Finite Group Databasebeta

← #29834875

Subgroups — #29834875

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#21455485C211yesmaximal, characteristic
2#21454416C211yesmaximal, characteristic, complemented
3#21448026C212yesmaximal, complemented
4#21448020C212yesmaximal, complemented
5#21448008C212yesmaximal, complemented
6#21447998C212yesmaximal, complemented
7#21447608C211yesmaximal, characteristic, complemented
8C2×#16839588C211yesmaximal, characteristic, complemented
9#21447531C211yesmaximal, characteristic, complemented
10#21447432C211yesmaximal, characteristic, complemented
11C61×#2138C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840340—21no—
14#16840242—21no—
15C2×#14519717C2211yescharacteristic
16#16840005—22no—
17#16840000C2212yes—
18#16839995C2212yes—
19#16839805—21no—
20#16839801—22no—
21#16839799C2212yes—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24#107×D61C2212yes—
25#16839763—21no—
26#16839589C2212yes—
27#16839588C2212yes—
28#16839588C2212yes—
29#16839587C2212yes—
30#16839566C2212yes—
31#16839563C2212yes—
32#16839541C2212yes—
33#16839541C2212yes—
34C2×#135×C61C2211yescharacteristic
35C22×C16×C61C2211yescharacteristic
36C61×#427C2211yescharacteristic
37C61×#358C2211yescharacteristic
38C61×#358C2211yescharacteristic
39C61×#357C2211yescharacteristic
40C61×#357C2211yescharacteristic
41C2×#11758222D411yescharacteristic
42#14519783—22no—
43#14519783—22no—
44C2×#11758194D411yescharacteristic
45C2×#11758189D412yes—
46C2×C8×D61C2312yes—
47#14519756—41no—
48#14519753—22no—
49#14519751—41no—
50#14519749—42no—
51#14519748—22no—
52C4×Dic122—41no—
53C4×D244—41no—
54#14519737—42no—
55#14519735—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C4⋊C4×D61—22no—
59#14519717—22no—
60C2×#11758185C2312yes—
61#14519711—21no—
62#14519711—21no—
63#14519711C2312yes—
64#14519711C2312yes—
65#14519710—22no—
66#14519708—22no—
67#14519701—22no—
68#14519698—22no—
69C2×#41×C61—21no—
70#135×C61—22no—
71#135×C61—22no—
72C2×Q16×C61C2311yescharacteristic
73C2×SD16×C61—21no—
74C2×D8×C61C2311yescharacteristic
75C22×C8×C61C2311yescharacteristic
76#117×C61—21no—
77C2×C16×C61—22no—
78C2×C16×C61C2311yescharacteristic
79C2×C16×C61C2311yescharacteristic
80#107×C61C2311yescharacteristic
81#107×C61C2311yescharacteristic
82#106×C61—21no—
83#41×C61—42no—
84#41×C61—42no—
85C2×Q8×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C2×C22⋊D61—42no—
91C2×C22⋊D61D812yes—
92C22×Dic61—41no—
93C22×Dic61—41no—
94#11758222—22no—
95#11758222—22no—
96C2×D244D811yescharacteristic
97C2×C4×D61—42no—
98C2×C4×D61C2×D412yes—
99C2×Dic122D811yescharacteristic
100Q16×C61—21no—
101Q16×C61—21no—
102SD16×C61—42no—
103D8×C61—21no—
104D8×C61—21no—
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61#4112yes—
107C2×C8×C61#4112yes—
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C22⋊C4×C61—41no—
112C42×C61—41no—
113C22⋊Dic61—42no—
114#11758198—42no—
115#11758198—42no—
116C4⋊Dic61—22no—
117C4⋊Dic61—22no—
118C4⋊Dic61—41no—
119C4⋊Dic61—41no—
120#11758196—42no—
121#11758196—42no—
122C4×Dic61—42no—
123#11758194—22no—
124C2×#92418C2×D412yes—
125#11758189—24no—
126C8×D61—22no—
127C8×D61—22no—
128C16×C61—21no—
129C16×C61—21no—
130#11758185—22no—
131C23×C61—41no—
132C23×C61C2×D811yescharacteristic
133C22×D61C2×D812yes—
134Q8×C61—41no—
135Q8×C61—41no—
136D4×C61—41no—
137D4×C61—41no—
138D4×C61—82no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—82no—
143C2×C4×C61#12312yes—
144C2×C4×C61C2×D811yescharacteristic
145C2×C4×C61#13712yes—
146C2×C4×C61C22×D411yescharacteristic
147C22⋊D61—24no—
148C22⋊D61—162no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—161no—
152C2×Dic61—161no—
153C2×Dic61C2×D812yes—
154D244—22no—
155C4×D61—22no—
156C4×D61—22no—
157C4×D61—162no—
158Dic122—22no—
159C8×C61#12312yes—
160C8×C61C2×#4111yescharacteristic
161C8×C61C2×#4111yescharacteristic
162#92418—22no—
163#33652—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—81no—
167C22×C61—81no—
168C22×C61—81no—
169C22×C61#48012yes—
170C22×C61C22×D811yescharacteristic
171C2×D61—22no—
172C2×D61—22no—
173C2×D61—22no—
174C4×C61—81no—
175C4×C61—81no—
176C4×C61—81no—
177C4×C61#48012yes—
178C4×C61C2×#12311yescharacteristic
179C4×C61C2×#13711yescharacteristic
180Dic61—22no—
181Dic61—82no—
182Dic61—82no—
183#3061—611no—
184#2913—611no—
185#2237—612no—
186#2236—612no—
187#2234—612no—
188#2232—612no—
189#2159—611no—
190C2×#366—611no—
191#2139—611no—
192#2138D6111yescharacteristic, complemented
193#2113—611no—
194C2×C61—22no—
195C2×C61—161no—
196C2×C61C2×#48011yescharacteristic
197C2×C61#341511yescharacteristic
198C2×C61#341711yescharacteristic
199D61—42no—
200C2×#135C2×D6111yescharacteristic
201C2×#131—611no—
202#537—1221no—
203#520—1221no—
204C2×M32—611no—
205C22×C16C2×D6111yescharacteristic
206#468—1222no—
207#467—612no—
208#466—612no—
209#428—1222no—
210#428—1221no—
211#427—612no—
212#427C2×D6111yescharacteristic
213C2×#107—611no—
214C2×#107—611no—
215C2×#107—612no—
216#417—1221no—
217#367—612no—
218#366—612no—
219#366—612no—
220#366—612no—
221#358—612no—
222#358C2×D6111yescharacteristic
223#358C2×D6111yescharacteristic
224#357—612no—
225#357C2×D6111yescharacteristic
226#357C2×D6111yescharacteristic
227C22⋊C16—612no—
228C22⋊C16—612no—
229C61#3365211yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×#41—611no—
232#135—22no—
233#135—22no—
234C2×Q16C22×D6111yescharacteristic
235C2×SD16—21no—
236C2×D8C22×D6111yescharacteristic
237#131—1222no—
238#131—1222no—
239C2×M16—611no—
240C2×M16—612no—
241C22×C8—612no—
242C22×C8C22×D6111yescharacteristic
243C22⋊Q8—1222no—
244C22⋊Q8—2441no—
245#121—1222no—
246#121—2441no—
247#121—2442no—
248C4×Q8—2441no—
249C4×D4—2441no—
250C4×D4—2442no—
251#117—21no—
252#117—1222no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256M32—1222no—
257C2×C16—22no—
258C2×C16—612no—
259C2×C16C22×D6111yescharacteristic
260C2×C16C22×D6111yescharacteristic
261#107—612no—
262#107—612no—
263#107—1222no—
264#107—1221no—
265#107—1221no—
266#107C22×D6111yescharacteristic
267#107C22×D6111yescharacteristic
268#106—21no—
269#106—1222no—
270#103—1222no—
271#102—1222no—
272#41—42no—
273#41—42no—
274#41—1222no—
275#41—1222no—
276C2×Q8—21no—
277C2×Q8—611no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—612no—
281C2×D4—611no—
282C2×D4—2442no—
283C22×C4—41no—
284C22×C4—612no—
285C22×C4—2442no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4D4×D6111yescharacteristic
289Q16—21no—
290Q16—21no—
291SD16—42no—
292D8—21no—
293D8—21no—
294M16—1222no—
295M16—1224no—
296C2×C8—612no—
297C2×C8—1222no—
298C2×C8—1222no—
299C2×C8#1175822212yes—
300C2×C8D4×D6111yescharacteristic
301C2×C8#1175822412yes—
302C2×C8C23×D6111yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—1222no—
306C4⋊C4—1222no—
307C4⋊C4—2442no—
308C4⋊C4—2442no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C22⋊C4—41no—
312C22⋊C4—2442no—
313C22⋊C4—2442no—
314C22⋊C4—2442no—
315C42—41no—
316C42—2442no—
317C16—21no—
318C16—21no—
319C16—1222no—
320C23—41no—
321C23—612no—
322C23D8×D6111yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—1222no—
326D4—41no—
327D4—41no—
328D4—82no—
329D4—1224no—
330D4—1222no—
331D4—9762no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—612no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—9762no—
340C2×C4—9761no—
341C2×C4—9761no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4#1451975912yes—
345C2×C4D8×D6111yescharacteristic
346C2×C4#1451980612yes—
347C2×C4C2×D4×D6111yescharacteristic
348C8—1222no—
349C8#1451975912yes—
350C8C2×#1175822411yescharacteristic
351C8C2×#1175822211yescharacteristic
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—1222no—
358C22—1222no—
359C22—1222no—
360C22#1684007412yes—
361C22C2×D8×D6111yescharacteristic
362C4—81no—
363C4—81no—
364C4—81no—
365C4—1222no—
366C4—4882no—
367C4—4882no—
368C4#1684007412yes—
369C4C2×#1451975911yescharacteristic
370C4C2×#1451980611yescharacteristic
371C2—22no—
372C2—161no—
373C2—2442no—
374C2C2×#1684007411yescharacteristic
375C2#2145895711yescharacteristic
376C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radical#21455485C2
hypercenter#2138D61
hypocenterC61#33652
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #21454416C2semidirect
3. #21448026C2semidirect
4. #21448020C2semidirect
5. #21448008C2semidirect
6. #21447998C2semidirect
7. #21447608C2semidirect
8. C2×#16839588C2semidirect
9. #21447531C2semidirect
10. #21447432C2semidirect
11. C61×#2138C2semidirect
192. #2138D61semidirect
229. C61#33652semidirect

Maximal subgroups

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

subgroupindex
#214554852
#214544162
#214480262
#214480202
#214480082
#214479982
#214476082
C2×#168395882
#214475312
#214474322
C61×#21382
#3365261