Finite Group Databasebeta

← #29665563

Subgroups — #29665563

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456746C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3#21455285C211yesmaximal, characteristic
4#21454859C211yesmaximal, characteristic, complemented
5#21445367C214yesmaximal, complemented
6#21445285C214yesmaximal, complemented
7#21444505C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C61×#1571C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840135C2212yescomplemented
15#16840113C2212yescomplemented
16#16840043—22no—
17#16840043C2214yes—
18#16839978—22no—
19#16839968—22no—
20#16839968C2212yes—
21#16839958C2212yes—
22#16839903—22no—
23#16839841C2214yes—
24#16839839—22no—
25#16839815C2211yescharacteristic
26C2×#14519711—21no—
27#16839764—21no—
28C2×#14519699—21no—
29#16839724—22no—
30#16839509C2212yescomplemented
31#16839507C2214yescomplemented
32#16839505C2212yescomplemented
33C61×#431C2211yescharacteristic
34C61×#423C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C61×#335C2214yescomplemented
37C2×#11758222—21no—
38C22×Dic122—21no—
39C22×D244D411yescharacteristic
40C22×C4×D61—21no—
41#14519805—44no—
42C2×#11758190—22no—
43C2×#11758190—24no—
44C2×D488—22no—
45C2×#11758189—24no—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777C2311yescharacteristic
50#14519777C2312yes—
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57C4×D244—21no—
58C4×D244—21no—
59C4×D244D412yes—
60C4×D244C2312yes—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×#11758198—21no—
65#14519711—24no—
66C4⋊#92418C2312yes—
67#14519702—24no—
68#14519702—22no—
69#14519699—24no—
70#14519699—22no—
71C22⋊#92418—21no—
72#14519689—24no—
73C4×#92418—21no—
74C2×M16×C61—21no—
75C22×C8×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61D412yes—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—21no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C2×C22⋊D61—41no—
89C22×Dic61—21no—
90C22×Dic61—41no—
91#11758222—42no—
92#11758222—42no—
93C2×D244—22no—
94C2×D244—22no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244C2×D412yes—
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—41no—
103C2×C4×D61—82no—
104C2×Dic122—22no—
105C2×Dic122—21no—
106C2×Dic122—41no—
107C2×Dic122—82no—
108M16×C61—42no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C2×C8×C61—42no—
112C2×C8×C61C2×D414yes—
113C42×C61C2×D411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61#4111yescharacteristic
116C42×C61C2411yescharacteristic
117#11758198—22no—
118#11758198—22no—
119#11758198—41no—
120#11758198—41no—
121C4⋊Dic61—21no—
122C4⋊Dic61—22no—
123C4⋊Dic61—21no—
124C4⋊Dic61—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—41no—
127C4⋊Dic61—41no—
128C4⋊Dic61C2×D412yes—
129C2×#92418—22no—
130D488—44no—
131#11758190—44no—
132#11758190—84no—
133#11758189—84no—
134C23×C61#12111yescharacteristic
135C22×D61—22no—
136C22×D61—42no—
137C22×D61—41no—
138C22×D61—81no—
139C22×D61—81no—
140C2×C4×C61—22no—
141C2×C4×C61—41no—
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)12yes—
146C2×C4×C61#13712yes—
147C2×C4×C61C22×D411yescharacteristic
148C2×C4×C61C22×D411yescharacteristic
149C2×C4×C61C2×#4111yescharacteristic
150C22⋊D61—82no—
151C2×Dic61—22no—
152C2×Dic61—42no—
153C2×Dic61—41no—
154C2×Dic61—81no—
155C2×Dic61—82no—
156D244—22no—
157D244—22no—
158D244—41no—
159D244—41no—
160D244—42no—
161D244—42no—
162D244—81no—
163C4×D61—44no—
164C4×D61—82no—
165C4×D61—82no—
166Dic122—41no—
167Dic122—41no—
168Dic122—44no—
169Dic122—82no—
170C8×C61—24no—
171C8×C61—42no—
172C8×C61—42no—
173#92418—42no—
174#19421—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61#46812yes—
178C22×C61C2×#12111yescharacteristic
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—84no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61#46812yes—
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191C4×C61C2×#13711yescharacteristic
192C4×C61#57612yes—
193Dic61—42no—
194Dic61—82no—
195Dic61—81no—
196#3206—611no—
197C2×#474—611no—
198#3036—611no—
199#2975—611no—
200#1699—614no—
201#1685—614no—
202#1571D6111yescharacteristic, complemented
203#1549—611no—
204#1540—611no—
205C2×C61—22no—
206C2×C61#304111yescharacteristic
207C2×C61C2×#46811yescharacteristic
208C2×C61#306911yescharacteristic
209D61—82no—
210D61—161no—
211C2×AGL(1,ℤ/8)—1221no—
212#532—611no—
213C2×C4⋊Q8—611no—
214C2×C4×D4—611no—
215#492—612no—
216#487—612no—
217#474—614no—
218#474—1222no—
219#463—1222no—
220#461—612no—
221#461—1222no—
222#459—612no—
223#448—1222no—
224#436—614no—
225#436—1222no—
226#431—611no—
227#431C2×D6111yescharacteristic
228C2×#107—1221no—
229#423C2×D6111yescharacteristic
230C2×C4⋊C8C2×D6111yescharacteristic
231#417—1221no—
232C2×#102—1221no—
233#407—1222no—
234#335C2×D6114yescomplemented
235#332—612no—
236#332—614no—
237#331—612no—
238C61#1942111yescharacteristic, complemented, Sylow
239C2×#41—1221no—
240C22×Q8—1221no—
241C22×D4—611no—
242C23×C4—1221no—
243AGL(1,ℤ/8)—2444no—
244C2×SD16—1222no—
245C2×SD16—1224no—
246C2×D8—1222no—
247C2×M16—21no—
248C2×M16—1224no—
249C22×C8—21no—
250C4⋊Q8—612no—
251C4⋊Q8—611no—
252C4⋊Q8—1221no—
253C4⋊Q8—1221no—
254C4⋊Q8—1221no—
255#127—611no—
256#124—1221no—
257#121—1221no—
258#121—1221no—
259C4×D4—612no—
260C4×D4—612no—
261C4×D4—1221no—
262C4×D4—1221no—
263C4×D4—1221no—
264C4×D4—1221no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42C22×D6111yescharacteristic
270#107—1224no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—612no—
274C4⋊C8D24412yes—
275C4⋊C8C22×D6112yes—
276C4⋊C8C22×D6112yes—
277#103—1222no—
278#103—1224no—
279#102—1222no—
280#102—1224no—
281C22⋊C8—1221no—
282C22⋊C8—1224no—
283C22⋊C8D24412yes—
284C4×C8—1221no—
285C4×C8C22×D6112yes—
286C24—1221no—
287#41—2442no—
288#41—2442no—
289C2×Q8—1221no—
290C2×Q8—1222no—
291C2×Q8—4882no—
292C2×Q8—2441no—
293C2×D4—612no—
294C2×D4—1222no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—1221no—
298C2×D4—1221no—
299C2×D4—1222no—
300C2×D4—2441no—
301C22×C4—1221no—
302C22×C4—1222no—
303C22×C4—1222no—
304C22×C4—4882no—
305C22×C4—2441no—
306C22×C4—2441no—
307C22×C4C2×D24411yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4#1175822611yescharacteristic
310SD16—4884no—
311SD16—2444no—
312D8—2444no—
313M16—42no—
314M16—4884no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—1222no—
319C2×C8C2×D24414yes—
320C4⋊C4—612no—
321C4⋊C4—1221no—
322C4⋊C4—1222no—
323C4⋊C4—1221no—
324C4⋊C4—1222no—
325C4⋊C4—1221no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—1222no—
329C22⋊C4—1222no—
330C22⋊C4—2441no—
331C22⋊C4—2441no—
332C42C2×D24411yescharacteristic
333C42D4×D6111yescharacteristic
334C42#1175822611yescharacteristic
335C42C23×D6111yescharacteristic
336C23—1222no—
337C23—4881no—
338C23—4881no—
339C23—2441no—
340C23—2442no—
341C23#1451975211yescharacteristic
342Q8—4882no—
343Q8—2441no—
344Q8—2441no—
345Q8—2444no—
346D4—1222no—
347D4—1222no—
348D4—4881no—
349D4—4882no—
350D4—2442no—
351D4—2441no—
352D4—2441no—
353D4—2442no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—1222no—
357C2×C4—4882no—
358C2×C4—4882no—
359C2×C4—4882no—
360C2×C4—4881no—
361C2×C4—2441no—
362C2×C4—2442no—
363C2×C4—2444no—
364C2×C4#1451975211yescharacteristic
365C2×C4#1451975211yescharacteristic
366C2×C4#1451975211yescharacteristic
367C2×C4#1451980112yes—
368C2×C4#1451980912yes—
369C2×C4C2×D4×D6111yescharacteristic
370C2×C4C22×D24411yescharacteristic
371C2×C4C2×#1175822611yescharacteristic
372C8—24no—
373C8—42no—
374C8—42no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—4884no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22#1684000912yes—
386C22C2×#1451975211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—4881no—
391C4—4882no—
392C4—2442no—
393C4#1684000912yes—
394C4C2×#1451980111yescharacteristic
395C4C2×#1451980911yescharacteristic
396C4#1684056712yes—
397C2—22no—
398C2—9761no—
399C2—4882no—
400C2#2145532311yescharacteristic
401C2C2×#1684000911yescharacteristic
402C2#2145555711yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1571C2
socleC22×C61C2×#121
prime radical#21455285C2
hypercenter#1571D61
hypocenterC61#19421
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
1. #21456746C2semidirect
2. C2×#16840043C2semidirect
4. #21454859C2semidirect
5. #21445367C2semidirect
6. #21445285C2semidirect
7. #21444505C2semidirect
8. #21444469C2semidirect
9. C61×#1571C2semidirect
14. #16840135C22semidirect
15. #16840113C22semidirect
30. #16839509C22semidirect
31. #16839507C22semidirect
32. #16839505C22semidirect
36. C61×#335C22semidirect
202. #1571D61semidirect
234. #335C2×D61semidirect
238. C61#19421semidirect

Maximal subgroups

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

subgroupindex
#214567462
C2×#168400432
#214552852
#214548592
#214453672
#214452852
#214445052
#214444692
C61×#15712
#1942161