×#41×C61D411yescharacteristic
71C2×#41×C61C2311yescharacteristic
72C22×Q8×C61—21no—
73C22×Q8×C61C2×C411yescharacteristic
74C22×Q8×C61D411yescharacteristic
75C22×Q8×C61D411yescharacteristic
76C2×M16×C61—21no—
77#117×C61—21no—
78#104×C61—42no—
79#103×C61—42no—
80#101×C61—21no—
81#100×C61—21no—
82#99×C61—21no—
83#99×C61—21no—
84C23×D61—21no—
85#41×C61—22no—
86#41×C61—22no—
87#41×C61—41no—
88#41×C61—41no—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92#41×C61—41no—
93#41×C61—44no—
94#41×C61—44no—
95#41×C61—44no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—41no—
104C2×Q8×C61—41no—
105C2×Q8×C61—44no—
106C2×Q8×C61C22⋊C411yescharacteristic
107C2×Q8×C61C22⋊C411yescharacteristic
108C2×Q8×C61C22⋊C411yescharacteristic
109C2×Q8×C61#4111yescharacteristic
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61C22⋊C411yescharacteristic
114C2×D4×C61C2×D411yescharacteristic
115C2×D4×C61C2×D411yescharacteristic
116C2×D4×C61#4111yescharacteristic
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—41no—
131C2×D244—41no—
132M16×C61—41no—
133M16×C61—41no—
134C2×C8×C61—41no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C42×C61—41no—
138C22⋊Dic61—21no—
139C22⋊Dic61—22no—
140C22⋊Dic61—22no—
141#11758201—42no—
142#11758201—42no—
143#11758201—42no—
144#11758199—42no—
145#11758198—41no—
146#11758198—41no—
147C4⋊Dic61—21no—
148C4⋊Dic61—44no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4×Dic61—21no—
152C4×Dic61—44no—
153C4×Dic61—41no—
154C4×Dic61—41no—
155#11758194—41no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—41no—
159C23×C61—21no—
160C23×C61C4×D411yescharacteristic
161C23×C61#12011yescharacteristic
162C23×C61#12111yescharacteristic
163C22×D61—41no—
164C22×D61—81no—
165C22×D61—161no—
166Q8×C61—24no—
167Q8×C61—21no—
168Q8×C61—21no—
169Q8×C61—21no—
170Q8×C61—21no—
171Q8×C61—41no—
172Q8×C61—42no—
173Q8×C61—42no—
174Q8×C61#9912yes—
175Q8×C61#9912yes—
176D4×C61—21no—
177D4×C61—21no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184C2×C4×C61—24no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—81no—
200C2×C4×C61C2×C22⋊C411yescharacteristic
201C2×C4×C61C4×D411yescharacteristic
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61#12311yescharacteristic
204C22⋊D61—161no—
205C22⋊D61—161no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209D244—41no—
210D244—41no—
211D244—81no—
212C8×C61—81no—
213#92418—41no—
214#92418—41no—
215#13103—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61—41no—
223C22×C61—81no—
224C22×C61#38611yescharacteristic
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—162no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—22no—
232C4×C61—22no—
233C4×C61—21no—
234C4×C61—41no—
235C4×C61—161no—
236C4×C61C2×#9911yescharacteristic
237C4×C61C2×#9911yescharacteristic
238Dic61—81no—
239Dic61—82no—
240Dic61—82no—
241#3010—611no—
242#2884—611no—
243#2128—611no—
244#2122—611no—
245#1796D6111yescharacteristic, complemented
246#1395—611no—
247#1391—611no—
248C2×C61—21no—
249C2×C61—41no—
250C2×C61—41no—
251C2×C61—41no—
252C2×C61—81no—
253C2×C61#189811yescharacteristic
254D61—161no—
255C2×2−1+4C2×D6111yescharacteristic
256C2×AGL(1,ℤ/8)—611no—
257C22×SD16—611no—
258#535—611no—
259#519—1221no—
260#449—1221no—
261#449—1221no—
262#448—1221no—
263#448—1221no—
264#421—21no—
265#419—21no—
266#419—611no—
267#416—611no—
268#413C2×D6111yescharacteristic
269#410—611no—
270#410—611no—
271#410—1222no—
272#410—1222no—
273#410—1221no—
274#410C2×D6111yescharacteristic
275#408—611no—
276#353—1222no—
277AΣL(1,𝔽16)—1222no—
278C61#1310311yescharacteristic, complemented, Sylow
2792−1+4—42no—
2802−1+4—42no—
281C2×#41—21no—
282C2×#41—21no—
283C2×#41—21no—
284C2×#41C4×D6111yescharacteristic
285C2×#41D24411yescharacteristic
286C2×#41C22⋊D6111yescharacteristic
287C2×#41C22×D6111yescharacteristic
288C22×Q8—21no—
289C22×Q8C4×D6111yescharacteristic
290C22×Q8D24411yescharacteristic
291C22×Q8C22⋊D6111yescharacteristic
292C22×D4—611no—
293AGL(1,ℤ/8)—2442no—
294C2×SD16—1221no—
295C2×SD16—1221no—
296C2×SD16—1221no—
297C2×SD16—2442no—
298C2×D8—1221no—
299C2×M16—21no—
300C2×M16—611no—
301C22×C8—611no—
302#127—1221no—
303#124—1221no—
304#121—1221no—
305#121—1221no—
306#120—1221no—
307#120—1221no—
308C4×Q8—2442no—
309C4×Q8—2442no—
310#117—21no—
311#117—611no—
312#117—1224no—
313#117—1221no—
314#117—1221no—
315#104—42no—
316#103—42no—
317#103—1221no—
318#103—1221no—
319#103—1221no—
320#102—1221no—
321#101—21no—
322#100—21no—
323#99—21no—
324#99—21no—
325#99—612no—
326#99—612no—
327#99—1224no—
328#99—1221no—
329#99—1221no—
330C22⋊C8—1221no—
331C22⋊C8—1221no—
332C24—1221no—
333#41—22no—
334#41—22no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—44no—
342#41—44no—
343#41—44no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—44no—
354C2×Q8#1175819811yescharacteristic
355C2×Q8#1175819811yescharacteristic
356C2×Q8#1175819811yescharacteristic
357C2×Q8#1175822411yescharacteristic
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—2441no—
364C2×D4—2441no—
365C2×D4—2441no—
366C2×D4—2441no—
367C2×D4#1175819811yescharacteristic
368C2×D4D4×D6111yescharacteristic
369C2×D4D4×D6111yescharacteristic
370C2×D4#1175822411yescharacteristic
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4C2×C4×D6111yescharacteristic
378C22×C4C2×D24411yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380SD16—2442no—
381SD16—2442no—
382SD16—2442no—
383D8—2442no—
384M16—41no—
385M16—41no—
386M16—2441no—
387C2×C8—41no—
388C2×C8—1221no—
389C2×C8—1221no—
390C2×C8—2441no—
391C4⋊C4—41no—
392C4⋊C4—1221no—
393C4⋊C4—2444no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C22⋊C4—41no—
397C22⋊C4—1221no—
398C22⋊C4—1222no—
399C22⋊C4—1222no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C42—41no—
403C42—1221no—
404C42—2444no—
405C42—2441no—
406C42—2441no—
407C23—21no—
408C23—9761no—
409C23—4881no—
410C23—2441no—
411C23#1451973711yescharacteristic
412C23#1451974711yescharacteristic
413C23#1451974911yescharacteristic
414Q8—24no—
415Q8—21no—
416Q8—21no—
417Q8—21no—
418Q8—21no—
419Q8—41no—
420Q8—42no—
421Q8—42no—
422Q8#1451969112yes—
423Q8#1451969112yes—
424D4—21no—
425D4—21no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—9761no—
433D4—9761no—
434D4—4881no—
435D4—2441no—
436D4—2441no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—81no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4C2×#1175819811yescharacteristic
457C2×C4#1451973711yescharacteristic
458C2×C4#1451974911yescharacteristic
459C2×C4#1451976111yescharacteristic
460C8—81no—
461C8—2441no—
462C8—2441no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—81no—
471C22—9762no—
472C22—4881no—
473C22—4881no—
474C22—4881no—
475C22#1683963911yescharacteristic
476C4—21no—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—161no—
483C4—4881no—
484C4—4882no—
485C4—4882no—
486C4C2×#1451969111yescharacteristic
487C4C2×#1451969111yescharacteristic
488C2—21no—
489C2—41no—
490C2—41no—
491C2—41no—
492C2—81no—
493C2—9761no—
494C2#2144627011yescharacteristic
subgroupG/H
centerC2#21446270
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#1796C2
socleC2×C61#1898
prime radical#21455105C2
cosocleC2×#41×C61C23
hypercenter#1796D61
hypocenterC61#13103
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #1796

Complements

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

NH ≅ G/Ntype
2. #21454283C2semidirect
3. #21447490C2semidirect
4. #21447470C2semidirect
5. #21443949C2semidirect
6. #21443939C2semidirect
7. C61×#1796C2semidirect
8. C2×#14519802C4semidirect
9. C22×#11758201C4semidirect
16. #16839771C4semidirect
17. #16839758C4semidirect
245. #1796D61semidirect
278. C61#13103semidirect

Maximal subgroups

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

subgroupindex
#214551052
#214542832
#214474902
#214474702
#214439492
#214439392
C61×#17962
#1310361