Finite Group Databasebeta

← #29665172

Subgroups — #29665172

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic, complemented
3C2×#16839954C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445227C214yesmaximal, complemented
6#21445159C214yesmaximal, complemented
7#21444591C211yesmaximal, characteristic, complemented
8#21444474C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C22×#11758199—21no—
11#16840319C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic
13#16840134C2212yescomplemented
14#16840108C2212yescomplemented
15#16840063—22no—
16#16840053C2212yescomplemented
17#16840043C2212yescomplemented
18#16839959—22no—
19#16839954—22no—
20#16839954C2214yes—
21#16839913—22no—
22C8×D244—22no—
23#16839827C2214yescomplemented
24#16839815C2211yescharacteristic
25#16839804—21no—
26C2×C4⋊#92418C2211yescharacteristic
27#16839768—21no—
28C2×#14519698—21no—
29C22⋊C8×D61—22no—
30#16839512C2214yescomplemented
31#16839510C2212yescomplemented
32#16839509C2212yescomplemented
33C2×#127×C61C2211yescharacteristic
34C61×#431C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C61×#331C2214yescomplemented
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39C2×#11758200—24no—
40C2×#11758199—24no—
41C2×#11758199—44no—
42C22×#92418—21no—
43C2×C8×D61—24no—
44#14519777C2311yescharacteristic
45#14519777C2311yescharacteristic
46#14519769—21no—
47#14519756—21no—
48#14519756—21no—
49C4×Dic122—21no—
50C4×Dic122—21no—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244D412yes—
54C4×D244C2312yes—
55#14519736—21no—
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758198—21no—
58#14519711—22no—
59#14519709—22no—
60C4⋊#92418—21no—
61C4⋊#92418D412yes—
62C4⋊#92418C2312yes—
63#14519699—22no—
64#14519699—22no—
65#14519698—24no—
66#14519698—24no—
67C22⋊#92418D412yes—
68#14519689—24no—
69C4×#92418C2312yes—
70C22×D4×C61—21no—
71C22×D4×C61—21no—
72C22×D4×C61—21no—
73C2×M16×C61—21no—
74#127×C61—21no—
75#127×C61—21no—
76#127×C61—21no—
77#127×C61C2312yes—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61—21no—
83C4×C8×C61—21no—
84C24×C61—41no—
85C23×D61—21no—
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—41no—
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C2×D4×C61—42no—
93C2×D4×C61—42no—
94C2×D4×C61—82no—
95C2×D4×C61—82no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C22×Dic61—21no—
100C2×D244—21no—
101C2×D244—22no—
102C2×D244SD1614yescomplemented
103C2×D244C2×D412yes—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—82no—
107C2×Dic122—21no—
108C2×Dic122—21no—
109M16×C61—42no—
110C2×C8×C61—22no—
111C2×C8×C61—22no—
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C42×C61C2411yescharacteristic
116C22⋊Dic61—41no—
117#11758200—84no—
118#11758199—48no—
119#11758198—22no—
120#11758198—22no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61C2×D412yes—
127#11758196—41no—
128#11758196—41no—
129C4×Dic61—41no—
130C2×#92418—22no—
131C2×#92418—42no—
132C2×#92418C2×D414yes—
133C8×D61—84no—
134C23×C61—42no—
135C23×C61—82no—
136C23×C61—82no—
137C23×C61—82no—
138C23×C61#12111yescharacteristic
139C22×D61—22no—
140C22×D61—24no—
141C22×D61—81no—
142C22×D61—81no—
143D4×C61—42no—
144D4×C61—42no—
145D4×C61—44no—
146D4×C61—82no—
147D4×C61—82no—
148D4×C61—82no—
149D4×C61—82no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61C2×SD1612yes—
156C2×C4×C61#13512yes—
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C2×#4111yescharacteristic
160C2×Dic61—22no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—81no—
164D244—22no—
165D244—24no—
166D244C2×SD1614yes—
167C4×D61—44no—
168C4×D61—82no—
169Dic122—42no—
170Dic122—81no—
171C8×C61—42no—
172C8×C61—42no—
173#92418—24no—
174#92418—42no—
175#19384—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—82no—
179C22×C61—82no—
180C22×C61—82no—
181C22×C61—164no—
182C22×C61#46512yes—
183C22×C61C2×#12111yescharacteristic
184C2×D61—24no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—84no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61#46512yes—
192C4×C61C22×SD1611yescharacteristic
193C4×C61C2×#13511yescharacteristic
194C4×C61#57612yes—
195Dic61—42no—
196Dic61—81no—
197Dic61—81no—
198#3208—611no—
199#3077—611no—
200C2×#459—611no—
201#2972—611no—
202#1677—614no—
203#1665—614no—
204#1570—611no—
205#1545D6111yescharacteristic, complemented
206#1542—611no—
207C2×C61—22no—
208C2×C61—162no—
209C2×C61#304111yescharacteristic
210C2×C61C2×#46511yescharacteristic
211C2×C61#306811yescharacteristic
212D61—44no—
213C22×D8—1221no—
214#533—611no—
215C2×#127C2×D6111yescharacteristic
216C2×C4×D4—611no—
217#492—612no—
218#486—612no—
219#478—1222no—
220#476—612no—
221#474—612no—
222#460—1222no—
223#459—614no—
224#459—1222no—
225#450—1222no—
226C8×D4—614no—
227C8×D4—1222no—
228#431—611no—
229#431C2×D6111yescharacteristic
230#428—1221no—
231#423C2×D6111yescharacteristic
232C2×C4⋊C8—611no—
233#418—1221no—
234C2×#102—1221no—
235C2×C22⋊C8—1222no—
236#334—614no—
237#333—612no—
238#332—612no—
239#331C2×D6114yescomplemented
240C61#1938411yescharacteristic, complemented, Sylow
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—611no—
245C23×C4—1221no—
246C2×SD16—1224no—
247C2×D8—1224no—
248C2×D8—2444no—
249C2×M16—21no—
250C22×C8—1224no—
251C22×C8—1221no—
252C4⋊Q8—611no—
253C4⋊Q8—611no—
254#127—21no—
255#127—21no—
256#127—21no—
257#127C22×D6112yes—
258#125—1221no—
259C22⋊Q8—1221no—
260C22⋊Q8—1221no—
261C4×Q8—1221no—
262C4×Q8—1221no—
263C4×D4—612no—
264C4×D4—612no—
265C4×D4—1221no—
266C4×D4—1221no—
267#117—1221no—
268C2×C4⋊C4—611no—
269C2×C22⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#107—1222no—
272#106—1222no—
273C4⋊C8—21no—
274C4⋊C8—612no—
275C4⋊C8—612no—
276C4⋊C8—1221no—
277C4⋊C8C22×D6112yes—
278C4⋊C8C22×D6112yes—
279#102—1224no—
280#102—1222no—
281#102—1222no—
282#102—1224no—
283C22⋊C8—21no—
284C22⋊C8—612no—
285C22⋊C8—1224no—
286C4×C8—21no—
287C4×C8—612no—
288C24—41no—
289C24—1221no—
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—612no—
303C2×D4—614no—
304C2×D4—1221no—
305C2×D4—1222no—
306C22×C4—1221no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—4882no—
310C22×C4#1175822211yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4D4×D6111yescharacteristic
313SD16—4884no—
314D8—2448no—
315M16—42no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—614no—
319C2×C8—1222no—
320C2×C8—4884no—
321C2×C8—2442no—
322C4⋊C4—612no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—1221no—
328C4⋊C4—2441no—
329C4⋊C4—2441no—
330C22⋊C4—1222no—
331C22⋊C4—1222no—
332C22⋊C4—2441no—
333C42—2441no—
334C42#1175822211yescharacteristic
335C42D4×D6111yescharacteristic
336C42D4×D6111yescharacteristic
337C42C23×D6111yescharacteristic
338C23—42no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—1224no—
343C23—1222no—
344C23—4881no—
345C23—4881no—
346C23#1451975011yescharacteristic
347Q8—4881no—
348Q8—2442no—
349D4—42no—
350D4—42no—
351D4—44no—
352D4—82no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—614no—
357D4—1224no—
358D4—1222no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—1222no—
362C2×C4—4882no—
363C2×C4—4881no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2444no—
367C2×C4#1451975011yescharacteristic
368C2×C4#1451975011yescharacteristic
369C2×C4#1451975011yescharacteristic
370C2×C4SD16×D6112yes—
371C2×C4#1451979612yes—
372C2×C4C2×D4×D6111yescharacteristic
373C2×C4C2×D4×D6111yescharacteristic
374C2×C4C2×#1175822211yescharacteristic
375C8—42no—
376C8—42no—
377C8—1224no—
378C8—2442no—
379C22—21no—
380C22—21no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—164no—
385C22—1224no—
386C22—4884no—
387C22—2442no—
388C22—2442no—
389C22#1683999012yes—
390C22C2×#1451975011yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—4881no—
395C4—4881no—
396C4—2442no—
397C4#1683999012yes—
398C4C2×SD16×D6111yescharacteristic
399C4C2×#1451979611yescharacteristic
400C4#1684056612yes—
401C2—22no—
402C2—162no—
403C2—2444no—
404C2#2145532111yescharacteristic
405C2C2×#1683999011yescharacteristic
406C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
prime radicalC2×#16839954C2
hypercenter#1545D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21455618C2semidirect
4. #21454842C2semidirect
5. #21445227C2semidirect
6. #21445159C2semidirect
7. #21444591C2semidirect
8. #21444474C2semidirect
9. C61×#1545C2semidirect
13. #16840134C22semidirect
14. #16840108C22semidirect
16. #16840053C22semidirect
17. #16840043C22semidirect
23. #16839827C22semidirect
30. #16839512C22semidirect
31. #16839510C22semidirect
32. #16839509C22semidirect
36. C61×#331C22semidirect
102. C2×D244SD16semidirect
205. #1545D61semidirect
239. #331C2×D61semidirect
240. C61#19384semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214556182
C2×#168399542
#214548422
#214452272
#214451592
#214445912
#214444742
C61×#15452
#1938461