Finite Group Databasebeta

← #29663918

Subgroups — #29663918

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#21455964C212yesmaximal, complemented
2C8×#11758222C211yesmaximal, characteristic, complemented
3#21445186C211yesmaximal, characteristic, complemented
4#21445181C211yesmaximal, characteristic, complemented
5#21445176C211yesmaximal, characteristic, complemented
6#21445171C211yesmaximal, characteristic, complemented
7#21445164C212yesmaximal, complemented
8#21445157C212yesmaximal, complemented
9#21444582C211yesmaximal, characteristic, complemented
10#21444524C212yesmaximal, complemented
11C61×#3147C211yesmaximal, characteristic, complemented
12C4×#11758222C2211yescharacteristic
13#16840083—22no—
14#16840044C2212yes—
15#16840039C2212yes—
16#16839959C2212yes—
17#16839954C2212yes—
18#16839905—22no—
19C8×Dic122C2211yescharacteristic
20C8×Dic122C2211yescharacteristic
21#16839872—22no—
22#16839862—22no—
23C8×D244C2211yescharacteristic
24C8×D244C2211yescharacteristic
25C8×C22⋊D61—21no—
26C8×C22⋊D61—21no—
27#16839819C2212yes—
28#16839815C2211yescharacteristic
29#16839813—21no—
30C4×C8×D61—21no—
31#16839515C2211yescharacteristic
32#16839515C2211yescharacteristic
33#16839513C2212yes—
34#16839508C2212yes—
35#16839508C2212yes—
36#16839506C2212yes—
37#16839506C2212yes—
38C2×#135×C61—21no—
39C2×#135×C61—21no—
40C61×#532C2212yes—
41C61×#495—21no—
42C61×#495—21no—
43C61×#494C2211yescharacteristic
44C61×#493C2211yescharacteristic
45C61×#492C2212yes—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222D411yescharacteristic
48C2×#11758200—22no—
49C2×#11758199—22no—
50C2×C8×D61—21no—
51C2×C8×D61—21no—
52C4×Dic122D411yescharacteristic
53C4×Dic122C2311yescharacteristic
54C4×D244D411yescharacteristic
55C4×D244C2311yescharacteristic
56C4×C22⋊D61—21no—
57C4×C22⋊D61—21no—
58#14519736D411yescharacteristic
59#14519734—21no—
60C42×D61—21no—
61#14519710—22no—
62#14519708—22no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65C4⋊#92418C2311yescharacteristic
66C4⋊#92418C2311yescharacteristic
67#14519706—21no—
68#14519706—21no—
69#14519700—22no—
70#14519698—22no—
71#14519698—22no—
72#14519697—22no—
73C22⋊#92418—21no—
74C22⋊#92418—22no—
75#14519689—21no—
76#14519689—21no—
77#14519688—22no—
78C4×#92418—21no—
79C8×Dic61—21no—
80C8×Dic61—21no—
81C2×#41×C61—22no—
82#135×C61—44no—
83#135×C61—44no—
84C2×Q16×C61—21no—
85C2×Q16×C61—21no—
86C2×SD16×C61—22no—
87C2×SD16×C61—22no—
88C2×D8×C61—21no—
89C2×D8×C61—21no—
90C22×C8×C61D411yescharacteristic
91C22×C8×C61D411yescharacteristic
92C4⋊Q8×C61C2312yes—
93#127×C61C2312yes—
94#124×C61—22no—
95#121×C61—22no—
96#121×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C2×C42×C61C2311yescharacteristic
100C4×C8×C61D411yescharacteristic
101C4×C8×C61D411yescharacteristic
102C4×C8×C61C2311yescharacteristic
103C4×C8×C61C2311yescharacteristic
104#41×C61—44no—
105#41×C61—44no—
106C2×Q8×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—21no—
116#11758222—22no—
117#11758222—22no—
118C2×D244C2×D411yescharacteristic
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×Dic122C2×D411yescharacteristic
123Q16×C61—42no—
124Q16×C61—42no—
125SD16×C61—44no—
126SD16×C61—44no—
127D8×C61—42no—
128D8×C61—42no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2×D411yescharacteristic
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C22⋊C4×C61—42no—
138C22⋊C4×C61—42no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143C22⋊Dic61—21no—
144#11758200—44no—
145#11758199—44no—
146#11758198—21no—
147#11758198—21no—
148C4⋊Dic61C2×D411yescharacteristic
149C4⋊Dic61C2×D411yescharacteristic
150#11758196—21no—
151#11758196—21no—
152C4×Dic61—21no—
153C4×Dic61—21no—
154C4×Dic61—21no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—22no—
159C8×D61—42no—
160C8×D61—42no—
161C23×C61—42no—
162C23×C61#12111yescharacteristic
163C22×D61—21no—
164Q8×C61—42no—
165Q8×C61—42no—
166D4×C61—42no—
167D4×C61—42no—
168D4×C61—42no—
169D4×C61—42no—
170D4×C61—82no—
171D4×C61—84no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—84no—
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C2×#4111yescharacteristic
185C22⋊D61—24no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189D244—21no—
190D244#13512yescomplemented
191C4×D61—24no—
192C4×D61—42no—
193C4×D61—41no—
194C4×D61—41no—
195Dic122—21no—
196Dic122#13512yescomplemented
197C8×C61—21no—
198C8×C61—21no—
199C8×C61AGL(1,ℤ/8)12yes—
200C8×C61AGL(1,ℤ/8)12yes—
201#92418—42no—
202#92418—42no—
203#92418—41no—
204#92418—41no—
205#19269—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—82no—
210C22×C61—82no—
211C22×C61—82no—
212C22×C61C2×#12111yescharacteristic
213C2×D61—22no—
214C2×D61—41no—
215C2×D61—41no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—82no—
221C4×C61C2×#13511yescharacteristic
222C4×C61C2×#13511yescharacteristic
223C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
224C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
225Dic61—22no—
226Dic61—41no—
227Dic61—41no—
228#3147D6111yescharacteristic, complemented
229#3115—612no—
230C8×#41—611no—
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—162no—
241C2×C61#303811yescharacteristic
242C2×C61#305011yescharacteristic
243C2×C61#306711yescharacteristic
244D61—42no—
245C2×#135—21no—
246C2×#135—21no—
247#532C2×D6112yes—
248C4×#41—611no—
249#495—21no—
250#495—21no—
251#494C2×D6111yescharacteristic
252#493C2×D6111yescharacteristic
253#492C2×D6112yes—
254#482—1222no—
255#475—612no—
256#474—612no—
257#460—612no—
258#459—612no—
259#449—1222no—
260C8×Q8—611no—
261C8×Q8—611no—
262#442—1222no—
263#440—1222no—
264C8×D4—611no—
265C8×D4—611no—
266C8×D4—1221no—
267C8×D4—1221no—
268#432—612no—
269#431—611no—
270#431—1221no—
271C2×C4×C8—1221no—
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—22no—
282C2×#41—611no—
283#135—44no—
284#135—44no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—1222no—
290C2×D8—21no—
291C2×D8—21no—
292C2×D8—1222no—
293C22×C8—1221no—
294C22×C8—1221no—
295C22×C8C22⋊D6111yescharacteristic
296C22×C8C22⋊D6111yescharacteristic
297C4⋊Q8C22×D6112yes—
298#127C22×D6112yes—
299#124—22no—
300#121—22no—
301#121—22no—
302C4×Q8—611no—
303C4×Q8—611no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—611no—
307C4×D4—611no—
308C4×D4—1221no—
309C4×D4—1221no—
310#117—611no—
311#117—1221no—
312C2×C42—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1222no—
315#106—1222no—
316C4⋊C8—611no—
317C4⋊C8—611no—
318C4⋊C8—612no—
319C4⋊C8—612no—
320C4⋊C8—1221no—
321C4⋊C8—1221no—
322#103—1222no—
323#102—1222no—
324#102—1222no—
325#102—1222no—
326C22⋊C8—1222no—
327C22⋊C8—1221no—
328C22⋊C8—1221no—
329C22⋊C8—1221no—
330#97—1222no—
331C4×C8—1221no—
332C4×C8—1221no—
333C4×C8—1221no—
334C4×C8C22⋊D6111yescharacteristic
335C4×C8C22⋊D6111yescharacteristic
336C4×C8C22×D6111yescharacteristic
337C4×C8C22×D6111yescharacteristic
338#41—44no—
339#41—44no—
340#41—1222no—
341#41—1222no—
342C2×Q8—22no—
343C2×Q8—611no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—42no—
348C2×D4—611no—
349C2×D4—1221no—
350C22×C4—42no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4D4×D6111yescharacteristic
355C22×C4#1175822411yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357Q16—42no—
358Q16—42no—
359SD16—44no—
360SD16—44no—
361SD16—2444no—
362D8—42no—
363D8—42no—
364D8—2444no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—1222no—
368C2×C8—1222no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—2442no—
372C2×C8—2442no—
373C2×C8C2×C22⋊D6111yescharacteristic
374C2×C8C2×C22⋊D6111yescharacteristic
375C2×C8C2×C22⋊D6111yescharacteristic
376C2×C8C2×C22⋊D6111yescharacteristic
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—611no—
380C4⋊C4—611no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—1221no—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C42—1221no—
389C42—1221no—
390C42—1221no—
391C42D4×D6111yescharacteristic
392C42#1175822411yescharacteristic
393C42C2×C22⋊D6111yescharacteristic
394C42C23×D6111yescharacteristic
395C23—42no—
396C23—1221no—
397C23#1451974811yescharacteristic
398Q8—42no—
399Q8—42no—
400Q8—612no—
401Q8—1221no—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—82no—
407D4—84no—
408D4—612no—
409D4—1224no—
410D4—1221no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4—1224no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4#1451974811yescharacteristic
426C2×C4#1451974811yescharacteristic
427C2×C4#1451974811yescharacteristic
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4C22×C22⋊D6111yescharacteristic
430C2×C4C2×#1175822411yescharacteristic
431C8—21no—
432C8—21no—
433C8—2442no—
434C8—2442no—
435C8—2441no—
436C8—2441no—
437C8#1451980312yes—
438C8#1451980312yes—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—1222no—
446C22—2441no—
447C22—2441no—
448C22C2×#1451974811yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—82no—
454C4—1222no—
455C4—2441no—
456C4—2441no—
457C4#135×D6111yescharacteristic
458C4#135×D6111yescharacteristic
459C4C2×#1451980311yescharacteristic
460C4C2×#1451980311yescharacteristic
461C2—41no—
462C2—162no—
463C2—2442no—
464C2#2145529511yescharacteristic
465C2#2145539611yescharacteristic
466C2#2145553511yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3147C2
socleC22×C61C2×#121
hypercenter#3147D61
hypocenterC61#19269
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #21455964C2semidirect
2. C8×#11758222C2semidirect
3. #21445186C2semidirect
4. #21445181C2semidirect
5. #21445176C2semidirect
6. #21445171C2semidirect
7. #21445164C2semidirect
8. #21445157C2semidirect
9. #21444582C2semidirect
10. #21444524C2semidirect
11. C61×#3147C2semidirect
190. D244#135semidirect
196. Dic122#135semidirect
228. #3147D61semidirect
280. C61#19269semidirect

Maximal subgroups

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

subgroupindex
#214559642
C8×#117582222
#214451862
#214451812
#214451762
#214451712
#214451642
#214451572
#214445822
#214445242
C61×#31472
#1926961