Finite Group Databasebeta

← #29710006

Subgroups — #29710006

Order 15616.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448335C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic, complemented
3#21447378C211yesmaximal, characteristic, complemented
4#21447347C211yesmaximal, characteristic, complemented
5#21447291C211yesmaximal, characteristic, complemented
6#21447246C212yesmaximal, complemented
7#21446999C212yesmaximal
8#21445711C212yesmaximal, complemented
9#102×Dic61C212yesmaximal, complemented
10#21445576C211yesmaximal, characteristic, complemented
11C61×#2071C211yesmaximal, characteristic, complemented
12C2×#14519748C2211yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14C2×#14519737—21no—
15C2×#14519735—21no—
16C2×C42×D61C2211yescharacteristic
17C2×#14519710C2212yes—
18C2×#14519710C2212yes—
19C2×#14519698C2212yes—
20C2×#14519689C2212yes—
21C2×C8×Dic61C2212yes—
22#16839660C2211yescharacteristic
23#16839656—21no—
24#16839638—21no—
25#16839632C2211yescharacteristic
26#16839627—21no—
27#16839619—21no—
28#16839619C2212yes—
29#16839619C2211yescharacteristic
30C4⋊C4×Dic61—21no—
31C4⋊C4×Dic61C2212yes—
32C4⋊C4×Dic61C2211yescharacteristic
33C22⋊C4×Dic61—21no—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38C2×#102×C61C2212yes—
39C61×#390C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C22×C22⋊D61—21no—
44C23×Dic61—21no—
45C22×C4×D61D411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2312yes—
49#14519748—22no—
50#14519748—22no—
51D4×Dic61—24no—
52#14519737—44no—
53#14519735—44no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758196—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—21no—
62C2×#11758198—22no—
63C2×#11758198—21no—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C42×D61—24no—
70#14519710—24no—
71#14519710—24no—
72#14519698—24no—
73#14519689—24no—
74C8×Dic61—24no—
75#14519684—21no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684—21no—
80C22×D4×C61C2311yescharacteristic
81C22×C8×C61C2312yes—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2312yes—
86C2×C22⋊C4×C61—21no—
87C2×C22⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89#102×C61—24no—
90C24×C61—21no—
91C23×D61C2×D411yescharacteristic
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61—21no—
95C2×D4×C61#4112yes—
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2411yescharacteristic
101C2×C22⋊D61—24no—
102C2×C22⋊D61—42no—
103C2×C22⋊D61—42no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—22no—
107C22×Dic61—42no—
108C22×Dic61—42no—
109C22×Dic61—42no—
110C22×Dic61C2×D411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112C22×Dic61C2×D411yescharacteristic
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—22no—
121C2×C4×D61—41no—
122C2×C4×D61D814yes—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—42no—
130C4⋊C4×C61—42no—
131C22⋊C4×C61—42no—
132C22⋊C4×C61—42no—
133C42×C61—22no—
134C22⋊Dic61—24no—
135C22⋊Dic61—42no—
136#11758198—44no—
137#11758198—42no—
138C4⋊Dic61—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—42no—
143C4⋊Dic61—42no—
144C4⋊Dic61#4112yes—
145#11758196—42no—
146C4×Dic61—22no—
147C4×Dic61—22no—
148C4×Dic61—44no—
149C4×Dic61—42no—
150C4×Dic61#4114yes—
151C2×#92418—22no—
152C2×#92418—22no—
153C2×#92418#4114yes—
154C23×C61—22no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61C22×D411yescharacteristic
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—21no—
168C22×D61C2×D812yes—
169D4×C61—24no—
170D4×C61—42no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61C2×D812yes—
182C2×C4×C61#13512yes—
183C2×C4×C61C2×#4112yes—
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61C2×#4112yes—
186C2×C4×C61C2×#4111yescharacteristic
187C22⋊D61—48no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—22no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—84no—
203C2×Dic61#12312yes—
204C2×Dic61#12312yes—
205C2×Dic61C2×D812yes—
206C2×Dic61#13712yes—
207C4×D61—28no—
208C4×D61—44no—
209C4×D61—42no—
210C4×D61—42no—
211C8×C61—42no—
212#92418—24no—
213#23754—611nomaximal, Sylow
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61C2×#12311yescharacteristic
222C22×C61#52511yescharacteristic
223C22×C61#53211yescharacteristic
224C22×C61C22×D811yescharacteristic
225C22×C61C2×#13511yescharacteristic
226C22×C61C2×#13711yescharacteristic
227C22×C61#57511yescharacteristic
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—41no—
232C2×D61—41no—
233C2×D61—41no—
234C2×D61—41no—
235C4×C61—22no—
236C4×C61—22no—
237C4×C61—82no—
238C4×C61—81no—
239C4×C61#52912yes—
240C4×C61#52912yes—
241Dic61—22no—
242Dic61—42no—
243Dic61—41no—
244Dic61—81no—
245Dic61—82no—
246Dic61#48014yes—
247C4×#121—611no—
248#2285—611no—
249#2099—611no—
250#2093—611no—
251#2083—611no—
252#2075—612no—
253#2071D6111yescharacteristic, complemented
254#2033—612no—
255C4×#107—612no—
256C4×#102—612no—
257C4×C22⋊C8—611no—
258C2×C61—82no—
259C2×C61#239511yescharacteristic
260C2×C61C2×#48011yescharacteristic
261C2×C61#309111yescharacteristic
262C2×C61#310011yescharacteristic
263C2×C61#310711yescharacteristic
264C2×C61#316911yescharacteristic
265C2×C61#317211yescharacteristic
266D61—42no—
267C2×#121—611no—
268C2×C4×D4—611no—
269C2×C4×D4—1221no—
270C2×#117—1221no—
271C22×C42—611no—
272C2×#107—612no—
273C2×#107—612no—
274C2×#102—612no—
275C2×#102C2×D6112yes—
276C2×C22⋊C8—612no—
277C2×C4×C8—612no—
278#390C2×D6111yescharacteristic
279#389—611no—
280#388—1221no—
281#386—1221no—
282#384—611no—
283#384C2×D6111yescharacteristic
284#382—1221no—
285#380—611no—
286#380—612no—
287#380—1221no—
288C4×C4⋊C4—612no—
289C4×C4⋊C4—611no—
290C4×C4⋊C4—1221no—
291C4×C22⋊C4—1221no—
292#340—612no—
293#340—612no—
294#340—612no—
295#340C2×D6112yes—
296#336—611no—
297#336C2×D6111yescharacteristic
298C61#2375411yescharacteristic, complemented, Sylow
299C22×D4—1221no—
300C22×D4C22×D6111yescharacteristic
301C23×C4—611no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—1221no—
305C22×C8—612no—
306C22×C8C22×D6112yes—
307#121—1222no—
308#121—1222no—
309C4×D4—1224no—
310C4×D4—2444no—
311#117—2444no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4C22×D6112yes—
321C2×C4⋊C4C22×D6111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1222no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1221no—
328C2×C42—611no—
329C2×C42—611no—
330C2×C42—611no—
331C2×C42—1224no—
332C2×C42—1222no—
333C2×C42—1221no—
334C2×C42C22×D6111yescharacteristic
335#107—1224no—
336#107—1224no—
337#102—24no—
338#102—1224no—
339C22⋊C8—1224no—
340C4×C8—1224no—
341#95—1221no—
342#95—1222no—
343#95—1221no—
344#95—1222no—
345#95—1221no—
346C24—21no—
347C24—611no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—1224no—
352C2×D4—2442no—
353C2×D4—2442no—
354C2×D4#1175822412yes—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—614no—
360C22×C4—611no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1221no—
365C22×C4—1222no—
366C22×C4—1222no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—1222no—
371C22×C4—1221no—
372C22×C4—2441no—
373C22×C4—2442no—
374C22×C4—2442no—
375C22×C4—2442no—
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4C23×D6111yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—614no—
383C2×C8—1222no—
384C2×C8—1222no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—612no—
391C4⋊C4—1221no—
392C4⋊C4—1222no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C4⋊C4—2442no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—1224no—
401C22⋊C4—2444no—
402C22⋊C4—2442no—
403C22⋊C4—2442no—
404C42—22no—
405C42—614no—
406C42—1222no—
407C42—1222no—
408C42—2444no—
409C42—2442no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—612no—
418C23—1221no—
419C23—1221no—
420C23—1221no—
421C23—1221no—
422C23—1221no—
423C23—1221no—
424C23C2×D4×D6111yescharacteristic
425D4—24no—
426D4—42no—
427D4—2448no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—612no—
442C2×C4—1228no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1221no—
447C2×C4—1222no—
448C2×C4—1221no—
449C2×C4—4884no—
450C2×C4—2444no—
451C2×C4—2441no—
452C2×C4—2442no—
453C2×C4—2441no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4—2442no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4D8×D6112yes—
462C2×C4#1451979812yes—
463C2×C4#41×D6112yes—
464C2×C4#41×D6112yes—
465C2×C4#41×D6111yescharacteristic
466C2×C4C2×#1175822411yescharacteristic
467C8—42no—
468C8—1224no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—1222no—
477C22—1222no—
478C22—1222no—
479C22—2441no—
480C22—2441no—
481C22—2441no—
482C22—2441no—
483C22#123×D6111yescharacteristic
484C22#1684027211yescharacteristic
485C22#1684031111yescharacteristic
486C22C2×D8×D6111yescharacteristic
487C22C2×#1451979811yescharacteristic
488C22#137×D6111yescharacteristic
489C22#1684055811yescharacteristic
490C4—22no—
491C4—22no—
492C4—81no—
493C4—82no—
494C4—614no—
495C4—1222no—
496C4—4882no—
497C4—4881no—
498C4—2441no—
499C4—2442no—
500C4#1684028912yes—
501C4#1684029512yes—
502C2—82no—
503C2—2442no—
504C2#2144908111yescharacteristic
505C2#480×D6111yescharacteristic
506C2#2145572911yescharacteristic
507C2#2145581811yescharacteristic
508C2#2145588711yescharacteristic
509C2#2145639711yescharacteristic
510C2#2145642811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenter#2071D61
hypocenterC61#23754
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #21448224C2semidirect
3. #21447378C2semidirect
4. #21447347C2semidirect
5. #21447291C2semidirect
6. #21447246C2semidirect
8. #21445711C2semidirect
9. #102×Dic61C2semidirect
10. #21445576C2semidirect
11. C61×#2071C2semidirect
253. #2071D61semidirect
298. C61#23754semidirect

Maximal subgroups

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

subgroupindex
#214483352
#214482242
#214473782
#214473472
#214472912
#214472462
#214469992
#214457112
#102×Dic612
#214455762
C61×#20712
#2375461