Finite Group Databasebeta

← #29719008

Subgroups — #29719008

Order 15616.

1064 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839901C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21450290C211yesmaximal, characteristic
4#21447227C212yesmaximal, complemented
5#21447134C212yesmaximal, complemented
6#21446245C211yesmaximal, characteristic
7#21446174C212yesmaximal, complemented
8#21445999C211yesmaximal, characteristic, complemented
9#21445903C211yesmaximal, characteristic, complemented
10#21445824C212yesmaximal, complemented
11Q8×C22⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C22×#11758201C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15C2×#14519761—21no—
16C2×#14519761—21no—
17C2×#14519751C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19#16839901—24no—
20C2×C4⋊#92418C2212yes—
21C2×#14519701—24no—
22C2×#14519701C2212yes—
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2212yes—
25C2×#14519701C2211yescharacteristic
26C2×#14519697C2212yes—
27C2×C22⋊#92418C2211yescharacteristic
28#16839699—22no—
29#16839664C2212yes—
30#16839632C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C23×Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2212yes—
37C61×#391—22no—
38C61×#391C2212yes—
39C61×#380—21no—
40C61×#380C2211yescharacteristic
41C4×C22⋊C4×C61—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C23×Dic61—21no—
46C2×#11758201—22no—
47C2×#11758201—22no—
48C2×#11758201—24no—
49C2×#11758201—42no—
50C22×#92418—22no—
51C22×#92418—42no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55#14519761—44no—
56#14519761—44no—
57#14519751—22no—
58#14519751—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C4⋊#92418—42no—
70#14519701—24no—
71#14519701—24no—
72#14519701—22no—
73#14519701—22no—
74#14519697—42no—
75C22⋊#92418—22no—
76C22×Q8×C61—21no—
77C22×Q8×C61—22no—
78C22×Q8×C61—24no—
79C22×Q8×C61—42no—
80C22×Q8×C61D414yes—
81C22×Q8×C61C2311yescharacteristic
82C22×Q8×C61C2311yescharacteristic
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85C4×Q8×C61—24no—
86C4×Q8×C61—42no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61C2312yes—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C42×C61—22no—
93C2×C42×C61C2312yes—
94#95×C61—22no—
95#95×C61—22no—
96#95×C61—21no—
97#95×C61D412yes—
98C24×C61C2×D411yescharacteristic
99C23×D61—21no—
100C2×Q8×C61—24no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—28no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—44no—
108C2×Q8×C61—44no—
109C2×Q8×C61—48no—
110C2×Q8×C61C2×D412yes—
111C2×Q8×C61C2×D412yes—
112C2×Q8×C61#4114yes—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—44no—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61—42no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—22no—
137C2×D244—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—44no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—42no—
143C22⋊C4×C61SD1614yescomplemented
144C42×C61—22no—
145C42×C61—22no—
146C42×C61—44no—
147#11758201—44no—
148#11758201—44no—
149#11758198—44no—
150#11758198—42no—
151#11758198—44no—
152C4⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—21no—
156C4⋊Dic61—44no—
157C4⋊Dic61—42no—
158C4⋊Dic61—44no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—44no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61#12112yes—
170C23×C61C2×SD1612yes—
171C23×C61AGL(1,ℤ/8)12yes—
172C23×C61C22×D411yescharacteristic
173C22×D61—41no—
174C22×D61—42no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—42no—
178Q8×C61—22no—
179Q8×C61—24no—
180Q8×C61—22no—
181Q8×C61—22no—
182Q8×C61—48no—
183Q8×C61—42no—
184Q8×C61#12314yescomplemented
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—48no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61#12112yes—
203C2×C4×C61C2×SD1614yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)12yes—
208C22⋊D61—84no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—84no—
216D244—41no—
217D244—41no—
218D244—42no—
219#92418—41no—
220#92418—42no—
221#92418—41no—
222#24528—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—24no—
225C22×C61—22no—
226C22×C61—24no—
227C22×C61#46014yes—
228C22×C61C2×#12111yescharacteristic
229C22×C61#54612yes—
230C22×C61C22×SD1611yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61#57812yes—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—82no—
236C2×D61—81no—
237C2×D61—82no—
238C4×C61—22no—
239C4×C61—24no—
240C4×C61—24no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61C2×#12311yescharacteristic
245C4×C61#53411yescharacteristic
246C4×C61#55112yes—
247Dic61—82no—
248Dic61—81no—
249C2×#448—611no—
250C22×#103—611no—
251#2501—611no—
252Q8×C22⋊C4D6111yescharacteristic, complemented
253#2072—612no—
254#2055—612no—
255#1893—611no—
256#1880—612no—
257#1850—611no—
258#1823—611no—
259#1803—612no—
260C2×C61—24no—
261C2×C61#241211yescharacteristic
262C2×C61C2×#46011yescharacteristic
263C2×C61#304411yescharacteristic
264C2×C61#329812yes—
265C2×C61#330812yes—
266D61—161no—
267C23×Q8C2×D6111yescharacteristic
268C22×SD16—611no—
269C22×SD16—611no—
270C23×C8—611no—
271C2×#123—1221no—
272C2×#123—1221no—
273C2×#121—611no—
274C2×C4×Q8C2×D6112yes—
275C22×C4⋊C4—611no—
276#448—1224no—
277C2×C4⋊C8—612no—
278C2×#103—612no—
279C2×#103—611no—
280C2×#103—612no—
281C2×#103—611no—
282C2×#103—1224no—
283C2×#102—612no—
284C2×C22⋊C8—611no—
285#399—1222no—
286#391—22no—
287#391C2×D6112yes—
288#390—612no—
289#384—612no—
290#380—21no—
291#380C2×D6111yescharacteristic
292C4×C22⋊C4—21no—
293C4×C22⋊C4C2×D6111yescharacteristic
294#340—612no—
295#340—612no—
296#340—612no—
297#336—612no—
298C61#2452811yescharacteristic, complemented, Sylow
299C22×Q8—22no—
300C22×Q8—21no—
301C22×Q8—24no—
302C22×Q8—42no—
303C22×Q8D24414yes—
304C22×Q8C22×D6111yescharacteristic
305C22×Q8C22×D6111yescharacteristic
306C22×D4—611no—
307C22×D4—1221no—
308C23×C4—21no—
309C23×C4—1221no—
310C23×C4C22×D6111yescharacteristic
311C2×SD16—1222no—
312C2×SD16—1222no—
313C2×SD16—1224no—
314C2×SD16—2442no—
315C22×C8—612no—
316C22×C8—611no—
317C22×C8—611no—
318C22×C8—1222no—
319C22×C8—2442no—
320#123—2444no—
321#123—2444no—
322#121—1222no—
323#121—1222no—
324C4×Q8—24no—
325C4×Q8—42no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4C22×D6112yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4C22⋊D6111yescharacteristic
340C2×C22⋊C4C22⋊D6111yescharacteristic
341C2×C42—22no—
342C2×C42C22×D6112yes—
343C4⋊C8—2442no—
344#103—1224no—
345#103—1224no—
346#103—1222no—
347#103—1222no—
348#102—2442no—
349C22⋊C8—1222no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95C22⋊D6112yes—
354C24—1221no—
355C24C2×C22⋊D6111yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—28no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8C2×D24412yes—
367C2×Q8C2×D24412yes—
368C2×Q8#1175822414yes—
369C2×D4—1221no—
370C2×D4—1222no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—2442no—
374C2×D4—2442no—
375C2×D4—2442no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—44no—
383C22×C4—1222no—
384C22×C4—1221no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4C2×D24412yes—
389C22×C4#1175822212yes—
390C22×C4C2×C22⋊D6112yes—
391C22×C4C2×C22⋊D6111yescharacteristic
392C22×C4C2×C22⋊D6112yes—
393C22×C4C23×D6111yescharacteristic
394SD16—2444no—
395SD16—2444no—
396C2×C8—1221no—
397C2×C8—1221no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—2444no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—44no—
407C4⋊C4—1222no—
408C4⋊C4—1221no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—2444no—
412C4⋊C4—2442no—
413C4⋊C4—2444no—
414C22⋊C4—22no—
415C22⋊C4—42no—
416C22⋊C4—2444no—
417C22⋊C4—2442no—
418C22⋊C4—2444no—
419C22⋊C4#1175820114yescomplemented
420C42—22no—
421C42—22no—
422C42—44no—
423C23—22no—
424C23—22no—
425C23—2441no—
426C23—2441no—
427C23—2441no—
428C23—2442no—
429C23—2442no—
430C23#1451975112yes—
431C23C2×#1175820112yes—
432C23#1451980212yes—
433C23C22×C22⋊D6111yescharacteristic
434Q8—22no—
435Q8—24no—
436Q8—22no—
437Q8—22no—
438Q8—48no—
439Q8—42no—
440Q8#1451976114yescomplemented
441D4—4884no—
442D4—2441no—
443D4—2441no—
444D4—2442no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—4884no—
463C2×C4—2441no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2441no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4#1451975112yes—
470C2×C4C2×#1175820114yes—
471C2×C4C22×D24411yescharacteristic
472C2×C4C2×#1175822412yes—
473C2×C4C2×#1175822211yescharacteristic
474C2×C4#1451982212yes—
475C8—2441no—
476C8—2442no—
477C8—2441no—
478C22—24no—
479C22—24no—
480C22—22no—
481C22—22no—
482C22—4882no—
483C22—4882no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22#1683996214yes—
488C22C2×#1451975111yescharacteristic
489C22#1684040612yes—
490C22C22×#1175820111yescharacteristic
491C22C2×#1451980211yescharacteristic
492C22#1684057712yes—
493C4—24no—
494C4—22no—
495C4—24no—
496C4—42no—
497C4—42no—
498C4—82no—
499C4—4882no—
500C4—4881no—
501C4C2×#1451976111yescharacteristic
502C4#1684032411yescharacteristic
503C4#1684043812yes—
504C2—24no—
505C2—9761no—
506C2#2144920911yescharacteristic
507C2C2×#1683996211yescharacteristic
508C2#2145534911yescharacteristic
509C2#2145768412yes—
510C2#2145781012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingQ8×C22⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterQ8×C22⋊C4D61
hypocenterC61#24528
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#14519701C2semidirect
4. #21447227C2semidirect
5. #21447134C2semidirect
7. #21446174C2semidirect
8. #21445999C2semidirect
9. #21445903C2semidirect
10. #21445824C2semidirect
11. Q8×C22⋊C4×C61C2semidirect
143. C22⋊C4×C61SD16semidirect
184. Q8×C61#123semidirect
252. Q8×C22⋊C4D61semidirect
298. C61#24528semidirect
419. C22⋊C4#11758201semidirect
440. Q8#14519761semidirect

Maximal subgroups

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

subgroupindex
C2×#168399012
C22×#145197012
#214502902
#214472272
#214471342
#214462452
#214461742
#214459992
#214459032
#214458242
Q8×C22⋊C4×C612
#2452861