Finite Group Databasebeta

← #29665169

Subgroups — #29665169

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#21455612C211yesmaximal, characteristic, complemented
2C2×#16839954C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic, complemented
4#21445225C214yesmaximal, complemented
5#21445157C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444487C211yesmaximal, characteristic, complemented
8#21444476C211yesmaximal, characteristic, complemented
9C61×#3208C211yesmaximal, characteristic, complemented
10C22×#11758199—21no—
11C2×C4×D244C2211yescharacteristic
12#16840059—22no—
13#16840049C2212yescomplemented
14#16840039C2212yescomplemented
15#16839954—22no—
16#16839954C2214yes—
17C8×D244C2214yescomplemented
18#16839827—22no—
19#16839815C2211yescharacteristic
20#16839805—21no—
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839766—21no—
24C2×#14519698—21no—
25C22⋊C8×D61—22no—
26#16839513C2214yescomplemented
27#16839511C2212yescomplemented
28#16839508C2212yescomplemented
29#16839506C2214yescomplemented
30C61×#533C2211yescharacteristic
31C2×#127×C61C2211yescharacteristic
32C61×#492C2212yescomplemented
33C61×#486C2212yescomplemented
34C61×#460—22no—
35C61×#450—22no—
36C61×#431C2211yescharacteristic
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39C2×#11758199—24no—
40C2×#11758199—44no—
41C2×#11758194—21no—
42C22×#92418—21no—
43C2×C8×D61—24no—
44C4×D244—21no—
45C4×D244—21no—
46C4×D244D412yes—
47C4×D244C2312yes—
48C2×C4⋊Dic61D411yescharacteristic
49C2×#11758198—21no—
50#14519710—22no—
51#14519708—22no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2312yes—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519698—24no—
58#14519697—22no—
59#14519697—22no—
60C22⋊#92418—21no—
61#14519689—24no—
62C4×#92418—21no—
63C22×D4×C61—21no—
64C22×D4×C61—21no—
65C22×D4×C61—21no—
66C2×SD16×C61—24no—
67C4⋊Q8×C61C2311yescharacteristic
68C4⋊Q8×C61C2311yescharacteristic
69#127×C61—21no—
70#127×C61—21no—
71#127×C61—21no—
72#127×C61C2312yes—
73#125×C61—21no—
74C22⋊Q8×C61—21no—
75C22⋊Q8×C61—21no—
76C4×Q8×C61—21no—
77C4×Q8×C61—21no—
78#117×C61—21no—
79C2×C42×C61C2311yescharacteristic
80C4⋊C8×C61D412yes—
81#102×C61—24no—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C24×C61—41no—
85C23×D61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—22no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C2×D4×C61—82no—
97C2×D4×C61—82no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C22×Dic61—21no—
102C2×D244—21no—
103C2×D244—22no—
104C2×D244SD1614yescomplemented
105C2×D244C2×D412yes—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—82no—
109SD16×C61—84no—
110C2×C8×C61C2×D414yes—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C42×C61—41no—
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122C42×C61C2411yescharacteristic
123#11758199—48no—
124#11758198—22no—
125#11758198—22no—
126C4⋊Dic61—21no—
127C4⋊Dic61C2×D412yes—
128#11758194—42no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—42no—
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—
143Q8×C61—42no—
144Q8×C61—81no—
145D4×C61—42no—
146D4×C61—42no—
147D4×C61—44no—
148D4×C61—82no—
149D4×C61—82no—
150D4×C61—82no—
151D4×C61—82no—
152C2×C4×C61—22no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61C2×SD1612yes—
160C2×C4×C61#13512yes—
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C2×Dic61—22no—
165C2×Dic61—81no—
166D244—22no—
167D244—24no—
168D244C2×SD1614yes—
169C4×D61—44no—
170C4×D61—82no—
171C8×C61—24no—
172#92418—42no—
173#92418—42no—
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—81no—
192C4×C61—81no—
193C4×C61#46512yes—
194C4×C61C22×SD1611yescharacteristic
195C4×C61C2×#13511yescharacteristic
196C4×C61#57612yes—
197Dic61—42no—
198#3208D6111yescharacteristic, complemented
199#3077—611no—
200C2×#459—611no—
201#2972—611no—
202#1677—614no—
203#1665—614no—
204#1570—611no—
205#1545—611no—
206#1542—611no—
207C2×C61—22no—
208C2×C61—162no—
209C2×C61#304111yescharacteristic
210C2×C61C2×#46511yescharacteristic
211C2×C61#306811yescharacteristic
212D61—44no—
213C22×D8—1221no—
214#533C2×D6111yescharacteristic
215C2×#127C2×D6111yescharacteristic
216C2×C4×D4—611no—
217#492C2×D6112yescomplemented
218#486C2×D6112yescomplemented
219#478—1222no—
220#476—612no—
221#474—612no—
222#460—22no—
223#459—614no—
224#459—1222no—
225#450—22no—
226C8×D4—614no—
227C8×D4—1222no—
228#431—611no—
229#431C2×D6111yescharacteristic
230#428—1221no—
231#423—611no—
232C2×C4⋊C8—611no—
233#418—1221no—
234C2×#102—1221no—
235C2×C22⋊C8—1222no—
236#334—614no—
237#333—612no—
238#332—612no—
239#331—614no—
240C61#1938411yescharacteristic, complemented, Sylow
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—611no—
245C23×C4—1221no—
246C2×SD16—24no—
247C2×D8—1224no—
248C2×D8—2444no—
249C2×M16—1221no—
250C22×C8—1221no—
251C22×C8—1224no—
252C4⋊Q8C22×D6111yescharacteristic
253C4⋊Q8C22×D6111yescharacteristic
254#127—21no—
255#127—21no—
256#127—21no—
257#127C22×D6112yes—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—612no—
264C4×D4—612no—
265C4×D4—1221no—
266C4×D4—1221no—
267#117—21no—
268C2×C4⋊C4—611no—
269C2×C22⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#107—1222no—
272#106—1222no—
273C4⋊C8—612no—
274C4⋊C8—612no—
275C4⋊C8—612no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278C4⋊C8C22⋊D6112yes—
279#102—24no—
280#102—1222no—
281#102—1222no—
282#102—1224no—
283C22⋊C8—1221no—
284C22⋊C8—1224no—
285C22⋊C8C22⋊D6112yes—
286C4×C8—1221no—
287C4×C8C22×D6112yes—
288C24—41no—
289C24—1221no—
290C2×Q8—21no—
291C2×Q8—21no—
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×C4D4×D6111yescharacteristic
311C22×C4#1175822411yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313SD16—84no—
314D8—2448no—
315M16—2442no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8—4884no—
320C2×C8—2442no—
321C2×C8C2×C22⋊D6114yes—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—612no—
329C4⋊C4—1221no—
330C22⋊C4—41no—
331C22⋊C4—1222no—
332C22⋊C4—1222no—
333C42—41no—
334C42D4×D6111yescharacteristic
335C42#1175822411yescharacteristic
336C42C2×C22⋊D6111yescharacteristic
337C42C23×D6111yescharacteristic
338C23—42no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—1224no—
343C23—1222no—
344C23—4881no—
345C23—4881no—
346C23#1451974811yescharacteristic
347Q8—42no—
348Q8—81no—
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—41no—
362C2×C4—41no—
363C2×C4—1222no—
364C2×C4—4882no—
365C2×C4—4881no—
366C2×C4—2444no—
367C2×C4#1451974811yescharacteristic
368C2×C4#1451974811yescharacteristic
369C2×C4#1451974811yescharacteristic
370C2×C4SD16×D6112yes—
371C2×C4#1451979812yes—
372C2×C4C2×D4×D6111yescharacteristic
373C2×C4C22×C22⋊D6111yescharacteristic
374C2×C4C2×#1175822411yescharacteristic
375C8—24no—
376C8—2442no—
377C8—2442no—
378C8—2442no—
379C22—21no—
380C22—21no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—164no—
385C22—1224no—
386C22—4884no—
387C22—2442no—
388C22—2442no—
389C22#1683998812yes—
390C22C2×#1451974811yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—2442no—
397C4#1683998812yes—
398C4C2×SD16×D6111yescharacteristic
399C4C2×#1451979811yescharacteristic
400C4#1684056512yes—
401C2—22no—
402C2—162no—
403C2—2444no—
404C2#2145531811yescharacteristic
405C2C2×#1683998811yescharacteristic
406C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3208C2
socleC22×C61C2×#121
prime radicalC2×#16839954C2
hypercenter#3208D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
3. #21454842C2semidirect
4. #21445225C2semidirect
5. #21445157C2semidirect
6. #21444590C2semidirect
7. #21444487C2semidirect
8. #21444476C2semidirect
9. C61×#3208C2semidirect
13. #16840049C22semidirect
14. #16840039C22semidirect
17. C8×D244C22semidirect
26. #16839513C22semidirect
27. #16839511C22semidirect
28. #16839508C22semidirect
29. #16839506C22semidirect
32. C61×#492C22semidirect
33. C61×#486C22semidirect
104. C2×D244SD16semidirect
198. #3208D61semidirect
217. #492C2×D61semidirect
218. #486C2×D61semidirect
240. C61#19384semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168399542
#214548422
#214452252
#214451572
#214445902
#214444872
#214444762
C61×#32082
#1938461