Finite Group Databasebeta

← #29834545

Subgroups — #29834545

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840180C211yesmaximal, characteristic, complemented
2#21455485C211yesmaximal, characteristic, complemented
3#21455463C211yesmaximal, characteristic, complemented
4#21447929C212yesmaximal, complemented
5#21447919C212yesmaximal, complemented
6#21447912C212yesmaximal, complemented
7#21447902C212yesmaximal, complemented
8#21447607C211yesmaximal, characteristic, complemented
9#21447526C211yesmaximal, characteristic, complemented
10C2×#16839563C211yesmaximal, characteristic, complemented
11C2×C61×#509C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840340—21no—
14#16840329—21no—
15#16840180—24no—
16C2×#14519716C2212yescomplemented
17C22×#11758185C2211yescharacteristic
18C2×C16×D61C2212yescomplemented
19#16840012—22no—
20#16840005—22no—
21#16840000C2212yescomplemented
22#16839995C2212yes—
23#16839995C2212yes—
24#16839995C2212yescomplemented
25#16839800C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27#16839767—21no—
28#16839763—21no—
29#16839590C2212yescomplemented
30#16839588C2212yescomplemented
31#16839566C2211yescharacteristic, complemented
32#16839566C2211yescharacteristic, complemented
33#16839563C2212yes—
34#16839563C2212yes—
35#16839563C2211yescharacteristic, complemented
36#16839563C2211yescharacteristic, complemented
37C2×#135×C61C2211yescharacteristic
38C22×D8×C61C2211yescharacteristic
39C61×#509—24no—
40C2×SD32×C61C2212yes—
41C2×D16×C61C2212yes—
42C2×M32×C61C2211yescharacteristic
43C2×#11758222D411yescharacteristic
44#14519783—22no—
45#14519783—22no—
46C2×#11758194D411yescharacteristic
47C2×#11758189D412yes—
48C2×C8×D61C2312yes—
49#14519773—41no—
50#14519764—41no—
51#14519756—41no—
52#14519753—22no—
53#14519751—41no—
54#14519749—42no—
55#14519748—22no—
56#14519748—22no—
57#14519748—22no—
58#14519746—42no—
59#14519736—21no—
60C2×C4⋊Dic61—21no—
61#14519716—24no—
62C2×#11758185—22no—
63C2×#11758185C2311yescharacteristic
64C2×#11758185C2311yescharacteristic
65C16×D61—24no—
66#14519711—21no—
67#14519711C2311yescharacteristic
68#14519711C2311yescharacteristic
69#14519711C2312yes—
70#14519709—21no—
71#14519701—22no—
72#14519698—22no—
73#14519698—22no—
74#14519698—22no—
75C2×#41×C61—21no—
76C22×D4×C61—21no—
77#135×C61—22no—
78#135×C61—22no—
79C2×Q16×C61C2311yescharacteristic
80C2×SD16×C61—21no—
81C2×D8×C61—22no—
82C2×D8×C61—21no—
83C2×D8×C61—22no—
84C2×D8×C61C2312yes—
85C2×D8×C61C2311yescharacteristic
86C22×C8×C61C2311yescharacteristic
87SD32×C61—24no—
88D16×C61—24no—
89M32×C61D414yescomplemented
90C2×C16×C61C2312yes—
91C24×C61—41no—
92#41×C61—42no—
93#41×C61—42no—
94C2×Q8×C61—21no—
95C2×D4×C61—22no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C2×D4×C61—42no—
99C2×D4×C61—41no—
100C2×D4×C61—42no—
101C22×C4×C61—41no—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—42no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—42no—
106C2×C22⋊D61D812yes—
107C22×Dic61—41no—
108#11758222—22no—
109#11758222—22no—
110C2×D244D811yescharacteristic
111C2×C4×D61C2×D412yes—
112C2×Dic122D811yescharacteristic
113Q16×C61—21no—
114Q16×C61—21no—
115SD16×C61—42no—
116D8×C61—21no—
117D8×C61—22no—
118D8×C61—22no—
119D8×C61—21no—
120D8×C61—42no—
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D412yes—
123C2×C8×C61#4112yes—
124C2×C8×C61C2411yescharacteristic
125C22⋊Dic61—41no—
126C22⋊Dic61—42no—
127C22⋊Dic61—42no—
128C22⋊Dic61—42no—
129#11758198—42no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—41no—
134#11758196—42no—
135C4×Dic61—41no—
136#11758194—22no—
137C2×#92418C2×D412yes—
138#11758189—24no—
139C8×D61—22no—
140C8×D61—22no—
141C16×C61C2×D414yes—
142#11758185—21no—
143#11758185—21no—
144C23×C61—41no—
145C23×C61—42no—
146C23×C61—82no—
147C23×C61—161no—
148C23×C61—161no—
149C23×C61C2×D811yescharacteristic
150C22×D61C2×D812yes—
151Q8×C61—41no—
152Q8×C61—41no—
153D4×C61—42no—
154D4×C61—41no—
155D4×C61—42no—
156D4×C61—41no—
157D4×C61—82no—
158D4×C61—82no—
159C2×C4×C61—41no—
160C2×C4×C61—82no—
161C2×C4×C61#12112yes—
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61#13512yes—
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—24no—
166C22⋊D61—162no—
167C22⋊D61—162no—
168C22⋊D61—162no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—42no—
172C2×Dic61—161no—
173C2×Dic61C2×D812yes—
174D244—22no—
175C4×D61—22no—
176C4×D61—22no—
177Dic122—22no—
178C8×C61#12112yes—
179C8×C61C22×D411yescharacteristic
180C8×C61C2×#4111yescharacteristic
181#92418—22no—
182#33612—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—81no—
186C22×C61—82no—
187C22×C61—82no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61—82no—
191C22×C61—164no—
192C22×C61#46612yes—
193C22×C61C22×D811yescharacteristic
194C2×D61—22no—
195C2×D61—22no—
196C2×D61—22no—
197C4×C61—81no—
198C4×C61#46612yes—
199C4×C61C2×#12111yescharacteristic
200C4×C61C2×#13511yescharacteristic
201Dic61—22no—
202Dic61—82no—
203Dic61—81no—
204Dic61—81no—
205C2×#509D6111yescharacteristic, complemented
206C2×#504—611no—
207#3061—611no—
208#3058—611no—
209#2220—612no—
210#2218—612no—
211#2217—612no—
212#2215—612no—
213#2159—611no—
214#2138—611no—
215C2×#357—611no—
216C2×C61—22no—
217C2×C61—161no—
218C2×C61—162no—
219C2×C61C2×#46611yescharacteristic
220C2×C61#341511yescharacteristic
221C2×C61#341611yescharacteristic
222D61—42no—
223C2×#135C2×D6111yescharacteristic
224C22×D8C2×D6111yescharacteristic
225C2×#131—611no—
226#537—1221no—
227#535—1221no—
228#509—24no—
229C2×SD32C2×D6112yes—
230C2×D16C2×D6112yes—
231#504—1224no—
232C2×M32—612no—
233C2×M32C2×D6111yescharacteristic
234C22×C16—611no—
235C22×C16—612no—
236#469—1222no—
237#468—1222no—
238#467—612no—
239#466—612no—
240#466—612no—
241#466—612no—
242#427—611no—
243C2×#107—611no—
244#418—1221no—
245#417—1221no—
246#367—612no—
247#366—612no—
248#358—611no—
249#358—611no—
250#357—612no—
251#357—612no—
252#357—611no—
253#357—611no—
254C61#3361211yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41—611no—
257C22×D4—21no—
258#135—22no—
259#135—22no—
260C2×Q16C22×D6111yescharacteristic
261C2×SD16—21no—
262C2×D8—21no—
263C2×D8—22no—
264C2×D8—22no—
265C2×D8C22×D6112yes—
266C2×D8C22×D6111yescharacteristic
267#131—1222no—
268#131—1222no—
269C2×M16—611no—
270C2×M16—612no—
271C22×C8—612no—
272C22×C8C22×D6111yescharacteristic
273#127—2441no—
274#124—2441no—
275C22⋊Q8—1222no—
276C22⋊Q8—2441no—
277#121—1222no—
278#121—1222no—
279#121—1222no—
280#121—2441no—
281#121—2442no—
282#120—2442no—
283#117—1221no—
284C2×C4⋊C4—1221no—
285SD32—24no—
286D16—24no—
287M32—1224no—
288M32C22⋊D6114yescomplemented
289C2×C16—611no—
290C2×C16—611no—
291C2×C16—1222no—
292C2×C16—1224no—
293C2×C16C22×D6112yes—
294#107—612no—
295#107—611no—
296#107—611no—
297#107—1221no—
298#106—1221no—
299#103—1222no—
300#102—1222no—
301#102—1222no—
302#102—1222no—
303C24—41no—
304#41—42no—
305#41—42no—
306#41—1222no—
307#41—1222no—
308C2×Q8—21no—
309C2×Q8—611no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—612no—
317C2×D4—611no—
318C2×D4—2442no—
319C2×D4—2442no—
320C2×D4—2442no—
321C22×C4—41no—
322C22×C4—612no—
323C22×C4—2441no—
324C22×C4D4×D6111yescharacteristic
325Q16—21no—
326Q16—21no—
327SD16—42no—
328D8—21no—
329D8—22no—
330D8—22no—
331D8—21no—
332D8—42no—
333M16—1222no—
334M16—1224no—
335C2×C8—612no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8D4×D6111yescharacteristic
339C2×C8#1175822412yes—
340C2×C8C2×C22⋊D6112yes—
341C2×C8C23×D6111yescharacteristic
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—1222no—
345C4⋊C4—2442no—
346C4⋊C4—2441no—
347C22⋊C4—2442no—
348C22⋊C4—2442no—
349C22⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2441no—
352C42—2441no—
353C16—1221no—
354C16—1221no—
355C16C2×C22⋊D6114yes—
356C23—41no—
357C23—42no—
358C23—82no—
359C23—161no—
360C23—161no—
361C23—612no—
362C23D8×D6111yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—1222no—
366D4—42no—
367D4—41no—
368D4—42no—
369D4—41no—
370D4—82no—
371D4—82no—
372D4—1224no—
373D4—1222no—
374D4—9762no—
375D4—9762no—
376D4—9762no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—612no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—9761no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2442no—
386C2×C4#1451974812yes—
387C2×C4D8×D6111yescharacteristic
388C2×C4#1451979612yes—
389C2×C4C2×D4×D6111yescharacteristic
390C8—1222no—
391C8#1451974812yes—
392C8C22×C22⋊D6111yescharacteristic
393C8C2×#1175822411yescharacteristic
394C22—21no—
395C22—21no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—82no—
402C22—164no—
403C22—1222no—
404C22—1222no—
405C22—1222no—
406C22#1683999512yes—
407C22C2×D8×D6111yescharacteristic
408C4—81no—
409C4—1222no—
410C4—4882no—
411C4—4881no—
412C4—4881no—
413C4#1683999512yes—
414C4C2×#1451974811yescharacteristic
415C4C2×#1451979611yescharacteristic
416C2—22no—
417C2—162no—
418C2—161no—
419C2—2442no—
420C2C2×#1683999511yescharacteristic
421C2#2145895911yescharacteristic
422C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33612
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16840180C2semidirect
2. #21455485C2semidirect
3. #21455463C2semidirect
4. #21447929C2semidirect
5. #21447919C2semidirect
6. #21447912C2semidirect
7. #21447902C2semidirect
8. #21447607C2semidirect
9. #21447526C2semidirect
10. C2×#16839563C2semidirect
11. C2×C61×#509C2semidirect
16. C2×#14519716C22semidirect
18. C2×C16×D61C22semidirect
21. #16840000C22semidirect
24. #16839995C22semidirect
29. #16839590C22semidirect
30. #16839588C22semidirect
31. #16839566C22semidirect
32. #16839566C22semidirect
35. #16839563C22semidirect
36. #16839563C22semidirect
89. M32×C61D4semidirect
205. C2×#509D61semidirect
254. C61#33612semidirect
288. M32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401802
#214554852
#214554632
#214479292
#214479192
#214479122
#214479022
#214476072
#214475262
C2×#168395632
C2×C61×#5092
#3361261