Finite Group Databasebeta

← #29665184

Subgroups — #29665184

Order 15616.

708 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#21456741C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21454843C211yesmaximal, characteristic, complemented
4#21445244C214yesmaximal, complemented
5#21445158C214yesmaximal, complemented
6#21444591C211yesmaximal, characteristic, complemented
7#21444505C211yesmaximal, characteristic, complemented
8#21444468C211yesmaximal, characteristic, complemented
9C61×#3036C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840134C2212yescomplemented
12#16840114C2212yescomplemented
13#16840042—22no—
14#16840042C2214yes—
15#16839967—22no—
16#16839900—22no—
17D4×#92418—22no—
18D4×#92418C2214yes—
19#16839815C2211yescharacteristic
20#16839815C2211yescharacteristic
21C2×#14519710—21no—
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519698—21no—
24C2×C22⋊#92418—22no—
25#16839512C2214yescomplemented
26#16839508C2212yescomplemented
27#16839507C2214yescomplemented
28#16839506C2212yescomplemented
29C2×AGL(1,ℤ/8)×C61—21no—
30C61×#532C2211yescharacteristic
31C2×C4×D4×C61C2211yescharacteristic
32C61×#463—22no—
33C61×#461C2212yes—
34C61×#459C2212yes—
35C61×#423C2211yescharacteristic
36C61×#417—21no—
37C22×Dic122—21no—
38C2×#11758200—24no—
39C22×#92418—21no—
40C22×#92418—24no—
41#14519777—21no—
42#14519777—21no—
43#14519777—21no—
44#14519777C2312yes—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—21no—
47#14519710—24no—
48C4⋊#92418—21no—
49C4⋊#92418D412yes—
50C4⋊#92418C2312yes—
51C4⋊#92418C2312yes—
52#14519700—24no—
53#14519698—24no—
54C22⋊#92418—21no—
55C22⋊#92418—24no—
56C22⋊#92418D412yes—
57C4×#92418—21no—
58C4×#92418C2312yes—
59C2×#41×C61—21no—
60C22×D4×C61D411yescharacteristic
61C23×C4×C61—21no—
62AGL(1,ℤ/8)×C61—44no—
63C2×SD16×C61—22no—
64C2×D8×C61—22no—
65C2×M16×C61—21no—
66C4⋊Q8×C61C2311yescharacteristic
67#127×C61C2311yescharacteristic
68#124×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61D412yes—
76C4×D4×C61C2312yes—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C22⋊C4×C61—21no—
79C2×C42×C61C2311yescharacteristic
80C4⋊C8×C61—21no—
81C4⋊C8×C61C2312yes—
82#103×C61—22no—
83#102×C61—22no—
84C24×C61—21no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—22no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—41no—
94C2×D4×C61SD1614yescomplemented
95C2×D4×C61C2×D412yes—
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—41no—
100C22×C4×C61—82no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×Dic61—41no—
105C2×Dic122—22no—
106C2×Dic122—41no—
107C2×Dic122—82no—
108SD16×C61—44no—
109D8×C61—44no—
110M16×C61—42no—
111C2×C8×C61—22no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61C2×D412yes—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—22no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61C2411yescharacteristic
124#11758200—84no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—42no—
132C2×#92418—84no—
133C2×#92418C2×D414yes—
134C23×C61—22no—
135C23×C61—24no—
136C23×C61—41no—
137C23×C61—81no—
138C23×C61—81no—
139C23×C61#12111yescharacteristic
140Q8×C61—41no—
141Q8×C61—41no—
142D4×C61—22no—
143D4×C61—24no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—42no—
147D4×C61—81no—
148D4×C61—82no—
149D4×C61C2×SD1614yes—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—41no—
153C2×C4×C61—44no—
154C2×C4×C61—41no—
155C2×C4×C61—82no—
156C2×C4×C61—81no—
157C2×C4×C61—82no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61C2×SD1612yes—
162C2×C4×C61#13512yes—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×Dic61—42no—
167C2×Dic61—82no—
168Dic122—44no—
169Dic122—82no—
170C8×C61—42no—
171#92418—24no—
172#92418—42no—
173#92418—42no—
174#19385—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—24no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—84no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61#46512yes—
185C22×C61C2×#12111yescharacteristic
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—42no—
190C4×C61—81no—
191C4×C61#46512yes—
192C4×C61C22×SD1611yescharacteristic
193C4×C61C2×#13511yescharacteristic
194C4×C61#57612yes—
195Dic61—82no—
196#3206—611no—
197C2×#474—611no—
198#3036D6111yescharacteristic
199#2972—611no—
200#1679—614no—
201#1665—614no—
202#1570—611no—
203#1549—611no—
204#1540—611no—
205C2×C61—22no—
206C2×C61—44no—
207C2×C61—161no—
208C2×C61#304111yescharacteristic
209C2×C61C2×#46511yescharacteristic
210C2×C61#306911yescharacteristic
211C2×AGL(1,ℤ/8)—21no—
212#532C2×D6111yescharacteristic
213C2×C4⋊Q8—611no—
214C2×C4×D4C2×D6111yescharacteristic
215#492—612no—
216#487—612no—
217#474—614no—
218#474—1222no—
219#463—22no—
220#461—1222no—
221#461C2×D6112yes—
222#459C2×D6112yes—
223#448—1222no—
224C8×D4—614no—
225C8×D4—1222no—
226#431—611no—
227#431—611no—
228C2×#107—1221no—
229#423C2×D6111yescharacteristic
230C2×C4⋊C8—611no—
231#417—21no—
232C2×#102—1221no—
233C2×C22⋊C8—1222no—
234#334—614no—
235#332—612no—
236#332—614no—
237#331—612no—
238C61#1938511yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×Q8—1221no—
241C22×D4C22⋊D6111yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—44no—
244C2×SD16—22no—
245C2×SD16—1224no—
246C2×D8—22no—
247C2×M16—21no—
248C22×C8—1224no—
249C22×C8—1221no—
250C4⋊Q8—612no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253C4⋊Q8—1221no—
254C4⋊Q8C22×D6111yescharacteristic
255#127C22×D6111yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4C22⋊D6112yes—
264C4×D4C22×D6112yes—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4C22⋊D6111yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D6111yescharacteristic
270#107—1224no—
271C4⋊C8—21no—
272C4⋊C8—612no—
273C4⋊C8—612no—
274C4⋊C8—612no—
275C4⋊C8—1221no—
276C4⋊C8C22×D6112yes—
277#103—22no—
278#103—1224no—
279#102—22no—
280#102—1224no—
281C22⋊C8—612no—
282C22⋊C8—1224no—
283C22⋊C8—1221no—
284C4×C8—612no—
285C4×C8—1221no—
286C24—21no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—1222no—
291C2×Q8—4882no—
292C2×Q8—2441no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—22no—
298C2×D4—41no—
299C2×D4#1175820014yescomplemented
300C2×D4C2×C22⋊D6112yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—2441no—
307C22×C4D4×D6111yescharacteristic
308C22×C4#1175822411yescharacteristic
309C22×C4C2×C22⋊D6111yescharacteristic
310SD16—44no—
311SD16—4884no—
312D8—44no—
313M16—42no—
314C2×C8—22no—
315C2×C8—614no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—4884no—
319C2×C8—2442no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—1222no—
324C4⋊C4—1222no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4C2×C22⋊D6112yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42D4×D6111yescharacteristic
333C42#1175822411yescharacteristic
334C42C2×C22⋊D6111yescharacteristic
335C42C23×D6111yescharacteristic
336C23—22no—
337C23—24no—
338C23—41no—
339C23—81no—
340C23—81no—
341C23#1451974911yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—4882no—
345Q8—2444no—
346D4—24no—
347D4—22no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—81no—
352D4—82no—
353D4C2×#1175820014yes—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—4882no—
363C2×C4—2442no—
364C2×C4#1451974911yescharacteristic
365C2×C4#1451974911yescharacteristic
366C2×C4#1451974911yescharacteristic
367C2×C4C2×#1175820012yes—
368C2×C4#1451979712yes—
369C2×C4C2×D4×D6111yescharacteristic
370C2×C4C22×C22⋊D6111yescharacteristic
371C2×C4C2×#1175822411yescharacteristic
372C8—42no—
373C8—1224no—
374C8—2442no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#1683998912yes—
386C22C2×#1451974911yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—4882no—
393C4#1683998912yes—
394C4C22×#1175820011yescharacteristic
395C4C2×#1451979711yescharacteristic
396C4#1684056612yes—
397C2—22no—
398C2—44no—
399C2—161no—
400C2#2145532411yescharacteristic
401C2C2×#1683998911yescharacteristic
402C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3036C2
socleC22×C61C2×#121
hypercenter#3036D61
hypocenterC61#19385
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #21456741C2semidirect
2. C2×#16840042C2semidirect
3. #21454843C2semidirect
4. #21445244C2semidirect
5. #21445158C2semidirect
6. #21444591C2semidirect
7. #21444505C2semidirect
8. #21444468C2semidirect
11. #16840134C22semidirect
12. #16840114C22semidirect
25. #16839512C22semidirect
26. #16839508C22semidirect
27. #16839507C22semidirect
28. #16839506C22semidirect
94. C2×D4×C61SD16semidirect
238. C61#19385semidirect
299. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567412
C2×#168400422
#214548432
#214452442
#214451582
#214445912
#214445052
#214444682
C61×#30362
#1938561