Finite Group Databasebeta

← #29697228

Subgroups — #29697228

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839759C211yesmaximal, characteristic
2#21448714C211yesmaximal, characteristic, complemented
3#21448656C211yesmaximal, characteristic
4#21446500C212yesmaximal, complemented
5#21446445C212yesmaximal, complemented
6#21446084C212yesmaximal, complemented
7#21445911C211yesmaximal, characteristic, complemented
8#21445757C211yesmaximal, characteristic
9#21445638C212yesmaximal, complemented
10C8×C22⋊Dic61C211yesmaximal, characteristic, complemented
11C61×#1807C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519742C2212yes—
14C2×#14519736C2211yescharacteristic
15#16839759C418yescomplemented
16C2×#14519702C2211yescharacteristic
17C2×#14519702C2211yescharacteristic
18C2×#14519700C2212yes—
19C2×#14519699C2211yescharacteristic
20C2×#14519699C2211yescharacteristic
21C2×#14519697C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×C8×Dic61C2212yes—
24#16839663C2212yes—
25#16839662—22no—
26#16839658C2212yes—
27#16839650—22no—
28#16839629—21no—
29#16839621—21no—
30C4⋊C4×Dic61C2212yes—
31#16839613—21no—
32C4×C22⋊Dic61C2211yescharacteristic
33#16839609—21no—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839516C2212yes—
37C23×C8×C61C2211yescharacteristic
38C2×#106×C61C2212yes—
39C61×#380C2211yescharacteristic
40C61×#380C2211yescharacteristic
41C61×#340C2212yes—
42C2×#11758222—22no—
43C2×#11758222—24no—
44C2×#11758222—42no—
45C2×#11758222C2×C414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50C22×#92418C2312yes—
51#14519742—24no—
52#14519742—42no—
53#14519736—22no—
54#14519736C2×C414yes—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61D412yes—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61C2312yes—
67#14519702C2×C414yes—
68#14519702C2×C414yes—
69#14519700—24no—
70#14519699C2×C414yes—
71#14519699C2×C414yes—
72#14519697—42no—
73C22⋊#92418—22no—
74C8×Dic61—42no—
75#14519684—22no—
76#14519684—22no—
77#14519684D411yescharacteristic
78#14519684D411yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C22×C8×C61—42no—
81C22×C8×C61C2×C414yes—
82C22×C8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84C2×C4⋊C4×C61C2312yes—
85C2×C4⋊C4×C61C2312yes—
86C2×C22⋊C4×C61—21no—
87C2×C22⋊C4×C61—21no—
88#106×C61—42no—
89#95×C61—21no—
90#95×C61—21no—
91C24×C61C2×D411yescharacteristic
92C23×D61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61C22×C412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4112yes—
100C22×C4×C61C2411yescharacteristic
101C2×C22⋊D61—24no—
102C2×C22⋊D61—44no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61C2×D412yes—
107C22×Dic61C2×D412yes—
108#11758222—28no—
109#11758222—44no—
110#11758222—48no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—21no—
114C2×D244C22×C412yes—
115C2×C4×D61—24no—
116C2×C4×D61—44no—
117C2×Dic122—21no—
118C2×Dic122—22no—
119C2×Dic122C22×C412yes—
120C2×Dic122#4114yes—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61—44no—
124C2×C8×C61C22×C412yes—
125C2×C8×C61C22×C412yes—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C22⋊C4×C61—81no—
131C22⋊C4×C61—81no—
132C22⋊Dic61—24no—
133C22⋊Dic61—24no—
134#11758198—44no—
135#11758198—82no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61C22×C412yes—
139C4⋊Dic61C22×C412yes—
140#11758196—44no—
141#11758196—44no—
142C4×Dic61—24no—
143C4×Dic61—22no—
144C4×Dic61—22no—
145C4×Dic61—44no—
146C4×Dic61—82no—
147C2×#92418—22no—
148C2×#92418—22no—
149C2×#92418—22no—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61C4×D412yes—
155C23×C61C22×D411yescharacteristic
156C22×D61—22no—
157C22×D61—42no—
158C22×D61—42no—
159C22×D61—41no—
160C22×D61—41no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—24no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61C4×D412yes—
172C2×C4×C61C23×C411yescharacteristic
173C2×C4×C61C2×#4112yes—
174C2×C4×C61C2×#4111yescharacteristic
175C2×C4×C61H(5,𝔽2)12yes—
176C22⋊D61—48no—
177C2×Dic61—22no—
178C2×Dic61—22no—
179C2×Dic61—24no—
180C2×Dic61—22no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—22no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61#13514yes—
190D244—22no—
191D244—22no—
192D244—42no—
193C4×D61—48no—
194Dic122—22no—
195Dic122—24no—
196Dic122#11714yes—
197C8×C61—22no—
198C8×C61—22no—
199#92418—42no—
200#22577—611nomaximal, Sylow
201C22×C61—22no—
202C22×C61—22no—
203C22×C61—22no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61C2×C4×D411yescharacteristic
207C22×C61#54612yes—
208C22×C61C2×#13512yes—
209C22×C61#57711yescharacteristic
210C22×C61#57711yescharacteristic
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—22no—
217C4×C61—82no—
218C4×C61—82no—
219C4×C61C2×#11711yescharacteristic
220C4×C61#51811yescharacteristic
221C4×C61#55112yes—
222Dic61—24no—
223Dic61—24no—
224Dic61—42no—
225Dic61—42no—
226Dic61—42no—
227C2×#416—611no—
228#2347—611no—
229#2341—611no—
230#1934—612no—
231#1926—612no—
232#1865—612no—
233#1825—611no—
234#1807D6111yescharacteristic, complemented
235#1788—611no—
236C4×#103—612no—
237C8×C22⋊C4—611no—
238C2×C61—42no—
239C2×C61#234211yescharacteristic
240C2×C61C4×#13511yescharacteristic
241C2×C61#295211yescharacteristic
242C2×C61#330112yes—
243C2×C61#330412yes—
244D61—82no—
245C22×#41—611no—
246C23×C8C2×D6111yescharacteristic
247C2×C4×Q8—612no—
248C2×#117—611no—
249C2×#106C2×D6112yes—
250#416—618no—
251C2×#103—611no—
252C2×#103—611no—
253C2×#103—612no—
254C2×#102—611no—
255C2×#102—611no—
256C2×#102—612no—
257C2×C22⋊C8—611no—
258C2×C4×C8—612no—
259#390—612no—
260#390—1222no—
261#389—612no—
262#388—1222no—
263#383—1221no—
264#380—1221no—
265#380C2×D6111yescharacteristic
266#380C2×D6111yescharacteristic
267C4×C4⋊C4—612no—
268C4×C4⋊C4—1221no—
269C4×C22⋊C4—611no—
270C4×C22⋊C4—1221no—
271#340—612no—
272#340—612no—
273#340C2×D6112yes—
274#336—612no—
275C61#2257711yescharacteristic, complemented, Sylow
276C2×#41—614no—
277C2×#41—1224no—
278C2×#41—1222no—
279C2×#41—2442no—
280C22×Q8—611no—
281C22×D4—611no—
282C22×D4—1221no—
283C23×C4—1221no—
284C23×C4C22×D6111yescharacteristic
285C22×C8—42no—
286C22×C8—612no—
287C22×C8C4×D6114yes—
288C22×C8C22×D6111yescharacteristic
289C22×C8C22×D6111yescharacteristic
290C4×Q8—1224no—
291C4×Q8—2442no—
292#117—614no—
293#117—1222no—
294C2×C4⋊C4—611no—
295C2×C4⋊C4—611no—
296C2×C4⋊C4—1222no—
297C2×C4⋊C4—1222no—
298C2×C4⋊C4C22×D6112yes—
299C2×C4⋊C4C22×D6112yes—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—612no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1222no—
305C2×C22⋊C4—1222no—
306C2×C42—612no—
307C2×C42—1221no—
308C2×C42—1222no—
309C2×C42—1222no—
310#106—42no—
311#103—614no—
312#103—614no—
313#103—1224no—
314#102—614no—
315#102—614no—
316#102—2442no—
317C22⋊C8—1222no—
318C4×C8—2442no—
319#95—21no—
320#95—21no—
321#95—611no—
322#95—611no—
323#95—1222no—
324#95—1222no—
325C24—1221no—
326C24D4×D6111yescharacteristic
327#41—1228no—
328#41—2444no—
329#41—2448no—
330C2×Q8—612no—
331C2×Q8—614no—
332C2×Q8—1221no—
333C2×Q8—1222no—
334C2×D4—612no—
335C2×D4—1224no—
336C2×D4—1222no—
337C2×D4—1221no—
338C2×D4—1222no—
339C2×D4—2444no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—612no—
345C22×C4—612no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—1224no—
350C22×C4—2444no—
351C22×C4C2×C4×D6112yes—
352C22×C4D4×D6111yescharacteristic
353C22×C4#1175822612yes—
354C22×C4C23×D6111yescharacteristic
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—44no—
358C2×C8—1222no—
359C2×C8—1222no—
360C2×C8—1222no—
361C2×C8C2×C4×D6112yes—
362C2×C8C2×C4×D6112yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—612no—
368C4⋊C4—612no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—2444no—
372C4⋊C4—2444no—
373C22⋊C4—81no—
374C22⋊C4—81no—
375C22⋊C4—1224no—
376C22⋊C4—1224no—
377C22⋊C4—4882no—
378C22⋊C4—2444no—
379C42—1222no—
380C42—1222no—
381C42—1224no—
382C42—4882no—
383C42—2444no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—1222no—
389C23—2442no—
390C23—2441no—
391C23—2441no—
392C23—2442no—
393C23#1451973912yes—
394C23C2×D4×D6111yescharacteristic
395Q8—614no—
396Q8—1222no—
397Q8—1224no—
398D4—1222no—
399D4—1222no—
400D4—2448no—
401D4—2442no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—614no—
413C2×C4—1222no—
414C2×C4—1222no—
415C2×C4—1222no—
416C2×C4—1224no—
417C2×C4—1222no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—2448no—
423C2×C4—2442no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4#1451973912yes—
427C2×C4C22×C4×D6111yescharacteristic
428C2×C4#41×D6112yes—
429C2×C4C2×#1175822611yescharacteristic
430C2×C4#1451982212yes—
431C8—22no—
432C8—22no—
433C8—2442no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—42no—
438C22—42no—
439C22—4882no—
440C22—4882no—
441C22—2442no—
442C22—2442no—
443C22—2442no—
444C22C2×#1451973911yescharacteristic
445C22#1684040112yes—
446C22#135×D6112yes—
447C22#1684056811yescharacteristic
448C22#1684056811yescharacteristic
449C4—22no—
450C4—82no—
451C4—82no—
452C4—1224no—
453C4—1224no—
454C4—2442no—
455C4—2442no—
456C4—2442no—
457C4#117×D6111yescharacteristic
458C4#1684023511yescharacteristic
459C4#1684043912yes—
460C2—42no—
461C2—4882no—
462C2#2144866511yescharacteristic
463C2#2145459011yescharacteristic
464C2#2145470211yescharacteristic
465C2#2145771612yes—
466C2#2145775312yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1807C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1807D61
hypocenterC61#22577
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. #21448714C2semidirect
4. #21446500C2semidirect
5. #21446445C2semidirect
6. #21446084C2semidirect
7. #21445911C2semidirect
9. #21445638C2semidirect
10. C8×C22⋊Dic61C2semidirect
11. C61×#1807C2semidirect
15. #16839759C4semidirect
234. #1807D61semidirect
275. C61#22577semidirect

Maximal subgroups

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

subgroupindex
C2×#168397592
#214487142
#214486562
#214465002
#214464452
#214460842
#214459112
#214457572
#214456382
C8×C22⋊Dic612
C61×#18072
#2257761