Finite Group Databasebeta

← #29663561

Subgroups — #29663561

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic
2#21456944C211yesmaximal, characteristic, complemented
3#21456213C211yesmaximal, characteristic, complemented
4#21445394C212yesmaximal, complemented
5#21445352C212yesmaximal, complemented
6#21445270C212yesmaximal, complemented
7#21445228C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9C2×#16839510C211yesmaximal, characteristic, complemented
10#21444469C211yesmaximal, characteristic, complemented
11C61×#2935C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840158—21no—
15#16840122—22no—
16#16840113C2212yes—
17#16840109C2212yes—
18#16840108—21no—
19#16840078—22no—
20#16840052C2212yes—
21#16840052C2212yes—
22#16840042C2212yes—
23#16840039—21no—
24#16839996—22no—
25#16839957C2212yes—
26#16839954—21no—
27#16839815C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839764—21no—
30#16839764—21no—
31#16839515C2212yes—
32#16839513C2212yes—
33#16839510C2212yes—
34#16839510C2212yes—
35#16839509C2211yescharacteristic
36#16839509C2211yescharacteristic
37#16839505C2211yescharacteristic
38#16839505C2211yescharacteristic
39C2×#131×C61—21no—
40C61×#532C2211yescharacteristic
41C61×#436C2212yes—
42C61×#435—22no—
43C8×D4×C61C2212yes—
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#408—21no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758199—22no—
50C22×#92418—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2311yescharacteristic
55#14519777C2312yes—
56#14519774C2311yescharacteristic
57#14519765—21no—
58#14519751—21no—
59#14519751—21no—
60C4×D244—21no—
61C4×D244—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64#14519710—22no—
65#14519708—22no—
66#14519708—22no—
67C4⋊#92418—21no—
68C4⋊#92418C2312yes—
69C4⋊#92418C2311yescharacteristic
70C4⋊#92418C2311yescharacteristic
71#14519702—22no—
72#14519702—22no—
73#14519699—22no—
74#14519699—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—22no—
78#14519697—22no—
79C22⋊#92418—21no—
80C4×#92418—21no—
81C2×#41×C61—21no—
82#131×C61—44no—
83C2×M16×C61—22no—
84C2×M16×C61D411yescharacteristic
85C2×M16×C61D411yescharacteristic
86C22×C8×C61—21no—
87C22×C8×C61—22no—
88C4⋊Q8×C61D411yescharacteristic
89#127×C61D411yescharacteristic
90#124×C61—21no—
91#121×C61—21no—
92#121×C61D412yes—
93C4×D4×C61—21no—
94C4×D4×C61C2312yes—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99C22⋊C8×C61—22no—
100C22⋊C8×C61—22no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103#41×C61—42no—
104#41×C61—42no—
105C2×Q8×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61C2×D412yes—
110C22×C4×C61—22no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—41no—
115C22×Dic61—41no—
116#11758222—42no—
117#11758222—42no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×C4×D61—41no—
122C2×Dic122—21no—
123C2×Dic122—22no—
124C2×Dic122—41no—
125C2×Dic122—82no—
126M16×C61—22no—
127M16×C61—22no—
128M16×C61—44no—
129C2×C8×C61—21no—
130C2×C8×C61—21no—
131C2×C8×C61—44no—
132C2×C8×C61—42no—
133C2×C8×C61C2×D412yes—
134C2×C8×C61C2×D412yes—
135C4⋊C4×C61—21no—
136C4⋊C4×C61C2×D412yes—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143#11758199—82no—
144#11758198—41no—
145#11758198—41no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—22no—
155C2×#92418—81no—
156C23×C61—22no—
157C23×C61#12111yescharacteristic
158C22×D61—41no—
159Q8×C61—42no—
160D4×C61—22no—
161D4×C61—22no—
162D4×C61—44no—
163D4×C61—42no—
164D4×C61—81no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#13512yes—
175C2×C4×C61AGL(1,ℤ/8)12yes—
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C22⋊D61—82no—
180C2×Dic61—42no—
181C2×Dic61—41no—
182C2×Dic61—82no—
183D244—41no—
184D244—41no—
185D244—41no—
186D244—41no—
187D244—81no—
188C4×D61—82no—
189Dic122—41no—
190Dic122—41no—
191Dic122—82no—
192Dic122—82no—
193C8×C61—22no—
194C8×C61—22no—
195C8×C61—41no—
196C8×C61—41no—
197#92418—41no—
198#92418—42no—
199#92418—41no—
200#19236—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61#46312yes—
207C22×C61C2×#12111yescharacteristic
208C2×D61—81no—
209C2×D61—81no—
210C2×D61—81no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—42no—
215C4×C61#46312yes—
216C4×C61C2×#13511yescharacteristic
217C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
218C4×C61#57711yescharacteristic
219C4×C61#57811yescharacteristic
220Dic61—81no—
221Dic61—82no—
222#3245—611no—
223#3227—611no—
224#3141—611no—
225#2935D6111yescharacteristic, complemented
226#1703—612no—
227#1697—612no—
228#1683—612no—
229#1677—612no—
230#1565—611no—
231C2×#333—611no—
232#1540—611no—
233C2×C61—22no—
234C2×C61—82no—
235C2×C61C2×#46311yescharacteristic
236C2×C61#306311yescharacteristic
237C2×C61#306811yescharacteristic
238D61—161no—
239C2×#131—21no—
240#532—611no—
241#532C2×D6111yescharacteristic
242C2×C4⋊Q8—611no—
243#498—1221no—
244#489—1222no—
245#487—612no—
246#486—612no—
247#486—1221no—
248#481—1222no—
249#476—612no—
250#476—612no—
251#474—612no—
252#474—1221no—
253#466—1222no—
254#459—612no—
255#459—1221no—
256#436C2×D6112yes—
257#435—22no—
258C8×D4C2×D6112yes—
259#431—611no—
260C2×C4⋊C8—611no—
261C2×C4⋊C8C2×D6111yescharacteristic
262#417—1221no—
263#417—1221no—
264#408—21no—
265C4×M16C2×D6111yescharacteristic
266#335—612no—
267#334—612no—
268#333—612no—
269#333—612no—
270#332—611no—
271#332—611no—
272#331—611no—
273#331—611no—
274C61#1923611yescharacteristic, complemented, Sylow
275C2×#41—21no—
276C2×#41—1221no—
277C22×Q8—1221no—
278C2×D8—1222no—
279#131—44no—
280C2×M16—22no—
281C2×M16C22⋊D6111yescharacteristic
282C2×M16C22⋊D6111yescharacteristic
283C22×C8—21no—
284C22×C8—22no—
285C22×C8—1221no—
286C4⋊Q8—612no—
287C4⋊Q8—611no—
288C4⋊Q8—1221no—
289C4⋊Q8—1221no—
290C4⋊Q8—1221no—
291C4⋊Q8D24411yescharacteristic
292#127—611no—
293#127D24411yescharacteristic
294#124—21no—
295#124—1221no—
296#121—21no—
297#121—1221no—
298#121—1221no—
299#121D24412yes—
300C4×D4—21no—
301C4×D4—1221no—
302C4×D4—1221no—
303C4×D4C22×D6112yes—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1221no—
306C2×C42C22×D6111yescharacteristic
307#107—1222no—
308#106—1222no—
309#106—1222no—
310C4⋊C8—21no—
311C4⋊C8—611no—
312C4⋊C8—611no—
313C4⋊C8—612no—
314C4⋊C8—1221no—
315C4⋊C8C22×D6111yescharacteristic
316C4⋊C8C22×D6111yescharacteristic
317#103—1222no—
318#103—1222no—
319#102—1222no—
320#102—1222no—
321#102—1222no—
322#102—1222no—
323#102—1222no—
324#102—1222no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8—1221no—
328#97C22⋊D6112yes—
329C4×C8—1221no—
330C4×C8C22×D6112yes—
331#41—42no—
332#41—42no—
333#41—2442no—
334#41—2442no—
335C2×Q8—21no—
336C2×Q8—1221no—
337C2×Q8—1222no—
338C2×Q8—4882no—
339C2×Q8—2441no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—2441no—
347C2×D4C2×D24412yes—
348C22×C4—22no—
349C22×C4—2441no—
350C22×C4—2441no—
351C22×C4C2×D24411yescharacteristic
352C22×C4#1175822211yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354D8—4882no—
355M16—22no—
356M16—22no—
357M16—44no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—44no—
361C2×C8—42no—
362C2×C8—1221no—
363C2×C8—1221no—
364C2×C8—1222no—
365C2×C8—4881no—
366C2×C8C2×C22⋊D6112yes—
367C2×C8C2×C22⋊D6112yes—
368C4⋊C4—21no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—2441no—
374C4⋊C4—2441no—
375C4⋊C4C2×D24412yes—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C42C2×D24411yescharacteristic
381C42#1175822211yescharacteristic
382C42C2×C22⋊D6111yescharacteristic
383C42C23×D6111yescharacteristic
384C23—22no—
385C23—2441no—
386C23#1451975111yescharacteristic
387Q8—42no—
388Q8—4882no—
389Q8—4882no—
390Q8—2441no—
391Q8—2441no—
392D4—22no—
393D4—22no—
394D4—44no—
395D4—42no—
396D4—81no—
397D4—4881no—
398D4—4882no—
399D4—2441no—
400D4—2441no—
401D4—2441no—
402D4—2441no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—4882no—
410C2×C4—4882no—
411C2×C4—2441no—
412C2×C4—2442no—
413C2×C4#1451975111yescharacteristic
414C2×C4#1451975111yescharacteristic
415C2×C4#1451975111yescharacteristic
416C2×C4#1451979712yes—
417C2×C4#1451980312yes—
418C2×C4C22×C22⋊D6111yescharacteristic
419C2×C4C22×D24411yescharacteristic
420C2×C4C2×#1175822211yescharacteristic
421C8—22no—
422C8—22no—
423C8—41no—
424C8—41no—
425C8—2441no—
426C8—2442no—
427C8—2441no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—4881no—
434C22—4881no—
435C22—4881no—
436C22#1683997712yes—
437C22C2×#1451975111yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—4881no—
443C4—4882no—
444C4#1683997712yes—
445C4C2×#1451979711yescharacteristic
446C4C2×#1451980311yescharacteristic
447C4#1684057411yescharacteristic
448C4#1684057911yescharacteristic
449C2—22no—
450C2—82no—
451C2—9761no—
452C2C2×#1683997711yescharacteristic
453C2#2145551011yescharacteristic
454C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2935C2
socleC22×C61C2×#121
prime radical#21457126C2
hypercenter#2935D61
hypocenterC61#19236
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #21456944C2semidirect
3. #21456213C2semidirect
4. #21445394C2semidirect
5. #21445352C2semidirect
6. #21445270C2semidirect
7. #21445228C2semidirect
8. #21444571C2semidirect
9. C2×#16839510C2semidirect
10. #21444469C2semidirect
11. C61×#2935C2semidirect
225. #2935D61semidirect
274. C61#19236semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214569442
#214562132
#214453942
#214453522
#214452702
#214452282
#214445712
C2×#168395102
#214444692
C61×#29352
#1923661