Finite Group Databasebeta

← #29696459

Subgroups — #29696459

Order 15616.

1154 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2C2×#16839839C211yesmaximal, characteristic
3#21448770C211yesmaximal, characteristic
4#21446779C211yesmaximal, characteristic, complemented
5#21446766C211yesmaximal, characteristic, complemented
6#21446249C212yesmaximal, complemented
7#21446128C212yesmaximal, complemented
8#21445904C212yesmaximal, complemented
9C4×#14519701C211yesmaximal, characteristic, complemented
10#21445537C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840472—24no—
14#16840472C418yescomplemented
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519755—22no—
18C2×#14519753C2212yes—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519735C2212yes—
22C2×C4⋊C4×D61C2212yes—
23#16839839—24no—
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519689C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839663—21no—
29#16839652—22no—
30#16839619C2212yes—
31C4⋊C4×Dic61—21no—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122—21no—
45C22×D244D411yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61C2312yes—
48C2×#11758189—24no—
49C2×#11758189—24no—
50C2×#11758189—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53#14519777—22no—
54#14519777C2×C414yes—
55#14519769—28no—
56#14519755—216no—
57#14519753—24no—
58#14519753C2×C418yes—
59C4×D244—28no—
60C4×D244C2×C414yes—
61C4×D244D414yes—
62#14519735—24no—
63#14519735—42no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—22no—
70C2×#11758196—22no—
71C4⋊C4×D61—24no—
72C4⋊C4×D61C2×C418yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198D412yes—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—22no—
79#14519701—22no—
80#14519701—22no—
81#14519689—24no—
82C4×#92418—22no—
83#14519684—21no—
84#14519684—22no—
85C22×Q8×C61C2311yescharacteristic
86C22×C8×C61C2312yes—
87C4×Q8×C61—22no—
88C4×Q8×C61C2×C414yes—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2312yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—22no—
97#103×C61—42no—
98C23×D61C2×D412yes—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61C22×C412yes—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61C2×D412yes—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244C22⋊C414yes—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—28no—
123C2×C4×D61—82no—
124C2×C4×D61—82no—
125C2×C4×D61C22×C414yes—
126C2×C4×D61C2×D414yes—
127C2×Dic122—24no—
128C2×Dic122—81no—
129C2×Dic122—81no—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—42no—
138C4⋊C4×C61—81no—
139C4⋊C4×C61C22×C412yes—
140C42×C61—24no—
141C42×C61C22×C412yes—
142C42×C61C2×D412yes—
143#11758198—28no—
144#11758198—44no—
145#11758198—82no—
146#11758198C22⋊C418yes—
147C4⋊Dic61—24no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61C22⋊C414yes—
153C4⋊Dic61C22×C414yes—
154C4⋊Dic61#4112yes—
155#11758196—28no—
156#11758196—28no—
157C4×Dic61—44no—
158C4×Dic61—42no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418#4112yes—
165#11758189—48no—
166C23×C61C22×D411yescharacteristic
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61C2×C22⋊C414yes—
174Q8×C61—22no—
175Q8×C61—22no—
176Q8×C61—41no—
177Q8×C61—41no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61C2×C22⋊C412yes—
190C2×C4×C61C2×C22⋊C412yes—
191C2×C4×C61#13712yes—
192C2×C4×C61#13712yes—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)11yescharacteristic
198C2×C4×C612−1+411yescharacteristic
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—24no—
202C2×Dic61—22no—
203C2×Dic61—41no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61C2×C22⋊C414yes—
211D244—28no—
212C4×D61—24no—
213C4×D61—24no—
214C4×D61—24no—
215C4×D61—24no—
216C4×D61—88no—
217Dic122—84no—
218C8×C61—42no—
219#92418—22no—
220#92418—41no—
221#22502—611nomaximal, Sylow
222C22×C61C22×C22⋊C411yescharacteristic
223C22×C61#54611yescharacteristic
224C22×C61#54711yescharacteristic
225C22×C61C2×#13711yescharacteristic
226C22×C61C2×#13711yescharacteristic
227C22×C61#57611yescharacteristic
228C22×C61#57711yescharacteristic
229C2×D61—24no—
230C2×D61—24no—
231C2×D61—24no—
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—42no—
236C4×C61—22no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—82no—
240C4×C61#41914yes—
241C4×C61C4×#4111yescharacteristic
242C4×C61#52011yescharacteristic
243C4×C61#54611yescharacteristic
244C4×C61#54711yescharacteristic
245Dic61—24no—
246Dic61—42no—
247Dic61—44no—
248Dic61—81no—
249C2×#554—611no—
250C2×#436—611no—
251#2354—611no—
252#1990—611no—
253#1987—611no—
254#1894—612no—
255#1873—612no—
256#1823—612no—
257#1803D6111yescharacteristic, complemented
258C4×#103—611no—
259#1736—612no—
260C2×C61#233911yescharacteristic
261C2×C61C2×#41911yescharacteristic
262C2×C61#289711yescharacteristic
263C2×C61#333111yescharacteristic
264C2×C61#333311yescharacteristic
265C2×C61#335611yescharacteristic
266C2×C61#335711yescharacteristic
267D61—44no—
268C22×M16—612no—
269#554—618no—
270#554—1224no—
271C2×C4⋊Q8—611no—
272C2×#125—1221no—
273C2×C22⋊Q8—612no—
274C2×C22⋊Q8—1222no—
275C2×C4×Q8C2×D6111yescharacteristic
276C2×C4×D4—611no—
277C2×C4×D4—1221no—
278C2×#117—612no—
279C22×C4⋊C4—612no—
280#436—1224no—
281C2×C4⋊C8C2×D6111yescharacteristic
282C2×#103—611no—
283C2×#103—611no—
284C2×#103C2×D6112yes—
285C2×C22⋊C8—612no—
286C2×C4×C8—611no—
287#390—1221no—
288#388—1222no—
289#384C2×D6111yescharacteristic
290#380—612no—
291C4×C4⋊C4—611no—
292C4×C4⋊C4—1221no—
293#340—612no—
294#340—612no—
295#340—611no—
296#340—612no—
297#340—612no—
298#340—611no—
299#340C2×D6112yes—
300C61#2250211yescharacteristic, complemented, Sylow
301C22×Q8—1221no—
302C22×Q8C22×D6111yescharacteristic
303C22×D4—611no—
304C23×C4—612no—
305C23×C4—1222no—
306C2×M16—1224no—
307C2×M16—1224no—
308C2×M16—2442no—
309C22×C8—611no—
310C22×C8—611no—
311C22×C8C22×D6112yes—
312C4⋊Q8—614no—
313C4⋊Q8—1222no—
314#125—1228no—
315C22⋊Q8—618no—
316C22⋊Q8—12216no—
317C22⋊Q8—1224no—
318C4×Q8—22no—
319C4×Q8C4×D6114yes—
320C4×D4—614no—
321C4×D4—614no—
322C4×D4—1228no—
323#117—1224no—
324#117—2442no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—618no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—1224no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4C22×D6112yes—
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C22⋊C4—612no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C42—21no—
344C2×C42—1222no—
345C2×C42—1221no—
346C2×C42C22×D6111yescharacteristic
347C4⋊C8—22no—
348#103—42no—
349#103—1222no—
350#103—1222no—
351C22⋊C8—1224no—
352C4×C8—1222no—
353#95—1222no—
354#95—1221no—
355C24—612no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—1224no—
362C2×Q8—4881no—
363C2×Q8—4881no—
364C2×Q8C2×C4×D6112yes—
365C2×D4—614no—
366C2×D4—1222no—
367C2×D4—1222no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—612no—
372C22×C4—614no—
373C22×C4—614no—
374C22×C4—1228no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1221no—
380C22×C4—1222no—
381C22×C4—1221no—
382C22×C4—4882no—
383C22×C4—4882no—
384C22×C4D4×D6111yescharacteristic
385C22×C4D4×D6111yescharacteristic
386C22×C4C23×D6111yescharacteristic
387M16—2448no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—612no—
392C2×C8—1221no—
393C2×C8—1221no—
394C2×C8—1221no—
395C2×C8—1221no—
396C2×C8—1221no—
397C4⋊C4—21no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—81no—
403C4⋊C4—614no—
404C4⋊C4—614no—
405C4⋊C4—612no—
406C4⋊C4—1228no—
407C4⋊C4—1228no—
408C4⋊C4—1222no—
409C4⋊C4—1221no—
410C4⋊C4—1224no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4C2×C4×D6112yes—
414C22⋊C4—618no—
415C22⋊C4—1228no—
416C22⋊C4—4882no—
417C22⋊C4—2444no—
418C42—24no—
419C42—2444no—
420C42—2442no—
421C42C2×C4×D6112yes—
422C42D4×D6112yes—
423C23—614no—
424C23—1222no—
425C23—1222no—
426C23—1222no—
427C23—1222no—
428C23—1222no—
429C23—1222no—
430C23C2×D4×D6111yescharacteristic
431Q8—22no—
432Q8—22no—
433Q8—41no—
434Q8—41no—
435Q8—4884no—
436D4—1228no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—614no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1224no—
452C2×C4—1224no—
453C2×C4—1224no—
454C2×C4—1224no—
455C2×C4—1224no—
456C2×C4—1222no—
457C2×C4—4888no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2442no—
463C2×C4—2441no—
464C2×C4—2442no—
465C2×C4C22⋊C4×D6112yes—
466C2×C4C22⋊C4×D6112yes—
467C2×C4#1451980612yes—
468C2×C4#1451980912yes—
469C2×C4C22×C4×D6111yescharacteristic
470C2×C4C2×D4×D6111yescharacteristic
471C2×C4#41×D6111yescharacteristic
472C2×C4#41×D6111yescharacteristic
473C2×C4#1451982211yescharacteristic
474C2×C4#1451982511yescharacteristic
475C8—42no—
476C8—1222no—
477C8—2441no—
478C22—1224no—
479C22—1224no—
480C22—1224no—
481C22—2442no—
482C22—2442no—
483C22—2442no—
484C22—2442no—
485C22C2×C22⋊C4×D6111yescharacteristic
486C22#1684040911yescharacteristic
487C22#1684041411yescharacteristic
488C22C2×#1451980611yescharacteristic
489C22C2×#1451980911yescharacteristic
490C22#1684056611yescharacteristic
491C22#1684057111yescharacteristic
492C4—22no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—1224no—
497C4—4881no—
498C4—2444no—
499C4—2442no—
500C4#1683976914yes—
501C4#1684023011yescharacteristic
502C4#1684024311yescharacteristic
503C4#1684040911yescharacteristic
504C4#1684041411yescharacteristic
505C2—2444no—
506C2#2144864711yescharacteristic
507C2C2×#1683976911yescharacteristic
508C2#2145434311yescharacteristic
509C2#2145813811yescharacteristic
510C2#2145816011yescharacteristic
511C2#2145843111yescharacteristic
512C2#2145844211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#22502
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #21446779C2semidirect
5. #21446766C2semidirect
6. #21446249C2semidirect
7. #21446128C2semidirect
8. #21445904C2semidirect
9. C4×#14519701C2semidirect
10. #21445537C2semidirect
11. C61×#1803C2semidirect
14. #16840472C4semidirect
257. #1803D61semidirect
300. C61#22502semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168398392
#214487702
#214467792
#214467662
#214462492
#214461282
#214459042
C4×#145197012
#214455372
C61×#18032
#2250261