Finite Group Databasebeta

← #29729872

Subgroups — #29729872

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840112C211yesmaximal, characteristic, complemented
2C2×#16839964C211yesmaximal, characteristic
3#21450768C211yesmaximal, characteristic
4#21447227C212yesmaximal, complemented
5#21447134C212yesmaximal
6#21446771C211yesmaximal, characteristic, complemented
7#21446538C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal
9#21445824C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic
11C61×#2365C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519757—21no—
15C2×#14519757—21no—
16C2×C4×D244C2211yescharacteristic
17#16840112—24no—
18#16839964—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×C4⋊#92418C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839699—22no—
28#16839664C2212yes—
29#16839632C2212yes—
30#16839524C2211yescharacteristic
31#16839524C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2212yes—
37C61×#391C2212yes—
38C61×#389—21no—
39C61×#387—22no—
40C4×C4⋊C4×C61—21no—
41C4×C4⋊C4×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C2×#11758201—24no—
47C2×#11758201—24no—
48C2×#11758201—42no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777—22no—
54#14519757—44no—
55#14519757—44no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C4⋊#92418—22no—
67C4⋊#92418—42no—
68#14519701—24no—
69#14519701—24no—
70#14519701—24no—
71#14519697—22no—
72#14519697—22no—
73C4×#92418—22no—
74C22×Q8×C61C2312yes—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61D414yes—
77C4×Q8×C61—24no—
78C4×Q8×C61—42no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61D411yescharacteristic
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C42×C61—21no—
87C2×C42×C61—22no—
88C2×C42×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90#95×C61—22no—
91#95×C61—21no—
92#95×C61—22no—
93#95×C61D412yes—
94C23×D61—21no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61—22no—
100C2×Q8×C61C2×D414yes—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—22no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244#4112yes—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×Dic122—42no—
121C2×Dic122—42no—
122C2×Dic122—42no—
123C4⋊C4×C61—24no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—44no—
130C4⋊C4×C61SD1614yescomplemented
131C4⋊C4×C61#4112yes—
132C42×C61—22no—
133C42×C61—22no—
134C42×C61—44no—
135C42×C61—42no—
136C42×C61C2×D412yes—
137C42×C61#4112yes—
138#11758201—48no—
139#11758198—44no—
140#11758198—42no—
141#11758198—44no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—44no—
147C4⋊Dic61—42no—
148C4⋊Dic61—44no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—22no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—22no—
155C2×#92418—21no—
156C2×#92418—22no—
157C2×#92418—21no—
158C23×C61C22×D411yescharacteristic
159C22×D61—22no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166Q8×C61—24no—
167Q8×C61—24no—
168Q8×C61—42no—
169Q8×C61—42no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61#12112yes—
187C2×C4×C61#12112yes—
188C2×C4×C61C2×SD1614yes—
189C2×C4×C61C2×SD1612yes—
190C2×C4×C61AGL(1,ℤ/8)12yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203D244—22no—
204D244—22no—
205D244—41no—
206D244—41no—
207C4×D61—84no—
208Dic122—84no—
209#92418—41no—
210#92418—42no—
211#92418—41no—
212#25320—611nomaximal, Sylow
213C22×C61C2×#12111yescharacteristic
214C22×C61#54611yescharacteristic
215C22×C61#54811yescharacteristic
216C22×C61C22×SD1611yescharacteristic
217C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
218C22×C61#57811yescharacteristic
219C22×C61#57811yescharacteristic
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—24no—
228C4×C61—22no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61—44no—
232C4×C61—81no—
233C4×C61#46014yes—
234C4×C61#52911yescharacteristic
235C4×C61#54211yescharacteristic
236C4×C61#54611yescharacteristic
237C4×C61#54711yescharacteristic
238Dic61—82no—
239Dic61—81no—
240C2×#487—611no—
241C2×#461—611no—
242#2542—611no—
243#2365D6111yescharacteristic, complemented
244#2072—612no—
245#2055—612no—
246#1988—611no—
247#1939—611no—
248#1880—612no—
249#1803—612no—
250C4×#102—611no—
251C2×C61#242711yescharacteristic
252C2×C61C2×#46011yescharacteristic
253C2×C61#304411yescharacteristic
254C2×C61#329811yescharacteristic
255C2×C61#333011yescharacteristic
256C2×C61#335911yescharacteristic
257C2×C61#336711yescharacteristic
258D61—82no—
259C22×SD16—612no—
260C2×C4⋊Q8—611no—
261C2×C4⋊Q8C2×D6111yescharacteristic
262C2×C22⋊Q8—1221no—
263C2×C22⋊Q8—1221no—
264C2×C4×Q8C2×D6112yes—
265C2×C4×D4—611no—
266#487—1224no—
267#461—1224no—
268C2×C4⋊C8—611no—
269C2×C4⋊C8—612no—
270C2×#103—612no—
271C2×#103—612no—
272C2×#103—612no—
273C2×#102—611no—
274C2×#102—611no—
275C2×C4×C8—611no—
276#399—1222no—
277#391C2×D6112yes—
278#390—612no—
279#389—21no—
280#387—22no—
281#384—612no—
282C4×C4⋊C4—21no—
283C4×C4⋊C4C2×D6111yescharacteristic
284C4×C4⋊C4C2×D6111yescharacteristic
285#340—612no—
286#340—611no—
287#340—612no—
288#340—611no—
289#336—612no—
290C61#2532011yescharacteristic, complemented, Sylow
291C22×Q8—1221no—
292C22×Q8C22×D6112yes—
293C22×D4—611no—
294C23×C4—1221no—
295C2×SD16—1224no—
296C2×SD16—1224no—
297C2×SD16—2442no—
298C22×C8—612no—
299C22×C8—611no—
300C22×C8—611no—
301C4⋊Q8—22no—
302C4⋊Q8—1222no—
303C4⋊Q8—1222no—
304C4⋊Q8D24414yes—
305C22⋊Q8—2444no—
306C22⋊Q8—2444no—
307C4×Q8—24no—
308C4×Q8—42no—
309C4×D4—1222no—
310C4×D4—1222no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4C22⋊D6111yescharacteristic
320C2×C4⋊C4C22⋊D6111yescharacteristic
321C2×C4⋊C4C22×D6111yescharacteristic
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1222no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42C22×D6112yes—
329C2×C42C22×D6111yescharacteristic
330C4⋊C8—1222no—
331C4⋊C8—2442no—
332#103—1224no—
333#103—1224no—
334#103—1224no—
335#102—1222no—
336#102—1222no—
337C4×C8—1222no—
338#95—22no—
339#95—21no—
340#95—22no—
341#95C22⋊D6112yes—
342C24—1221no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—2442no—
349C2×Q8—2442no—
350C2×Q8—2442no—
351C2×Q8C2×D24414yes—
352C2×D4—612no—
353C2×D4—1221no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—1222no—
362C22×C4—1221no—
363C22×C4—2442no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4C2×C22⋊D6112yes—
367C22×C4C2×C22⋊D6111yescharacteristic
368C22×C4C2×C22⋊D6112yes—
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C23×D6111yescharacteristic
371SD16—2448no—
372C2×C8—1221no—
373C2×C8—1221no—
374C2×C8—1222no—
375C2×C8—1221no—
376C2×C8—1222no—
377C2×C8—1221no—
378C2×C8—1221no—
379C2×C8—1222no—
380C2×C8—1221no—
381C4⋊C4—21no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—44no—
388C4⋊C4—1221no—
389C4⋊C4—1222no—
390C4⋊C4—1221no—
391C4⋊C4—1222no—
392C4⋊C4—2444no—
393C4⋊C4—2442no—
394C4⋊C4—2444no—
395C4⋊C4#1175820114yescomplemented
396C4⋊C4#1175822412yes—
397C22⋊C4—2444no—
398C22⋊C4—2442no—
399C22⋊C4—2444no—
400C42—22no—
401C42—22no—
402C42—44no—
403C42—42no—
404C42C2×D24412yes—
405C42#1175822212yes—
406C23—1222no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23—2441no—
413C23C22×C22⋊D6111yescharacteristic
414Q8—24no—
415Q8—24no—
416Q8—42no—
417Q8—42no—
418Q8—4884no—
419D4—1222no—
420D4—1222no—
421D4—2441no—
422D4—2441no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—4884no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2442no—
445C2×C4—2441no—
446C2×C4#1451975112yes—
447C2×C4#1451975112yes—
448C2×C4C2×#1175820114yes—
449C2×C4C2×#1175820112yes—
450C2×C4#1451980212yes—
451C2×C4C22×D24411yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4C2×#1175822411yescharacteristic
454C2×C4C2×#1175822211yescharacteristic
455C2×C4#1451982211yescharacteristic
456C2×C4#1451982411yescharacteristic
457C8—2441no—
458C8—2442no—
459C8—2441no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22—4881no—
464C22—2442no—
465C22—2442no—
466C22—2442no—
467C22C2×#1451975111yescharacteristic
468C22#1684040611yescharacteristic
469C22#1684041911yescharacteristic
470C22C22×#1175820111yescharacteristic
471C22C2×#1451980211yescharacteristic
472C22#1684057711yescharacteristic
473C22#1684057711yescharacteristic
474C4—24no—
475C4—22no—
476C4—44no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—4881no—
481C4—4882no—
482C4#1683996214yes—
483C4#1684029411yescharacteristic
484C4#1684037711yescharacteristic
485C4#1684040711yescharacteristic
486C4#1684041611yescharacteristic
487C2—4882no—
488C2#2144939011yescharacteristic
489C2C2×#1683996211yescharacteristic
490C2#2145534911yescharacteristic
491C2#2145768411yescharacteristic
492C2#2145812411yescharacteristic
493C2#2145845711yescharacteristic
494C2#2145853311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2365C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2365D61
hypocenterC61#25320
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
1. C2×#16840112C2semidirect
4. #21447227C2semidirect
6. #21446771C2semidirect
7. #21446538C2semidirect
9. #21445824C2semidirect
11. C61×#2365C2semidirect
130. C4⋊C4×C61SD16semidirect
243. #2365D61semidirect
290. C61#25320semidirect
395. C4⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168401122
C2×#168399642
#214507682
#214472272
#214471342
#214467712
#214465382
#214461742
#214458242
C4×#145196972
C61×#23652
#2532061