Finite Group Databasebeta

← #29835011

Subgroups — #29835011

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×#14519800C211yesmaximal, characteristic
3#21448030C212yesmaximal, complemented
4#21447953C211yesmaximal, characteristic, complemented
5#21447869C211yesmaximal, characteristic, complemented
6#21447865C211yesmaximal, characteristic, complemented
7#21447762C212yesmaximal
8#21447675C212yesmaximal, complemented
9#21447670C212yesmaximal, complemented
10#21447617C211yesmaximal, characteristic, complemented
11C61×#2225C211yesmaximal, characteristic, complemented
12C2×#14519800C2211yescharacteristic
13#16840317—21no—
14C4×#11758226—21no—
15C2×#14519723C2212yes—
16C2×#14519722C2212yes—
17#16840142C2211yescharacteristic
18#16840114—21no—
19#16840004—21no—
20#16840004C2212yes—
21Q16×Dic61C2211yescharacteristic
22#16839856—21no—
23#16839856C2212yes—
24C4×#11758201—21no—
25#16839847—21no—
26C4×D488C2211yescharacteristic
27#16839844C2211yescharacteristic
28#16839799C2211yescharacteristic
29#16839757—21no—
30C4×C8×D61C2211yescharacteristic
31#16839587C2212yes—
32C4⋊#11758185C2211yescharacteristic
33#16839567C2212yes—
34#16839565C2212yes—
35#16839562C2212yes—
36#16839541C2212yes—
37C16×Dic61C2212yes—
38C61×#500C2211yescharacteristic
39C4×Q16×C61C2211yescharacteristic
40C61×#367C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×#11758226—21no—
44#14519800—22no—
45#14519800—22no—
46C2×Dic244—21no—
47C2×Dic244C2312yes—
48C2×#11758201—21no—
49C2×D488C2311yescharacteristic
50C2×C8×D61D411yescharacteristic
51C2×C8×D61C2311yescharacteristic
52#14519777—21no—
53#14519768—41no—
54#14519757—22no—
55#14519753—41no—
56Q8×Dic61—21no—
57Q8×Dic61—41no—
58#14519742—22no—
59C4×D244—21no—
60C4×D244—41no—
61#14519739—21no—
62#14519739—41no—
63#14519735—21no—
64#14519735—41no—
65#14519735—41no—
66C42×D61—41no—
67C42×D61D411yescharacteristic
68#14519723—24no—
69#14519722—24no—
70C2×#11758185C2312yes—
71#14519711C2311yescharacteristic
72#14519708—21no—
73#14519701—21no—
74#14519700—22no—
75#14519700—22no—
76#14519699—21no—
77#14519697—21no—
78C4×#92418D411yescharacteristic
79C8×Dic61D411yescharacteristic
80C8×Dic61C2311yescharacteristic
81C2×Q16×C61C2311yescharacteristic
82C4⋊Q8×C61—21no—
83C4×Q8×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—21no—
91C2×Q8×C61—41no—
92#11758226—24no—
93#11758226—42no—
94C2×D244—21no—
95C2×D244—41no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61—41no—
99C2×C4×D61D811yescharacteristic
100C2×C4×D61D811yescharacteristic
101C2×C4×D61C2×D411yescharacteristic
102C2×Dic122—22no—
103Q16×C61—21no—
104Q16×C61#4112yes—
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2411yescharacteristic
107C4⋊C4×C61—21no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C42×C61—41no—
112C42×C61C2×D411yescharacteristic
113#11758201—24no—
114#11758198—41no—
115#11758198—42no—
116#11758198—41no—
117#11758198—41no—
118C4⋊Dic61—21no—
119C4⋊Dic61—41no—
120C4⋊Dic61—41no—
121C4⋊Dic61—41no—
122C4⋊Dic61—42no—
123#11758196—42no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126C4×Dic61—42no—
127C4×Dic61D811yescharacteristic
128C4×Dic61D811yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132Dic244—22no—
133Dic244—22no—
134Dic244—42no—
135D488—21no—
136D488#4112yes—
137C8×D61—21no—
138C8×D61—21no—
139C8×D61—41no—
140C16×C61—22no—
141#11758185—22no—
142C22×D61—41no—
143C22×D61C2×D811yescharacteristic
144Q8×C61—22no—
145Q8×C61—41no—
146Q8×C61—82no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C2×Dic61—41no—
154C2×Dic61—42no—
155C2×Dic61C2×D811yescharacteristic
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×D811yescharacteristic
158D244—22no—
159D244—44no—
160D244—41no—
161C4×D61—21no—
162C4×D61—21no—
163C4×D61—22no—
164C4×D61—22no—
165C4×D61—44no—
166C4×D61—82no—
167C4×D61—82no—
168Dic122—42no—
169Dic122—42no—
170C8×C61—21no—
171C8×C61C2×#4111yescharacteristic
172C8×C61C2×#4111yescharacteristic
173#92418—21no—
174#92418—21no—
175#33667—611nomaximal, Sylow
176C22×C61C22×D811yescharacteristic
177C2×D61—21no—
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—42no—
181C2×D61—81no—
182C2×D61—81no—
183C4×C61—21no—
184C4×C61—42no—
185C4×C61—82no—
186C4×C61—81no—
187C4×C61#50812yes—
188C4×C61#52511yescharacteristic
189C4×C61#57511yescharacteristic
190Dic61—21no—
191Dic61—21no—
192Dic61—82no—
193Dic61—81no—
194Dic61#51012yes—
195#3170—611no—
196C4×#135—611no—
197#2237—612no—
198#2225D6111yescharacteristic, complemented
199#2224—611no—
200#2209—611no—
201#2208—611no—
202#2189—612no—
203C4×Q32—612no—
204C4×SD32—612no—
205#2162—611no—
206C2×C61#310011yescharacteristic
207C2×C61C2×#50811yescharacteristic
208C2×C61C2×#51011yescharacteristic
209D61—41no—
210D61—82no—
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—611no—
222C4×Q16—612no—
223C4×Q16—1221no—
224C4×Q16C2×D6111yescharacteristic
225C4×SD16—1221no—
226C4×SD16—1221no—
227C4×D8—611no—
228C4×D8—611no—
229#427—611no—
230#416—1221no—
231C2×C4×C8—611no—
232#367C2×D6112yes—
233#366—612no—
234C4⋊C16—611no—
235C4⋊C16C2×D6111yescharacteristic
236#358—612no—
237#358—612no—
238#358C2×D6112yes—
239#357—612no—
240C22⋊C16—612no—
241C4×C16—612no—
242C61#3366711yescharacteristic, complemented, Sylow
243C2×#41—1221no—
244#135—1222no—
245#135—1222no—
246C2×Q16—612no—
247C2×Q16—1221no—
248C2×Q16C22×D6111yescharacteristic
249C2×SD16—1221no—
250C2×D8—611no—
251C22×C8—611no—
252C22×C8—611no—
253C4⋊Q8—21no—
254C4⋊Q8—1221no—
255#125—2441no—
256C22⋊Q8—1222no—
257C22⋊Q8—2441no—
258C4×Q8—21no—
259C4×Q8—1222no—
260C4×Q8—1221no—
261C4×Q8—2441no—
262C4×D4—1221no—
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—21no—
281#103—1222no—
282#103—1222no—
283#103—1221no—
284#102—1221no—
285#102—1221no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C4×C8C22×D6111yescharacteristic
290#41—1224no—
291#41—2442no—
292C2×Q8—21no—
293C2×Q8—41no—
294C2×Q8—1222no—
295C2×D4—1221no—
296C2×D4—2441no—
297C22×C4—611no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4—2441no—
303Q16—21no—
304Q16—1222no—
305Q16—1222no—
306Q16—2442no—
307Q16#1175822612yes—
308SD16—1224no—
309D8—612no—
310D8—1221no—
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—41no—
322C4⋊C4—1221no—
323C4⋊C4—2442no—
324C4⋊C4—2441no—
325C4⋊C4—2442no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—2441no—
329C22⋊C4—2441no—
330C22⋊C4—2442no—
331C22⋊C4—2441no—
332C42—41no—
333C42—611no—
334C42—611no—
335C42—611no—
336C42—2441no—
337C42—2441no—
338C42—2442no—
339C42D4×D6111yescharacteristic
340C16—22no—
341C16—1222no—
342C23—611no—
343C23—2441no—
344Q8—22no—
345Q8—41no—
346Q8—82no—
347Q8—2442no—
348Q8—2442no—
349D4—1222no—
350D4—2444no—
351D4—2441no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—611no—
356C2×C4—611no—
357C2×C4—611no—
358C2×C4—1222no—
359C2×C4—1222no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—4882no—
363C2×C4—4882no—
364C2×C4—2444no—
365C2×C4—2441no—
366C2×C4—2442no—
367C2×C4D8×D6111yescharacteristic
368C2×C4D8×D6111yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C8—21no—
371C8—1221no—
372C8—1221no—
373C8#41×D6111yescharacteristic
374C8C2×#1175822611yescharacteristic
375C22—1221no—
376C22—1221no—
377C22—1221no—
378C22—4881no—
379C22—4881no—
380C22—2442no—
381C22C2×D8×D6111yescharacteristic
382C4—21no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4—612no—
387C4—1221no—
388C4—1221no—
389C4—4882no—
390C4—4881no—
391C4#1684019312yes—
392C4#1684027511yescharacteristic
393C4#1684056111yescharacteristic
394C2—4882no—
395C2—2441no—
396C2#2145582111yescharacteristic
397C2C2×#1684019311yescharacteristic
398C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2225C2
socleC22×C61C22×D8
prime radicalC2×#14519800C22
hypercenter#2225D61
hypocenterC61#33667
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
1. #21456406C2semidirect
3. #21448030C2semidirect
4. #21447953C2semidirect
5. #21447869C2semidirect
6. #21447865C2semidirect
8. #21447675C2semidirect
9. #21447670C2semidirect
10. #21447617C2semidirect
11. C61×#2225C2semidirect
198. #2225D61semidirect
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×#145198002
#214480302
#214479532
#214478692
#214478652
#214477622
#214476752
#214476702
#214476172
C61×#22252
#3366761