Finite Group Databasebeta

← #29779759

Subgroups — #29779759

Order 15616.

742 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
1#21454180C211yesmaximal, characteristic, complemented
2C4×#14519766C211yesmaximal, characteristic
3#21447389C212yesmaximal, complemented
4#21447056C212yesmaximal
5#21446517C212yesmaximal
6#21446511C212yesmaximal, complemented
7#21445824C211yesmaximal, characteristic, complemented
8#21445820C211yesmaximal, characteristic
9#21445816C211yesmaximal, characteristic
10#21445674C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C2×#14519768—21no—
13C2×#14519766C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15C2×C4⋊#92418C2211yescharacteristic
16C2×#14519701C2212yes—
17C2×#14519699C2212yes—
18C2×#14519697C2212yes—
19C2×#14519687C2212yes—
20#16839669C2211yescharacteristic
21#16839663C2211yescharacteristic
22#16839661—21no—
23#16839661C2212yes—
24#16839658—21no—
25#16839658C2212yes—
26#16839652—21no—
27#16839652—21no—
28#16839632C2211yescharacteristic
29#16839627—21no—
30#16839627C2211yescharacteristic
31C4×#11758198—21no—
32C4×#11758198—21no—
33C42×Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C22×#92418C2312yes—
46#14519768—44no—
47#14519766—22no—
48#14519766—22no—
49C4×D244—22no—
50C4×D244—22no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×#11758196—22no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68C4⋊#92418—22no—
69#14519701—42no—
70#14519699—42no—
71#14519697—24no—
72#14519687—42no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79C22×Q8×C61C2311yescharacteristic
80C22×C8×C61C2312yes—
81C4×Q8×C61—22no—
82C4×Q8×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—22no—
91#103×C61—24no—
92C23×D61—21no—
93C2×Q8×C61—21no—
94C2×Q8×C61—21no—
95C2×Q8×C61—21no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61#4112yes—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2411yescharacteristic
105C22×Dic61—21no—
106C22×Dic61—22no—
107C22×Dic61C2×D411yescharacteristic
108C22×Dic61C2×D411yescharacteristic
109C22×Dic61C2×D411yescharacteristic
110C22×Dic61C2×D411yescharacteristic
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244#4112yes—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C8×C61—22no—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—42no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C42×C61—42no—
131C42×C61#4112yes—
132C42×C61#4112yes—
133#11758198—24no—
134#11758198—24no—
135#11758198—42no—
136#11758198—42no—
137#11758198—42no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—44no—
143C4⋊Dic61—42no—
144C4⋊Dic61—42no—
145#11758196—44no—
146C4×Dic61—22no—
147C4×Dic61—22no—
148C4×Dic61—21no—
149C4×Dic61—21no—
150C4×Dic61—21no—
151C4×Dic61—21no—
152C4×Dic61—41no—
153C4×Dic61—41no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C23×C61C22×D411yescharacteristic
158C22×D61—22no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165Q8×C61—22no—
166Q8×C61—22no—
167Q8×C61—41no—
168Q8×C61—41no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—22no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61#12412yes—
181C2×C4×C61#12412yes—
182C2×C4×C61AGL(1,ℤ/8)12yes—
183C2×C4×C61#13712yes—
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61H(5,𝔽2)11yescharacteristic
189C2×C4×C612−1+411yescharacteristic
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208D244—22no—
209D244—22no—
210D244—41no—
211D244—41no—
212C4×D61—84no—
213C8×C61—42no—
214#92418—42no—
215#29131—611nomaximal, Sylow
216C22×C61#52511yescharacteristic
217C22×C61C2×#12411yescharacteristic
218C22×C61#53711yescharacteristic
219C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
220C22×C61C2×#13711yescharacteristic
221C22×C61#57511yescharacteristic
222C22×C61#57511yescharacteristic
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—42no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—22no—
231C4×C61—42no—
232C4×C61—82no—
233C4×C61—81no—
234C4×C61#48914yes—
235C4×C61#55211yescharacteristic
236C4×C61#55211yescharacteristic
237C4×C61#55311yescharacteristic
238C4×C61#55311yescharacteristic
239Dic61—41no—
240Dic61—41no—
241Dic61—41no—
242Dic61—41no—
243Dic61—81no—
244Dic61—82no—
245#2859—611no—
246C4×#124—611no—
247#2101—612no—
248#2042—612no—
249#1936—612no—
250#1935—612no—
251#1803—611no—
252#1803D6111yescharacteristic, complemented
253#1802—611no—
254#1802—611no—
255#1767—611no—
256C2×C61#256211yescharacteristic
257C2×C61#311511yescharacteristic
258C2×C61#311611yescharacteristic
259C2×C61C2×#48911yescharacteristic
260C2×C61#314211yescharacteristic
261C2×C61#325111yescharacteristic
262C2×C61#325211yescharacteristic
263D61—82no—
264C2×#125—1221no—
265C2×#124—611no—
266C2×C4×Q8C2×D6111yescharacteristic
267C2×C4×D4—611no—
268C2×C4⋊C8—611no—
269C2×C4⋊C8C2×D6111yescharacteristic
270C2×#103—612no—
271C2×#103C2×D6112yes—
272C2×#102—612no—
273C2×#102—612no—
274C2×#97—612no—
275#391—611no—
276#390—611no—
277#389—612no—
278#389—612no—
279#389—1221no—
280#389—1221no—
281#388—1221no—
282#388—1221no—
283#384—611no—
284#384C2×D6111yescharacteristic
285#382—611no—
286#382—1221no—
287C4×C22⋊C4—1221no—
288C4×C22⋊C4—1221no—
289C43—611no—
290#340—612no—
291#340—612no—
292#340—612no—
293#340C2×D6112yes—
294#336—612no—
295C61#2913111yescharacteristic, complemented, Sylow
296C22×Q8C22×D6111yescharacteristic
297C22×D4—611no—
298C23×C4—1221no—
299C22×C8—612no—
300C22×C8C22×D6112yes—
301#125—2444no—
302#124—1222no—
303#124—1222no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×D4—1222no—
307C4×D4—1222no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—612no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C4⋊C4C22×D6112yes—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C42—21no—
325C2×C42—611no—
326C2×C42—611no—
327C2×C42—611no—
328C2×C42—611no—
329C2×C42—611no—
330C2×C42—611no—
331C2×C42C22×D6111yescharacteristic
332C4⋊C8—22no—
333C4⋊C8—1222no—
334#103—24no—
335#103—2442no—
336#102—1224no—
337#102—2442no—
338#97—2442no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342#95—1221no—
343#95—1222no—
344#95—1222no—
345C24—1221no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8#1175822412yes—
352C2×D4—612no—
353C2×D4—1221no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—1221no—
366C22×C4—1222no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4—2442no—
370C22×C4D4×D6111yescharacteristic
371C22×C4D4×D6111yescharacteristic
372C22×C4C23×D6111yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—1222no—
377C2×C8—1222no—
378C2×C8—1222no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C4⋊C4—1222no—
390C4⋊C4—2442no—
391C4⋊C4—2444no—
392C4⋊C4—2444no—
393C4⋊C4—2442no—
394C22⋊C4—1224no—
395C22⋊C4—1224no—
396C22⋊C4—2442no—
397C22⋊C4—2442no—
398C22⋊C4—2442no—
399C42—42no—
400C42—1222no—
401C42—1221no—
402C42—1221no—
403C42—1221no—
404C42—1221no—
405C42—1222no—
406C42—2441no—
407C42—2441no—
408C42#1175822212yes—
409C42#1175822412yes—
410C23—1222no—
411C23—2441no—
412C23—2441no—
413C23—2441no—
414C23—2441no—
415C23—2441no—
416C23—2441no—
417C23C2×D4×D6111yescharacteristic
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422D4—1222no—
423D4—1222no—
424D4—2441no—
425D4—2441no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—1221no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—4884no—
450C2×C4—2441no—
451C2×C4—2442no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4#1451976312yes—
457C2×C4#1451976312yes—
458C2×C4#1451980412yes—
459C2×C4#1451980912yes—
460C2×C4#41×D6111yescharacteristic
461C2×C4C2×#1175822411yescharacteristic
462C2×C4C2×#1175822411yescharacteristic
463C2×C4C2×#1175822211yescharacteristic
464C2×C4#1451982311yescharacteristic
465C2×C4#1451982511yescharacteristic
466C8—42no—
467C8—2442no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22—2442no—
473C22—2442no—
474C22—2442no—
475C22#1684027211yescharacteristic
476C22C2×#1451976311yescharacteristic
477C22#1684034211yescharacteristic
478C22C2×#1451980411yescharacteristic
479C22C2×#1451980911yescharacteristic
480C22#1684055811yescharacteristic
481C22#1684056111yescharacteristic
482C4—22no—
483C4—42no—
484C4—81no—
485C4—82no—
486C4—4881no—
487C4—4882no—
488C4—2441no—
489C4—2441no—
490C4—2441no—
491C4—2441no—
492C4#1684012314yes—
493C4#1684044611yescharacteristic
494C4#1684044711yescharacteristic
495C4#1684045611yescharacteristic
496C4#1684046411yescharacteristic
497C2—4882no—
498C2#2145098211yescharacteristic
499C2#2145596911yescharacteristic
500C2#2145598011yescharacteristic
501C2C2×#1684012311yescharacteristic
502C2#2145621711yescharacteristic
503C2#2145718411yescharacteristic
504C2#2145719511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1803D61
hypocenterC61#29131
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. #21454180C2semidirect
3. #21447389C2semidirect
6. #21446511C2semidirect
7. #21445824C2semidirect
10. #21445674C2semidirect
11. C61×#1803C2semidirect
252. #1803D61semidirect
295. C61#29131semidirect

Maximal subgroups

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

subgroupindex
#214541802
C4×#145197662
#214473892
#214470562
#214465172
#214465112
#214458242
#214458202
#214458162
#214456742
C61×#18032
#2913161