Finite Group Databasebeta

← #29663136

Subgroups — #29663136

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457136C211yesmaximal, characteristic
2#21457131C211yesmaximal, characteristic
3#21456329C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic
5#21445341C212yesmaximal, complemented
6#21445331C212yesmaximal, complemented
7#21445197C212yesmaximal, complemented
8#21445193C212yesmaximal, complemented
9C2×#16839512C211yesmaximal, characteristic, complemented
10#21444510C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15#16840165C2211yescharacteristic
16#16840165C2211yescharacteristic
17#16840161—21no—
18#16840161—21no—
19#16840158C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840053C2212yes—
22#16840049C2212yes—
23#16840048—21no—
24#16840047—21no—
25#16839993—22no—
26#16839991—22no—
27#16839968C2212yes—
28#16839964C2212yes—
29#16839963—21no—
30#16839962—21no—
31#16839839—21no—
32#16839839C2212yes—
33#16839827—21no—
34#16839827C2212yes—
35#16839798—21no—
36#16839798—21no—
37C2×C4⋊#92418C2211yescharacteristic
38#16839730—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839512C2212yes—
41#16839512C2212yes—
42#16839510C2212yes—
43#16839507C2212yes—
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yes—
48C61×#332C2212yes—
49C2×#11758222—21no—
50C2×#11758190—22no—
51C2×#11758201—22no—
52#14519785—44no—
53C2×#11758189—22no—
54C22×#92418—21no—
55C22×#92418D411yescharacteristic
56C22×#92418D411yescharacteristic
57C2×C8×D61—22no—
58#14519777D411yescharacteristic
59#14519777C2311yescharacteristic
60#14519777C2311yescharacteristic
61#14519774D411yescharacteristic
62#14519769—21no—
63#14519765—21no—
64#14519756—21no—
65#14519756—21no—
66#14519751—21no—
67#14519751D412yes—
68C4×Dic122—21no—
69C4×Dic122—21no—
70C4×D244—21no—
71C4×D244C2312yes—
72#14519736—21no—
73#14519710—21no—
74#14519710—21no—
75#14519710—21no—
76#14519710—21no—
77#14519709—22no—
78#14519708—22no—
79#14519708—21no—
80#14519708—21no—
81#14519708—21no—
82#14519708—21no—
83C4⋊#92418—21no—
84C4⋊#92418C2312yes—
85#14519702—22no—
86#14519701—22no—
87#14519699—22no—
88#14519697—22no—
89#14519689—22no—
90#14519689—22no—
91C4×#92418D411yescharacteristic
92C4×#92418D411yescharacteristic
93C4×#92418C2311yescharacteristic
94C4×#92418C2311yescharacteristic
95C2×M16×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61C2311yescharacteristic
98C4⋊Q8×C61C2311yescharacteristic
99#125×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104#117×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—22no—
116#11758222—42no—
117#11758222—42no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244C2×D412yes—
121C2×C4×D61—22no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122—21no—
125M16×C61—42no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—81no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61#4111yescharacteristic
140C42×C61C2411yescharacteristic
141C22⋊Dic61—41no—
142#11758201—82no—
143#11758198—22no—
144#11758198—22no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C2×D412yes—
151#11758196—41no—
152#11758196—41no—
153C4×Dic61—41no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—81no—
157C2×#92418SD1614yescomplemented
158C2×#92418C2×D411yescharacteristic
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162#11758190—82no—
163#11758189—44no—
164C8×D61—44no—
165C23×C61#12111yescharacteristic
166C22×D61—22no—
167Q8×C61—41no—
168Q8×C61—41no—
169Q8×C61—81no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C2×SD1612yes—
179C2×C4×C61#13712yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C22⋊D61—44no—
184C2×Dic61—22no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187D244—22no—
188D244—22no—
189D244—42no—
190D244—42no—
191C4×D61—42no—
192C4×D61—42no—
193Dic122—42no—
194Dic122—41no—
195Dic122—41no—
196Dic122—81no—
197C8×C61—42no—
198C8×C61—42no—
199#92418—21no—
200#92418—21no—
201#92418—42no—
202#92418C2×SD1612yes—
203#92418C2×SD1612yes—
204#19194—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61#46112yes—
208C22×C61C2×#12111yescharacteristic
209C2×D61—42no—
210C2×D61—42no—
211C2×D61—42no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—81no—
216C4×C61—81no—
217C4×C61#46112yes—
218C4×C61C22×SD1611yescharacteristic
219C4×C61C2×#13711yescharacteristic
220C4×C61#57711yescharacteristic
221C4×C61#57711yescharacteristic
222Dic61—42no—
223Dic61—81no—
224Dic61—81no—
225#3246—611no—
226#3246—611no—
227#3160—611no—
228#2931—611no—
229#1695—612no—
230#1693—612no—
231#1671—612no—
232#1671—612no—
233C2×#334—611no—
234#1550—611no—
235#1550D6111yescharacteristic, complemented
236C2×C61—22no—
237C2×C61C2×#46111yescharacteristic
238C2×C61#306011yescharacteristic
239C2×C61#306711yescharacteristic
240D61—82no—
241C2×#131—1221no—
242#533—611no—
243#533C2×D6111yescharacteristic
244#532—611no—
245#500—611no—
246#500—611no—
247#499—1221no—
248#499—1221no—
249#498—611no—
250#498—611no—
251#476—612no—
252#476—612no—
253#475—1221no—
254#475—1221no—
255#465—1222no—
256#465—1222no—
257#461—612no—
258#461—612no—
259#460—1221no—
260#460—1221no—
261#436—612no—
262#436—1221no—
263C8×D4—612no—
264C8×D4—1221no—
265#427—1221no—
266#427—1221no—
267#423C2×D6111yescharacteristic
268C2×C4⋊C8—611no—
269C2×C4⋊C8C2×D6111yescharacteristic
270#408—1221no—
271C2×C4×C8—611no—
272#334—612no—
273#334—612no—
274#333—612no—
275#333C2×D6112yes—
276#332—612no—
277#332C2×D6112yes—
278C61#1919411yescharacteristic, complemented, Sylow
279C2×#41—1221no—
280C2×SD16—1222no—
281C2×SD16—1222no—
282#131—2444no—
283C2×M16—21no—
284C2×M16—1222no—
285C22×C8—21no—
286C22×C8—611no—
287C22×C8—611no—
288C22×C8—1222no—
289C22×C8—1221no—
290C4⋊Q8—611no—
291C4⋊Q8—611no—
292C4⋊Q8—611no—
293C4⋊Q8C22×D6111yescharacteristic
294C4⋊Q8C22×D6111yescharacteristic
295#127—611no—
296#125—21no—
297#125—1221no—
298#124—1221no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—1221no—
302C22⋊Q8—1221no—
303#121—612no—
304#121—1221no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—1221no—
308C4×Q8—1221no—
309C4×D4—612no—
310C4×D4—1221no—
311#117—21no—
312#117—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1221no—
315#107—1221no—
316#107—1221no—
317#107—1221no—
318#106—1222no—
319#106—1221no—
320#106—1221no—
321#106—1221no—
322#106—1221no—
323#106—1222no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—612no—
327C4⋊C8—1221no—
328C4⋊C8C22×D6112yes—
329C4⋊C8C22×D6112yes—
330#103—1222no—
331#103—1222no—
332#102—1222no—
333#102—1222no—
334C22⋊C8—1222no—
335C22⋊C8—1222no—
336C4×C8—611no—
337C4×C8—611no—
338C4×C8—611no—
339C4×C8—611no—
340#41—2442no—
341#41—2442no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—1221no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×D4—612no—
348C2×D4—1222no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—1222no—
352C22×C4#1175822211yescharacteristic
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355SD16—4882no—
356SD16—4882no—
357M16—42no—
358M16—2444no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—81no—
362C2×C8—611no—
363C2×C8—611no—
364C2×C8—614no—
365C2×C8—611no—
366C2×C8—611no—
367C2×C8—1222no—
368C2×C8—1222no—
369C2×C8—4881no—
370C2×C8—2444no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—612no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—2441no—
384C4⋊C4—2441no—
385C22⋊C4—41no—
386C22⋊C4—1222no—
387C22⋊C4—1222no—
388C22⋊C4—2441no—
389C42—41no—
390C42—2441no—
391C42#1175822211yescharacteristic
392C42D4×D6111yescharacteristic
393C42D4×D6111yescharacteristic
394C42C23×D6111yescharacteristic
395C23—1222no—
396C23#1451975011yescharacteristic
397Q8—41no—
398Q8—41no—
399Q8—81no—
400Q8—4881no—
401Q8—2441no—
402Q8—2441no—
403Q8—2442no—
404D4—1222no—
405D4—1222no—
406D4—2442no—
407D4—2444no—
408D4—2442no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—1222no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4#1451975011yescharacteristic
420C2×C4#1451975011yescharacteristic
421C2×C4#1451975011yescharacteristic
422C2×C4SD16×D6112yes—
423C2×C4#1451980912yes—
424C2×C4C2×D4×D6111yescharacteristic
425C2×C4C2×D4×D6111yescharacteristic
426C2×C4C2×#1175822211yescharacteristic
427C8—42no—
428C8—42no—
429C8—612no—
430C8—612no—
431C8—1221no—
432C8—1221no—
433C8—2442no—
434C22—21no—
435C22—21no—
436C22—2442no—
437C22—2442no—
438C22—2442no—
439C22#1683996612yes—
440C22C2×#1451975011yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—81no—
445C4—81no—
446C4—4881no—
447C4—4881no—
448C4—2442no—
449C4#1683996612yes—
450C4C2×SD16×D6111yescharacteristic
451C4C2×#1451980911yescharacteristic
452C4#1684057111yescharacteristic
453C4#1684057211yescharacteristic
454C2—22no—
455C2—4882no—
456C2C2×#1683996611yescharacteristic
457C2#2145548011yescharacteristic
458C2#2145553811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1550D61
hypocenterC61#19194
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
3. #21456329C2semidirect
5. #21445341C2semidirect
6. #21445331C2semidirect
7. #21445197C2semidirect
8. #21445193C2semidirect
9. C2×#16839512C2semidirect
10. #21444510C2semidirect
11. C61×#1550C2semidirect
157. C2×#92418SD16semidirect
235. #1550D61semidirect
278. C61#19194semidirect

Maximal subgroups

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

subgroupindex
#214571362
#214571312
#214563292
#214545302
#214453412
#214453312
#214451972
#214451932
C2×#168395122
#214445102
C61×#15502
#1919461