Finite Group Databasebeta

← #29692138

Subgroups — #29692138

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×#16840053C211yesmaximal, characteristic
2C4×#14519777C211yesmaximal, characteristic, complemented
3#21448283C211yesmaximal, characteristic
4#21446766C211yesmaximal, characteristic, complemented
5#21446578C211yesmaximal, characteristic, complemented
6#21446562C212yesmaximal, complemented
7#21446516C212yesmaximal
8#21446440C212yesmaximal
9#21446405C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic
11C61×#1767C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×C4×Dic122—21no—
14C2×C4×D244C2211yescharacteristic
15C2×#14519734—21no—
16#16840053C418yescomplemented
17C2×#14519711C2212yes—
18C2×#14519709C2212yes—
19C2×#14519699C2212yes—
20C2×#14519699C2212yes—
21#16839668—21no—
22#16839664C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839660—21no—
25#16839660C2211yescharacteristic
26#16839660C2211yescharacteristic
27#16839660C2212yes—
28#16839653—21no—
29#16839653—21no—
30#16839632C2211yescharacteristic
31C4×C4⋊Dic61—21no—
32C4×C4⋊Dic61C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×#11758198—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#97×C61C2212yes—
42C43×C61C2211yescharacteristic
43C61×#336C2212yes—
44C22×Dic122—21no—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47#14519777—22no—
48#14519777C2×C414yes—
49C4×Dic122—44no—
50C4×D244—22no—
51C4×D244C2×C414yes—
52#14519734—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—21no—
62C2×#11758196—21no—
63C2×#11758196—22no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61—22no—
71#14519711—42no—
72#14519709C2×C418yes—
73#14519699—42no—
74#14519699C2×C418yes—
75#14519684—21no—
76#14519684—21no—
77#14519684—22no—
78#14519684—21no—
79#14519684—22no—
80C22×C8×C61C2312yes—
81C22×C8×C61C2312yes—
82C2×C42×C61D411yescharacteristic
83C2×C42×C61D411yescharacteristic
84C2×C42×C61D411yescharacteristic
85C2×C42×C61D411yescharacteristic
86C2×C42×C61C2311yescharacteristic
87C2×C42×C61C2311yescharacteristic
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—22no—
90C4⋊C8×C61C2×C414yes—
91#97×C61—42no—
92C23×D61—21no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—22no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244C22×C412yes—
111C2×C4×D61—24no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×Dic122—24no—
115C2×Dic122—42no—
116C2×Dic122—42no—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C2×C8×C61—22no—
120C2×C8×C61—22no—
121C2×C8×C61—22no—
122C2×C8×C61C22×C414yes—
123C42×C61—22no—
124C42×C61—22no—
125C42×C61—22no—
126C42×C61—21no—
127C42×C61—21no—
128C42×C61—21no—
129C42×C61—21no—
130C42×C61—41no—
131C42×C61C22×C412yes—
132C42×C61#4112yes—
133#11758198—24no—
134#11758198—42no—
135#11758198—42no—
136#11758198—42no—
137#11758198—42no—
138C4⋊Dic61—24no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—42no—
148C4⋊Dic61—42no—
149C4⋊Dic61C22×C414yes—
150C4⋊Dic61C22×C412yes—
151#11758196—42no—
152#11758196—42no—
153#11758196—44no—
154C4×Dic61—42no—
155C4×Dic61—44no—
156C23×C61C22×D411yescharacteristic
157C22×D61—22no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61C4×D412yes—
178C2×C4×C61C4×D412yes—
179C2×C4×C61AGL(1,ℤ/8)12yes—
180C2×C4×C61AGL(1,ℤ/8)12yes—
181C2×C4×C61#13712yes—
182C2×C4×C61C23×C411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61H(5,𝔽2)11yescharacteristic
185C2×C4×C612−1+411yescharacteristic
186C2×C4×C612−1+411yescharacteristic
187C2×C4×C612−1+411yescharacteristic
188C2×Dic61—24no—
189C2×Dic61—22no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203D244—22no—
204D244—22no—
205D244—41no—
206D244—41no—
207C4×D61—48no—
208Dic122—48no—
209C8×C61—24no—
210C8×C61—42no—
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—22no—
228C4×C61—41no—
229C4×C61—41no—
230C4×C61—41no—
231C4×C61#44014yes—
232C4×C61#51911yescharacteristic
233C4×C61#52011yescharacteristic
234C4×C61#54111yescharacteristic
235C4×C61#54211yescharacteristic
236Dic61—42no—
237Dic61—44no—
238Dic61—81no—
239Dic61—82no—
240Dic61—81no—
241C2×#476—611no—
242C4×C4⋊Q8—611no—
243#2293—611no—
244#1987—611no—
245#1947—611no—
246#1944—612no—
247#1936—612no—
248#1925—612no—
249#1920—612no—
250#1802—611no—
251#1767D6111yescharacteristic, complemented
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⋊Q8—611no—
261C2×C4×Q8—1221no—
262C2×C4×D4—611no—
263C2×#117—1221no—
264#476—618no—
265C2×#107—612no—
266C2×#106—612no—
267C2×C4⋊C8C2×D6111yescharacteristic
268C2×C4⋊C8C2×D6111yescharacteristic
269C2×#102—612no—
270C2×#102—612no—
271C2×#97C2×D6112yes—
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—611no—
284C4×C4⋊C4—1221no—
285C4×C22⋊C4—1221no—
286C43C2×D6111yescharacteristic
287#340—612no—
288#340—612no—
289#340—612no—
290#340—612no—
291#336C2×D6112yes—
292C61#2211711yescharacteristic, complemented, Sylow
293C22×Q8—1221no—
294C22×D4—611no—
295C23×C4—1221no—
296C22×C8C22×D6112yes—
297C22×C8C22×D6112yes—
298C4⋊Q8—614no—
299C4⋊Q8—1222no—
300C4×Q8—2444no—
301C4×D4—614no—
302C4×D4—1222no—
303#117—2444no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—612no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1221no—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C42—1221no—
321C2×C42—1222no—
322C2×C42D24411yescharacteristic
323C2×C42D24411yescharacteristic
324C2×C42D24411yescharacteristic
325C2×C42D24411yescharacteristic
326C2×C42C22×D6111yescharacteristic
327C2×C42C22×D6111yescharacteristic
328C2×C42C22×D6111yescharacteristic
329#107—2442no—
330#106—618no—
331C4⋊C8—22no—
332C4⋊C8C4×D6114yes—
333#102—618no—
334#102—2442no—
335#97—42no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339#95—1222no—
340#95—1222no—
341C24—1221no—
342C2×Q8—1224no—
343C2×Q8—2442no—
344C2×Q8—2442no—
345C2×D4—612no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—1222no—
352C22×C4—1221no—
353C22×C4—1224no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4C2×D24411yescharacteristic
360C22×C4C2×D24411yescharacteristic
361C22×C4C2×D24411yescharacteristic
362C22×C4C2×D24411yescharacteristic
363C22×C4C2×D24411yescharacteristic
364C22×C4C2×D24411yescharacteristic
365C22×C4C23×D6111yescharacteristic
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8C2×C4×D6114yes—
372C4⋊C4—614no—
373C4⋊C4—612no—
374C4⋊C4—1221no—
375C4⋊C4—1222no—
376C4⋊C4—1224no—
377C4⋊C4—1222no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1222no—
382C4⋊C4—1221no—
383C4⋊C4—2442no—
384C4⋊C4—2442no—
385C4⋊C4—2444no—
386C4⋊C4—2442no—
387C4⋊C4—2442no—
388C22⋊C4—1224no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4—2442no—
392C22⋊C4—2442no—
393C42—22no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—22no—
399C42—22no—
400C42—41no—
401C42—2442no—
402C42—2444no—
403C42C2×C4×D6112yes—
404C42#1175822212yes—
405C23—1222no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23C22×D24411yescharacteristic
413Q8—2448no—
414D4—1222no—
415D4—1222no—
416D4—2441no—
417D4—2441no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—1222no—
432C2×C4—1224no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2442no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2442no—
441C2×C4—2441no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2442no—
446C2×C4—2448no—
447C2×C4C4×D24412yes—
448C2×C4C4×D24412yes—
449C2×C4#1451980512yes—
450C2×C4#1451980512yes—
451C2×C4#1451981012yes—
452C2×C4C22×C4×D6111yescharacteristic
453C2×C4C2×#1175822211yescharacteristic
454C2×C4#1451982211yescharacteristic
455C2×C4#1451982411yescharacteristic
456C2×C4#1451982411yescharacteristic
457C2×C4#1451982511yescharacteristic
458C8—24no—
459C8—42no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22—4881no—
464C22—2442no—
465C22—2442no—
466C22—2442no—
467C22C2×C4×D24411yescharacteristic
468C22#1684033811yescharacteristic
469C22#1684034311yescharacteristic
470C22C2×#1451980511yescharacteristic
471C22C2×#1451980511yescharacteristic
472C22C2×#1451981011yescharacteristic
473C22#1684056211yescharacteristic
474C4—22no—
475C4—22no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—4881no—
480C4—4882no—
481C4—4881no—
482C4—2442no—
483C4—2444no—
484C4#1683986414yes—
485C4#1684023811yescharacteristic
486C4#1684024111yescharacteristic
487C4#1684037011yescharacteristic
488C4#1684037711yescharacteristic
489C2—4882no—
490C2#2144850211yescharacteristic
491C2C2×#1683986411yescharacteristic
492C2C4×#1451980511yescharacteristic
493C2#2145701911yescharacteristic
494C2#2145704211yescharacteristic
495C2#2145719311yescharacteristic
496C2#2145720011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1767C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1767D61
hypocenterC61#22117
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. C4×#14519777C2semidirect
4. #21446766C2semidirect
5. #21446578C2semidirect
6. #21446562C2semidirect
9. #21446405C2semidirect
11. C61×#1767C2semidirect
16. #16840053C4semidirect
251. #1767D61semidirect
292. C61#22117semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
C4×#145197772
#214482832
#214467662
#214465782
#214465622
#214465162
#214464402
#214464052
#214458202
C61×#17672
#2211761