Finite Group Databasebeta

← #29834667

Subgroups — #29834667

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456331C211yesmaximal, characteristic, complemented
2C2×#16839844C211yesmaximal, characteristic
3#21447979C212yesmaximal, complemented
4#21447975C212yesmaximal, complemented
5#21447965C212yesmaximal, complemented
6#21447955C212yesmaximal, complemented
7#21447605C211yesmaximal, characteristic, complemented
8#21447574C211yesmaximal, characteristic, complemented
9#21447536C211yesmaximal, characteristic
10#21447534C211yesmaximal, characteristic
11C61×#2159C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13#16840315—21no—
14C2×#14519739—21no—
15C2×#14519717C2211yescharacteristic
16#16840166C2211yescharacteristic, complemented
17#16840166C2211yescharacteristic, complemented
18#16840163C2211yescharacteristic, complemented
19#16840163C2211yescharacteristic, complemented
20#16840162—21no—
21#16840159—21no—
22#16839844—21no—
23#16839844—21no—
24#16839844C2212yes—
25#16839844C2212yes—
26#16839800C2211yescharacteristic
27#16839798—21no—
28C2×#14519697—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839589C2212yescomplemented
31#16839587C2212yescomplemented
32#16839580C2212yescomplemented
33#16839580C2212yescomplemented
34#16839564C2212yes—
35#16839564C2212yes—
36#16839562C2212yes—
37#16839562C2212yes—
38C2×M32×C61C2211yescharacteristic
39C61×#427C2211yescharacteristic
40C2×#107×C61C2211yescharacteristic
41C61×#367C2212yescomplemented
42C61×#366C2212yescomplemented
43C22×D244—21no—
44C22×C4×D61—41no—
45C2×D488—21no—
46C2×D488—22no—
47C2×D488Q814yescomplemented
48C2×D488C2312yes—
49C22×#92418D411yescharacteristic
50#14519776—21no—
51#14519776—21no—
52#14519768—21no—
53#14519768—21no—
54#14519754—41no—
55#14519754—41no—
56C4×Dic122—41no—
57C4×Dic122—41no—
58#14519739—22no—
59#14519739—22no—
60#14519739—41no—
61#14519739—41no—
62#14519736—21no—
63C2×#11758198—41no—
64C2×C4×Dic61D411yescharacteristic
65#14519717—22no—
66C2×#11758185C2312yes—
67#14519711C2311yescharacteristic
68#14519711C2311yescharacteristic
69#14519710—21no—
70#14519710—21no—
71#14519709—21no—
72#14519708—21no—
73#14519708—21no—
74#14519697—22no—
75#14519697—22no—
76C8×Dic61D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C8×Dic61C2311yescharacteristic
80C22×C8×C61C2311yescharacteristic
81#117×C61—21no—
82C2×C4⋊C4×C61—21no—
83M32×C61—22no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2311yescharacteristic
87#107×C61C2312yes—
88#107×C61C2311yescharacteristic
89#106×C61—21no—
90C23×D61—41no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61D811yescharacteristic
94C22×Dic61D811yescharacteristic
95C2×D244—24no—
96C2×D244—22no—
97C2×D244—41no—
98C2×D244—42no—
99C2×C4×D61—42no—
100C2×C4×D61—42no—
101C2×C4×D61—84no—
102C2×Dic122—41no—
103C2×Dic122—41no—
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2×Q812yes—
106C2×C8×C61#4112yes—
107C2×C8×C61C2411yescharacteristic
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61—41no—
114C22⋊Dic61—41no—
115#11758198—42no—
116#11758198—42no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—41no—
120C4⋊Dic61—41no—
121#11758196—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—41no—
126C4×Dic61D811yescharacteristic
127C4×Dic61D811yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418—22no—
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133D488—24no—
134D488—22no—
135D488C2×Q814yes—
136C16×C61—22no—
137#11758185—22no—
138C23×C61C2×D811yescharacteristic
139C22×D61—42no—
140C22×D61—44no—
141C22×D61—82no—
142C22×D61—82no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—82no—
147C2×C4×C61C22⋊Q812yes—
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61#13712yes—
150C2×C4×C61C22×D411yescharacteristic
151C2×Dic61—22no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61C2×D811yescharacteristic
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×D811yescharacteristic
158C2×Dic61C2×D811yescharacteristic
159D244—24no—
160D244—42no—
161D244—44no—
162C4×D61—84no—
163C4×D61—84no—
164Dic122—82no—
165Dic122—82no—
166C8×C61C22⋊Q812yes—
167C8×C61C22×Q811yescharacteristic
168C8×C61C2×#4111yescharacteristic
169#92418—21no—
170#92418—21no—
171#33627—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61#47412yes—
175C22×C61C22×D811yescharacteristic
176C2×D61—44no—
177C2×D61—88no—
178C2×D61—82no—
179C2×D61—82no—
180C4×C61—81no—
181C4×C61—82no—
182C4×C61—81no—
183C4×C61#47412yes—
184C4×C61C2×C22⋊Q811yescharacteristic
185C4×C61C2×#13711yescharacteristic
186Dic61—21no—
187Dic61—21no—
188Dic61—21no—
189Dic61—81no—
190Dic61—81no—
191Dic61#50912yes—
192#3160—611no—
193C2×C4×D8—611no—
194#2228—612no—
195#2228—612no—
196#2226—612no—
197#2224—612no—
198#2159—611no—
199#2159D6111yescharacteristic, complemented
200#2150—611no—
201#2140—611no—
202#2140—611no—
203C2×C61—22no—
204C2×C61C2×#47411yescharacteristic
205C2×C61C2×#50911yescharacteristic
206C2×C61#341411yescharacteristic
207D61—84no—
208C22×D8—611no—
209#533—1221no—
210C2×C4×D4—1221no—
211C2×M32—611no—
212C2×M32C2×D6111yescharacteristic
213#500—611no—
214#500—611no—
215#499—611no—
216#499—611no—
217#499—1221no—
218#498—1221no—
219C4×D8—612no—
220C4×D8—612no—
221C4×D8—1221no—
222C4×D8—1221no—
223#427—611no—
224#427—1221no—
225#427C2×D6111yescharacteristic
226C2×#107C2×D6111yescharacteristic
227C2×#102—1221no—
228C2×C4×C8—611no—
229#367—612no—
230#367C2×D6112yescomplemented
231#366—612no—
232#366C2×D6112yescomplemented
233C4⋊C16—612no—
234C4⋊C16—612no—
235#357—612no—
236#357—612no—
237#357—612no—
238#357—612no—
239C61#3362711yescharacteristic, complemented, Sylow
240C22×D4—1221no—
241C23×C4—2441no—
242C2×D8—614no—
243C2×D8—612no—
244C2×D8—1222no—
245C2×D8—1221no—
246C22×C8—611no—
247C22×C8C22×D6111yescharacteristic
248C4⋊Q8—1221no—
249C4⋊Q8—1221no—
250#125—1221no—
251#125—1221no—
252C22⋊Q8—2441no—
253C22⋊Q8—2441no—
254C4×Q8—2441no—
255C4×Q8—2441no—
256C4×D4—1222no—
257C4×D4—1222no—
258C4×D4—2441no—
259C4×D4—2441no—
260#117—21no—
261#117—1221no—
262C2×C4⋊C4—21no—
263C2×C22⋊C4—2441no—
264C2×C42—611no—
265M32—22no—
266M32—1222no—
267C2×C16—612no—
268C2×C16C22×D6112yes—
269#107—21no—
270#107—611no—
271#107—611no—
272#107—1221no—
273#107—1221no—
274#107C22×D6111yescharacteristic
275#107C22×D6111yescharacteristic
276#107C22×D6112yes—
277#106—21no—
278#106—1221no—
279#106—1221no—
280#106—1221no—
281#102—1222no—
282#102—1222no—
283C4×C8—611no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287C24—2441no—
288C2×Q8—2441no—
289C2×Q8—2441no—
290C2×D4—1222no—
291C2×D4—1224no—
292C2×D4—2442no—
293C2×D4—2441no—
294C22×C4—41no—
295C22×C4—611no—
296C22×C4—611no—
297C22×C4—4884no—
298C22×C4—2442no—
299C22×C4—2442no—
300C22×C4D4×D6111yescharacteristic
301D8—614no—
302D8—1222no—
303D8—1224no—
304C2×C8—611no—
305C2×C8—611no—
306C2×C8—1222no—
307C2×C8#1175822212yes—
308C2×C8D4×D6111yescharacteristic
309C2×C8Q8×D6112yes—
310C2×C8C23×D6111yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—1221no—
316C4⋊C4—1221no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C22⋊C4—41no—
324C22⋊C4—2442no—
325C22⋊C4—2442no—
326C22⋊C4—2441no—
327C42—41no—
328C42—611no—
329C42—611no—
330C42—611no—
331C42—611no—
332C42—2441no—
333C16—22no—
334C16—1222no—
335C23—4882no—
336C23—4882no—
337C23—2442no—
338C23—2444no—
339C23D8×D6111yescharacteristic
340Q8—4882no—
341Q8—4882no—
342D4—1224no—
343D4—2444no—
344D4—2442no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—611no—
353C2×C4—1222no—
354C2×C4—4884no—
355C2×C4—4884no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4#1451975512yes—
360C2×C4D8×D6111yescharacteristic
361C2×C4#1451980912yes—
362C2×C4C2×D4×D6111yescharacteristic
363C8—1221no—
364C8—1221no—
365C8#1451975512yes—
366C8C2×Q8×D6111yescharacteristic
367C8C2×#1175822211yescharacteristic
368C22—21no—
369C22—21no—
370C22—4888no—
371C22—4882no—
372C22—4882no—
373C22—2444no—
374C22#1684004112yes—
375C22C2×D8×D6111yescharacteristic
376C4—81no—
377C4—82no—
378C4—81no—
379C4—612no—
380C4—1221no—
381C4—1221no—
382C4—1221no—
383C4—4881no—
384C4—4881no—
385C4#1684004112yes—
386C4C2×#1451975511yescharacteristic
387C4C2×#1451980911yescharacteristic
388C2—22no—
389C2—4884no—
390C2C2×#1684004111yescharacteristic
391C2#509×D6111yescharacteristic
392C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radicalC22×D488C22
hypercenter#2159D61
hypocenterC61#33627
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. #21456331C2semidirect
3. #21447979C2semidirect
4. #21447975C2semidirect
5. #21447965C2semidirect
6. #21447955C2semidirect
7. #21447605C2semidirect
8. #21447574C2semidirect
11. C61×#2159C2semidirect
16. #16840166C22semidirect
17. #16840166C22semidirect
18. #16840163C22semidirect
19. #16840163C22semidirect
30. #16839589C22semidirect
31. #16839587C22semidirect
32. #16839580C22semidirect
33. #16839580C22semidirect
41. C61×#367C22semidirect
42. C61×#366C22semidirect
47. C2×D488Q8semidirect
199. #2159D61semidirect
230. #367C2×D61semidirect
232. #366C2×D61semidirect
239. C61#33627semidirect

Maximal subgroups

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

subgroupindex
#214563312
C2×#168398442
#214479792
#214479752
#214479652
#214479552
#214476052
#214475742
#214475362
#214475342
C61×#21592
#3362761