Finite Group Databasebeta

← #29835010

Subgroups — #29835010

Order 15616.

552 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456406C211yesmaximal, characteristic, complemented
2C4×#14519796C211yesmaximal, characteristic
3#21448029C212yesmaximal, complemented
4#21447958C211yesmaximal, characteristic, complemented
5#21447870C211yesmaximal, characteristic, complemented
6#21447864C211yesmaximal, characteristic, complemented
7#21447760C212yesmaximal
8#21447677C212yesmaximal, complemented
9#21447674C212yesmaximal, complemented
10#21447617C211yesmaximal, characteristic, complemented
11C61×#2224C211yesmaximal, characteristic, complemented
12C2×#14519796C2211yescharacteristic
13#16840317—21no—
14C4×#11758224—21no—
15C2×Dic488C2212yes—
16C2×#14519721C2212yes—
17#16840142C2211yescharacteristic
18#16840114—21no—
19#16840004—21no—
20#16840004C2212yes—
21C4×Dic244C2211yescharacteristic
22#16839856—21no—
23#16839856C2212yes—
24#16839856C2211yescharacteristic
25#16839853—21no—
26C4×#11758200—21no—
27D8×Dic61C2211yescharacteristic
28#16839799C2211yescharacteristic
29#16839755—21no—
30C4×C8×D61C2211yescharacteristic
31#16839589C2212yes—
32C4⋊#11758185C2211yescharacteristic
33#16839567C2212yes—
34#16839565C2212yes—
35#16839565C2212yes—
36#16839541C2212yes—
37C16×Dic61C2212yes—
38C61×#500C2211yescharacteristic
39C4×D8×C61C2211yescharacteristic
40C61×#366C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#357C2212yes—
43C2×#11758224—21no—
44#14519796—22no—
45#14519796—22no—
46C2×Dic244—21no—
47C2×Dic244C2312yes—
48C2×Dic244C2311yescharacteristic
49C2×#11758200—21no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61C2311yescharacteristic
52#14519777—21no—
53#14519768—41no—
54#14519757—22no—
55#14519753—41no—
56Q8×Dic61—41no—
57C4×Dic122—21no—
58#14519742—21no—
59#14519742—22no—
60D4×Dic61—21no—
61C4×C22⋊D61—41no—
62#14519737—41no—
63#14519735—21no—
64#14519735—41no—
65#14519733—41no—
66C2×C4×Dic61—41no—
67C42×D61D411yescharacteristic
68Dic488—24no—
69#14519721—24no—
70C2×#11758185C2312yes—
71#14519711C2311yescharacteristic
72#14519708—21no—
73#14519702—21no—
74#14519700—22no—
75#14519700—22no—
76#14519700—21no—
77#14519698—21no—
78C4×#92418D411yescharacteristic
79C8×Dic61D411yescharacteristic
80C8×Dic61C2311yescharacteristic
81C2×D8×C61C2311yescharacteristic
82C4⋊Q8×C61—21no—
83C4×D4×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2312yes—
87#107×C61C2311yescharacteristic
88#102×C61—21no—
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—41no—
91C2×D4×C61—21no—
92C22×C4×C61—41no—
93C2×C22⋊D61—41no—
94C22×Dic61—41no—
95C22×Dic61—41no—
96#11758224—24no—
97#11758224—42no—
98C2×C4×D61D811yescharacteristic
99C2×C4×D61D811yescharacteristic
100C2×C4×D61C2×D411yescharacteristic
101C2×Dic122—21no—
102C2×Dic122—22no—
103D8×C61—21no—
104D8×C61#4112yes—
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2411yescharacteristic
107C4⋊C4×C61—21no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61C2×D411yescharacteristic
112C22⋊Dic61—41no—
113#11758200—24no—
114#11758198—41no—
115#11758198—42no—
116C4⋊Dic61—21no—
117C4⋊Dic61—41no—
118C4⋊Dic61—41no—
119C4⋊Dic61—42no—
120#11758196—41no—
121#11758196—41no—
122#11758196—42no—
123C4×Dic61—41no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126C4×Dic61—42no—
127C4×Dic61D811yescharacteristic
128C4×Dic61D811yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132Dic244—21no—
133Dic244—22no—
134Dic244—22no—
135Dic244—42no—
136Dic244#4112yes—
137C8×D61—21no—
138C8×D61—21no—
139C8×D61—41no—
140C16×C61—22no—
141#11758185—22no—
142C23×C61—41no—
143C22×D61C2×D811yescharacteristic
144Q8×C61—82no—
145D4×C61—22no—
146D4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—82no—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C22⋊D61—44no—
154C2×Dic61—44no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—42no—
158C2×Dic61—82no—
159C2×Dic61C2×D811yescharacteristic
160C2×Dic61C2×D811yescharacteristic
161C2×Dic61C2×D811yescharacteristic
162C4×D61—21no—
163C4×D61—21no—
164C4×D61—22no—
165C4×D61—22no—
166Dic122—22no—
167Dic122—41no—
168Dic122—42no—
169Dic122—42no—
170C8×C61—21no—
171C8×C61C2×#4111yescharacteristic
172C8×C61C2×#4111yescharacteristic
173#92418—21no—
174#92418—21no—
175#33667—611nomaximal, Sylow
176C22×C61—42no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61C22×D811yescharacteristic
180C2×D61—21no—
181C2×D61—21no—
182C2×D61—21no—
183C4×C61—21no—
184C4×C61—81no—
185C4×C61—82no—
186C4×C61#50812yes—
187C4×C61#52511yescharacteristic
188C4×C61#57511yescharacteristic
189Dic61—21no—
190Dic61—21no—
191Dic61—42no—
192Dic61—82no—
193Dic61—81no—
194Dic61#51012yes—
195#3170—611no—
196C4×#135—611no—
197#2237—612no—
198#2225—611no—
199#2224D6111yescharacteristic, complemented
200#2209—611no—
201#2208—611no—
202#2189—612no—
203C4×Q32—612no—
204C4×SD32—612no—
205#2162—611no—
206C2×C61—82no—
207C2×C61#310011yescharacteristic
208C2×C61C2×#50811yescharacteristic
209C2×C61C2×#51011yescharacteristic
210D61—41no—
211C2×#135—611no—
212#533—1221no—
213C4×#41—1221no—
214C2×Q32—612no—
215C2×SD32—612no—
216#500C2×D6111yescharacteristic
217#494—611no—
218#487—1221no—
219#467—612no—
220#467—1221no—
221C4×Q16—612no—
222C4×Q16—611no—
223C4×Q16—611no—
224C4×Q16—1221no—
225C4×SD16—1221no—
226C4×SD16—1221no—
227C4×D8—611no—
228C4×D8C2×D6111yescharacteristic
229#427—611no—
230#416—1221no—
231C2×C4×C8—611no—
232#367—612no—
233#366C2×D6112yes—
234C4⋊C16—611no—
235C4⋊C16C2×D6111yescharacteristic
236#358—612no—
237#358—612no—
238#358—612no—
239#357C2×D6112yes—
240C22⋊C16—612no—
241C4×C16—612no—
242C61#3366711yescharacteristic, complemented, Sylow
243C2×#41—1221no—
244#135—1222no—
245#135—1222no—
246C2×Q16—611no—
247C2×Q16—612no—
248C2×Q16—1221no—
249C2×SD16—1221no—
250C2×D8C22×D6111yescharacteristic
251C22×C8—611no—
252C22×C8—611no—
253C4⋊Q8—21no—
254C4⋊Q8—1221no—
255#125—2441no—
256C22⋊Q8—1222no—
257C22⋊Q8—2441no—
258C4×Q8—1221no—
259C4×Q8—1222no—
260C4×Q8—1221no—
261C4×Q8—2441no—
262C4×D4—21no—
263C4×D4—1221no—
264C4×D4—2441no—
265C4×D4—2441no—
266#117—1221no—
267#117—2441no—
268#117—2441no—
269C2×C42—611no—
270C2×C42—2441no—
271Q32—1224no—
272SD32—1224no—
273C2×C16—612no—
274C2×C16C22×D6112yes—
275#107—21no—
276#107—611no—
277#107C22×D6112yes—
278#107C22×D6111yescharacteristic
279#106—1221no—
280#103—1221no—
281#103—1222no—
282#103—1222no—
283#103—1221no—
284#102—21no—
285#102—1221no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8C22×D6111yescharacteristic
290#41—1224no—
291#41—2442no—
292C2×Q8—41no—
293C2×Q8—1221no—
294C2×Q8—1222no—
295C2×D4—21no—
296C2×D4—2441no—
297C22×C4—41no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—611no—
301C22×C4—2441no—
302C22×C4—2441no—
303Q16—612no—
304Q16—1222no—
305Q16—1221no—
306Q16—1222no—
307Q16—2442no—
308SD16—1224no—
309D8—21no—
310D8#1175822612yes—
311C2×C8—611no—
312C2×C8—611no—
313C2×C8—1221no—
314C2×C8—1221no—
315C2×C8—2441no—
316C2×C8D4×D6111yescharacteristic
317C2×C8C23×D6111yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—1221no—
322C4⋊C4—2441no—
323C4⋊C4—2442no—
324C4⋊C4—2442no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—41no—
329C22⋊C4—2441no—
330C22⋊C4—2442no—
331C22⋊C4—2441no—
332C42—611no—
333C42—611no—
334C42—611no—
335C42—2441no—
336C42—2441no—
337C42—2442no—
338C42—2441no—
339C42D4×D6111yescharacteristic
340C16—22no—
341C16—1222no—
342C23—41no—
343C23—611no—
344Q8—82no—
345Q8—1222no—
346Q8—2442no—
347Q8—2442no—
348Q8—2441no—
349D4—22no—
350D4—41no—
351D4—2444no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—611no—
356C2×C4—611no—
357C2×C4—611no—
358C2×C4—1222no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1222no—
362C2×C4—4882no—
363C2×C4—2444no—
364C2×C4—2441no—
365C2×C4—2442no—
366C2×C4—2441no—
367C2×C4D8×D6111yescharacteristic
368C2×C4D8×D6111yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C8—21no—
371C8—1221no—
372C8—1221no—
373C8#41×D6111yescharacteristic
374C8C2×#1175822611yescharacteristic
375C22—42no—
376C22—81no—
377C22—81no—
378C22—1221no—
379C22—1221no—
380C22—1221no—
381C22C2×D8×D6111yescharacteristic
382C4—21no—
383C4—81no—
384C4—82no—
385C4—612no—
386C4—1221no—
387C4—1221no—
388C4—4881no—
389C4—4882no—
390C4—2442no—
391C4#1684019112yes—
392C4#1684027511yescharacteristic
393C4#1684056111yescharacteristic
394C2—82no—
395C2—2441no—
396C2#2145582111yescharacteristic
397C2C2×#1684019111yescharacteristic
398C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2224C2
socleC22×C61C22×D8
prime radicalC2×#14519796C22
hypercenter#2224D61
hypocenterC61#33667
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2224

Complements

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

NH ≅ G/Ntype
1. #21456406C2semidirect
3. #21448029C2semidirect
4. #21447958C2semidirect
5. #21447870C2semidirect
6. #21447864C2semidirect
8. #21447677C2semidirect
9. #21447674C2semidirect
10. #21447617C2semidirect
11. C61×#2224C2semidirect
199. #2224D61semidirect
242. C61#33667semidirect

Maximal subgroups

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

subgroupindex
#214564062
C4×#145197962
#214480292
#214479582
#214478702
#214478642
#214477602
#214476772
#214476742
#214476172
C61×#22242
#3366761