Finite Group Databasebeta

← #29697133

Subgroups — #29697133

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839756C211yesmaximal, characteristic
2#21448650C211yesmaximal, characteristic
3#21446508C212yesmaximal, complemented
4#21446453C212yesmaximal, complemented
5#21446079C212yesmaximal, complemented
6#21445914C211yesmaximal, characteristic, complemented
7#21445842C211yesmaximal, characteristic, complemented
8#21445757C211yesmaximal, characteristic
9#21445647C212yesmaximal, complemented
10#21445597C211yesmaximal, characteristic, complemented
11C61×#2347C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×C4×D244C2212yes—
15C2×#14519708C2212yes—
16#16839756C418yescomplemented
17C2×#14519700C2212yes—
18C2×#14519700C2211yescharacteristic
19C2×#14519700C2211yescharacteristic
20C2×#14519697C2212yes—
21C2×#14519697C2211yescharacteristic
22C2×#14519697C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24C2×#14519688C2212yes—
25#16839668C2212yes—
26#16839638—22no—
27#16839624—21no—
28#16839622C2211yescharacteristic
29#95×D61—21no—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C2×#117×C61C2211yescharacteristic
35C61×#389C2212yes—
36C61×#388—22no—
37C61×#383—21no—
38C61×#380C2211yescharacteristic
39C4×C4⋊C4×C61—21no—
40C4×C4⋊C4×C61C2212yes—
41C4×C22⋊C4×C61C2211yescharacteristic
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×C4×D61—22no—
51C22×#92418—42no—
52C22×#92418C2×C414yescomplemented
53C22×#92418C2312yes—
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56C4×D244—24no—
57C4×D244—42no—
58C2×C4⋊Dic61C2312yes—
59C2×C22⋊Dic61—21no—
60C2×#11758198—22no—
61#14519708—42no—
62#14519700—42no—
63#14519700C2×C414yescomplemented
64#14519700C2×C414yescomplemented
65#14519697—24no—
66#14519697C2×C414yes—
67#14519697C2×C414yes—
68C22⋊#92418—22no—
69#14519688—42no—
70#14519684—21no—
71#14519684—22no—
72#14519684—22no—
73C23×C4×C61C2311yescharacteristic
74#117×C61—22no—
75#117×C61C2×C414yescomplemented
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61C2312yes—
78C2×C4⋊C4×C61C2311yescharacteristic
79C2×C4⋊C4×C61C2311yescharacteristic
80C2×C22⋊C4×C61—21no—
81C2×C22⋊C4×C61—21no—
82C2×C22⋊C4×C61D412yes—
83C2×C42×C61—21no—
84C2×C42×C61—22no—
85C2×C42×C61C2312yes—
86#95×C61—21no—
87#95×C61—22no—
88#95×C61—22no—
89#95×C61D411yescharacteristic
90#95×C61D411yescharacteristic
91C24×C61C2×D411yescharacteristic
92C23×D61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61C22×C412yes—
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61#4112yes—
103C22×C4×C61C2411yescharacteristic
104C2×C22⋊D61—24no—
105C2×C22⋊D61—44no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108#11758222—28no—
109#11758222—44no—
110#11758222—48no—
111C2×D244—21no—
112C2×D244—22no—
113C2×D244C22×C412yes—
114C2×D244#4114yes—
115C2×C4×D61—24no—
116C2×C4×D61—44no—
117C2×C4×D61—44no—
118C2×C4×D61—44no—
119C2×C4×D61—82no—
120C2×Dic122—22no—
121C2×Dic122—22no—
122C2×Dic122—21no—
123C2×Dic122C22×C412yes—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—44no—
129C4⋊C4×C61C22×C412yes—
130C4⋊C4×C61C22×C412yes—
131C22⋊C4×C61—24no—
132C22⋊C4×C61—42no—
133C22⋊C4×C61—81no—
134C42×C61—24no—
135C42×C61—22no—
136C42×C61—22no—
137C42×C61—82no—
138C22⋊Dic61—81no—
139#11758198—44no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C2×#92418—22no—
143C2×#92418—22no—
144C2×#92418—22no—
145C2×#92418—22no—
146C2×#92418—22no—
147C2×#92418—44no—
148C2×#92418C22×C412yes—
149C2×#92418C22×C412yes—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61C4×D412yes—
155C23×C61C22×D411yescharacteristic
156C22×D61—22no—
157C22×D61—24no—
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—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61C4×D412yes—
180C2×C4×C61C23×C411yescharacteristic
181C2×C4×C61C2×#4112yes—
182C2×C4×C61C2×#4111yescharacteristic
183C2×C4×C61H(5,𝔽2)12yes—
184C22⋊D61—48no—
185C2×Dic61—22no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190D244—22no—
191D244—24no—
192D244#11714yes—
193C4×D61—48no—
194C4×D61—88no—
195Dic122—22no—
196Dic122—22no—
197Dic122—42no—
198#92418—22no—
199#92418—22no—
200#92418—42no—
201#22568—611nomaximal, Sylow
202C22×C61—22no—
203C22×C61—22no—
204C22×C61—22no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61C2×C4×D411yescharacteristic
208C22×C61#54612yes—
209C22×C61#57512yes—
210C22×C61#57711yescharacteristic
211C22×C61#57711yescharacteristic
212C2×D61—24no—
213C2×D61—42no—
214C2×D61—42no—
215C2×D61—44no—
216C2×D61—44no—
217C4×C61—22no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61—82no—
223C4×C61C2×#11711yescharacteristic
224C4×C61#51811yescharacteristic
225C4×C61#55112yes—
226Dic61—42no—
227Dic61—82no—
228C2×#416—611no—
229#2347D6111yescharacteristic, complemented
230#2339—611no—
231#1935—612no—
232#1927—612no—
233#1864—612no—
234#1825—611no—
235#1807—611no—
236#1788—611no—
237#1762—612no—
238#1752—611no—
239C2×C61—42no—
240C2×C61#234211yescharacteristic
241C2×C61#294311yescharacteristic
242C2×C61#295211yescharacteristic
243C2×C61#330712yes—
244C2×C61#331112yes—
245D61—44no—
246C22×#41—611no—
247C23×C8—611no—
248C2×C4×D4—612no—
249C2×#117C2×D6111yescharacteristic
250C2×#106—612no—
251#416—618no—
252C2×#103—611no—
253C2×#103—611no—
254C2×#103—612no—
255C2×#102—611no—
256C2×#102—611no—
257C2×#102—612no—
258C2×C22⋊C8—611no—
259C2×#97—612no—
260#391—612no—
261#389C2×D6112yes—
262#388—22no—
263#386—1222no—
264#383—21no—
265#381—1221no—
266#380—611no—
267#380C2×D6111yescharacteristic
268C4×C4⋊C4—21no—
269C4×C4⋊C4C2×D6112yes—
270C4×C22⋊C4C2×D6111yescharacteristic
271C2×#95—1221no—
272#340—612no—
273#340—612no—
274#340—612no—
275#336—612no—
276C61#2256811yescharacteristic, complemented, Sylow
277C2×#41—614no—
278C2×#41—1224no—
279C2×#41—1222no—
280C2×#41—2442no—
281C22×Q8—611no—
282C22×D4—611no—
283C22×D4—1221no—
284C23×C4—1221no—
285C23×C4—1222no—
286C23×C4C22×D6111yescharacteristic
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—614no—
290C22×C8—612no—
291C22×C8—2442no—
292C4×D4—1224no—
293C4×D4—2442no—
294#117—22no—
295#117C4×D6114yescomplemented
296C2×C4⋊C4—22no—
297C2×C4⋊C4—612no—
298C2×C4⋊C4C22×D6112yes—
299C2×C4⋊C4C22×D6111yescharacteristic
300C2×C4⋊C4C22×D6111yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1222no—
305C2×C22⋊C4C22⋊D6112yes—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42C22×D6112yes—
309#106—2442no—
310#103—614no—
311#103—614no—
312#103—2442no—
313#102—614no—
314#102—614no—
315#102—1224no—
316C22⋊C8—1222no—
317#97—2442no—
318#95—21no—
319#95—22no—
320#95—22no—
321#95—1221no—
322#95—1222no—
323#95—1222no—
324#95C22⋊D6111yescharacteristic
325#95C22⋊D6111yescharacteristic
326C24—1221no—
327C24C2×C22⋊D6111yescharacteristic
328#41—1228no—
329#41—2444no—
330#41—2448no—
331C2×Q8—612no—
332C2×Q8—1221no—
333C2×Q8—1222no—
334C2×Q8—1222no—
335C2×D4—612no—
336C2×D4—614no—
337C2×D4—1224no—
338C2×D4—1221no—
339C2×D4—1222no—
340C2×D4—2444no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—1221no—
347C22×C4—1224no—
348C22×C4—1222no—
349C22×C4—4882no—
350C22×C4—2444no—
351C22×C4—2444no—
352C22×C4—2444no—
353C22×C4C2×C4×D6112yes—
354C22×C4#1175822212yes—
355C22×C4C2×C22⋊D6112yes—
356C22×C4C2×C22⋊D6112yes—
357C22×C4C2×C22⋊D6111yescharacteristic
358C22×C4C23×D6111yescharacteristic
359C2×C8—612no—
360C2×C8—612no—
361C2×C8—1222no—
362C2×C8—1222no—
363C2×C8—1222no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—2444no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—44no—
372C4⋊C4—1222no—
373C4⋊C4—1222no—
374C4⋊C4C2×C4×D6112yes—
375C4⋊C4C2×C4×D6112yes—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—81no—
379C22⋊C4—4881no—
380C22⋊C4—2444no—
381C42—22no—
382C42—22no—
383C42—24no—
384C42—82no—
385C23—22no—
386C23—21no—
387C23—21no—
388C23—22no—
389C23—1222no—
390C23—1224no—
391C23—2442no—
392C23—2441no—
393C23—2441no—
394C23C4×C22⋊D6112yes—
395C23C22×C22⋊D6111yescharacteristic
396Q8—1222no—
397Q8—1222no—
398Q8—2442no—
399D4—614no—
400D4—1222no—
401D4—1224no—
402D4—2448no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—1222no—
422C2×C4—4888no—
423C2×C4—2448no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2442no—
428C2×C4C4×C22⋊D6112yes—
429C2×C4C22×C4×D6111yescharacteristic
430C2×C4#41×D6112yes—
431C2×C4C2×#1175822211yescharacteristic
432C2×C4#1451982312yes—
433C8—1222no—
434C8—1222no—
435C8—2442no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—42no—
441C22—1224no—
442C22—2442no—
443C22—2442no—
444C22—2444no—
445C22—2444no—
446C22C2×C4×C22⋊D6111yescharacteristic
447C22#1684040012yes—
448C22#1684056012yes—
449C22#1684057011yescharacteristic
450C22#1684057311yescharacteristic
451C4—22no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—82no—
457C4—4882no—
458C4—2442no—
459C4#117×D6111yescharacteristic
460C4#1684023611yescharacteristic
461C4#1684043612yes—
462C2—42no—
463C2—2444no—
464C2#2144866911yescharacteristic
465C2#2145461711yescharacteristic
466C2#2145470011yescharacteristic
467C2#2145778912yes—
468C2#2145784912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2347C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#2347D61
hypocenterC61#22568
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
3. #21446508C2semidirect
4. #21446453C2semidirect
5. #21446079C2semidirect
6. #21445914C2semidirect
7. #21445842C2semidirect
9. #21445647C2semidirect
10. #21445597C2semidirect
11. C61×#2347C2semidirect
16. #16839756C4semidirect
52. C22×#92418C2×C4semidirect
63. #14519700C2×C4semidirect
64. #14519700C2×C4semidirect
75. #117×C61C2×C4semidirect
229. #2347D61semidirect
276. C61#22568semidirect
295. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168397562
#214486502
#214465082
#214464532
#214460792
#214459142
#214458422
#214457572
#214456472
#214455972
C61×#23472
#2256861