Finite Group Databasebeta

← #29834451

Subgroups — #29834451

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840192C211yesmaximal, characteristic, complemented
2C22×C16×D61C211yesmaximal, characteristic, complemented
3#21455401C211yesmaximal, characteristic, complemented
4#21455396C211yesmaximal, characteristic, complemented
5#21447918C212yesmaximal, complemented
6#21447914C212yesmaximal, complemented
7#21447909C212yesmaximal, complemented
8#21447904C212yesmaximal, complemented
9#21447601C211yesmaximal, characteristic, complemented
10#21447527C211yesmaximal, characteristic, complemented
11C61×#2138C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840274—21no—
15#16840269—21no—
16#16840192—24no—
17C2×#14519723C2211yescharacteristic, complemented
18C2×#14519721C2211yescharacteristic, complemented
19C2×#14519720C2211yescharacteristic, complemented
20C2×#14519718C2211yescharacteristic, complemented
21C22×#11758185C2211yescharacteristic
22C2×C16×D61—24no—
23C2×C16×D61C2212yes—
24C2×C16×D61C2212yes—
25#16840004C2212yes—
26#16840000C2212yes—
27#16839999C2212yes—
28#16839995C2212yes—
29#16839992—22no—
30#16839988—22no—
31#16839800C2211yescharacteristic
32#16839758—21no—
33#16839756—21no—
34#16839589C2212yescomplemented
35#16839587C2212yescomplemented
36#16839567C2212yescomplemented
37#16839564C2212yescomplemented
38C2×#135×C61C2211yescharacteristic
39C22×C16×C61C2211yescharacteristic
40C61×#427C2211yescharacteristic
41C61×#358C2212yescomplemented
42C61×#357C2212yescomplemented
43C2×#11758222—21no—
44C22×C4×D61D411yescharacteristic
45#14519799—22no—
46#14519799—22no—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50C22×#92418D411yescharacteristic
51C2×C8×D61—22no—
52C2×C8×D61—22no—
53C2×C8×D61D412yes—
54C2×C8×D61C2312yes—
55#14519761—41no—
56#14519758—41no—
57#14519757—22no—
58#14519753—22no—
59#14519752—22no—
60#14519748—22no—
61#14519747—41no—
62#14519745—41no—
63C4×D244—42no—
64C4×C22⋊D61—42no—
65#14519736—21no—
66#14519723—22no—
67#14519721—22no—
68#14519720—22no—
69#14519718—22no—
70C2×#11758185D414yescomplemented
71C2×#11758185C2311yescharacteristic
72C2×#11758185C2311yescharacteristic
73C16×D61—24no—
74C16×D61—24no—
75#14519711C2312yes—
76#14519709—21no—
77#14519701—22no—
78#14519700—22no—
79#14519698—22no—
80#14519697—22no—
81C2×#41×C61—21no—
82#135×C61—22no—
83#135×C61—22no—
84C2×Q16×C61C2311yescharacteristic
85C2×SD16×C61—21no—
86C2×D8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88#117×C61—21no—
89C2×C16×C61—22no—
90C2×C16×C61C2312yes—
91#107×C61C2312yes—
92#106×C61—21no—
93C23×D61D811yescharacteristic
94#41×C61—42no—
95#41×C61—42no—
96C2×Q8×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102C2×C22⋊D61—42no—
103C22×Dic61D811yescharacteristic
104#11758222—42no—
105#11758222—42no—
106C2×D244—21no—
107C2×D244—42no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—41no—
111C2×C4×D61D812yes—
112C2×C4×D61C2×D412yes—
113C2×Dic122—21no—
114Q16×C61—21no—
115Q16×C61—21no—
116SD16×C61—42no—
117D8×C61—21no—
118D8×C61—21no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D412yes—
121C2×C8×C61#4112yes—
122C2×C8×C61C2411yescharacteristic
123C4⋊C4×C61—22no—
124C22⋊C4×C61—41no—
125C42×C61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—42no—
128#11758198—42no—
129#11758198—42no—
130#11758198—42no—
131C4⋊Dic61—22no—
132C4⋊Dic61—42no—
133#11758196—42no—
134C4×Dic61—41no—
135C2×#92418—22no—
136C2×#92418C2×D412yes—
137Dic244—21no—
138Dic244—21no—
139D488—21no—
140D488—21no—
141#11758190—42no—
142C8×D61—24no—
143C8×D61—22no—
144C8×D61—22no—
145C16×C61—22no—
146#11758185C2×D412yes—
147#11758185C2×D412yes—
148C23×C61—41no—
149C23×C61C2×D811yescharacteristic
150C22×D61—22no—
151C22×D61—24no—
152C22×D61—41no—
153C22×D61C2×D812yes—
154Q8×C61—41no—
155Q8×C61—41no—
156D4×C61—41no—
157D4×C61—41no—
158D4×C61—82no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—82no—
162C2×C4×C61#12112yes—
163C2×C4×C61C2×D811yescharacteristic
164C2×C4×C61#13512yes—
165C2×C4×C61C22×D411yescharacteristic
166C22⋊D61—82no—
167C22⋊D61—84no—
168C2×Dic61—22no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61C2×D812yes—
172D244—41no—
173D244—41no—
174D244—84no—
175C4×D61—22no—
176C4×D61—22no—
177C4×D61—24no—
178C4×D61—82no—
179Dic122—41no—
180Dic122—41no—
181C8×C61#12112yes—
182C8×C61C22×D411yescharacteristic
183C8×C61C2×#4111yescharacteristic
184#92418—22no—
185#33600—611nomaximal, Sylow
186C22×C61—22no—
187C22×C61—82no—
188C22×C61—81no—
189C22×C61#46612yes—
190C22×C61C22×D811yescharacteristic
191C2×D61—28no—
192C2×D61—22no—
193C2×D61—22no—
194C2×D61—82no—
195C2×D61—81no—
196C2×D61#50814yescomplemented
197C4×C61—82no—
198C4×C61—81no—
199C4×C61#46612yes—
200C4×C61C2×#12111yescharacteristic
201C4×C61C2×#13511yescharacteristic
202Dic61—22no—
203Dic61—82no—
204Dic61—81no—
205C2×#508—611no—
206C23×C16—611no—
207#3050—611no—
208#3050—611no—
209#2217—612no—
210#2217—612no—
211#2216—612no—
212#2215—612no—
213#2158—611no—
214#2138—611no—
215#2138D6111yescharacteristic, complemented
216C2×C61—22no—
217C2×C61—161no—
218C2×C61C2×#46611yescharacteristic
219C2×C61C2×#50812yes—
220D61—24no—
221D61—161no—
222C2×#135—611no—
223C2×#135C2×D6111yescharacteristic
224C23×C8—611no—
225#525—1221no—
226#525—1221no—
227#508—1224no—
228C2×Q32—611no—
229C2×SD32—611no—
230C2×SD32—611no—
231C2×D16—611no—
232C22×C16—611no—
233C22×C16—612no—
234C22×C16—612no—
235C22×C16—1224no—
236C22×C16C2×D6111yescharacteristic
237#467—612no—
238#467—612no—
239#466—612no—
240#466—612no—
241#465—1222no—
242#465—1222no—
243#427—611no—
244#427C2×D6111yescharacteristic
245#416—1221no—
246#416—1221no—
247#367—612no—
248#366—612no—
249#358—612no—
250#358C2×D6112yescomplemented
251#357—612no—
252#357C2×D6112yescomplemented
253C61#3360011yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C2×#41—1221no—
256C23×C4—611no—
257#135—22no—
258#135—22no—
259#135—1222no—
260#135—1222no—
261C2×Q16—611no—
262C2×Q16C22×D6111yescharacteristic
263C2×SD16—21no—
264C2×SD16—1221no—
265C2×D8—611no—
266C2×D8C22×D6111yescharacteristic
267C22×C8—611no—
268C22×C8—612no—
269C22×C8—612no—
270C22×C8—1222no—
271C22×C8—1222no—
272C22×C8C22×D6111yescharacteristic
273#123—2441no—
274#123—2441no—
275C22⋊Q8—1222no—
276C22⋊Q8—1222no—
277#121—1222no—
278#121—1222no—
279#120—2441no—
280#120—2441no—
281C4×D4—2442no—
282C4×D4—2442no—
283#117—21no—
284#117—1221no—
285Q32—1222no—
286SD32—1222no—
287SD32—1222no—
288D16—1222no—
289C2×C16—22no—
290C2×C16—614no—
291C2×C16—611no—
292C2×C16—611no—
293C2×C16—1224no—
294C2×C16—1224no—
295C2×C16C22×D6112yes—
296#107—612no—
297#107C22×D6112yes—
298#106—21no—
299#106—1221no—
300#103—1222no—
301#103—1222no—
302#102—1222no—
303#102—1222no—
304C24—611no—
305#41—42no—
306#41—42no—
307#41—2442no—
308#41—2442no—
309C2×Q8—21no—
310C2×Q8—1221no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—1221no—
314C2×D4—2442no—
315C2×D4—2442no—
316C2×D4—2441no—
317C22×C4—41no—
318C22×C4—611no—
319C22×C4—612no—
320C22×C4—612no—
321C22×C4—1222no—
322C22×C4—1222no—
323C22×C4—2441no—
324C22×C4D4×D6111yescharacteristic
325Q16—21no—
326Q16—21no—
327Q16—1221no—
328Q16—1221no—
329SD16—42no—
330SD16—2442no—
331D8—21no—
332D8—21no—
333D8—1221no—
334D8—1221no—
335C2×C8—612no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8—1222no—
339C2×C8—1224no—
340C2×C8#1175822212yes—
341C2×C8D4×D6112yes—
342C2×C8D4×D6111yescharacteristic
343C2×C8C23×D6111yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—1222no—
346C4⋊C4—2442no—
347C4⋊C4—2442no—
348C22⋊C4—41no—
349C22⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2442no—
352C22⋊C4—2442no—
353C22⋊C4—2441no—
354C42—41no—
355C42—2441no—
356C16—22no—
357C16—612no—
358C16—612no—
359C23—41no—
360C23—612no—
361C23—1224no—
362C23—1222no—
363C23—2441no—
364C23D8×D6111yescharacteristic
365Q8—41no—
366Q8—41no—
367Q8—2441no—
368Q8—2441no—
369D4—41no—
370D4—41no—
371D4—82no—
372D4—4884no—
373D4—4884no—
374D4—4882no—
375D4—2441no—
376D4—2441no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4—612no—
381C2×C4—1222no—
382C2×C4—1224no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—4882no—
386C2×C4—2442no—
387C2×C4—2441no—
388C2×C4#1451975012yes—
389C2×C4D8×D6111yescharacteristic
390C2×C4#1451979812yes—
391C2×C4C2×D4×D6111yescharacteristic
392C8—1222no—
393C8#1451975012yes—
394C8C2×D4×D6111yescharacteristic
395C8C2×#1175822211yescharacteristic
396C22—22no—
397C22—81no—
398C22—82no—
399C22—614no—
400C22—1228no—
401C22—1222no—
402C22—1222no—
403C22—4881no—
404C22—4882no—
405C22#1683999712yes—
406C22C2×D8×D6111yescharacteristic
407C4—81no—
408C4—82no—
409C4—1222no—
410C4—4881no—
411C4—4882no—
412C4#1683999712yes—
413C4C2×#1451975011yescharacteristic
414C4C2×#1451979811yescharacteristic
415C2—22no—
416C2—161no—
417C2—1224no—
418C2—9761no—
419C2C2×#1683999711yescharacteristic
420C2#508×D6112yes—
subgroupG/H
centerC2×C4D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
hypercenter#2138D61
hypocenterC61#33600
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. C2×#16840192C2semidirect
2. C22×C16×D61C2semidirect
3. #21455401C2semidirect
4. #21455396C2semidirect
5. #21447918C2semidirect
6. #21447914C2semidirect
7. #21447909C2semidirect
8. #21447904C2semidirect
9. #21447601C2semidirect
10. #21447527C2semidirect
11. C61×#2138C2semidirect
17. C2×#14519723C22semidirect
18. C2×#14519721C22semidirect
19. C2×#14519720C22semidirect
20. C2×#14519718C22semidirect
34. #16839589C22semidirect
35. #16839587C22semidirect
36. #16839567C22semidirect
37. #16839564C22semidirect
41. C61×#358C22semidirect
42. C61×#357C22semidirect
70. C2×#11758185D4semidirect
196. C2×D61#508semidirect
215. #2138D61semidirect
250. #358C2×D61semidirect
252. #357C2×D61semidirect
253. C61#33600semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
C22×C16×D612
#214554012
#214553962
#214479182
#214479142
#214479092
#214479042
#214476012
#214475272
C61×#21382
#3360061