Finite Group Databasebeta

← #29647870

Subgroups — #29647870

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459668C211yesmaximal, characteristic
2#21454859C211yesmaximal, characteristic, complemented
3#21448727C211yesmaximal, characteristic, complemented
4#21444382C211yesmaximal, characteristic, complemented
5#21444359C214yesmaximal, complemented
6#21444320C214yesmaximal, complemented
7#21444319C211yesmaximal, characteristic, complemented
8#21444314C211yesmaximal, characteristic, complemented
9C61×#1517C211yesmaximal, characteristic, complemented
10#16840494—22no—
11#16840486C412yescomplemented
12#16840477C412yescomplemented
13#16840458—22no—
14#16840432C412yescomplemented
15#16840416C412yescomplemented
16#16840349—22no—
17#16840326—21no—
18#16840319C2211yescharacteristic
19#16840272C2214yes—
20#16840245—21no—
21C2×C4×D244C2211yescharacteristic
22C2×#14519739—21no—
23#16839841—22no—
24#16839839—22no—
25#16839815C2211yescharacteristic
26#16839724C2214yes—
27#16839658—21no—
28#16839636—22no—
29#16839621C2212yes—
30C4×C4⋊Dic61C2211yescharacteristic
31#95×D61C2212yes—
32#16839500C2212yes—
33#16839500C2212yes—
34#16839498C2214yes—
35#16839496C2212yes—
36#16839496C2212yes—
37C61×#532C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#382C2211yescharacteristic
40C61×#327C2214yes—
41C2×#11758224C2×C414yescomplemented
42C22×D244D411yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61C2312yes—
45C2×#11758189—24no—
46#14519777C2×C411yescharacteristic, complemented
47#14519777C2×C411yescharacteristic, complemented
48#14519773—22no—
49#14519772—24no—
50#14519769—21no—
51#14519766—22no—
52#14519763—24no—
53#14519761—24no—
54#14519759—24no—
55#14519757C2×C414yescomplemented
56#14519756—21no—
57#14519756C2312yes—
58#14519751—22no—
59#14519751—22no—
60#14519749—24no—
61#14519748C2×C414yescomplemented
62#14519745—24no—
63Q8×Dic61—22no—
64C4×Dic122—21no—
65C4×Dic122C2×C412yescomplemented
66C4×D244C2×C412yes—
67C4×D244C2×C412yes—
68C4×D244D412yes—
69C4×D244D412yes—
70D4×Dic61—24no—
71D4×Dic61—22no—
72#14519739—44no—
73#14519737—24no—
74#14519736—21no—
75#14519736—22no—
76#14519735C2×C414yescomplemented
77C2×C4⋊Dic61—22no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×#11758196—22no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83C2×C4×Dic61—21no—
84C4⋊#92418—21no—
85C22⋊#92418C2312yes—
86#14519689—24no—
87C4×#92418—21no—
88#14519684—22no—
89#14519684C2311yescharacteristic
90#14519684C2311yescharacteristic
91C2×#41×C61—21no—
92C4⋊Q8×C61C2×C411yescharacteristic, complemented
93#127×C61C2×C411yescharacteristic, complemented
94#124×C61—21no—
95#121×C61—21no—
96#121×C61C2312yes—
97C4×D4×C61—21no—
98C4×D4×C61C2×C412yescomplemented
99C2×C4⋊C4×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—21no—
102C22⋊C8×C61C2312yes—
103C4×C8×C61—21no—
104#95×C61—21no—
105#95×C61C2312yes—
106C23×D61C2×D412yes—
107#41×C61—42no—
108#41×C61—42no—
109C2×Q8×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61C22×C412yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C22×Dic61—21no—
122C22×Dic61—24no—
123C22×Dic61—21no—
124C22×Dic61C2×D412yes—
125#11758224—84no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244C22⋊C412yes—
129C2×D244C22⋊C412yes—
130C2×C4×D61—44no—
131C2×C4×D61—44no—
132C2×C4×D61—82no—
133C2×C4×D61—82no—
134C2×C4×D61C22×C414yes—
135C2×C4×D61C2×D414yes—
136C2×Dic122—21no—
137C2×Dic122C22×C412yes—
138C2×C8×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—42no—
141C4⋊C4×C61C22×C412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C42×C61C22×C411yescharacteristic
145C42×C61C22×C411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C22⋊Dic61—22no—
149C22⋊Dic61—24no—
150#11758198—24no—
151#11758198—24no—
152#11758198—44no—
153#11758198C22⋊C414yes—
154#11758198C22⋊C414yes—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—24no—
159C4⋊Dic61—42no—
160C4⋊Dic61C22⋊C412yes—
161C4⋊Dic61C22⋊C412yes—
162C4⋊Dic61C22×C412yes—
163#11758196—22no—
164#11758196—22no—
165#11758196—82no—
166C4×Dic61—22no—
167C4×Dic61—24no—
168C4×Dic61—42no—
169C2×#92418—22no—
170#11758189—84no—
171C23×C61—22no—
172C23×C61C22×D411yescharacteristic
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61C2×C22⋊C414yes—
178Q8×C61—42no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—81no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61AGL(1,ℤ/8)12yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C23×C411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—84no—
200C22⋊D61—84no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—84no—
211C2×Dic61C2×C22⋊C414yes—
212D244—24no—
213D244—24no—
214C4×D61—44no—
215C4×D61—44no—
216C4×D61—88no—
217Dic122—42no—
218Dic122—81no—
219C8×C61—42no—
220#92418—42no—
221#17630—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61C22×C22⋊C411yescharacteristic
228C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C2×D61—24no—
231C2×D61—24no—
232C2×D61—24no—
233C2×D61—44no—
234C2×D61—44no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—42no—
240C4×C61—82no—
241C4×C61#41712yes—
242C4×C61#41712yes—
243Dic61—24no—
244Dic61—42no—
245Dic61—42no—
246Dic61—42no—
247Dic61—81no—
248#3506—611no—
249#2975—611no—
250#2349—611no—
251#1518—611no—
252#1517D6111yescharacteristic, complemented
253#1513—614no—
254#1504—614no—
255#1503—611no—
256#1502—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61—82no—
260C2×C61#288311yescharacteristic
261C2×C61C2×#41711yescharacteristic
262C2×C61#290111yescharacteristic
263D61—44no—
264#559—1222no—
265#556—612no—
266#555—612no—
267#553—1222no—
268#550—612no—
269#547—612no—
270#538—1222no—
271#535—1221no—
272#533—611no—
273#532C2×D6111yescharacteristic
274#525—614no—
275#520—1221no—
276C2×C4×D4—611no—
277C2×C4×D4—1221no—
278#436—1222no—
279#436—1222no—
280#431—611no—
281#431C2×D6111yescharacteristic
282#407—614no—
283#389—1221no—
284#385—1222no—
285#382C2×D6111yescharacteristic
286#380—612no—
287C4×C4⋊C4—611no—
288C2×#95—612no—
289#328—612no—
290#328—612no—
291#328—614no—
292#327—612no—
293#327—612no—
294#327C2×D6114yes—
295C61#1763011yescharacteristic, complemented, Sylow
296C2×#41—21no—
297C2×#41—614no—
298C22×D4—611no—
299C23×C4—612no—
300C23×C4—1222no—
301C2×M16—1224no—
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304C4⋊Q8C4×D6111yescharacteristic, complemented
305#127—1222no—
306#127C4×D6111yescharacteristic, complemented
307#126—1224no—
308#125—1221no—
309#124—21no—
310#124—1224no—
311#124—1222no—
312#123—1224no—
313#123—1224no—
314C22⋊Q8—612no—
315C22⋊Q8—614no—
316C22⋊Q8—1221no—
317#121—21no—
318#121—614no—
319#121—1224no—
320#121—1222no—
321#121—1222no—
322#121C22×D6112yes—
323#120—1224no—
324C4×Q8—612no—
325C4×Q8—1222no—
326C4×Q8—1221no—
327C4×D4—21no—
328C4×D4—612no—
329C4×D4—612no—
330C4×D4—612no—
331C4×D4—612no—
332C4×D4—1224no—
333C4×D4—1222no—
334C4×D4—1224no—
335C4×D4—2444no—
336C4×D4C4×D6112yescomplemented
337#117—614no—
338#117—1222no—
339#117—1221no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C22⋊C4—612no—
346C2×C22⋊C4—1222no—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349C4⋊C8—21no—
350C4⋊C8—1221no—
351C22⋊C8—612no—
352C22⋊C8—1224no—
353C22⋊C8C22×D6112yes—
354C4×C8—21no—
355C4×C8—1221no—
356#95—21no—
357#95—611no—
358#95—611no—
359#95—1222no—
360#95C22×D6112yes—
361C24—612no—
362#41—42no—
363#41—42no—
364#41—4884no—
365C2×Q8—21no—
366C2×Q8—612no—
367C2×Q8—1221no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—612no—
372C2×D4—612no—
373C2×D4—1224no—
374C2×D4—1224no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4C2×C4×D6112yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—612no—
381C22×C4—614no—
382C22×C4—614no—
383C22×C4—1224no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—4882no—
387C22×C4—4882no—
388C22×C4—2444no—
389C22×C4—2444no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C23×D6111yescharacteristic
393M16—4884no—
394C2×C8—22no—
395C2×C8—1222no—
396C4⋊C4—21no—
397C4⋊C4—42no—
398C4⋊C4—612no—
399C4⋊C4—612no—
400C4⋊C4—612no—
401C4⋊C4—1224no—
402C4⋊C4—1222no—
403C4⋊C4—1221no—
404C4⋊C4—1222no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—4882no—
408C4⋊C4—2442no—
409C4⋊C4C2×C4×D6112yes—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—614no—
413C22⋊C4—614no—
414C22⋊C4—1224no—
415C22⋊C4—1224no—
416C22⋊C4—1224no—
417C22⋊C4—1222no—
418C22⋊C4—2444no—
419C42—1224no—
420C42—1222no—
421C42—2442no—
422C42C2×C4×D6111yescharacteristic
423C42C2×C4×D6111yescharacteristic
424C42D4×D6111yescharacteristic
425C42D4×D6111yescharacteristic
426C23—22no—
427C23—614no—
428C23—1222no—
429C23—1222no—
430C23—2442no—
431C23—2442no—
432C23C2×D4×D6111yescharacteristic
433Q8—42no—
434Q8—4881no—
435Q8—2442no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—81no—
441D4—1224no—
442D4—1224no—
443D4—4884no—
444D4—4884no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—614no—
453C2×C4—1222no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—4888no—
457C2×C4—4884no—
458C2×C4—2444no—
459C2×C4—2444no—
460C2×C4—2442no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4C22⋊C4×D6111yescharacteristic
467C2×C4C22⋊C4×D6111yescharacteristic
468C2×C4C22⋊C4×D6111yescharacteristic
469C2×C4C22⋊C4×D6111yescharacteristic
470C2×C4#1451980112yes—
471C2×C4#1451980612yes—
472C2×C4C22×C4×D6111yescharacteristic
473C2×C4C2×D4×D6111yescharacteristic
474C8—42no—
475C8—2442no—
476C22—21no—
477C22—21no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—1224no—
482C22—1224no—
483C22—1224no—
484C22—2444no—
485C22—2444no—
486C22C2×C22⋊C4×D6111yescharacteristic
487C22C2×#1451980111yescharacteristic
488C22C2×#1451980611yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—82no—
495C4—1224no—
496C4—4881no—
497C4—2442no—
498C4—2442no—
499C4—2442no—
500C4#1683976012yes—
501C4#1683976012yes—
502C2—21no—
503C2—21no—
504C2—82no—
505C2—2444no—
506C2#2145427811yescharacteristic
507C2C2×#1683976011yescharacteristic
508C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1517C2
socleC22×C61C22×C22⋊C4
prime radical#21459668C2
cosocleC22×C4×C61C24
hypercenter#1517D61
hypocenterC61#17630
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #21454859C2semidirect
3. #21448727C2semidirect
4. #21444382C2semidirect
5. #21444359C2semidirect
6. #21444320C2semidirect
7. #21444319C2semidirect
8. #21444314C2semidirect
9. C61×#1517C2semidirect
11. #16840486C4semidirect
12. #16840477C4semidirect
14. #16840432C4semidirect
15. #16840416C4semidirect
41. C2×#11758224C2×C4semidirect
46. #14519777C2×C4semidirect
47. #14519777C2×C4semidirect
55. #14519757C2×C4semidirect
61. #14519748C2×C4semidirect
65. C4×Dic122C2×C4semidirect
76. #14519735C2×C4semidirect
92. C4⋊Q8×C61C2×C4semidirect
93. #127×C61C2×C4semidirect
98. C4×D4×C61C2×C4semidirect
252. #1517D61semidirect
295. C61#17630semidirect
304. C4⋊Q8C4×D61semidirect
306. #127C4×D61semidirect
336. C4×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596682
#214548592
#214487272
#214443822
#214443592
#214443202
#214443192
#214443142
C61×#15172
#1763061