Finite Group Databasebeta

← #29665274

Subgroups — #29665274

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455914C211yesmaximal, characteristic
2#21455823C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic
4#21445267C212yesmaximal, complemented
5#21445239C212yesmaximal, complemented
6#21445205C212yesmaximal, complemented
7#21445191C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444529C211yesmaximal, characteristic, complemented
10#21444484C211yesmaximal, characteristic, complemented
11C61×#3231C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13#16840095—21no—
14#16840090—21no—
15#16840049C2212yes—
16#16840039C2212yes—
17#16839964C2212yes—
18#16839964C2212yes—
19#16839912—21no—
20#16839912—21no—
21#16839860—22no—
22C4×#11758201—22no—
23C8×D244C2212yes—
24C8×D244C2212yes—
25#16839827—22no—
26#16839819C2211yescharacteristic
27#16839815C2211yescharacteristic
28#16839815C2211yescharacteristic
29C22⋊C8×D61—22no—
30#16839515C2212yescomplemented
31#16839513C2212yescomplemented
32#16839511C2212yescomplemented
33#16839508C2212yescomplemented
34#16839508C2212yescomplemented
35#16839508C2212yescomplemented
36C61×#533C2211yescharacteristic
37C61×#533C2211yescharacteristic
38C61×#500C2211yescharacteristic, complemented
39C61×#498C2211yescharacteristic, complemented
40C61×#487C2211yescharacteristic, complemented
41C61×#487C2211yescharacteristic, complemented
42C61×#485—22no—
43C61×#479—22no—
44C61×#431C2211yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49C2×C8×D61—22no—
50C2×C8×D61—22no—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244D412yes—
54C4×D244C2312yes—
55C2×C4⋊Dic61D411yescharacteristic
56C2×#11758198—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519701—22no—
64#14519697—22no—
65#14519697—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69#14519689—22no—
70#14519689—22no—
71#14519688—21no—
72C4×#92418—21no—
73C4×#92418—21no—
74C4⋊Q8×C61C2311yescharacteristic
75C4⋊Q8×C61C2311yescharacteristic
76C4⋊Q8×C61C2311yescharacteristic
77C4⋊Q8×C61C2311yescharacteristic
78#125×C61—21no—
79#125×C61—21no—
80C22⋊Q8×C61—21no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61—21no—
83C22⋊Q8×C61—21no—
84C4×Q8×C61—21no—
85C4×Q8×C61—21no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88#117×C61—21no—
89#117×C61—21no—
90C2×C42×C61C2311yescharacteristic
91#107×C61—22no—
92#106×C61—22no—
93C4⋊C8×C61D412yes—
94#103×C61—22no—
95#103×C61—22no—
96C22⋊C8×C61D412yes—
97C4×C8×C61C2311yescharacteristic
98C4×C8×C61C2311yescharacteristic
99C23×D61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×Dic61—21no—
108C2×D244—21no—
109C2×D244—24no—
110C2×D244C2×D412yes—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—82no—
114C2×C8×C61C2×D412yes—
115C2×C8×C61C2×D412yes—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—41no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—41no—
128C22⋊C4×C61—41no—
129C22⋊C4×C61—41no—
130C42×C61—41no—
131C42×C61—41no—
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C42×C61#4111yescharacteristic
135C42×C61C2411yescharacteristic
136#11758201—44no—
137#11758201—44no—
138#11758198—22no—
139#11758198—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61C2×D412yes—
142C2×#92418—22no—
143C2×#92418—22no—
144C2×#92418—22no—
145C8×D61—82no—
146C8×D61—82no—
147C23×C61#12111yescharacteristic
148C22×D61—22no—
149C22×D61—24no—
150C22×D61—81no—
151C22×D61—81no—
152Q8×C61—42no—
153Q8×C61—42no—
154Q8×C61—42no—
155Q8×C61—81no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C2×Dic61—22no—
170C2×Dic61—81no—
171D244—24no—
172D244—22no—
173D244#13514yes—
174C4×D61—42no—
175C4×D61—42no—
176C4×D61—82no—
177C8×C61—22no—
178C8×C61—22no—
179#92418—42no—
180#92418—42no—
181#92418—42no—
182#19395—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61C2×#12111yescharacteristic
187C2×D61—24no—
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—84no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61—81no—
198C4×C61—81no—
199C4×C61C2×#13512yes—
200C4×C61#57712yes—
201Dic61—42no—
202#3231D6111yescharacteristic, complemented
203#3110—611no—
204#3101—611no—
205#2972—611no—
206#1683—612no—
207#1679—612no—
208#1673—612no—
209#1671—612no—
210#1572—611no—
211#1554—611no—
212#1544—611no—
213C2×C61—41no—
214C2×C61#304211yescharacteristic
215C2×C61#305011yescharacteristic
216C2×C61#306911yescharacteristic
217D61—44no—
218#533C2×D6111yescharacteristic
219#533C2×D6111yescharacteristic
220C2×C4×D4—611no—
221#500C2×D6111yescharacteristic, complemented
222#498C2×D6111yescharacteristic, complemented
223#487C2×D6111yescharacteristic, complemented
224#487C2×D6111yescharacteristic, complemented
225#485—22no—
226#484—1221no—
227#483—1221no—
228#479—22no—
229#476—612no—
230#474—612no—
231#461—612no—
232#461—612no—
233#450—1221no—
234#450—1221no—
235#440—1222no—
236C4×SD16—1222no—
237C8×D4—612no—
238C8×D4—612no—
239C8×D4—1222no—
240#432—611no—
241#431—611no—
242#431—611no—
243#431C2×D6111yescharacteristic
244C2×C22⋊C8—1222no—
245#335—612no—
246#334—612no—
247#333—612no—
248#332—612no—
249#332—612no—
250#332—612no—
251C61#1939511yescharacteristic, complemented, Sylow
252C22×D4—611no—
253C23×C4—1221no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C22×C8—1222no—
257C22×C8—1222no—
258C4⋊Q8C22×D6111yescharacteristic
259C4⋊Q8C22×D6111yescharacteristic
260C4⋊Q8C22×D6111yescharacteristic
261C4⋊Q8C22×D6111yescharacteristic
262#125—21no—
263#125—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—612no—
273C4×D4—612no—
274C4×D4—1221no—
275C4×D4—1221no—
276#117—21no—
277#117—21no—
278C2×C4⋊C4—611no—
279C2×C22⋊C4—1221no—
280C2×C42C22×D6111yescharacteristic
281#107—22no—
282#107—1222no—
283#106—22no—
284#106—1222no—
285C4⋊C8—612no—
286C4⋊C8—612no—
287C4⋊C8—612no—
288C4⋊C8C22⋊D6112yes—
289#103—22no—
290#103—22no—
291#103—1222no—
292#103—1222no—
293#102—1222no—
294#102—1222no—
295C22⋊C8—1221no—
296C22⋊C8—1221no—
297C22⋊C8—1221no—
298C22⋊C8—1222no—
299C22⋊C8—1222no—
300C22⋊C8C22⋊D6112yes—
301#97—1221no—
302C4×C8—1221no—
303C4×C8—1221no—
304C4×C8C22×D6111yescharacteristic
305C4×C8C22×D6111yescharacteristic
306C24—1221no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×D4—612no—
312C2×D4—1224no—
313C2×D4—1221no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—4882no—
318C22×C4D4×D6111yescharacteristic
319C22×C4#1175822411yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321SD16—2444no—
322SD16—2444no—
323C2×C8—1222no—
324C2×C8—1222no—
325C2×C8—1222no—
326C2×C8—4882no—
327C2×C8—4882no—
328C2×C8C2×C22⋊D6112yes—
329C2×C8C2×C22⋊D6112yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—612no—
343C4⋊C4—1221no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—1222no—
347C22⋊C4—1222no—
348C42—41no—
349C42—41no—
350C42D4×D6111yescharacteristic
351C42#1175822411yescharacteristic
352C42C2×C22⋊D6111yescharacteristic
353C42C23×D6111yescharacteristic
354C23—1224no—
355C23—1222no—
356C23—4881no—
357C23—4881no—
358C23#1451974811yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—81no—
363D4—614no—
364D4—1224no—
365D4—1222no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—1222no—
374C2×C4—4882no—
375C2×C4—4881no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4#1451974811yescharacteristic
379C2×C4#1451974811yescharacteristic
380C2×C4#1451974811yescharacteristic
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C22×C22⋊D6111yescharacteristic
383C2×C4C2×#1175822411yescharacteristic
384C8—22no—
385C8—22no—
386C8—2442no—
387C8—2442no—
388C8—2442no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—1224no—
393C22—4884no—
394C22—2442no—
395C22—2442no—
396C22C2×#1451974811yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—2442no—
406C4#135×D6112yes—
407C4#1684057012yes—
408C2—41no—
409C2—2444no—
410C2#2145532711yescharacteristic
411C2#2145539611yescharacteristic
412C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3231C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#3231D61
hypocenterC61#19395
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3231

Complements

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

NH ≅ G/Ntype
4. #21445267C2semidirect
5. #21445239C2semidirect
6. #21445205C2semidirect
7. #21445191C2semidirect
8. #21444599C2semidirect
9. #21444529C2semidirect
10. #21444484C2semidirect
11. C61×#3231C2semidirect
30. #16839515C22semidirect
31. #16839513C22semidirect
32. #16839511C22semidirect
33. #16839508C22semidirect
34. #16839508C22semidirect
35. #16839508C22semidirect
38. C61×#500C22semidirect
39. C61×#498C22semidirect
40. C61×#487C22semidirect
41. C61×#487C22semidirect
202. #3231D61semidirect
221. #500C2×D61semidirect
222. #498C2×D61semidirect
223. #487C2×D61semidirect
224. #487C2×D61semidirect
251. C61#19395semidirect

Maximal subgroups

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

subgroupindex
#214559142
#214558232
#214548422
#214452672
#214452392
#214452052
#214451912
#214445992
#214445292
#214444842
C61×#32312
#1939561