Finite Group Databasebeta

← #29663504

Subgroups — #29663504

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic
2#21456276C211yesmaximal, characteristic, complemented
3#21454563C211yesmaximal, characteristic, complemented
4#21445306C212yesmaximal, complemented
5#21445286C212yesmaximal, complemented
6#21445182C212yesmaximal, complemented
7#21445162C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444466C211yesmaximal, characteristic, complemented
10C2×#16839508C211yesmaximal, characteristic, complemented
11C61×#3226C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13C2×#14519808—21no—
14C2×#14519784—21no—
15#16840312C2211yescharacteristic
16#16840154—22no—
17#16840141C2212yes—
18#16840136C2212yes—
19#16840042C2212yes—
20#16840039—21no—
21#16839996—22no—
22#16839957C2212yes—
23#16839954—21no—
24#16839840C2212yes—
25#16839834—22no—
26D4×#92418C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839729—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yes—
32#16839512C2212yes—
33#16839508C2212yes—
34#16839508C2212yes—
35#16839507C2211yescharacteristic
36#16839507C2211yescharacteristic
37#16839505C2211yescharacteristic
38#16839505C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#500—21no—
42C61×#486—21no—
43C61×#480—22no—
44C61×#474C2212yes—
45C61×#474C2212yes—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48#14519808—44no—
49#14519808—44no—
50C2×#11758202—22no—
51C2×#11758202—22no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758199—22no—
55#14519784—44no—
56C2×#11758194—22no—
57C2×#11758194D411yescharacteristic
58C2×#11758194D411yescharacteristic
59C22×#92418—21no—
60C22×#92418—21no—
61C22×#92418—22no—
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519710—22no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2311yescharacteristic
75#14519698—22no—
76#14519697—22no—
77C22⋊#92418—22no—
78C22⋊#92418—22no—
79#14519688D412yes—
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61C2312yes—
88#127×C61D411yescharacteristic
89#124×C61—21no—
90#121×C61—21no—
91#121×C61D412yes—
92C4×D4×C61—21no—
93C4×D4×C61C2312yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#107×C61—22no—
98#107×C61—22no—
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2311yescharacteristic
101#102×C61—22no—
102#102×C61—22no—
103C22⋊C8×C61—21no—
104C4×C8×C61—21no—
105#41×C61—42no—
106#41×C61—42no—
107C2×Q8×C61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—41no—
110C2×Q8×C61—42no—
111C2×Q8×C61—42no—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—22no—
117C22×C4×C61—41no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×C22⋊D61—41no—
122#11758222—42no—
123#11758222—42no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61C2×D412yes—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143#11758202—44no—
144#11758202—44no—
145#11758201—44no—
146#11758201—44no—
147#11758199—82no—
148#11758198—41no—
149#11758198—41no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152#11758194—22no—
153#11758194—44no—
154#11758194D814yescomplemented
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—42no—
159C2×#92418—44no—
160C2×#92418—81no—
161C2×#92418C2×D412yes—
162C2×#92418C2×D412yes—
163C23×C61—22no—
164C23×C61#12111yescharacteristic
165C22×D61—41no—
166Q8×C61—42no—
167Q8×C61—42no—
168Q8×C61—42no—
169Q8×C61—82no—
170D4×C61—22no—
171D4×C61—22no—
172D4×C61—44no—
173D4×C61—42no—
174D4×C61—81no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—82no—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61C2×D812yes—
187C2×C4×C61AGL(1,ℤ/8)12yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C22⋊D61—82no—
192C2×Dic61—41no—
193D244—41no—
194D244—41no—
195D244—41no—
196D244—41no—
197D244—81no—
198C4×D61—82no—
199Dic122—41no—
200Dic122—41no—
201C8×C61—41no—
202C8×C61—41no—
203#92418—22no—
204#92418—41no—
205#92418—41no—
206#92418—42no—
207#92418C2×D814yes—
208#19231—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61#45912yes—
215C22×C61C2×#12111yescharacteristic
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—81no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—42no—
223C4×C61—82no—
224C4×C61#45912yes—
225C4×C61C22×D811yescharacteristic
226C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C61#57711yescharacteristic
228C4×C61#57811yescharacteristic
229Dic61—81no—
230#3245—611no—
231#3226D6111yescharacteristic, complemented
232#3152—611no—
233#2935—611no—
234#1689—612no—
235#1685—612no—
236#1669—612no—
237#1665—612no—
238#1565—611no—
239#1540—611no—
240C2×#332—611no—
241C2×C61—22no—
242C2×C61—82no—
243C2×C61C2×#45911yescharacteristic
244C2×C61#306311yescharacteristic
245C2×C61#306911yescharacteristic
246D61—161no—
247C2×#137—1221no—
248C2×#137—1221no—
249C2×#131—1221no—
250#532—611no—
251#532C2×D6111yescharacteristic
252C2×C4⋊Q8C2×D6111yescharacteristic
253#500—21no—
254#497—1222no—
255#494—612no—
256#492—612no—
257#486—21no—
258#480—22no—
259#474—612no—
260#474—1221no—
261#474C2×D6112yes—
262#474C2×D6112yes—
263#466—1222no—
264#459—612no—
265#459—1221no—
266#436—612no—
267#435—1222no—
268C8×D4—612no—
269#431C2×D6111yescharacteristic
270C2×C4⋊C8—611no—
271C2×C4⋊C8—611no—
272#408—1221no—
273C4×M16—611no—
274#335—612no—
275#334—612no—
276#332—612no—
277#332—612no—
278#332—611no—
279#332—611no—
280#331—611no—
281#331—611no—
282C61#1923111yescharacteristic, complemented, Sylow
283C2×#41—21no—
284C2×#41—1221no—
285C22×Q8—21no—
286#137—2444no—
287#137—2444no—
288C2×Q16—1222no—
289C2×Q16—1222no—
290C2×SD16—1222no—
291C2×SD16—1222no—
292C2×D8—1222no—
293#131—2444no—
294C2×M16—611no—
295C2×M16—611no—
296C2×M16—1222no—
297C22×C8—1222no—
298C22×C8—1221no—
299C22×C8—1221no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—611no—
304C4⋊Q8C22⋊D6111yescharacteristic
305C4⋊Q8C22×D6112yes—
306#127—611no—
307#127C22⋊D6111yescharacteristic
308#124—21no—
309#124—1221no—
310#121—21no—
311#121—1221no—
312#121—1221no—
313#121C22⋊D6112yes—
314C4×D4—21no—
315C4×D4—1221no—
316C4×D4—1221no—
317C4×D4C22×D6112yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C42C22×D6111yescharacteristic
321#107—22no—
322#107—22no—
323#107—1222no—
324C4⋊C8—611no—
325C4⋊C8—611no—
326C4⋊C8—612no—
327C4⋊C8—1221no—
328C4⋊C8—1221no—
329C4⋊C8C22×D6111yescharacteristic
330C4⋊C8C22×D6111yescharacteristic
331#102—22no—
332#102—22no—
333#102—1222no—
334#102—1222no—
335C22⋊C8—21no—
336C22⋊C8—1222no—
337C22⋊C8—1222no—
338#97—612no—
339C4×C8—21no—
340C4×C8—612no—
341#41—42no—
342#41—42no—
343#41—2442no—
344#41—2442no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—1221no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—2441no—
358C2×D4C2×C22⋊D6112yes—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—2441no—
362C22×C4D4×D6111yescharacteristic
363C22×C4#1175822411yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365Q16—2444no—
366Q16—2444no—
367SD16—2444no—
368SD16—2444no—
369D8—4882no—
370M16—614no—
371M16—1222no—
372M16—2444no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—612no—
376C2×C8—612no—
377C2×C8—1221no—
378C2×C8—1221no—
379C2×C8—1222no—
380C2×C8—4881no—
381C2×C8—2444no—
382C2×C8—2442no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4C2×C22⋊D6112yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C42D4×D6111yescharacteristic
396C42#1175822411yescharacteristic
397C42C2×C22⋊D6111yescharacteristic
398C42C23×D6111yescharacteristic
399C23—22no—
400C23—2441no—
401C23#1451974911yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—42no—
405Q8—82no—
406Q8—2441no—
407Q8—2441no—
408D4—22no—
409D4—22no—
410D4—44no—
411D4—42no—
412D4—81no—
413D4—4881no—
414D4—4882no—
415D4—2441no—
416D4—2441no—
417D4—2441no—
418D4—2441no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—4882no—
428C2×C4—2441no—
429C2×C4#1451974911yescharacteristic
430C2×C4#1451974911yescharacteristic
431C2×C4#1451974911yescharacteristic
432C2×C4D8×D6112yes—
433C2×C4#1451980112yes—
434C2×C4C2×D4×D6111yescharacteristic
435C2×C4C22×C22⋊D6111yescharacteristic
436C2×C4C2×#1175822411yescharacteristic
437C8—41no—
438C8—41no—
439C8—614no—
440C8—1222no—
441C8—2441no—
442C8—2441no—
443C8—2442no—
444C22—21no—
445C22—21no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—4881no—
450C22—4881no—
451C22—4881no—
452C22#1683995512yes—
453C22C2×#1451974911yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—42no—
458C4—82no—
459C4—4881no—
460C4#1683995512yes—
461C4C2×D8×D6111yescharacteristic
462C4C2×#1451980111yescharacteristic
463C4#1684057011yescharacteristic
464C4#1684057711yescharacteristic
465C2—22no—
466C2—82no—
467C2—9761no—
468C2C2×#1683995511yescharacteristic
469C2#2145550611yescharacteristic
470C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3226C2
socleC22×C61C2×#121
prime radical#21457126C2
hypercenter#3226D61
hypocenterC61#19231
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
2. #21456276C2semidirect
3. #21454563C2semidirect
4. #21445306C2semidirect
5. #21445286C2semidirect
6. #21445182C2semidirect
7. #21445162C2semidirect
8. #21444572C2semidirect
9. #21444466C2semidirect
10. C2×#16839508C2semidirect
11. C61×#3226C2semidirect
154. #11758194D8semidirect
231. #3226D61semidirect
282. C61#19231semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214562762
#214545632
#214453062
#214452862
#214451822
#214451622
#214445722
#214444662
C2×#168395082
C61×#32262
#1923161