Finite Group Databasebeta

← #29574829

Subgroups — #29574829

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839968C211yesmaximal, characteristic
2C4⋊C4×D244C211yesmaximal, characteristic
3#21446780C211yesmaximal, characteristic, complemented
4#21446771C211yesmaximal, characteristic, complemented
5C2×#16839491C211yesmaximal, characteristic
6#21443703C211yesmaximal, characteristic, complemented
7C61×#1279C211yesmaximal, characteristic, complemented
8C22×#11758190—21no—
9C2×#14519777C2211yescharacteristic
10C2×C4×D244—21no—
11C2×C4×D244C2211yescharacteristic
12C2×C4⋊C4×D61—21no—
13#16839968—22no—
14#16839968C414yescomplemented
15C2×C4⋊#92418C2211yescharacteristic
16C2×#14519702—21no—
17#16839660—21no—
18#16839636—22no—
19#16839632C2211yescharacteristic
20#16839619—21no—
21#16839525—22no—
22#16839524—21no—
23#16839524—21no—
24#16839491—22no—
25C2×C4⋊C8×C61C2211yescharacteristic
26C2×C4×C8×C61C2211yescharacteristic
27C4×C4⋊C4×C61C2211yescharacteristic
28C22×Dic122—21no—
29C22×D244D411yescharacteristic
30C22×C4×D61—21no—
31C22×C4×D61—22no—
32C2×#11758190—24no—
33C2×#11758190—42no—
34C2×#11758190—42no—
35C22×#92418—21no—
36#14519777—21no—
37#14519777—21no—
38#14519777—21no—
39#14519777C2×C412yescomplemented
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—42no—
43C4×D244—42no—
44C4×D244C2×C412yes—
45C4×D244D412yes—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61D411yescharacteristic
51C2×C4⋊Dic61D412yes—
52C2×#11758196—22no—
53C4⋊C4×D61—42no—
54C4⋊C4×D61—42no—
55C2×#11758198—21no—
56C2×#11758198—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59#14519702—24no—
60#14519684—22no—
61C22×C8×C61—21no—
62C22×C8×C61—21no—
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C4⋊C4×C61D411yescharacteristic
65C2×C42×C61—21no—
66C2×C42×C61C2311yescharacteristic
67C4⋊C8×C61—21no—
68C4⋊C8×C61C2×C412yescomplemented
69C4×C8×C61—21no—
70C4×C8×C61—21no—
71#95×C61—21no—
72C23×D61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—22no—
80C22×Dic61—41no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244—21no—
84C2×D244C22⋊C412yes—
85C2×D244D812yes—
86C2×D244SD1612yes—
87C2×C4×D61—22no—
88C2×C4×D61—22no—
89C2×C4×D61—41no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61—41no—
93C2×C4×D61—44no—
94C2×C4×D61—44no—
95C2×C4×D61—82no—
96C2×Dic122—22no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C2×C8×C61—41no—
105C2×C8×C61—41no—
106C2×C8×C61—41no—
107C2×C8×C61—41no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61SD1612yescomplemented
112C42×C61—41no—
113C42×C61—41no—
114C42×C61C22×C411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61#4111yescharacteristic
117C42×C61#4111yescharacteristic
118#11758198—22no—
119#11758198—22no—
120#11758198—44no—
121C4⋊Dic61—21no—
122C4⋊Dic61—22no—
123C4⋊Dic61—22no—
124C4⋊Dic61—41no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C4⋊Dic61—42no—
131C4⋊Dic61C22⋊C412yes—
132#11758196—44no—
133C2×#92418—41no—
134C2×#92418—41no—
135C2×#92418—41no—
136#11758190—48no—
137C23×C61#12011yescharacteristic
138C22×D61—22no—
139C22×D61—22no—
140C22×D61—22no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12311yescharacteristic
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61C2×SD1611yescharacteristic
161C2×C4×C61C2×SD1611yescharacteristic
162C2×C4×C61#13511yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61#13711yescharacteristic
166C2×C4×C61#13711yescharacteristic
167C2×Dic61—22no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176D244—22no—
177D244—22no—
178D244—22no—
179D244#10214yes—
180C4×D61—42no—
181C4×D61—42no—
182C4×D61—42no—
183C4×D61—42no—
184C4×D61—88no—
185Dic122—42no—
186Dic122—42no—
187Dic122—81no—
188Dic122—81no—
189C8×C61—42no—
190C8×C61—42no—
191#92418—81no—
192#8985—611nomaximal, Sylow
193C22×C61#38611yescharacteristic
194C22×C61#44711yescharacteristic
195C22×C61#44811yescharacteristic
196C22×C61#45011yescharacteristic
197C22×C61#45211yescharacteristic
198C22×C61#45511yescharacteristic
199C22×C61#45611yescharacteristic
200C2×D61—24no—
201C2×D61—42no—
202C2×D61—42no—
203C2×D61—42no—
204C2×D61—42no—
205C2×D61—42no—
206C2×D61—42no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—42no—
210C4×C61—81no—
211C4×C61C2×#10211yescharacteristic
212C4×C61#41711yescharacteristic
213C4×C61C4×SD1611yescharacteristic
214C4×C61#44011yescharacteristic
215C4×C61#46111yescharacteristic
216C4×C61#46311yescharacteristic
217C4×C61#48011yescharacteristic
218C4×C61#48111yescharacteristic
219Dic61—42no—
220Dic61—82no—
221Dic61—82no—
222C2×#461—611no—
223D4×C4⋊C4—611no—
224#1990—611no—
225#1988—611no—
226C2×D4⋊C8—611no—
227#1296—611no—
228#1279D6111yescharacteristic, complemented
229C2×C61#163911yescharacteristic
230C2×C61#164311yescharacteristic
231C2×C61#164711yescharacteristic
232C2×C61#165111yescharacteristic
233C2×C61#187411yescharacteristic
234C2×C61#187711yescharacteristic
235C2×C61#188211yescharacteristic
236D61—44no—
237C22×SD16—1221no—
238C2×C4⋊Q8—611no—
239C2×C4×D4—611no—
240C2×C4×D4—1221no—
241C22×C4⋊C4—1221no—
242#461—614no—
243#461—1222no—
244C2×C4⋊C8—611no—
245C2×C4⋊C8C2×D6111yescharacteristic
246C2×#103—1221no—
247C2×C4×C8C2×D6111yescharacteristic
248#389—1221no—
249#385—1222no—
250#384—611no—
251#380—1221no—
252C4×C4⋊C4C2×D6111yescharacteristic
253#340—1222no—
254#340—1221no—
255#340—1221no—
256D4⋊C8—1222no—
257C61#898511yescharacteristic, complemented, Sylow
258C22×Q8—1221no—
259C22×D4—611no—
260C23×C4—1222no—
261C23×C4—1221no—
262C2×SD16—1224no—
263C2×SD16—2442no—
264C2×SD16—2442no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—1221no—
268C4⋊Q8—612no—
269C4⋊Q8—1221no—
270C4⋊Q8—1221no—
271C4⋊Q8—1221no—
272C4×D4—612no—
273C4×D4—612no—
274C4×D4—1221no—
275C4×D4—1221no—
276C4×D4—2442no—
277C4×D4—2442no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—612no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4—2442no—
286C2×C4⋊C4—2442no—
287C2×C4⋊C4D24411yescharacteristic
288C2×C4⋊C4D24411yescharacteristic
289C2×C22⋊C4—1221no—
290C2×C22⋊C4—1222no—
291C2×C42—21no—
292C2×C42C22×D6111yescharacteristic
293C4⋊C8—21no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296C4⋊C8C4×D6112yescomplemented
297#103—1224no—
298C4×C8—21no—
299C4×C8—21no—
300#95—21no—
301#95—1222no—
302C24—1221no—
303C2×Q8—1222no—
304C2×Q8—2441no—
305C2×Q8—2441no—
306C2×Q8—2441no—
307C2×Q8—2441no—
308C2×Q8—2441no—
309C2×D4—612no—
310C2×D4—612no—
311C2×D4—612no—
312C2×D4—1221no—
313C2×D4—1221no—
314C2×D4—1221no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—1221no—
318C22×C4—1222no—
319C22×C4—1222no—
320C22×C4—1222no—
321C22×C4—4882no—
322C22×C4—2444no—
323C22×C4—2444no—
324C22×C4—2441no—
325C22×C4—2441no—
326C22×C4—2441no—
327C22×C4—2441no—
328C22×C4—2441no—
329C22×C4C2×D24411yescharacteristic
330C22×C4D4×D6111yescharacteristic
331C22×C4D4×D6111yescharacteristic
332SD16—2448no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—2441no—
340C2×C8—2441no—
341C2×C8—2441no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—612no—
346C4⋊C4—1221no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—2442no—
350C4⋊C4—2444no—
351C4⋊C4—2441no—
352C4⋊C4—2441no—
353C4⋊C4—2441no—
354C4⋊C4—2441no—
355C4⋊C4—2441no—
356C4⋊C4—2441no—
357C4⋊C4#1175819012yescomplemented
358C22⋊C4—1222no—
359C22⋊C4—1222no—
360C22⋊C4—2444no—
361C42—41no—
362C42—41no—
363C42C2×C4×D6111yescharacteristic
364C42#1175822211yescharacteristic
365C42D4×D6111yescharacteristic
366C42#1175822611yescharacteristic
367C23—1222no—
368C23—1222no—
369C23—1222no—
370C23—2441no—
371C23—2441no—
372C23—2441no—
373C23—2441no—
374C23#1451974711yescharacteristic
375Q8—4881no—
376Q8—4881no—
377Q8—2442no—
378Q8—2442no—
379D4—614no—
380D4—1222no—
381D4—1222no—
382D4—1222no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—1222no—
392C2×C4—4888no—
393C2×C4—4881no—
394C2×C4—4881no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2442no—
402C2×C4—2442no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4C22⋊C4×D6111yescharacteristic
406C2×C4#1451973911yescharacteristic
407C2×C4C4×D24411yescharacteristic
408C2×C4#1451975011yescharacteristic
409C2×C4#1451975211yescharacteristic
410C2×C4#1451976211yescharacteristic
411C2×C4D8×D6111yescharacteristic
412C2×C4SD16×D6111yescharacteristic
413C2×C4C2×#1175819011yescharacteristic
414C2×C4#1451979911yescharacteristic
415C2×C4#1451980411yescharacteristic
416C2×C4#1451980511yescharacteristic
417C2×C4#1451980911yescharacteristic
418C2×C4#1451981011yescharacteristic
419C8—42no—
420C8—42no—
421C8—4881no—
422C22—1224no—
423C22—2442no—
424C22—2442no—
425C22—2442no—
426C22—2442no—
427C22—2442no—
428C22—2442no—
429C22#1683964011yescharacteristic
430C22#1683989711yescharacteristic
431C22#1683990211yescharacteristic
432C22#1683991511yescharacteristic
433C22#1683992611yescharacteristic
434C22#1683994111yescharacteristic
435C22#1683994611yescharacteristic
436C4—21no—
437C4—21no—
438C4—42no—
439C4—81no—
440C4—4882no—
441C4—4882no—
442C4—2442no—
443C4#102×D6111yescharacteristic
444C4#1683976211yescharacteristic
445C4C4×#1175819011yescharacteristic
446C4#1683986411yescharacteristic
447C4#1683996811yescharacteristic
448C4#1683997811yescharacteristic
449C4#1684007711yescharacteristic
450C4#1684008211yescharacteristic
451C2—2444no—
452C2#2144500111yescharacteristic
453C2#2144502911yescharacteristic
454C2#2144505711yescharacteristic
455C2#2144508511yescharacteristic
456C2#2144613511yescharacteristic
457C2#2144615211yescharacteristic
458C2#2144618911yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1279C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1279D61
hypocenterC61#8985
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #21446780C2semidirect
4. #21446771C2semidirect
6. #21443703C2semidirect
7. C61×#1279C2semidirect
14. #16839968C4semidirect
39. #14519777C2×C4semidirect
68. C4⋊C8×C61C2×C4semidirect
111. C4⋊C4×C61SD16semidirect
228. #1279D61semidirect
257. C61#8985semidirect
296. C4⋊C8C4×D61semidirect
357. C4⋊C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168399682
C4⋊C4×D2442
#214467802
#214467712
C2×#168394912
#214437032
C61×#12792
#898561