Finite Group Databasebeta

← #29663924

Subgroups — #29663924

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456247C211yesmaximal, characteristic, complemented
2#21455970C212yesmaximal, complemented
3#21445189C211yesmaximal, characteristic, complemented
4#21445184C211yesmaximal, characteristic, complemented
5#21445179C211yesmaximal, characteristic, complemented
6#21445174C211yesmaximal, characteristic, complemented
7#21445167C212yesmaximal, complemented
8#21445160C212yesmaximal, complemented
9#21444582C211yesmaximal, characteristic, complemented
10#21444522C212yesmaximal, complemented
11C8×#41×C61C211yesmaximal, characteristic, complemented
12C2×#14519799—21no—
13C2×#14519799—21no—
14#16840312C2212yes—
15#16840146—21no—
16#16840146—21no—
17#16840142C2211yescharacteristic
18#16840139C2211yescharacteristic
19#16840135C2212yes—
20#16840086—22no—
21#16840047C2212yes—
22#16840042C2212yes—
23#16839962C2212yes—
24#16839957C2212yes—
25#16839907—22no—
26#16839872—22no—
27#16839860—22no—
28#16839819C2212yes—
29#16839515C2211yescharacteristic
30#16839515C2211yescharacteristic
31#16839513C2212yes—
32#16839509C2212yes—
33#16839509C2212yes—
34#16839505C2212yes—
35#16839505C2212yes—
36C4×#41×C61C2211yescharacteristic
37C8×Q8×C61C2211yescharacteristic
38C8×Q8×C61C2211yescharacteristic
39C8×D4×C61—21no—
40C8×D4×C61—21no—
41C8×D4×C61C2211yescharacteristic
42C8×D4×C61C2211yescharacteristic
43C61×#431—21no—
44C61×#431C2211yescharacteristic
45C2×C4×C8×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222—22no—
48#14519799—44no—
49#14519799—44no—
50C2×Dic244—21no—
51C2×Dic244—21no—
52C2×#11758190—22no—
53C2×#11758190—22no—
54C2×#11758201—22no—
55C2×D488—21no—
56C2×D488—21no—
57C2×#11758199—22no—
58#14519777C2312yes—
59#14519774C2312yes—
60#14519765—22no—
61#14519751—22no—
62#14519751—22no—
63C4×D244—22no—
64C4×D244—22no—
65#14519710—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519701—22no—
70#14519698—22no—
71#14519697—22no—
72#14519697—22no—
73C22⋊#92418—22no—
74#14519688—22no—
75C2×#41×C61D411yescharacteristic
76C22×C8×C61—21no—
77C22×C8×C61—21no—
78C22×C8×C61D411yescharacteristic
79C22×C8×C61D411yescharacteristic
80C4×Q8×C61D411yescharacteristic
81C4×Q8×C61C2311yescharacteristic
82C4×D4×C61—21no—
83C4×D4×C61—21no—
84C4×D4×C61D411yescharacteristic
85C4×D4×C61C2311yescharacteristic
86#117×C61—21no—
87#117×C61D411yescharacteristic
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—21no—
91C4⋊C8×C61—21no—
92C4⋊C8×C61C2311yescharacteristic
93C4⋊C8×C61C2311yescharacteristic
94C22⋊C8×C61—21no—
95C22⋊C8×C61—21no—
96C22⋊C8×C61—21no—
97C4×C8×C61—21no—
98C4×C8×C61—21no—
99C4×C8×C61—21no—
100C4×C8×C61D411yescharacteristic
101C4×C8×C61D411yescharacteristic
102C4×C8×C61C2311yescharacteristic
103C4×C8×C61C2311yescharacteristic
104#41×C61—22no—
105#41×C61—22no—
106C2×Q8×C61C2×D411yescharacteristic
107C2×D4×C61—21no—
108C2×D4×C61C2×D411yescharacteristic
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—42no—
116#11758222—44no—
117#11758222—44no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—22no—
121C2×C4×D61—42no—
122C2×Dic122—22no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—21no—
126C2×C8×C61—21no—
127C2×C8×C61—42no—
128C2×C8×C61—42no—
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D411yescharacteristic
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61C2×D411yescharacteristic
136C4⋊C4×C61C2×D411yescharacteristic
137C22⋊C4×C61—21no—
138C22⋊C4×C61—21no—
139C22⋊C4×C61—21no—
140C42×C61—21no—
141C42×C61—21no—
142C42×C61—21no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61#4111yescharacteristic
146C42×C61C2411yescharacteristic
147#11758201—44no—
148#11758199—44no—
149#11758198—42no—
150#11758198—42no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C2×#92418—22no—
154C2×#92418—22no—
155Dic244—42no—
156Dic244—42no—
157D488—42no—
158D488—42no—
159#11758190—44no—
160#11758190—44no—
161C23×C61—21no—
162C23×C61#12111yescharacteristic
163C22×D61—42no—
164Q8×C61—21no—
165Q8×C61#13512yescomplemented
166D4×C61—24no—
167D4×C61—21no—
168D4×C61#13512yescomplemented
169C2×C4×C61—24no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C22⋊D61—84no—
188C2×Dic61—42no—
189D244—42no—
190D244—42no—
191D244—42no—
192D244—42no—
193D244—82no—
194C4×D61—84no—
195Dic122—42no—
196Dic122—42no—
197C8×C61—21no—
198C8×C61—21no—
199C8×C61—41no—
200C8×C61—41no—
201C8×C61AGL(1,ℤ/8)12yes—
202C8×C61AGL(1,ℤ/8)12yes—
203#92418—42no—
204#92418—42no—
205#19269—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—22no—
209C22×C61—21no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61C2×#12111yescharacteristic
213C2×D61—82no—
214C2×D61—82no—
215C2×D61—82no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—22no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—41no—
222C4×C61—41no—
223C4×C61C2×#13511yescharacteristic
224C4×C61C2×#13511yescharacteristic
225C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
226C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
227Dic61—82no—
228#3147—611no—
229#3115—612no—
230C8×#41D6111yescharacteristic, complemented
231#1670—611no—
232#1669—611no—
233#1668—611no—
234#1667—611no—
235#1666—612no—
236#1665—612no—
237#1568—611no—
238#1553—612no—
239C2×C61—41no—
240C2×C61—42no—
241C2×C61#303811yescharacteristic
242C2×C61#305011yescharacteristic
243C2×C61#306711yescharacteristic
244D61—162no—
245C2×#135—1221no—
246C2×#135—1221no—
247#532—612no—
248C4×#41C2×D6111yescharacteristic
249#495—1221no—
250#495—1221no—
251#494—611no—
252#493—611no—
253#492—612no—
254#482—1222no—
255#475—612no—
256#474—612no—
257#460—612no—
258#459—612no—
259#449—1222no—
260C8×Q8C2×D6111yescharacteristic
261C8×Q8C2×D6111yescharacteristic
262#442—1222no—
263#440—1222no—
264C8×D4—21no—
265C8×D4—21no—
266C8×D4C2×D6111yescharacteristic
267C8×D4C2×D6111yescharacteristic
268#432—612no—
269#431—21no—
270#431C2×D6111yescharacteristic
271C2×C4×C8—21no—
272C2×C4×C8C2×D6111yescharacteristic
273#335—611no—
274#335—611no—
275#334—612no—
276#332—612no—
277#332—612no—
278#331—612no—
279#331—612no—
280C61#1926911yescharacteristic, complemented, Sylow
281C2×#41—1222no—
282C2×#41D24411yescharacteristic
283#135—2444no—
284#135—2444no—
285C2×Q16—1221no—
286C2×Q16—1221no—
287C2×SD16—1222no—
288C2×SD16—1222no—
289C2×SD16—1222no—
290C2×D8—1222no—
291C2×D8—1221no—
292C2×D8—1221no—
293C22×C8—21no—
294C22×C8—21no—
295C22×C8C22⋊D6111yescharacteristic
296C22×C8C22⋊D6111yescharacteristic
297C4⋊Q8—612no—
298#127—612no—
299#124—1222no—
300#121—1222no—
301#121—1222no—
302C4×Q8D24411yescharacteristic
303C4×Q8C22×D6111yescharacteristic
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—1222no—
307C4×D4—1222no—
308C4×D4D24411yescharacteristic
309C4×D4C22×D6111yescharacteristic
310#117—21no—
311#117D24411yescharacteristic
312C2×C42—21no—
313C2×C42C22×D6111yescharacteristic
314#107—1222no—
315#106—1222no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—612no—
319C4⋊C8—612no—
320C4⋊C8C22×D6111yescharacteristic
321C4⋊C8C22×D6111yescharacteristic
322#103—1222no—
323#102—1222no—
324#102—1222no—
325#102—1222no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—1222no—
330#97—1222no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8C22⋊D6111yescharacteristic
335C4×C8C22⋊D6111yescharacteristic
336C4×C8C22×D6111yescharacteristic
337C4×C8C22×D6111yescharacteristic
338#41—22no—
339#41—22no—
340#41—2444no—
341#41—2444no—
342C2×Q8—1222no—
343C2×Q8C2×D24411yescharacteristic
344C2×D4—21no—
345C2×D4—1222no—
346C2×D4—1222no—
347C2×D4—1222no—
348C2×D4—2442no—
349C2×D4C2×D24411yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—2442no—
354C22×C4C2×D24411yescharacteristic
355C22×C4#1175822211yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357Q16—2442no—
358Q16—2442no—
359SD16—2444no—
360SD16—2444no—
361SD16—2444no—
362D8—2442no—
363D8—2444no—
364D8—2442no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—1222no—
372C2×C8—1222no—
373C2×C8C2×C22⋊D6111yescharacteristic
374C2×C8C2×C22⋊D6111yescharacteristic
375C2×C8C2×C22⋊D6111yescharacteristic
376C2×C8C2×C22⋊D6111yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—1222no—
380C4⋊C4—1222no—
381C4⋊C4C2×D24411yescharacteristic
382C4⋊C4C2×D24411yescharacteristic
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—2442no—
387C22⋊C4—2442no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42C2×D24411yescharacteristic
392C42#1175822211yescharacteristic
393C42C2×C22⋊D6111yescharacteristic
394C42C23×D6111yescharacteristic
395C23—21no—
396C23—2442no—
397C23#1451975111yescharacteristic
398Q8—21no—
399Q8—2442no—
400Q8—2442no—
401Q8#1451979912yescomplemented
402D4—24no—
403D4—21no—
404D4—4882no—
405D4—4884no—
406D4—2442no—
407D4—2442no—
408D4—2442no—
409D4—2442no—
410D4#1451979912yescomplemented
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—4884no—
424C2×C4—2442no—
425C2×C4#1451975111yescharacteristic
426C2×C4#1451975111yescharacteristic
427C2×C4#1451975111yescharacteristic
428C2×C4C22×C22⋊D6111yescharacteristic
429C2×C4C22×D24411yescharacteristic
430C2×C4C2×#1175822211yescharacteristic
431C8—21no—
432C8—21no—
433C8—41no—
434C8—41no—
435C8—2442no—
436C8—2442no—
437C8#1451980212yes—
438C8#1451980212yes—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—21no—
443C22—41no—
444C22—41no—
445C22—4882no—
446C22—4882no—
447C22—4882no—
448C22C2×#1451975111yescharacteristic
449C4—21no—
450C4—22no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—41no—
455C4—41no—
456C4—4882no—
457C4C2×#1451979911yescharacteristic
458C4C2×#1451979911yescharacteristic
459C4C2×#1451980211yescharacteristic
460C4C2×#1451980211yescharacteristic
461C2—41no—
462C2—42no—
463C2—9762no—
464C2#2145529911yescharacteristic
465C2#2145540011yescharacteristic
466C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×#41×C61C2
socleC22×C61C2×#121
hypercenterC8×#41D61
hypocenterC61#19269
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. #21456247C2semidirect
2. #21455970C2semidirect
3. #21445189C2semidirect
4. #21445184C2semidirect
5. #21445179C2semidirect
6. #21445174C2semidirect
7. #21445167C2semidirect
8. #21445160C2semidirect
9. #21444582C2semidirect
10. #21444522C2semidirect
11. C8×#41×C61C2semidirect
165. Q8×C61#135semidirect
168. D4×C61#135semidirect
230. C8×#41D61semidirect
280. C61#19269semidirect
401. Q8#14519799semidirect
410. D4#14519799semidirect

Maximal subgroups

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

subgroupindex
#214562472
#214559702
#214451892
#214451842
#214451792
#214451742
#214451672
#214451602
#214445822
#214445222
C8×#41×C612
#1926961