Finite Group Databasebeta

← #29731594

Subgroups — #29731594

Order 15616.

778 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×#16840165C211yesmaximal, characteristic, complemented
2C2×#16840049C211yesmaximal, characteristic
3#21450988C211yesmaximal, characteristic
4#21447300C212yesmaximal, complemented
5#21447204C212yesmaximal
6#21446767C211yesmaximal, characteristic, complemented
7#21446587C212yesmaximal, complemented
8#21446538C211yesmaximal, characteristic, complemented
9#21446439C212yesmaximal
10C4×#14519697C211yesmaximal, characteristic
11C61×#2369C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840165—24no—
15#16840049—24no—
16C2×#14519710C2212yes—
17C2×#14519710C2212yes—
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519697C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×C4×#92418C2211yescharacteristic
25#16839675—22no—
26#16839664C2212yes—
27#16839628C2212yes—
28#16839619—21no—
29#16839619—21no—
30#16839524C2211yescharacteristic
31#16839524C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#391—22no—
37C61×#389—21no—
38C61×#389C2212yes—
39C4×C4⋊C4×C61—21no—
40C4×C4⋊C4×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2212yes—
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×#92418C2312yes—
47C22×#92418C2311yescharacteristic
48C22×#92418C2311yescharacteristic
49#14519777—22no—
50#14519777—22no—
51C4×D244—22no—
52C4×D244—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×#11758198—21no—
57C2×#11758198—22no—
58C2×#11758198—22no—
59#14519710—24no—
60#14519710—24no—
61#14519708—24no—
62C4⋊#92418—22no—
63C4⋊#92418—24no—
64#14519697—24no—
65#14519697—22no—
66#14519697Q814yescomplemented
67C4×#92418—22no—
68#14519684—22no—
69#14519684—22no—
70C22×Q8×C61—21no—
71C4⋊Q8×C61—22no—
72C4⋊Q8×C61D414yes—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C4⋊C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61C2312yes—
79C2×C4⋊C4×C61C2311yescharacteristic
80C2×C42×C61—21no—
81C2×C42×C61—22no—
82C2×C42×C61C2312yes—
83C2×C42×C61C2311yescharacteristic
84#95×C61—22no—
85#95×C61—22no—
86#95×C61—21no—
87#95×C61D412yes—
88C23×D61—21no—
89C2×Q8×C61—24no—
90C2×Q8×C61—81no—
91C2×Q8×C61—81no—
92C22×C4×C61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—22no—
102C2×D244—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244C2×Q812yes—
108C2×C4×D61—42no—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C2×Dic122—42no—
112C2×Dic122—81no—
113C2×Dic122—81no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—82no—
122C4⋊C4×C61C2×D414yes—
123C4⋊C4×C61C2×Q812yes—
124C42×C61—22no—
125C42×C61—22no—
126C42×C61—42no—
127C42×C61—82no—
128C42×C61C2×D412yes—
129C42×C61#4112yes—
130#11758198—42no—
131#11758198—82no—
132#11758198—82no—
133C4⋊Dic61—21no—
134C4⋊Dic61—22no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—42no—
138C2×#92418—21no—
139C2×#92418—21no—
140C2×#92418—21no—
141C2×#92418—21no—
142C2×#92418—22no—
143C2×#92418—22no—
144C2×#92418—22no—
145C2×#92418C2×Q812yes—
146C2×#92418#4112yes—
147C23×C61C22×D411yescharacteristic
148C22×D61—22no—
149C22×D61—41no—
150C22×D61—41no—
151C22×D61—41no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61—41no—
155Q8×C61—84no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—24no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61#12112yes—
174C2×C4×C61#12112yes—
175C2×C4×C61#13512yes—
176C2×C4×C61#13712yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×Q811yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C2×C4×C612−1+411yescharacteristic
182C2×C4×C612−1+411yescharacteristic
183C2×Dic61—42no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189D244—22no—
190D244—22no—
191D244—41no—
192D244—41no—
193C4×D61—84no—
194Dic122—84no—
195#92418—22no—
196#92418—22no—
197#92418—42no—
198#25450—611nomaximal, Sylow
199C22×C61C2×#12111yescharacteristic
200C22×C61#54711yescharacteristic
201C22×C61D4×Q811yescharacteristic
202C22×C61C2×#13511yescharacteristic
203C22×C61C2×#13711yescharacteristic
204C22×C61#57611yescharacteristic
205C22×C61#57611yescharacteristic
206C2×D61—42no—
207C2×D61—42no—
208C2×D61—42no—
209C2×D61—81no—
210C2×D61—81no—
211C2×D61—81no—
212C2×D61—81no—
213C4×C61—22no—
214C4×C61—42no—
215C4×C61—42no—
216C4×C61—42no—
217C4×C61—44no—
218C4×C61—81no—
219C4×C61#46414yes—
220C4×C61#53311yescharacteristic
221C4×C61#54111yescharacteristic
222C4×C61#54811yescharacteristic
223C4×C61D4×Q811yescharacteristic
224Dic61—81no—
225Dic61—82no—
226C2×#500—611no—
227C2×#476—611no—
228#2563—611no—
229#2369D6111yescharacteristic, complemented
230#2085—612no—
231#2067—612no—
232#1987—611no—
233#1949—612no—
234#1939—611no—
235#1925—612no—
236C4×#102—611no—
237C2×C61#242911yescharacteristic
238C2×C61C2×#46411yescharacteristic
239C2×C61#304111yescharacteristic
240C2×C61#334111yescharacteristic
241C2×C61#335711yescharacteristic
242C2×C61#338011yescharacteristic
243C2×C61#338511yescharacteristic
244D61—82no—
245C2×C4⋊Q8—611no—
246C2×C4⋊Q8C2×D6111yescharacteristic
247C2×C4×D4—611no—
248#500—1224no—
249#476—1224no—
250C2×#107—612no—
251C2×#107—612no—
252C2×#106—612no—
253C2×C4⋊C8—612no—
254C2×C4⋊C8—611no—
255C2×#102—611no—
256C2×#102—611no—
257C2×#102—612no—
258C2×C4×C8—611no—
259#393—1222no—
260#391—22no—
261#390—612no—
262#389—21no—
263#389C2×D6112yes—
264#382—612no—
265#380—1221no—
266#380—1221no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4C2×D6112yes—
269C4×C4⋊C4C2×D6111yescharacteristic
270C4×C4⋊C4C2×D6111yescharacteristic
271#340—612no—
272#340—611no—
273#340—612no—
274#340—611no—
275#336—612no—
276C61#2545011yescharacteristic, complemented, Sylow
277C22×Q8—21no—
278C22×Q8—1221no—
279C22×D4—611no—
280C23×C4—1221no—
281C22×C8—611no—
282C22×C8—611no—
283C22×C8—612no—
284C4⋊Q8—22no—
285C4⋊Q8—1222no—
286C4⋊Q8—1222no—
287C4⋊Q8D24414yes—
288C4×D4—1222no—
289C4×D4—1222no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—612no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4C22⋊D6111yescharacteristic
296C2×C4⋊C4C22⋊D6111yescharacteristic
297C2×C4⋊C4C22×D6111yescharacteristic
298C2×C4⋊C4C22×D6112yes—
299C2×C4⋊C4C22×D6111yescharacteristic
300C2×C22⋊C4—1222no—
301C2×C22⋊C4—1222no—
302C2×C22⋊C4—1221no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42C22×D6112yes—
306C2×C42C22×D6111yescharacteristic
307#107—1224no—
308#107—1224no—
309#106—1224no—
310C4⋊C8—1224no—
311C4⋊C8—1222no—
312#102—614no—
313#102—1222no—
314#102—1224no—
315C4×C8—1222no—
316#95—22no—
317#95—22no—
318#95—21no—
319#95—1222no—
320#95—1222no—
321#95C22⋊D6112yes—
322C24—1221no—
323C2×Q8—24no—
324C2×Q8—81no—
325C2×Q8—81no—
326C2×Q8—4881no—
327C2×Q8—4881no—
328C2×Q8—2442no—
329C2×D4—612no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—1221no—
339C22×C4—1222no—
340C22×C4—2442no—
341C22×C4—2442no—
342C22×C4—2442no—
343C22×C4C2×C22⋊D6112yes—
344C22×C4C2×C22⋊D6112yes—
345C22×C4C2×C22⋊D6111yescharacteristic
346C22×C4C2×C22⋊D6111yescharacteristic
347C22×C4C23×D6111yescharacteristic
348C2×C8—612no—
349C2×C8—612no—
350C2×C8—1221no—
351C2×C8—1221no—
352C2×C8—1221no—
353C2×C8—1221no—
354C2×C8—1222no—
355C2×C8—1222no—
356C2×C8—1222no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—82no—
365C4⋊C4—1221no—
366C4⋊C4—1222no—
367C4⋊C4—1221no—
368C4⋊C4—1222no—
369C4⋊C4—2442no—
370C4⋊C4C2×D24414yes—
371C4⋊C4Q8×D6112yes—
372C22⋊C4—4882no—
373C22⋊C4—4882no—
374C22⋊C4—2442no—
375C42—22no—
376C42—22no—
377C42—42no—
378C42—82no—
379C42C2×D24412yes—
380C42#1175822212yes—
381C23—1222no—
382C23—2441no—
383C23—2441no—
384C23—2441no—
385C23—2441no—
386C23—2441no—
387C23—2441no—
388C23C22×C22⋊D6111yescharacteristic
389Q8—84no—
390Q8—4884no—
391D4—1222no—
392D4—1222no—
393D4—2441no—
394D4—2441no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—4884no—
413C2×C4—2441no—
414C2×C4—2442no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4#1451975112yes—
420C2×C4#1451975112yes—
421C2×C4#1451979712yes—
422C2×C4#1451980812yes—
423C2×C4C22×D24411yescharacteristic
424C2×C4C2×Q8×D6111yescharacteristic
425C2×C4#41×D6111yescharacteristic
426C2×C4C2×#1175822211yescharacteristic
427C2×C4#1451982411yescharacteristic
428C2×C4#1451982511yescharacteristic
429C8—1222no—
430C8—1222no—
431C8—2442no—
432C22—4881no—
433C22—4881no—
434C22—4881no—
435C22—4881no—
436C22—2442no—
437C22—2442no—
438C22—2442no—
439C22C2×#1451975111yescharacteristic
440C22#1684041111yescharacteristic
441C22Q8×C22⋊D6111yescharacteristic
442C22C2×#1451979711yescharacteristic
443C22C2×#1451980811yescharacteristic
444C22#1684056511yescharacteristic
445C22#1684056511yescharacteristic
446C4—22no—
447C4—42no—
448C4—42no—
449C4—44no—
450C4—42no—
451C4—81no—
452C4—4881no—
453C4—4882no—
454C4#1683998414yes—
455C4#1684031611yescharacteristic
456C4#1684037011yescharacteristic
457C4#1684042411yescharacteristic
458C4Q8×D24411yescharacteristic
459C2—4882no—
460C2#2144941211yescharacteristic
461C2C2×#1683998411yescharacteristic
462C2#2145532211yescharacteristic
463C2#2145823711yescharacteristic
464C2#2145844511yescharacteristic
465C2#2145868211yescharacteristic
466C2#2145872511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2369C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2369D61
hypocenterC61#25450
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
1. C2×#16840165C2semidirect
4. #21447300C2semidirect
6. #21446767C2semidirect
7. #21446587C2semidirect
8. #21446538C2semidirect
11. C61×#2369C2semidirect
66. #14519697Q8semidirect
229. #2369D61semidirect
276. C61#25450semidirect

Maximal subgroups

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

subgroupindex
C2×#168401652
C2×#168400492
#214509882
#214473002
#214472042
#214467672
#214465872
#214465382
#214464392
C4×#145196972
C61×#23692
#2545061