Finite Group Databasebeta

← #29692140

Subgroups — #29692140

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic
2C4⋊Q8×Dic61C211yesmaximal, characteristic, complemented
3#21448280C211yesmaximal, characteristic
4#21446577C211yesmaximal, characteristic, complemented
5#21446561C212yesmaximal, complemented
6#21446517C212yesmaximal
7#21446437C212yesmaximal
8#21446402C212yesmaximal, complemented
9#21445820C211yesmaximal, characteristic
10#21445674C211yesmaximal, characteristic, complemented
11C61×#1987C211yesmaximal, characteristic, complemented
12C2×Q8×Dic61—21no—
13C2×C4×D244C2211yescharacteristic
14C2×#14519734—21no—
15#16840049C418yescomplemented
16C2×#14519710C2212yes—
17C2×#14519708C2212yes—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519699C2212yes—
20C2×#14519697C2212yes—
21C2×#14519687C2212yes—
22#16839669—21no—
23#16839663C2211yescharacteristic
24#16839663C2211yescharacteristic
25#16839661—21no—
26#16839661C2212yes—
27#16839658C2211yescharacteristic
28#16839658C2211yescharacteristic
29#16839651—21no—
30#16839651—21no—
31#16839632C2211yescharacteristic
32C4⋊C4×Dic61—21no—
33C4⋊C4×Dic61C2212yes—
34C4×#11758198—21no—
35C42×Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×#92418C2312yes—
47Q8×Dic61—44no—
48C4×D244—22no—
49C4×D244C2×C414yes—
50#14519734—44no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61C2312yes—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70#14519710—42no—
71#14519708C2×C418yes—
72C4⋊#92418C2×C414yes—
73#14519699—42no—
74#14519697C2×C418yes—
75#14519687—42no—
76#14519684—21no—
77#14519684—21no—
78#14519684—22no—
79#14519684—22no—
80C22×Q8×C61—21no—
81C22×C8×C61C2312yes—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61C2×C414yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—22no—
91#95×C61—21no—
92C23×D61—21no—
93C2×Q8×C61—24no—
94C2×Q8×C61—42no—
95C2×Q8×C61—42no—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—22no—
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244C22×C412yes—
114C2×C4×D61—24no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C2×C8×C61—22no—
120C4⋊C4×C61—24no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61C22×C414yes—
127C42×C61—42no—
128C42×C61C22×C412yes—
129C42×C61#4112yes—
130#11758198—24no—
131#11758198—42no—
132#11758198—42no—
133#11758198—42no—
134#11758198—42no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—22no—
138C4⋊Dic61—44no—
139C4⋊Dic61—42no—
140C4⋊Dic61—42no—
141C4⋊Dic61C22×C412yes—
142#11758196—42no—
143#11758196—42no—
144C4×Dic61—22no—
145C4×Dic61—22no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C4×Dic61—21no—
149C4×Dic61—21no—
150C4×Dic61—22no—
151C4×Dic61—41no—
152C4×Dic61—44no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418C22×C414yes—
156C23×C61C22×D411yescharacteristic
157C22×D61—22no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164Q8×C61—48no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—24no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61C4×D412yes—
177C2×C4×C61C4×D412yes—
178C2×C4×C61AGL(1,ℤ/8)12yes—
179C2×C4×C61#13712yes—
180C2×C4×C61C23×C411yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61H(5,𝔽2)11yescharacteristic
183C2×C4×C612−1+411yescharacteristic
184C2×C4×C612−1+411yescharacteristic
185C2×C4×C612−1+411yescharacteristic
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—22no—
189C2×Dic61—21no—
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—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61AGL(1,ℤ/8)12yes—
204D244—22no—
205D244—22no—
206D244—41no—
207D244—41no—
208C4×D61—48no—
209C8×C61—42no—
210#92418—24no—
211#22117—611nomaximal, Sylow
212C22×C61C2×C4×D411yescharacteristic
213C22×C61#53611yescharacteristic
214C22×C61#53711yescharacteristic
215C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
216C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
217C22×C61C2×#13711yescharacteristic
218C22×C61#57511yescharacteristic
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C4×C61—22no—
227C4×C61—44no—
228C4×C61—81no—
229C4×C61—81no—
230C4×C61#44014yes—
231C4×C61#51911yescharacteristic
232C4×C61#52011yescharacteristic
233C4×C61#54111yescharacteristic
234C4×C61#54211yescharacteristic
235Dic61—22no—
236Dic61—42no—
237Dic61—41no—
238Dic61—41no—
239Dic61—41no—
240Dic61—82no—
241C2×#476—611no—
242C4×C4⋊Q8—611no—
243#2293—611no—
244#1987D6111yescharacteristic, complemented
245#1947—611no—
246#1944—612no—
247#1936—612no—
248#1925—612no—
249#1920—612no—
250#1802—611no—
251#1767—611no—
252C2×C61#232311yescharacteristic
253C2×C61C2×#44011yescharacteristic
254C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
255C2×C61#323511yescharacteristic
256C2×C61#323811yescharacteristic
257C2×C61#325211yescharacteristic
258C2×C61#325311yescharacteristic
259D61—82no—
260C2×C4⋊Q8C2×D6111yescharacteristic
261C2×C4×Q8—1221no—
262C2×C4×D4—611no—
263C2×#117—1221no—
264#476—618no—
265C2×#107—612no—
266C2×#106—612no—
267C2×C4⋊C8—611no—
268C2×C4⋊C8C2×D6111yescharacteristic
269C2×#102—612no—
270C2×#102—612no—
271C2×#97—612no—
272#391—1221no—
273#390—611no—
274#390—611no—
275#389—612no—
276#389—611no—
277#389—611no—
278#389—1221no—
279#388—1221no—
280#388—1221no—
281#384—611no—
282C4×C4⋊C4—612no—
283C4×C4⋊C4—1221no—
284C4×C4⋊C4C2×D6111yescharacteristic
285C4×C22⋊C4—1221no—
286C43—611no—
287#340—612no—
288#340—612no—
289#340C2×D6112yes—
290#340C2×D6112yes—
291#336—612no—
292C61#2211711yescharacteristic, complemented, Sylow
293C22×Q8—21no—
294C22×D4—611no—
295C23×C4—1221no—
296C22×C8—612no—
297C22×C8C22×D6112yes—
298C4⋊Q8—22no—
299C4⋊Q8C4×D6114yes—
300C4×Q8—2444no—
301C4×D4—614no—
302C4×D4—1222no—
303#117—2444no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—1221no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4C22×D6111yescharacteristic
313C2×C4⋊C4C22×D6111yescharacteristic
314C2×C4⋊C4C22×D6112yes—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C42—21no—
321C2×C42—611no—
322C2×C42—611no—
323C2×C42—611no—
324C2×C42—611no—
325C2×C42—611no—
326C2×C42—611no—
327C2×C42—1222no—
328C2×C42C22×D6111yescharacteristic
329#107—2442no—
330#106—618no—
331C4⋊C8—22no—
332C4⋊C8—614no—
333#102—618no—
334#102—2442no—
335#97—2442no—
336#95—21no—
337#95—1221no—
338#95—1221no—
339#95—1222no—
340#95—1222no—
341C24—1221no—
342C2×Q8—24no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×D4—612no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—1221no—
359C22×C4—1224no—
360C22×C4—1222no—
361C22×C4—2442no—
362C22×C4—2442no—
363C22×C4D4×D6111yescharacteristic
364C22×C4D4×D6111yescharacteristic
365C22×C4C23×D6111yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—614no—
370C2×C8—1222no—
371C2×C8—1222no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—612no—
379C4⋊C4—1221no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—2442no—
383C4⋊C4—2442no—
384C4⋊C4—2442no—
385C4⋊C4—2444no—
386C4⋊C4—2442no—
387C4⋊C4C2×C4×D6114yes—
388C22⋊C4—1224no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4—2442no—
392C22⋊C4—2442no—
393C42—42no—
394C42—1222no—
395C42—1222no—
396C42—1221no—
397C42—1221no—
398C42—1221no—
399C42—1221no—
400C42—1222no—
401C42—2441no—
402C42—2444no—
403C42C2×C4×D6112yes—
404C42#1175822412yes—
405C23—1222no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23C2×D4×D6111yescharacteristic
413Q8—48no—
414D4—1222no—
415D4—1222no—
416D4—2441no—
417D4—2441no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—612no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—1222no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—1221no—
437C2×C4—1221no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2448no—
448C2×C4#1451973712yes—
449C2×C4#1451973712yes—
450C2×C4#1451980412yes—
451C2×C4#1451980912yes—
452C2×C4C22×C4×D6111yescharacteristic
453C2×C4C2×#1175822411yescharacteristic
454C2×C4#1451982311yescharacteristic
455C2×C4#1451982411yescharacteristic
456C2×C4#1451982511yescharacteristic
457C2×C4#1451982511yescharacteristic
458C8—42no—
459C8—1224no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22—4881no—
464C22—2442no—
465C22—2442no—
466C22—2442no—
467C22C2×#1451973711yescharacteristic
468C22#1684033511yescharacteristic
469C22#1684034211yescharacteristic
470C22AGL(1,ℤ/8)×D6111yescharacteristic
471C22C2×#1451980411yescharacteristic
472C22C2×#1451980911yescharacteristic
473C22#1684055811yescharacteristic
474C4—22no—
475C4—44no—
476C4—81no—
477C4—81no—
478C4—1222no—
479C4—4882no—
480C4—2442no—
481C4—2441no—
482C4—2441no—
483C4—2441no—
484C4#1683986514yes—
485C4#1684023811yescharacteristic
486C4#1684024411yescharacteristic
487C4#1684037311yescharacteristic
488C4#1684038111yescharacteristic
489C2—4882no—
490C2#2144850411yescharacteristic
491C2C2×#1683986511yescharacteristic
492C2#2145462311yescharacteristic
493C2#2145701511yescharacteristic
494C2#2145704411yescharacteristic
495C2#2145719511yescharacteristic
496C2#2145719911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1987C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1987D61
hypocenterC61#22117
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×Dic61C2semidirect
4. #21446577C2semidirect
5. #21446561C2semidirect
8. #21446402C2semidirect
10. #21445674C2semidirect
11. C61×#1987C2semidirect
15. #16840049C4semidirect
244. #1987D61semidirect
292. C61#22117semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C4⋊Q8×Dic612
#214482802
#214465772
#214465612
#214465172
#214464372
#214464022
#214458202
#214456742
C61×#19872
#2211761