Finite Group Databasebeta

← #29688136

Subgroups — #29688136

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840122C211yesmaximal, characteristic
2#21448386C211yesmaximal, characteristic, complemented
3#21448358C211yesmaximal, characteristic
4#21446791C211yesmaximal, characteristic, complemented
5#21446633C211yesmaximal, characteristic
6#21446500C212yesmaximal
7#21446493C212yesmaximal, complemented
8#21445648C212yesmaximal, complemented
9#21445643C212yesmaximal
10#97×Dic61C211yesmaximal, characteristic, complemented
11C61×#1836C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519765C2211yescharacteristic
14C2×#14519742—21no—
15C2×#14519742C2211yescharacteristic
16C2×#14519739C2211yescharacteristic
17#16840122C418yescomplemented
18C2×#14519702C2212yes—
19C2×#14519700C2212yes—
20C2×#14519699C2212yes—
21C2×#14519697C2212yes—
22C2×#14519688C2211yescharacteristic
23C2×#14519687C2212yes—
24C2×C8×Dic61C2212yes—
25#16839667—21no—
26#16839667C2211yescharacteristic
27#16839663C2211yescharacteristic
28#16839658—21no—
29#16839658C2212yes—
30#16839651—21no—
31#16839651—21no—
32C4⋊C4×Dic61—21no—
33C4⋊C4×Dic61C2212yes—
34#16839609—21no—
35#16839609—21no—
36C42×Dic61C2211yescharacteristic
37#16839524C2212yes—
38#16839524C2212yes—
39C2×#97×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#382C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C22×Dic122—21no—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×#92418C2312yes—
49#14519777—22no—
50#14519777C2×C414yescomplemented
51#14519765—22no—
52#14519765C2×C414yes—
53#14519742—24no—
54#14519742—44no—
55#14519739—24no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2312yes—
58C2×#11758196—22no—
59C2×#11758196—21no—
60C2×#11758198—21no—
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×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61C2311yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73#14519702C2×C418yescomplemented
74#14519700—24no—
75#14519699C2×C418yes—
76#14519697—24no—
77#14519688—22no—
78#14519687—24no—
79C8×Dic61—24no—
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84C22×C8×C61C2312yes—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91#97×C61C2×C414yescomplemented
92#95×C61—21no—
93#95×C61—21no—
94C23×D61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2411yescharacteristic
101C22×Dic61—22no—
102C22×Dic61—21no—
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—21no—
110C2×D244C22×C412yes—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×Dic122—22no—
115C2×Dic122—22no—
116C2×Dic122—24no—
117C2×Dic122—21no—
118C2×Dic122—42no—
119C2×Dic122—42no—
120C2×Dic122C22×C412yes—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61C22×C414yes—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C42×C61C22×C412yes—
131C42×C61#4112yes—
132#11758198—24no—
133#11758198—24no—
134#11758198—42no—
135#11758198—42no—
136#11758198—42no—
137C4⋊Dic61—24no—
138C4⋊Dic61—22no—
139C4⋊Dic61C22×C414yes—
140#11758196—42no—
141#11758196—44no—
142C4×Dic61—22no—
143C4×Dic61—22no—
144C4×Dic61—22no—
145C4×Dic61—41no—
146C4×Dic61—41no—
147C4×Dic61—44no—
148C4×Dic61—42no—
149C4×Dic61#4114yes—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418#4114yes—
153C23×C61C22×D411yescharacteristic
154C22×D61—22no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61C4×D412yes—
175C2×C4×C61C4×D412yes—
176C2×C4×C61C23×C411yescharacteristic
177C2×C4×C61C2×#4112yes—
178C2×C4×C61C2×#4112yes—
179C2×C4×C61C2×#4111yescharacteristic
180C2×Dic61—21no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—24no—
189C2×Dic61—22no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61#12412yes—
195C2×Dic61#12412yes—
196C2×Dic61AGL(1,ℤ/8)12yes—
197C2×Dic61#13712yes—
198D244—24no—
199D244—42no—
200C4×D61—84no—
201Dic122—24no—
202Dic122—48no—
203Dic122—42no—
204C8×C61—24no—
205#92418—24no—
206#21655—611nomaximal, Sylow
207C22×C61C2×C4×D411yescharacteristic
208C22×C61C2×#12411yescharacteristic
209C22×C61#53211yescharacteristic
210C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
211C22×C61C2×#13711yescharacteristic
212C22×C61#57511yescharacteristic
213C22×C61#57511yescharacteristic
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—42no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C2×D61—81no—
221C4×C61—22no—
222C4×C61—22no—
223C4×C61—81no—
224C4×C61—81no—
225C4×C61—82no—
226C4×C61C4×#4112yes—
227C4×C61#52912yes—
228Dic61—22no—
229Dic61—22no—
230Dic61—22no—
231Dic61—22no—
232Dic61—44no—
233Dic61—42no—
234C2×#489—611no—
235C4×C4⋊Q8—611no—
236C4×#124—611no—
237#1993—611no—
238#1959—611no—
239#1934—612no—
240#1933—612no—
241#1836D6111yescharacteristic, complemented
242#1763—612no—
243#1762—612no—
244C4×#97—611no—
245C2×C61C4×#12411yescharacteristic
246C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
247C2×C61C4×#13711yescharacteristic
248C2×C61#314011yescharacteristic
249C2×C61#314111yescharacteristic
250C2×C61#318011yescharacteristic
251C2×C61#318111yescharacteristic
252D61—82no—
253C2×C4⋊Q8—611no—
254C2×#124—611no—
255C2×C4×Q8—611no—
256C2×C4×Q8—1221no—
257C2×C4×D4—611no—
258#489—618no—
259C2×#103—612no—
260C2×#103—612no—
261C2×#102—612no—
262C2×#102—612no—
263C2×#97—612no—
264C2×#97—611no—
265C2×#97C2×D6111yescharacteristic
266C2×C4×C8—612no—
267#391—611no—
268#391—1221no—
269#390—611no—
270#389—612no—
271#389—1221no—
272#388—1221no—
273#388—1221no—
274#384C2×D6111yescharacteristic
275#382C2×D6111yescharacteristic
276C4×C4⋊C4—612no—
277C4×C4⋊C4—1221no—
278C4×C22⋊C4—1221no—
279C4×C22⋊C4—1221no—
280C43—611no—
281#340—612no—
282#340—612no—
283#340C2×D6112yes—
284#340C2×D6112yes—
285C61#2165511yescharacteristic, complemented, Sylow
286C22×Q8—611no—
287C22×Q8—1221no—
288C22×D4—611no—
289C23×C4—1221no—
290C22×C8—612no—
291C22×C8C22×D6112yes—
292C4⋊Q8—614no—
293C4⋊Q8—1222no—
294#124—614no—
295#124—1222no—
296C4×Q8—1224no—
297C4×Q8—2444no—
298C4×D4—1224no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—612no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1221no—
304C2×C4⋊C4—1222no—
305C2×C4⋊C4C22×D6111yescharacteristic
306C2×C4⋊C4C22×D6111yescharacteristic
307C2×C4⋊C4C22×D6112yes—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4—1221no—
313C2×C42—611no—
314C2×C42—611no—
315C2×C42—611no—
316C2×C42—611no—
317C2×C42—611no—
318C2×C42—611no—
319C2×C42—1221no—
320C2×C42—1222no—
321C2×C42C22×D6111yescharacteristic
322#103—618no—
323#103—1224no—
324#102—618no—
325#102—1224no—
326#97—1224no—
327#97—1222no—
328#97C4×D6114yescomplemented
329C4×C8—1224no—
330#95—21no—
331#95—21no—
332#95—1221no—
333#95—1221no—
334#95—1222no—
335#95—1222no—
336C24—1221no—
337C2×Q8—612no—
338C2×Q8—1221no—
339C2×Q8—1224no—
340C2×Q8—1222no—
341C2×Q8—1222no—
342C2×Q8—2442no—
343C2×Q8—2442no—
344C2×D4—612no—
345C2×D4—1222no—
346C2×D4—1222no—
347C2×D4—1221no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—611no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4—2442no—
360C22×C4D4×D6111yescharacteristic
361C22×C4D4×D6111yescharacteristic
362C22×C4C23×D6111yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—614no—
366C2×C8—1222no—
367C2×C8—1222no—
368C2×C8C2×C4×D6114yes—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—614no—
376C4⋊C4—1222no—
377C4⋊C4—1224no—
378C4⋊C4—2442no—
379C4⋊C4—2444no—
380C22⋊C4—1224no—
381C22⋊C4—1224no—
382C22⋊C4—2442no—
383C22⋊C4—2442no—
384C22⋊C4—2442no—
385C42—614no—
386C42—1222no—
387C42—1222no—
388C42—1222no—
389C42—2441no—
390C42—2441no—
391C42—2442no—
392C42—2444no—
393C42C2×C4×D6112yes—
394C42#1175822612yes—
395C23—1222no—
396C23—2441no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23C2×D4×D6111yescharacteristic
403Q8—1224no—
404Q8—2448no—
405Q8—2442no—
406D4—1224no—
407D4—2442no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—612no—
422C2×C4—612no—
423C2×C4—612no—
424C2×C4—612no—
425C2×C4—1221no—
426C2×C4—1221no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—1221no—
430C2×C4—1221no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—1222no—
434C2×C4—1224no—
435C2×C4—4884no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2441no—
440C2×C4#1451973912yes—
441C2×C4#1451973912yes—
442C2×C4C22×C4×D6111yescharacteristic
443C2×C4#41×D6112yes—
444C2×C4#41×D6112yes—
445C2×C4C2×#1175822611yescharacteristic
446C8—24no—
447C8—1224no—
448C22—4881no—
449C22—4881no—
450C22—4881no—
451C22—4881no—
452C22—2442no—
453C22—2442no—
454C22—2442no—
455C22C2×#1451973911yescharacteristic
456C22#124×D6111yescharacteristic
457C22#1684031111yescharacteristic
458C22AGL(1,ℤ/8)×D6111yescharacteristic
459C22#137×D6111yescharacteristic
460C22#1684055811yescharacteristic
461C22#1684056111yescharacteristic
462C4—22no—
463C4—22no—
464C4—81no—
465C4—81no—
466C4—82no—
467C4—1222no—
468C4—1222no—
469C4—1222no—
470C4—1222no—
471C4—2444no—
472C4—2442no—
473C4#1684023012yes—
474C4#1684029712yes—
475C2—4882no—
476C2#2144835811yescharacteristic
477C2#2145462611yescharacteristic
478C2#2145463711yescharacteristic
479C2#2145620411yescharacteristic
480C2#2145621111yescharacteristic
481C2#2145651611yescharacteristic
482C2#2145652711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1836C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1836D61
hypocenterC61#21655
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
2. #21448386C2semidirect
4. #21446791C2semidirect
7. #21446493C2semidirect
8. #21445648C2semidirect
10. #97×Dic61C2semidirect
11. C61×#1836C2semidirect
17. #16840122C4semidirect
50. #14519777C2×C4semidirect
73. #14519702C2×C4semidirect
91. #97×C61C2×C4semidirect
241. #1836D61semidirect
285. C61#21655semidirect
328. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401222
#214483862
#214483582
#214467912
#214466332
#214465002
#214464932
#214456482
#214456432
#97×Dic612
C61×#18362
#2165561