Finite Group Databasebeta

← #29729881

Subgroups — #29729881

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×#16839964C211yesmaximal, characteristic
2#21450773C211yesmaximal, characteristic
3#21448879C211yesmaximal, characteristic, complemented
4#21447230C212yesmaximal, complemented
5#21447133C212yesmaximal
6#21446772C211yesmaximal, characteristic, complemented
7#21446537C211yesmaximal, characteristic, complemented
8#21446172C212yesmaximal
9#21445822C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic
11C2×C61×#487C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519753—21no—
14C2×#14519753—21no—
15C2×Q8×Dic61C2212yes—
16C2×C4×D244C2211yescharacteristic
17#16839964—24no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519706C2212yes—
20C2×#14519701C2212yes—
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23#16839697—22no—
24#16839669C2212yes—
25#16839663C2212yes—
26#16839661—21no—
27#16839648—22no—
28#16839631C2212yes—
29C4⋊C4×Dic61—21no—
30C4×C4⋊Dic61C2211yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic
32#16839525C2211yescharacteristic
33#16839525C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839516C2212yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×C4⋊Q8×C61C2211yescharacteristic
39C61×#487—24no—
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C4×D61—21no—
45C2×#11758201—24no—
46C2×#11758201—24no—
47C2×#11758201—42no—
48C22×#92418C2312yes—
49#14519753—44no—
50#14519753—44no—
51Q8×Dic61—24no—
52Q8×Dic61—42no—
53C4×D244—22no—
54C4×D244—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61D411yescharacteristic
57C2×C4⋊Dic61D411yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61C2312yes—
69C4⋊#92418—22no—
70#14519706—42no—
71#14519701—24no—
72#14519699—22no—
73#14519699—22no—
74#14519684—21no—
75#14519684—22no—
76#14519684—22no—
77#14519684D412yes—
78C22×Q8×C61—21no—
79C22×Q8×C61C2312yes—
80C22×C8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61D414yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91#103×C61—24no—
92#103×C61—24no—
93C4×C8×C61—22no—
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×C61—42no—
101C2×Q8×C61—42no—
102C2×Q8×C61—42no—
103C2×Q8×C61C2×D414yes—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61C2×D412yes—
113C22×Dic61C2×D412yes—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244#4112yes—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C2×C4×D61—42no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61—21no—
127C2×C8×C61—21no—
128C2×C8×C61—21no—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—42no—
134C42×C61C2×D412yes—
135C42×C61#4112yes—
136#11758201—48no—
137#11758198—44no—
138#11758198—42no—
139#11758198—44no—
140C4⋊Dic61—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—44no—
147C4⋊Dic61SD1614yescomplemented
148C4⋊Dic61#4112yes—
149#11758196—44no—
150#11758196—44no—
151C4×Dic61—22no—
152C4×Dic61—22no—
153C4×Dic61—44no—
154C4×Dic61—42no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
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—
170Q8×C61—84no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—24no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61#12112yes—
180C2×C4×C61#12112yes—
181C2×C4×C61C2×SD1612yes—
182C2×C4×C61AGL(1,ℤ/8)12yes—
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61H(5,𝔽2)11yescharacteristic
188C2×C4×C612−1+411yescharacteristic
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61C2×SD1614yes—
204D244—22no—
205D244—22no—
206D244—41no—
207D244—41no—
208C4×D61—84no—
209C8×C61—41no—
210C8×C61—41no—
211#92418—42no—
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—22no—
228C4×C61—44no—
229C4×C61—82no—
230C4×C61#46014yes—
231C4×C61#52911yescharacteristic
232C4×C61#54211yescharacteristic
233C4×C61#54611yescharacteristic
234C4×C61#54711yescharacteristic
235Dic61—24no—
236Dic61—42no—
237Dic61—42no—
238Dic61—81no—
239Dic61—81no—
240C2×#487D6111yescharacteristic, complemented
241C2×#461—611no—
242#2542—611no—
243#2365—611no—
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⋊Q8C2×D6111yescharacteristic
261C2×C4⋊Q8C2×D6111yescharacteristic
262C2×C22⋊Q8—1221no—
263C2×C22⋊Q8—1221no—
264C2×C4×Q8—612no—
265C2×C4×D4—611no—
266#487—24no—
267#461—1224no—
268C2×C4⋊C8—611no—
269C2×C4⋊C8—612no—
270C2×#103—612no—
271C2×#103C2×D6112yes—
272C2×#103C2×D6112yes—
273C2×#102—611no—
274C2×#102—611no—
275C2×C4×C8C2×D6111yescharacteristic
276#399—1222no—
277#391—612no—
278#390—612no—
279#389—1221no—
280#387—1222no—
281#384—612no—
282C4×C4⋊C4—611no—
283C4×C4⋊C4—611no—
284C4×C4⋊C4—1221no—
285#340—611no—
286#340—611no—
287#340—612no—
288#340—612no—
289#336—612no—
290C61#2532011yescharacteristic, complemented, Sylow
291C22×Q8—21no—
292C22×Q8C22×D6112yes—
293C22×D4—611no—
294C23×C4—1221no—
295C2×SD16—1224no—
296C2×SD16—1224no—
297C2×SD16—2442no—
298C22×C8—612no—
299C22×C8C22×D6111yescharacteristic
300C22×C8C22×D6111yescharacteristic
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8C22⋊D6114yes—
305C22⋊Q8—2444no—
306C22⋊Q8—2444no—
307C4×Q8—1224no—
308C4×Q8—2442no—
309C4×D4—1222no—
310C4×D4—1222no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4C22×D6112yes—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1221no—
326C2×C42—612no—
327C2×C42—1222no—
328C2×C42—1221no—
329C2×C42C22×D6111yescharacteristic
330C4⋊C8—1222no—
331C4⋊C8—2442no—
332#103—24no—
333#103—24no—
334#103—1224no—
335#102—1222no—
336#102—1222no—
337C4×C8—22no—
338#95—612no—
339#95—1222no—
340#95—1222no—
341#95—1221no—
342C24—1221no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8C2×C22⋊D6114yes—
352C2×D4—612no—
353C2×D4—1221no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—612no—
361C22×C4—612no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4D4×D6111yescharacteristic
369C22×C4D4×D6111yescharacteristic
370C22×C4C23×D6111yescharacteristic
371SD16—2448no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—1222no—
379C2×C8—1222no—
380C2×C8—1222no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—42no—
386C4⋊C4—614no—
387C4⋊C4—612no—
388C4⋊C4—1221no—
389C4⋊C4—1222no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—2444no—
395C4⋊C4—2444no—
396C4⋊C4—2444no—
397C22⋊C4—2444no—
398C22⋊C4—2444no—
399C22⋊C4—2442no—
400C42—1222no—
401C42—1222no—
402C42—2444no—
403C42—2442no—
404C42#1175822412yes—
405C42C2×C22⋊D6112yes—
406C23—1222no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23—2441no—
413C23C2×D4×D6111yescharacteristic
414Q8—24no—
415Q8—24no—
416Q8—42no—
417Q8—42no—
418Q8—84no—
419D4—1222no—
420D4—1222no—
421D4—2441no—
422D4—2441no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—614no—
432C2×C4—1222no—
433C2×C4—1222no—
434C2×C4—1222no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—4884no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4#1451974912yes—
448C2×C4#1451974912yes—
449C2×C4SD16×D6112yes—
450C2×C4#1451980412yes—
451C2×C4C22×C22⋊D6111yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4#41×D6111yescharacteristic
454C2×C4C2×#1175822411yescharacteristic
455C2×C4#1451982311yescharacteristic
456C2×C4#1451982411yescharacteristic
457C8—41no—
458C8—41no—
459C8—2442no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22—4881no—
464C22—2442no—
465C22—2442no—
466C22—2442no—
467C22C2×#1451974911yescharacteristic
468C22#1684040511yescharacteristic
469C22#1684042011yescharacteristic
470C22C2×SD16×D6111yescharacteristic
471C22C2×#1451980411yescharacteristic
472C22#1684057511yescharacteristic
473C22#1684057511yescharacteristic
474C4—22no—
475C4—44no—
476C4—82no—
477C4—1224no—
478C4—4881no—
479C4—4881no—
480C4—2442no—
481C4—2442no—
482C4#1683996014yes—
483C4#1684029111yescharacteristic
484C4#1684038111yescharacteristic
485C4#1684040011yescharacteristic
486C4#1684041111yescharacteristic
487C2—4882no—
488C2#2144939911yescharacteristic
489C2C2×#1683996011yescharacteristic
490C2#2145534611yescharacteristic
491C2#2145768311yescharacteristic
492C2#2145812311yescharacteristic
493C2#2145845811yescharacteristic
494C2#2145854211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#487C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#487D61
hypocenterC61#25320
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
3. #21448879C2semidirect
4. #21447230C2semidirect
6. #21446772C2semidirect
7. #21446537C2semidirect
9. #21445822C2semidirect
11. C2×C61×#487C2semidirect
147. C4⋊Dic61SD16semidirect
240. C2×#487D61semidirect
290. C61#25320semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
#214507732
#214488792
#214472302
#214471332
#214467722
#214465372
#214461722
#214458222
C4×#145196992
C2×C61×#4872
#2532061