Finite Group Databasebeta

← #29710008

Subgroups — #29710008

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#21448327C211yesmaximal, characteristic
2#21448224C211yesmaximal, characteristic, complemented
3#21447377C211yesmaximal, characteristic, complemented
4#21447346C211yesmaximal, characteristic, complemented
5#21447244C212yesmaximal, complemented
6#21447223C211yesmaximal, characteristic, complemented
7#21447003C212yesmaximal
8#107×Dic61C212yesmaximal, complemented
9#21445633C212yesmaximal, complemented
10#21445576C211yesmaximal, characteristic, complemented
11C61×#2083C211yesmaximal, characteristic, complemented
12C2×#14519752C2211yescharacteristic
13C2×#14519739—21no—
14C2×#14519739C2211yescharacteristic
15C2×#14519735—21no—
16C2×C42×D61C2211yescharacteristic
17C2×#14519711C2212yes—
18C2×#14519699C2212yes—
19C2×#14519697C2212yes—
20C2×#14519689C2212yes—
21C2×C8×Dic61C2212yes—
22#16839664C2211yescharacteristic
23#16839652—21no—
24#16839640—21no—
25#16839632C2211yescharacteristic
26#16839627—21no—
27#16839619—21no—
28#16839619C2212yes—
29#16839619C2211yescharacteristic
30C4⋊C4×Dic61—21no—
31C4⋊C4×Dic61C2212yes—
32C4⋊C4×Dic61C2211yescharacteristic
33#16839609—21no—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38C2×#107×C61C2212yes—
39C61×#389C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C22×D244—21no—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50#14519752—22no—
51#14519752—22no—
52#14519739—24no—
53#14519739—44no—
54#14519735—44no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758198—21no—
60C2×#11758198—22no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C42×D61—24no—
71#14519711—24no—
72#14519699—24no—
73#14519697—24no—
74#14519689—24no—
75C8×Dic61—24no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684—21no—
80C22×C8×C61C2312yes—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2312yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C42×C61C2311yescharacteristic
88#107×C61—24no—
89#95×C61—21no—
90C23×D61—21no—
91C23×D61C2×D411yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—22no—
100C22×Dic61C2×D411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—24no—
106C2×D244—21no—
107C2×D244—42no—
108C2×D244—42no—
109C2×D244#4112yes—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—22no—
118C2×C4×D61—41no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C2×C4×D61D814yes—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61#4114yes—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—42no—
131C4⋊C4×C61—42no—
132C4⋊C4×C61—42no—
133C4⋊C4×C61#4112yes—
134C42×C61—22no—
135#11758198—24no—
136#11758198—44no—
137#11758198—42no—
138#11758198—42no—
139#11758198—42no—
140#11758198—42no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—42no—
145C4⋊Dic61—42no—
146C4×Dic61—22no—
147C4×Dic61—22no—
148C4×Dic61—44no—
149C4×Dic61—42no—
150C4×Dic61#4114yes—
151C2×#92418—22no—
152C2×#92418—22no—
153C2×#92418—22no—
154C23×C61C22×D411yescharacteristic
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—21no—
161C22×D61—22no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61C2×D812yes—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—22no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61C2×D812yes—
183C2×C4×C61#13512yes—
184C2×C4×C61C2×#4112yes—
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61C2×#4112yes—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61#12312yes—
200C2×Dic61#12312yes—
201C2×Dic61C2×D812yes—
202C2×Dic61#13712yes—
203D244—24no—
204D244—48no—
205D244—42no—
206C4×D61—28no—
207C4×D61—44no—
208C4×D61—42no—
209C4×D61—42no—
210C4×D61—84no—
211C8×C61—24no—
212#92418—42no—
213#23754—611nomaximal, Sylow
214C22×C61C2×#12311yescharacteristic
215C22×C61#52511yescharacteristic
216C22×C61#53211yescharacteristic
217C22×C61C22×D811yescharacteristic
218C22×C61C2×#13511yescharacteristic
219C22×C61C2×#13711yescharacteristic
220C22×C61#57511yescharacteristic
221C2×D61—22no—
222C2×D61—22no—
223C2×D61—22no—
224C2×D61—41no—
225C2×D61—41no—
226C2×D61—41no—
227C2×D61—41no—
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C4×C61—22no—
236C4×C61—22no—
237C4×C61—42no—
238C4×C61—81no—
239C4×C61—82no—
240C4×C61#52912yes—
241C4×C61#52912yes—
242Dic61—22no—
243Dic61—41no—
244Dic61—82no—
245Dic61—81no—
246Dic61#48014yes—
247C4×#121—611no—
248#2285—611no—
249#2099—611no—
250#2093—611no—
251#2083D6111yescharacteristic, complemented
252#2075—612no—
253#2071—611no—
254#2033—612no—
255C4×#107—612no—
256C4×#102—612no—
257C4×C22⋊C8—611no—
258C2×C61#239511yescharacteristic
259C2×C61C2×#48011yescharacteristic
260C2×C61#309111yescharacteristic
261C2×C61#310011yescharacteristic
262C2×C61#310711yescharacteristic
263C2×C61#316911yescharacteristic
264C2×C61#317211yescharacteristic
265D61—42no—
266D61—82no—
267C2×#121—611no—
268C2×C4×D4—611no—
269C2×C4×D4—1221no—
270C2×#117—1221no—
271C22×C42—611no—
272C2×#107—612no—
273C2×#107C2×D6112yes—
274C2×#102—612no—
275C2×#102—612no—
276C2×C22⋊C8—612no—
277C2×C4×C8—612no—
278#390—611no—
279#389C2×D6111yescharacteristic
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—611no—
300C22×D4—1221no—
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—21no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4C22×D6111yescharacteristic
320C2×C4⋊C4C22×D6112yes—
321C2×C4⋊C4C22×D6111yescharacteristic
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1221no—
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—24no—
336#107—1224no—
337#102—1224no—
338#102—1224no—
339C22⋊C8—1224no—
340C4×C8—1224no—
341#95—21no—
342#95—1221no—
343#95—1222no—
344#95—1221no—
345#95—1222no—
346C24—611no—
347C24—1221no—
348C2×D4—612no—
349C2×D4—1224no—
350C2×D4—1222no—
351C2×D4—1221no—
352C2×D4—1222no—
353C2×D4—2442no—
354C2×D4—2442no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—611no—
359C22×C4—611no—
360C22×C4—614no—
361C22×C4—611no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1222no—
365C22×C4—1222no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4—1222no—
370C22×C4—1221no—
371C22×C4—1222no—
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—1222no—
382C2×C8—1222no—
383C2×C8—1222no—
384C2×C8#1175822614yes—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—1222no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C4⋊C4#1175822412yes—
398C22⋊C4—1224no—
399C22⋊C4—2442no—
400C22⋊C4—2444no—
401C22⋊C4—2442no—
402C22⋊C4—2442no—
403C22⋊C4—2442no—
404C42—22no—
405C42—614no—
406C42—1222no—
407C42—1222no—
408C42—2444no—
409C42—2442no—
410C23—612no—
411C23—1221no—
412C23—1221no—
413C23—1221no—
414C23—1221no—
415C23—1221no—
416C23—1221no—
417C23—1222no—
418C23—2441no—
419C23—2441no—
420C23—2441no—
421C23—2441no—
422C23—2441no—
423C23—2441no—
424C23C2×D4×D6111yescharacteristic
425D4—1224no—
426D4—2448no—
427D4—2442no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—612no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—612no—
445C2×C4—1228no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—4884no—
452C2×C4—2444no—
453C2×C4—2441no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4D8×D6112yes—
462C2×C4#1451980012yes—
463C2×C4#41×D6111yescharacteristic
464C2×C4#41×D6112yes—
465C2×C4C2×#1175822411yescharacteristic
466C2×C4C2×#1175822612yes—
467C8—24no—
468C8—2442no—
469C22—1222no—
470C22—1222no—
471C22—1222no—
472C22—4881no—
473C22—4881no—
474C22—4881no—
475C22—4881no—
476C22—2442no—
477C22—2442no—
478C22—2442no—
479C22—2441no—
480C22—2441no—
481C22—2441no—
482C22—2441no—
483C22#123×D6111yescharacteristic
484C22#1684027211yescharacteristic
485C22#1684031311yescharacteristic
486C22C2×D8×D6111yescharacteristic
487C22C2×#1451980011yescharacteristic
488C22#137×D6111yescharacteristic
489C22#1684055811yescharacteristic
490C4—22no—
491C4—22no—
492C4—42no—
493C4—81no—
494C4—82no—
495C4—614no—
496C4—1222no—
497C4—4881no—
498C4—4882no—
499C4—2441no—
500C4#1684029712yes—
501C4#1684029812yes—
502C2—4882no—
503C2—2442no—
504C2#2144908311yescharacteristic
505C2#480×D6111yescharacteristic
506C2#2145573511yescharacteristic
507C2#2145581811yescharacteristic
508C2#2145588911yescharacteristic
509C2#2145639911yescharacteristic
510C2#2145642711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2083C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#2083D61
hypocenterC61#23754
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2083

Complements

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

NH ≅ G/Ntype
2. #21448224C2semidirect
3. #21447377C2semidirect
4. #21447346C2semidirect
5. #21447244C2semidirect
6. #21447223C2semidirect
8. #107×Dic61C2semidirect
9. #21445633C2semidirect
10. #21445576C2semidirect
11. C61×#2083C2semidirect
251. #2083D61semidirect
298. C61#23754semidirect

Maximal subgroups

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

subgroupindex
#214483272
#214482242
#214473772
#214473462
#214472442
#214472232
#214470032
#107×Dic612
#214456332
#214455762
C61×#20832
#2375461