Finite Group Databasebeta

← #29707323

Subgroups — #29707323

Order 15616.

1282 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519768C211yesmaximal, characteristic, complemented
2C2×#16839764C211yesmaximal, characteristic
3C2×#16839761C211yesmaximal, characteristic
4#21448307C211yesmaximal, characteristic
5#21446969C214yesmaximal, complemented
6#21446964C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445512C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519768—24no—
13C2×#14519768—22no—
14C2×#14519768—24no—
15C2×#14519768C2214yes—
16C22×C4⋊Dic61—21no—
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758196—21no—
19C22×C4×Dic61C2211yescharacteristic
20#16839764—24no—
21#16839761—24no—
22C2×#14519702C2212yes—
23C2×#14519700C2212yes—
24C2×#14519699C2212yes—
25C2×#14519697C2212yes—
26#16839667C2212yes—
27#16839663C2212yes—
28#16839639—21no—
29#16839639—21no—
30C4×#11758198—22no—
31#16839524C2214yes—
32#16839516C2214yes—
33C22×M16×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C22×C4⋊C4×C61C2211yescharacteristic
36C61×#340C2214yes—
37C2×#11758222—22no—
38C2×#11758222—22no—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222Q814yes—
43C22×Dic122C2311yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—21no—
46C23×Dic61—21no—
47C23×Dic61D411yescharacteristic
48C23×Dic61D411yescharacteristic
49C22×C4×D61—21no—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194—22no—
53C22×#92418C2312yes—
54#14519768—216no—
55#14519768—28no—
56#14519768—28no—
57C2×C4⋊Dic61—28no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—24no—
61C2×C4⋊Dic61D414yes—
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—28no—
64C2×#11758196—24no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61D412yes—
70C2×C4×Dic61C2312yes—
71#14519702—24no—
72#14519700—24no—
73#14519699—24no—
74#14519697—24no—
75#14519684—22no—
76#14519684—22no—
77C23×C4×C61—21no—
78C23×C4×C61C2311yescharacteristic
79C2×M16×C61—22no—
80C2×M16×C61—22no—
81C2×M16×C61—22no—
82C22×C8×C61C2312yes—
83#117×C61—22no—
84#117×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61D414yes—
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2312yes—
90C2×C22⋊C4×C61—21no—
91C2×C42×C61—21no—
92C24×C61C2×D411yescharacteristic
93C23×D61—21no—
94C22×C4×C61—24no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—44no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×Q812yes—
102C22×C4×C61#4112yes—
103C22×C4×C61#4112yes—
104C22×C4×C61#4112yes—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—24no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—42no—
109C22×Dic61—24no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—22no—
114C22×Dic61—42no—
115C22×Dic61—42no—
116C22×Dic61—44no—
117C22×Dic61C2×D412yes—
118C22×Dic61C2×D412yes—
119#11758222—28no—
120#11758222—28no—
121#11758222—44no—
122#11758222—44no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244C2×Q812yes—
129C2×C4×D61—24no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—21no—
137C2×Dic122C2×Q812yes—
138M16×C61—44no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C2×D414yes—
147C22⋊C4×C61—42no—
148C42×C61—42no—
149#11758198—44no—
150#11758198—44no—
151C4⋊Dic61—216no—
152C4⋊Dic61—24no—
153C4⋊Dic61—22no—
154C4⋊Dic61C2×D414yes—
155#11758196—216no—
156C4×Dic61—24no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159#11758194—44no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61#12012yes—
166C23×C61C22⋊Q812yes—
167C23×C61#12412yes—
168C23×C61C22×D411yescharacteristic
169C22×D61—22no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173C22×D61—42no—
174C2×C4×C61—24no—
175C2×C4×C61—48no—
176C2×C4×C61—44no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61#12012yes—
181C2×C4×C61C22⋊Q814yes—
182C2×C4×C61C22⋊Q814yes—
183C2×C4×C61C22⋊Q812yes—
184C2×C4×C61#12414yes—
185C2×C4×C61#12414yes—
186C2×C4×C61#12412yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×Q811yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C22⋊D61—48no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—24no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—44no—
203C2×Dic61—44no—
204C2×Dic61—48no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—44no—
208D244—22no—
209D244—22no—
210D244—41no—
211D244—41no—
212C4×D61—48no—
213Dic122—22no—
214Dic122—22no—
215Dic122—41no—
216Dic122—41no—
217C8×C61—42no—
218#92418—42no—
219#23453—611nomaximal, Sylow
220C22×C61—24no—
221C22×C61—24no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61#39314yes—
225C22×C61C2×#12011yescharacteristic
226C22×C61C2×C22⋊Q811yescharacteristic
227C22×C61C2×#12411yescharacteristic
228C22×C61#57512yes—
229C22×C61#57512yes—
230C2×D61—44no—
231C2×D61—42no—
232C2×D61—82no—
233C2×D61—81no—
234C2×D61—81no—
235C4×C61—44no—
236C4×C61—82no—
237C4×C61#39314yes—
238C4×C61C2×C22⋊Q811yescharacteristic
239C4×C61C2×C22⋊Q811yescharacteristic
240C4×C61C2×#12411yescharacteristic
241C4×C61C2×#12411yescharacteristic
242Dic61—42no—
243Dic61—42no—
244Dic61—42no—
245Dic61—44no—
246C22×#125—611no—
247C2×#417—611no—
248C2×#417—611no—
249#2297—611no—
250#2026—614no—
251#2025—614no—
252#1736—611no—
253#1736D6111yescharacteristic, complemented
254#1728—611no—
255C2×C61—24no—
256C2×C61C2×#39311yescharacteristic
257C2×C61#300511yescharacteristic
258C2×C61#300511yescharacteristic
259C2×C61#308012yes—
260C2×C61#314212yes—
261D61—82no—
262C22×#41—611no—
263C22×M16—611no—
264C22×M16C2×D6111yescharacteristic
265C2×#125—614no—
266C2×#125—1222no—
267C2×#125—1224no—
268C2×#125—1224no—
269C2×#117C2×D6111yescharacteristic
270C22×C4⋊C4—611no—
271C22×C4⋊C4—1221no—
272C22×C4⋊C4—1221no—
273C22×C4⋊C4C2×D6111yescharacteristic
274C22×C42—611no—
275#417—1224no—
276#417—1224no—
277C2×#103—612no—
278C2×#103—612no—
279C2×#102—612no—
280C2×#102—612no—
281#391—612no—
282#390—612no—
283#386—1221no—
284#386—1221no—
285C4×C22⋊C4—1222no—
286#340—614no—
287#340C2×D6114yes—
288#336—614no—
289C61#2345311yescharacteristic, complemented, Sylow
290C2×#41—614no—
291C2×#41—1222no—
292C2×#41—1222no—
293C2×#41—1222no—
294C2×#41—1222no—
295C2×#41—1222no—
296C22×Q8—611no—
297C22×D4—611no—
298C22×D4—1221no—
299C23×C4—21no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4—1221no—
303C23×C4—1221no—
304C23×C4C22×D6111yescharacteristic
305C2×M16—22no—
306C2×M16—22no—
307C2×M16—22no—
308C2×M16—1222no—
309C2×M16—1222no—
310C2×M16—1222no—
311C22×C8—612no—
312C22×C8C22×D6112yes—
313#125—1228no—
314#125—12216no—
315#125—1228no—
316#117—22no—
317#117—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—614no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—1224no—
323C2×C4⋊C4—1224no—
324C2×C4⋊C4—1228no—
325C2×C4⋊C4—1228no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4D24414yes—
329C2×C4⋊C4C22×D6112yes—
330C2×C4⋊C4C22×D6112yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4—1222no—
334C2×C42—21no—
335C2×C42—612no—
336C2×C42—612no—
337C2×C42—1222no—
338C2×C42—1222no—
339#103—1224no—
340#103—1224no—
341#102—1224no—
342#102—1224no—
343#95—1222no—
344#95—1222no—
345C24—1221no—
346C24D4×D6111yescharacteristic
347#41—1228no—
348#41—1228no—
349#41—2444no—
350#41—2444no—
351C2×Q8—612no—
352C2×Q8—1221no—
353C2×Q8—1221no—
354C2×Q8—1221no—
355C2×Q8—1221no—
356C2×Q8—1221no—
357C2×D4—612no—
358C2×D4—1224no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1221no—
364C2×D4—2442no—
365C2×D4—2442no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—44no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—1221no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1224no—
377C22×C4—1224no—
378C22×C4—2442no—
379C22×C4—2442no—
380C22×C4—2444no—
381C22×C4—2442no—
382C22×C4—2442no—
383C22×C4C2×D24412yes—
384C22×C4#1175822212yes—
385C22×C4#1175822212yes—
386C22×C4D4×D6112yes—
387C22×C4D4×D6111yescharacteristic
388C22×C4#1175822412yes—
389C22×C4Q8×D6112yes—
390C22×C4C23×D6111yescharacteristic
391M16—44no—
392M16—2444no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1222no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—614no—
404C4⋊C4—12216no—
405C4⋊C4—12216no—
406C4⋊C4—1222no—
407C4⋊C4—1224no—
408C4⋊C4C2×D24414yes—
409C22⋊C4—42no—
410C22⋊C4—2444no—
411C22⋊C4—2444no—
412C42—42no—
413C42—1222no—
414C42—1222no—
415C42—1224no—
416C23—22no—
417C23—22no—
418C23—1222no—
419C23—2441no—
420C23—2441no—
421C23—2442no—
422C23—2442no—
423C23#1451974712yes—
424C23#1451975512yes—
425C23#1451976312yes—
426C23C2×D4×D6111yescharacteristic
427Q8—1222no—
428Q8—1222no—
429Q8—2441no—
430Q8—2441no—
431D4—1222no—
432D4—1222no—
433D4—2448no—
434D4—2441no—
435D4—2441no—
436C2×C4—24no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1224no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—2444no—
451C2×C4—2444no—
452C2×C4—2448no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2444no—
456C2×C4—2448no—
457C2×C4#1451974712yes—
458C2×C4#1451975514yes—
459C2×C4#1451975512yes—
460C2×C4#1451975714yes—
461C2×C4#1451976314yes—
462C2×C4#1451976312yes—
463C2×C4#1451976514yes—
464C2×C4C2×D4×D6111yescharacteristic
465C2×C4C22×D24411yescharacteristic
466C2×C4C2×Q8×D6111yescharacteristic
467C2×C4C2×#1175822411yescharacteristic
468C2×C4C2×#1175822211yescharacteristic
469C2×C4C2×#1175822211yescharacteristic
470C8—42no—
471C8—2442no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—4882no—
477C22—4881no—
478C22—4881no—
479C22—2444no—
480C22—2442no—
481C22#1683967514yes—
482C22C2×#1451974711yescharacteristic
483C22C2×#1451975511yescharacteristic
484C22C2×#1451976311yescharacteristic
485C22#1684055812yes—
486C22#1684056112yes—
487C4—44no—
488C4—82no—
489C4—2442no—
490C4—2442no—
491C4—2442no—
492C4—2444no—
493C4#1683967514yes—
494C4C2×#1451975511yescharacteristic
495C4C2×#1451975711yescharacteristic
496C4C2×#1451976311yescharacteristic
497C4C2×#1451976511yescharacteristic
498C2—24no—
499C2—4882no—
500C2C2×#1683967511yescharacteristic
501C2#2145506311yescharacteristic
502C2#2145506511yescharacteristic
503C2#2145564012yes—
504C2#2145621712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1736D61
hypocenterC61#23453
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
1. C22×#14519768C2semidirect
5. #21446969C2semidirect
6. #21446964C2semidirect
7. #21445540C2semidirect
8. #21445512C2semidirect
9. C61×#1736C2semidirect
253. #1736D61semidirect
289. C61#23453semidirect

Maximal subgroups

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

subgroupindex
C22×#145197682
C2×#168397642
C2×#168397612
#214483072
#214469692
#214469642
#214455402
#214455122
C61×#17362
#2345361