Finite Group Databasebeta

← #29834877

Subgroups — #29834877

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455489C211yesmaximal, characteristic
2#21448027C212yesmaximal, complemented
3#21448021C212yesmaximal, complemented
4#21448010C212yesmaximal, complemented
5#21447999C212yesmaximal, complemented
6#21447608C211yesmaximal, characteristic, complemented
7C2×#16839587C211yesmaximal, characteristic, complemented
8#21447529C211yesmaximal, characteristic, complemented
9#21447524C211yesmaximal, characteristic, complemented
10#21447433C211yesmaximal, characteristic, complemented
11C61×#2913C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840340—21no—
14C2×#14519717C2211yescharacteristic
15C22×#11758185C2211yescharacteristic
16#16840008—22no—
17#16840003C2212yes—
18#16839998C2212yes—
19C2×#14519711C2211yescharacteristic
20#16839764—21no—
21#16839589C2212yes—
22#16839587C2212yes—
23#16839587C2212yes—
24#16839587C2212yes—
25#16839565C2211yescharacteristic
26#16839565C2212yes—
27#16839565C2211yescharacteristic
28#16839562C2211yescharacteristic
29#16839562C2212yes—
30#16839562C2211yescharacteristic
31C22⋊#11758185C2212yes—
32C22⋊#11758185C2212yes—
33C2×#131×C61C2211yescharacteristic
34C61×#520—21no—
35C61×#428—22no—
36C61×#428—21no—
37C61×#427C2212yes—
38C61×#427C2211yescharacteristic
39C2×#107×C61C2212yes—
40C2×#107×C61C2211yescharacteristic
41C2×#11758222—21no—
42#14519799—22no—
43#14519799—22no—
44C2×Dic244C2311yescharacteristic
45C2×#11758190—21no—
46C2×D488C2311yescharacteristic
47#14519756—22no—
48#14519753—41no—
49#14519751—22no—
50#14519749—42no—
51#14519748—41no—
52C2×C4⋊Dic61—21no—
53#14519717—22no—
54C2×#11758185—22no—
55C2×#11758185C2311yescharacteristic
56C2×#11758185C2311yescharacteristic
57C2×#11758185C2312yes—
58#14519711—21no—
59#14519711C2312yes—
60#14519702—22no—
61#14519699—22no—
62C2×#41×C61D411yescharacteristic
63#131×C61—22no—
64#131×C61—22no—
65C2×M16×C61D412yes—
66C2×M16×C61D411yescharacteristic
67C22×C8×C61C2312yes—
68C22×C8×C61C2311yescharacteristic
69C4×Q8×C61—41no—
70C4×D4×C61—41no—
71C4×D4×C61—42no—
72#117×C61—22no—
73#117×C61—21no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—21no—
76#107×C61—22no—
77#107×C61—21no—
78#107×C61C2311yescharacteristic
79#107×C61C2312yes—
80#107×C61C2311yescharacteristic
81#106×C61—22no—
82#106×C61—21no—
83#41×C61—22no—
84#41×C61—22no—
85C2×Q8×C61D811yescharacteristic
86C2×D4×C61D811yescharacteristic
87C2×D4×C61D812yes—
88C22×C4×C61—42no—
89C22×C4×C61—41no—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C2×C22⋊D61—41no—
93C2×C22⋊D61—42no—
94C22×Dic61—41no—
95#11758222—42no—
96#11758222—42no—
97C2×D244—21no—
98C2×C4×D61—41no—
99C2×Dic122—21no—
100M16×C61—22no—
101M16×C61—24no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C2×C8×C61C2×D412yes—
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61#4112yes—
107C2×C8×C61#4112yes—
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—42no—
113C4⋊C4×C61—41no—
114C22⋊C4×C61—42no—
115C22⋊C4×C61—41no—
116C42×C61—42no—
117C42×C61—41no—
118C22⋊Dic61—42no—
119#11758198—42no—
120C4⋊Dic61—22no—
121C4⋊Dic61—41no—
122#11758196—42no—
123Dic244—21no—
124Dic244—21no—
125D488—21no—
126D488—21no—
127#11758190—42no—
128#11758185—21no—
129#11758185—21no—
130#11758185—22no—
131C23×C61C2×D812yes—
132C23×C61C2×D811yescharacteristic
133C22×D61—41no—
134Q8×C61—22no—
135D4×C61—24no—
136D4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—41no—
142C2×C4×C61—161no—
143C2×C4×C61—162no—
144C2×C4×C61#12312yes—
145C2×C4×C61C2×D812yes—
146C2×C4×C61C2×D811yescharacteristic
147C2×C4×C61#13712yes—
148C2×C4×C61C22×D411yescharacteristic
149C22⋊D61—82no—
150C22⋊D61—162no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153C2×Dic61—161no—
154D244—41no—
155D244—41no—
156C4×D61—82no—
157Dic122—41no—
158Dic122—41no—
159C8×C61—22no—
160C8×C61#12312yes—
161C8×C61C2×#4111yescharacteristic
162C8×C61C2×#4111yescharacteristic
163#33652—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—22no—
167C22×C61—22no—
168C22×C61—22no—
169C22×C61#48012yes—
170C22×C61C22×D811yescharacteristic
171C2×D61—81no—
172C2×D61—81no—
173C2×D61—81no—
174C4×C61—22no—
175C4×C61—82no—
176C4×C61—81no—
177C4×C61—81no—
178C4×C61#48012yes—
179C4×C61C2×#12311yescharacteristic
180C4×C61C2×#13711yescharacteristic
181Dic61—81no—
182Dic61—82no—
183#3061—611no—
184#2913D6111yescharacteristic, complemented
185#2237—612no—
186#2236—612no—
187#2234—612no—
188#2232—612no—
189#2159—611no—
190C2×#366—611no—
191#2139—611no—
192#2138—611no—
193#2113—611no—
194C2×C61—22no—
195C2×C61—42no—
196C2×C61C2×#48011yescharacteristic
197C2×C61#341511yescharacteristic
198C2×C61#341711yescharacteristic
199D61—161no—
200C2×#135—611no—
201C2×#131C2×D6111yescharacteristic
202#537—1221no—
203#520—21no—
204C2×M32—611no—
205C22×C16—611no—
206#468—1222no—
207#467—612no—
208#466—612no—
209#428—21no—
210#428—22no—
211#427C2×D6111yescharacteristic
212#427C2×D6112yes—
213C2×#107—611no—
214C2×#107C2×D6111yescharacteristic
215C2×#107C2×D6112yes—
216#417—1221no—
217#367—612no—
218#366—612no—
219#366—612no—
220#366—612no—
221#358—611no—
222#358—611no—
223#358—612no—
224#357—611no—
225#357—611no—
226#357—612no—
227C22⋊C16—612no—
228C22⋊C16—612no—
229C61#3365211yescharacteristic, complemented, Sylow
230C2×#41—1221no—
231C2×#41C22⋊D6111yescharacteristic
232#135—1222no—
233#135—1222no—
234C2×Q16—611no—
235C2×SD16—1221no—
236C2×D8—611no—
237#131—22no—
238#131—22no—
239C2×M16C22⋊D6111yescharacteristic
240C2×M16C22⋊D6112yes—
241C22×C8C22×D6112yes—
242C22×C8C22×D6111yescharacteristic
243C22⋊Q8—1222no—
244C22⋊Q8—2441no—
245#121—1222no—
246#121—2441no—
247#121—2442no—
248C4×Q8—41no—
249C4×D4—42no—
250C4×D4—41no—
251#117—22no—
252#117—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1221no—
256M32—1222no—
257C2×C16—611no—
258C2×C16—611no—
259C2×C16—612no—
260C2×C16—1222no—
261#107—22no—
262#107—21no—
263#107—612no—
264#107—1221no—
265#107C22×D6111yescharacteristic
266#107C22×D6112yes—
267#107C22×D6111yescharacteristic
268#106—22no—
269#106—21no—
270#103—1222no—
271#102—1222no—
272#41—22no—
273#41—22no—
274#41—2442no—
275#41—2442no—
276C2×Q8—1221no—
277C2×Q8#1175819911yescharacteristic
278C2×D4—1221no—
279C2×D4—2442no—
280C2×D4—2441no—
281C2×D4#1175819912yes—
282C2×D4#1175819911yescharacteristic
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—2441no—
286C22×C4—2441no—
287C22×C4C2×C22⋊D6112yes—
288C22×C4C2×C22⋊D6111yescharacteristic
289Q16—1221no—
290Q16—1221no—
291SD16—2442no—
292D8—1221no—
293D8—1221no—
294M16—22no—
295M16—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8#1175822212yes—
299C2×C8#1175822212yes—
300C2×C8C2×C22⋊D6112yes—
301C2×C8C2×C22⋊D6111yescharacteristic
302C2×C8C23×D6111yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—1222no—
309C4⋊C4—2441no—
310C4⋊C4—2442no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—2442no—
314C22⋊C4—2442no—
315C42—42no—
316C42—41no—
317C16—1221no—
318C16—1221no—
319C16—1222no—
320C23—2441no—
321C23C2×#1175819912yes—
322C23C2×#1175819911yescharacteristic
323Q8—22no—
324Q8—2441no—
325Q8—2441no—
326D4—24no—
327D4—22no—
328D4—9762no—
329D4—4882no—
330D4—2441no—
331D4—2441no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—161no—
338C2×C4—162no—
339C2×C4—9761no—
340C2×C4—4882no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4#1451976012yes—
344C2×C4C2×#1175819912yes—
345C2×C4C2×#1175819911yescharacteristic
346C2×C4#1451980812yes—
347C2×C4C22×C22⋊D6111yescharacteristic
348C8—22no—
349C8#1451976012yes—
350C8C2×#1175822211yescharacteristic
351C8C2×#1175822211yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—4881no—
358C22—4881no—
359C22—4881no—
360C22#1684007512yes—
361C22C22×#1175819911yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4—4882no—
367C4—4881no—
368C4#1684007512yes—
369C4C2×#1451976011yescharacteristic
370C4C2×#1451980811yescharacteristic
371C2—22no—
372C2—42no—
373C2—9761no—
374C2C2×#1684007511yescharacteristic
375C2#2145895811yescharacteristic
376C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2913C2
socleC22×C61C22×D8
prime radical#21455489C2
hypercenter#2913D61
hypocenterC61#33652
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #21448027C2semidirect
3. #21448021C2semidirect
4. #21448010C2semidirect
5. #21447999C2semidirect
6. #21447608C2semidirect
7. C2×#16839587C2semidirect
8. #21447529C2semidirect
9. #21447524C2semidirect
10. #21447433C2semidirect
11. C61×#2913C2semidirect
184. #2913D61semidirect
229. C61#33652semidirect

Maximal subgroups

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

subgroupindex
#214554892
#214480272
#214480212
#214480102
#214479992
#214476082
C2×#168395872
#214475292
#214475242
#214474332
C61×#29132
#3365261