Finite Group Databasebeta

← #29836975

Subgroups — #29836975

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458243C211yesmaximal, characteristic, complemented
2#21457578C211yesmaximal, characteristic, complemented
3#21448043C211yesmaximal, characteristic, complemented
4#21448012C212yesmaximal, complemented
5#21447971C211yesmaximal, characteristic, complemented
6#21447904C212yesmaximal, complemented
7#21447854C211yesmaximal, characteristic, complemented
8#21447741C212yesmaximal, complemented
9C16×D244C211yesmaximal, characteristic, complemented
10#21447525C212yesmaximal, complemented
11C4×SD32×C61C211yesmaximal, characteristic, complemented
12C2×#14519800C2212yes—
13#16840477—21no—
14#16840434—21no—
15C2×#14519718C2212yes—
16C2×C16×D61C2212yes—
17#16840167C2211yescharacteristic
18#16840139C2211yescharacteristic
19#16840084—22no—
20#16840052—21no—
21#16839999C2212yes—
22#16839995C2212yes—
23#16839962—21no—
24#16839910—22no—
25C8×D244C2211yescharacteristic
26#16839799C2212yes—
27#16839587C2212yes—
28C4⋊#11758185C2211yescharacteristic
29#16839566C2212yes—
30#16839564C2212yes—
31#16839564C2212yes—
32#16839562C2212yes—
33#16839541C2212yes—
34C2×SD32×C61C2211yescharacteristic
35C4×Q16×C61C2211yescharacteristic
36C4×D8×C61C2211yescharacteristic
37C61×#367C2211yescharacteristic
38C61×#358C2211yescharacteristic
39C61×#357C2211yescharacteristic
40C4×C16×C61C2211yescharacteristic
41C2×#11758226—22no—
42#14519800—24no—
43#14519800—42no—
44C2×#11758201—22no—
45C2×D488—21no—
46C2×D488C2312yes—
47C2×C8×D61C2312yes—
48#14519777—21no—
49#14519774—21no—
50#14519774—41no—
51#14519763—42no—
52#14519752—22no—
53#14519752—42no—
54#14519751—41no—
55#14519748—22no—
56C4×D244—41no—
57C4×D244D411yescharacteristic
58#14519736—41no—
59#14519735—22no—
60#14519718—42no—
61C2×#11758185C2312yes—
62C16×D61—42no—
63#14519711—21no—
64#14519711C2312yes—
65#14519708—22no—
66C4⋊#92418D411yescharacteristic
67#14519698—22no—
68#14519697—22no—
69#14519689D412yes—
70C2×Q16×C61C2311yescharacteristic
71C2×D8×C61C2311yescharacteristic
72C4×Q8×C61—21no—
73C4×D4×C61—21no—
74SD32×C61—22no—
75C2×C16×C61C2311yescharacteristic
76C2×C16×C61C2311yescharacteristic
77#107×C61C2311yescharacteristic
78#107×C61C2311yescharacteristic
79#103×C61—21no—
80#102×C61—21no—
81C4×C8×C61C2311yescharacteristic
82C2×Q8×C61—21no—
83C2×D4×C61—21no—
84C22×C4×C61—41no—
85C2×C22⋊D61—42no—
86#11758226—44no—
87#11758226—82no—
88C2×D244—22no—
89C2×D244—42no—
90C2×D244—41no—
91C2×D244—42no—
92C2×D244—41no—
93C2×D244D811yescharacteristic
94C2×C4×D61—42no—
95C2×C4×D61C2×D412yes—
96C2×Dic122—41no—
97Q16×C61—21no—
98Q16×C61#4112yes—
99D8×C61—21no—
100D8×C61—21no—
101C2×C8×C61C2×D411yescharacteristic
102C2×C8×C61C2411yescharacteristic
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C42×C61—41no—
108C42×C61C2×D411yescharacteristic
109C22⋊Dic61—42no—
110#11758201—44no—
111#11758198—42no—
112#11758198—42no—
113#11758198D812yes—
114C4⋊Dic61—22no—
115C4⋊Dic61—41no—
116C4⋊Dic61D811yescharacteristic
117C4×Dic61—42no—
118C2×#92418C2×D412yes—
119D488—22no—
120D488—22no—
121D488—81no—
122C8×D61—22no—
123C8×D61—22no—
124C16×C61—21no—
125C16×C61—21no—
126#11758185—22no—
127C23×C61—41no—
128C22×D61—42no—
129C22×D61C2×D812yes—
130Q8×C61—22no—
131Q8×C61—41no—
132D4×C61—41no—
133D4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—82no—
138C2×C4×C61C2×D811yescharacteristic
139C2×C4×C61C2×D811yescharacteristic
140C2×C4×C61C22×D411yescharacteristic
141C22⋊D61—162no—
142C2×Dic61—42no—
143C2×Dic61C2×D812yes—
144D244—22no—
145D244—42no—
146D244—42no—
147D244—84no—
148D244—82no—
149D244—161no—
150D244—162no—
151C4×D61—22no—
152C4×D61—22no—
153C4×D61—84no—
154Dic122—161no—
155C8×C61—21no—
156C8×C61C2×#4111yescharacteristic
157C8×C61H(5,𝔽2)11yescharacteristic
158#92418—22no—
159#33872—611nomaximal, Sylow
160C22×C61—81no—
161C22×C61—81no—
162C22×C61—81no—
163C22×C61C22×D811yescharacteristic
164C2×D61—22no—
165C2×D61—22no—
166C2×D61—22no—
167C2×D61—82no—
168C2×D61—82no—
169C2×D61—82no—
170C4×C61—21no—
171C4×C61—42no—
172C4×C61—81no—
173C4×C61—81no—
174C4×C61#50812yes—
175C4×C61#54611yescharacteristic
176C4×C61#57711yescharacteristic
177Dic61—22no—
178Dic61—82no—
179#3341—611no—
180#3287—611no—
181#2240—611no—
182#2234—612no—
183#2227—611no—
184#2215—612no—
185#2206—611no—
186#2186—612no—
187C4×SD32D6111yescharacteristic, complemented
188D4×C16—611no—
189#2138—612no—
190C2×C61—161no—
191C2×C61#329411yescharacteristic
192C2×C61C2×#50811yescharacteristic
193C2×C61#341511yescharacteristic
194D61—42no—
195D61—162no—
196C2×#135—612no—
197#555—1221no—
198#550—1221no—
199C2×SD32C2×D6111yescharacteristic
200C2×D16—612no—
201C22×C16—612no—
202#500—611no—
203#493—611no—
204#482—1222no—
205#476—1221no—
206#466—612no—
207#466—612no—
208#460—1221no—
209#449—1222no—
210C4×Q16C2×D6111yescharacteristic
211C4×D8C2×D6111yescharacteristic
212C8×D4—611no—
213#427—612no—
214#367C2×D6111yescharacteristic
215#366—612no—
216C4⋊C16—611no—
217#358—612no—
218#358C2×D6111yescharacteristic
219#357—612no—
220#357—612no—
221#357—612no—
222#357C2×D6111yescharacteristic
223C22⋊C16—612no—
224C4×C16C2×D6111yescharacteristic
225C61#3387211yescharacteristic, complemented, Sylow
226C2×#41—1222no—
227#135—1224no—
228#135—2442no—
229C2×Q16C22×D6111yescharacteristic
230C2×SD16—1222no—
231C2×D8—612no—
232C2×D8—1221no—
233C2×D8C22×D6111yescharacteristic
234C22×C8—612no—
235C4⋊Q8—1221no—
236#127—1221no—
237#127—2441no—
238#124—2442no—
239#121—1222no—
240#121—1222no—
241#121—2442no—
242#121—2441no—
243C4×Q8—21no—
244C4×D4—21no—
245C4×D4—611no—
246C4×D4—2441no—
247#117—1222no—
248#117—2441no—
249SD32—22no—
250D16—2442no—
251C2×C16—612no—
252C2×C16—2442no—
253C2×C16C22×D6111yescharacteristic
254C2×C16C22×D6111yescharacteristic
255#107—612no—
256#107—1221no—
257#107C22×D6111yescharacteristic
258#107C22×D6111yescharacteristic
259#106—1222no—
260C4⋊C8—611no—
261#103—21no—
262#102—21no—
263#102—1222no—
264#102—1222no—
265C22⋊C8—612no—
266C4×C8C22×D6111yescharacteristic
267#41—4882no—
268#41—2444no—
269C2×Q8—21no—
270C2×Q8—2441no—
271C2×D4—21no—
272C2×D4—611no—
273C2×D4—1222no—
274C2×D4—2442no—
275C2×D4—2442no—
276C2×D4—2442no—
277C2×D4—2441no—
278C2×D4—2441no—
279C22×C4—41no—
280C22×C4—612no—
281C22×C4—2442no—
282Q16—21no—
283Q16#1175822212yes—
284SD16—2444no—
285D8—21no—
286D8—21no—
287D8—1222no—
288D8—1222no—
289D8—4881no—
290C2×C8—612no—
291C2×C8—1222no—
292C2×C8—1222no—
293C2×C8D4×D6111yescharacteristic
294C2×C8C23×D6111yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—611no—
299C4⋊C4—1222no—
300C4⋊C4—2441no—
301C22⋊C4—41no—
302C22⋊C4—612no—
303C22⋊C4—2442no—
304C22⋊C4—2442no—
305C22⋊C4—2442no—
306C42—41no—
307C42—2442no—
308C42D4×D6111yescharacteristic
309C16—21no—
310C16—21no—
311C16—1222no—
312C23—41no—
313C23—612no—
314C23—2442no—
315Q8—22no—
316Q8—41no—
317Q8—9761no—
318D4—41no—
319D4—41no—
320D4—1222no—
321D4—9761no—
322D4—9762no—
323D4—9762no—
324D4—4884no—
325D4—4882no—
326D4—2442no—
327D4—2442no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—612no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—4884no—
336C2×C4—2442no—
337C2×C4D8×D6111yescharacteristic
338C2×C4D8×D6111yescharacteristic
339C2×C4C2×D4×D6111yescharacteristic
340C8—21no—
341C8—1222no—
342C8C2×#1175822211yescharacteristic
343C8#1451982311yescharacteristic
344C22—81no—
345C22—81no—
346C22—81no—
347C22—1222no—
348C22—1222no—
349C22—1222no—
350C22—4882no—
351C22—4882no—
352C22—4882no—
353C22C2×D8×D6111yescharacteristic
354C4—21no—
355C4—42no—
356C4—81no—
357C4—81no—
358C4—1222no—
359C4—4882no—
360C4#1684019312yes—
361C4#1684040811yescharacteristic
362C4#1684056911yescharacteristic
363C2—161no—
364C2—9762no—
365C2—2442no—
366C2#2145763411yescharacteristic
367C2C2×#1684019311yescharacteristic
368C2#2145895711yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC4×SD32×C61C2
socleC22×C61C22×D8
hypercenterC4×SD32D61
hypocenterC61#33872
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
1. #21458243C2semidirect
2. #21457578C2semidirect
3. #21448043C2semidirect
4. #21448012C2semidirect
5. #21447971C2semidirect
6. #21447904C2semidirect
7. #21447854C2semidirect
8. #21447741C2semidirect
9. C16×D244C2semidirect
10. #21447525C2semidirect
11. C4×SD32×C61C2semidirect
187. C4×SD32D61semidirect
225. C61#33872semidirect

Maximal subgroups

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

subgroupindex
#214582432
#214575782
#214480432
#214480122
#214479712
#214479042
#214478542
#214477412
C16×D2442
#214475252
C4×SD32×C612
#3387261