Finite Group Databasebeta

← #29834944

Subgroups — #29834944

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456403C211yesmaximal, characteristic, complemented
2C4×D8×D61C211yesmaximal, characteristic
3#21448015C212yesmaximal, complemented
4#21447967C211yesmaximal, characteristic, complemented
5#21447966C211yesmaximal, characteristic, complemented
6#21447861C211yesmaximal, characteristic, complemented
7#21447746C212yesmaximal
8SD32×Dic61C212yesmaximal, complemented
9#21447668C212yesmaximal, complemented
10#21447617C211yesmaximal, characteristic, complemented
11C61×#2208C211yesmaximal, characteristic, complemented
12C2×D8×D61C2211yescharacteristic
13#16840317—21no—
14C4×D4×D61—21no—
15C2×#14519720C2212yes—
16#16840167C2211yescharacteristic, complemented
17#16840112—21no—
18#16840000—21no—
19#16840000C2212yes—
20Q16×Dic61—21no—
21Q16×Dic61C2212yescomplemented
22C4×D488C2211yescharacteristic
23D8×Dic61C2211yescharacteristic
24#16839844C2211yescharacteristic
25C4×#11758199—21no—
26#16839842—21no—
27#107×D61C2211yescharacteristic
28#102×D61—21no—
29C4×C8×D61C2211yescharacteristic
30#16839590C2212yescomplemented
31#16839589C2212yescomplemented
32C4⋊#11758185C2211yescharacteristic, complemented
33#16839566C2212yescomplemented
34#16839564C2212yes—
35#16839563C2212yes—
36#16839541C2212yes—
37C16×Dic61C2212yescomplemented
38C2×SD32×C61C2212yes—
39C61×#494C2211yescharacteristic, complemented
40C4×D8×C61C2211yescharacteristic
41C61×C4⋊C16C2211yescharacteristic, complemented
42C61×#358C2212yescomplemented
43C2×D4×D61—21no—
44C22×C4×D61—41no—
45C2×D488C2311yescharacteristic
46C2×#11758199—21no—
47D8×D61—22no—
48D8×D61—22no—
49C2×C8×D61D411yescharacteristic
50C2×C8×D61C2311yescharacteristic
51#14519777—21no—
52#14519768—41no—
53#14519757—41no—
54#14519753—22no—
55Q8×Dic61—22no—
56#14519742—41no—
57C4×D244—21no—
58D4×Dic61—21no—
59#14519739—21no—
60C4×C22⋊D61—41no—
61#14519737—41no—
62#14519735—41no—
63C4⋊C4×D61—21no—
64C22⋊C4×D61—41no—
65C42×D61D411yescharacteristic
66#14519720—24no—
67C2×#11758185C2312yes—
68#14519711—21no—
69#14519711C2312yes—
70#14519711C2311yescharacteristic
71#14519710—21no—
72#14519701—22no—
73#14519701—22no—
74#14519699—21no—
75#14519698—21no—
76#14519697—21no—
77C4×#92418D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×Q16×C61—21no—
81C2×Q16×C61C2312yes—
82C2×D8×C61C2311yescharacteristic
83C4⋊Q8×C61—21no—
84C4×D4×C61—21no—
85SD32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#102×C61—21no—
89C4×C8×C61C2311yescharacteristic
90C23×D61—41no—
91C2×Q8×C61—22no—
92C2×D4×C61—21no—
93C22×C4×C61—41no—
94C2×C22⋊D61—41no—
95C22×Dic61—41no—
96D4×D61—24no—
97D4×D61—42no—
98C2×D244—21no—
99C2×C4×D61—41no—
100C2×C4×D61—41no—
101C2×C4×D61—41no—
102C2×C4×D61—84no—
103C2×C4×D61D811yescharacteristic
104C2×C4×D61D811yescharacteristic
105C2×C4×D61C2×D411yescharacteristic
106C2×Dic122—41no—
107Q16×C61—22no—
108Q16×C61—22no—
109Q16×C61—42no—
110D8×C61—21no—
111D8×C61#4112yes—
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2411yescharacteristic
114C4⋊C4×C61—21no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119C22⋊Dic61—41no—
120#11758199—24no—
121#11758198—42no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—41no—
127C4⋊Dic61—42no—
128#11758196—41no—
129#11758196—42no—
130C4×Dic61—42no—
131C4×Dic61D811yescharacteristic
132C4×Dic61D811yescharacteristic
133C4×Dic61C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136D488—21no—
137D488#4112yes—
138C8×D61—21no—
139C8×D61—21no—
140C8×D61—41no—
141C16×C61—22no—
142#11758185—22no—
143C23×C61—41no—
144C22×D61—44no—
145C22×D61—41no—
146C22×D61—82no—
147C22×D61—82no—
148C22×D61C2×D811yescharacteristic
149Q8×C61—42no—
150Q8×C61—42no—
151D4×C61—22no—
152D4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—82no—
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C22⋊D61—44no—
160C2×Dic61—42no—
161C2×Dic61—41no—
162C2×Dic61—82no—
163C2×Dic61C2×D811yescharacteristic
164C2×Dic61C2×D811yescharacteristic
165C2×Dic61C2×D811yescharacteristic
166D244—22no—
167D244—41no—
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—41no—
171C4×D61—82no—
172C4×D61—82no—
173C4×D61—82no—
174C4×D61SD3214yescomplemented
175Dic122—82no—
176C8×C61—21no—
177C8×C61C2×#4111yescharacteristic
178C8×C61C2×#4111yescharacteristic
179#92418—21no—
180#92418—21no—
181#33661—611nomaximal, Sylow
182C22×C61—42no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61C22×D811yescharacteristic
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—42no—
189C2×D61—88no—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61C2×SD3212yes—
193C4×C61—21no—
194C4×C61—82no—
195C4×C61—81no—
196C4×C61C2×SD3212yes—
197C4×C61#52511yescharacteristic
198C4×C61#57511yescharacteristic
199Dic61—21no—
200Dic61—21no—
201Dic61—81no—
202Dic61—82no—
203Dic61C2×SD3212yes—
204#3170—611no—
205C2×C4×D8—611no—
206#2235—612no—
207#2226—611no—
208#2226—611no—
209#2208—611no—
210#2208D6111yescharacteristic, complemented
211#2187—612no—
212C4×SD32—612no—
213C4×SD32—612no—
214#2162—611no—
215C2×C61—82no—
216C2×C61#310011yescharacteristic
217C2×C61C22×SD3211yescharacteristic
218C2×C61#341411yescharacteristic
219D61—22no—
220D61—82no—
221C22×D8—611no—
222#533—1221no—
223C2×C4×D4—1221no—
224C2×SD32—612no—
225C2×SD32C2×D6112yes—
226#500—611no—
227#494C2×D6111yescharacteristic, complemented
228#487—1221no—
229#467—612no—
230#467—1221no—
231C4×Q16—612no—
232C4×Q16—1221no—
233C4×D8—611no—
234C4×D8—611no—
235C4×D8—611no—
236C4×D8—1221no—
237C4×D8—1221no—
238C4×D8C2×D6111yescharacteristic
239C2×#107—611no—
240C2×#102—1221no—
241C2×C4×C8—611no—
242#367—612no—
243#367—612no—
244C4⋊C16—611no—
245C4⋊C16C2×D6111yescharacteristic, complemented
246#358—612no—
247#358C2×D6112yescomplemented
248#357—612no—
249#357—612no—
250C22⋊C16—612no—
251C4×C16—612no—
252C61#3366111yescharacteristic, complemented, Sylow
253C22×D4—1221no—
254C23×C4—2441no—
255C2×Q16—21no—
256C2×Q16C22×D6112yes—
257C2×D8—611no—
258C2×D8—1221no—
259C2×D8—1222no—
260C2×D8—1222no—
261C2×D8C22×D6111yescharacteristic
262C22×C8—611no—
263C22×C8—611no—
264C4⋊Q8—21no—
265C4⋊Q8—1221no—
266#125—2441no—
267C22⋊Q8—1222no—
268C22⋊Q8—2441no—
269C4×Q8—1222no—
270C4×Q8—2441no—
271C4×D4—21no—
272C4×D4—1221no—
273C4×D4—1221no—
274C4×D4—1221no—
275C4×D4—2441no—
276C4×D4—2441no—
277#117—2441no—
278C2×C4⋊C4—1221no—
279C2×C22⋊C4—2441no—
280C2×C42—611no—
281SD32—24no—
282SD32—1224no—
283C2×C16—612no—
284C2×C16C22×D6112yes—
285#107—612no—
286#107—611no—
287#107—1221no—
288#107—1221no—
289#107C22×D6111yescharacteristic
290#103—1222no—
291#103—1222no—
292#102—21no—
293#102—1221no—
294#102—1221no—
295#102—1221no—
296C4×C8—611no—
297C4×C8—611no—
298C4×C8—611no—
299C4×C8C22×D6111yescharacteristic
300C24—2441no—
301C2×Q8—22no—
302C2×Q8—2441no—
303C2×D4—21no—
304C2×D4—1224no—
305C2×D4—1221no—
306C2×D4—2441no—
307C2×D4—2442no—
308C22×C4—41no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—4884no—
313C22×C4—2441no—
314C22×C4—2441no—
315C22×C4—2441no—
316C22×C4—2441no—
317Q16—22no—
318Q16—22no—
319Q16—42no—
320D8—21no—
321D8—612no—
322D8—1224no—
323D8—1221no—
324D8#1175822412yes—
325C2×C8—611no—
326C2×C8—611no—
327C2×C8—1221no—
328C2×C8—1221no—
329C2×C8—2441no—
330C2×C8D4×D6111yescharacteristic
331C2×C8C23×D6111yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—1221no—
336C4⋊C4—1222no—
337C4⋊C4—2441no—
338C4⋊C4—2442no—
339C4⋊C4—2442no—
340C4⋊C4—2441no—
341C22⋊C4—41no—
342C22⋊C4—2441no—
343C22⋊C4—2442no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C42—611no—
347C42—611no—
348C42—611no—
349C42—2442no—
350C42D4×D6111yescharacteristic
351C16—22no—
352C16—1222no—
353C23—41no—
354C23—611no—
355C23—4882no—
356C23—4882no—
357C23—2444no—
358C23—2441no—
359Q8—42no—
360Q8—42no—
361Q8—4882no—
362D4—22no—
363D4—41no—
364D4—1222no—
365D4—2444no—
366D4—2441no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—82no—
370C2×C4—611no—
371C2×C4—614no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—4882no—
377C2×C4—4882no—
378C2×C4—4882no—
379C2×C4—4882no—
380C2×C4—2441no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4D8×D6111yescharacteristic
384C2×C4D8×D6111yescharacteristic
385C2×C4C2×D4×D6111yescharacteristic
386C8—21no—
387C8—1221no—
388C8—1221no—
389C8#41×D6111yescharacteristic
390C8C2×#1175822411yescharacteristic
391C22—42no—
392C22—81no—
393C22—81no—
394C22—612no—
395C22—1221no—
396C22—1221no—
397C22—4888no—
398C22—4881no—
399C22—4881no—
400C22—2442no—
401C22C2×D8×D6111yescharacteristic
402C4—21no—
403C4—82no—
404C4—81no—
405C4—612no—
406C4—1221no—
407C4—1221no—
408C4—4882no—
409C4—4881no—
410C4SD32×D6112yes—
411C4#1684027211yescharacteristic
412C4#1684055811yescharacteristic
413C2—82no—
414C2—1222no—
415C2—4882no—
416C2#2145581811yescharacteristic
417C2C2×SD32×D6111yescharacteristic
418C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2208C2
socleC22×C61C22×D8
prime radicalC2×D8×D61C22
hypercenter#2208D61
hypocenterC61#33661
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
1. #21456403C2semidirect
3. #21448015C2semidirect
4. #21447967C2semidirect
5. #21447966C2semidirect
6. #21447861C2semidirect
8. SD32×Dic61C2semidirect
9. #21447668C2semidirect
10. #21447617C2semidirect
11. C61×#2208C2semidirect
16. #16840167C22semidirect
21. Q16×Dic61C22semidirect
30. #16839590C22semidirect
31. #16839589C22semidirect
32. C4⋊#11758185C22semidirect
33. #16839566C22semidirect
37. C16×Dic61C22semidirect
39. C61×#494C22semidirect
41. C61×C4⋊C16C22semidirect
42. C61×#358C22semidirect
174. C4×D61SD32semidirect
210. #2208D61semidirect
227. #494C2×D61semidirect
245. C4⋊C16C2×D61semidirect
247. #358C2×D61semidirect
252. C61#33661semidirect

Maximal subgroups

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

subgroupindex
#214564032
C4×D8×D612
#214480152
#214479672
#214479662
#214478612
#214477462
SD32×Dic612
#214476682
#214476172
C61×#22082
#3366161