Finite Group Databasebeta

← #29689170

Subgroups — #29689170

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459464C211yesmaximal, characteristic
2#21454277C212yesmaximal, complemented
3#21446360C214yesmaximal, complemented
4#21446231C214yesmaximal, complemented
5#21445927C211yesmaximal, characteristic, complemented
6#21445735C212yesmaximal, complemented
7C61×#1828C211yesmaximal, characteristic, complemented
8C2×#14519822C2211yescharacteristic
9C22×#11758224C2211yescharacteristic
10#16840416—24no—
11#16840407—24no—
12#16840338C412yescomplemented
13#16840324C412yescomplemented
14#16840324C2212yes—
15#16840274C412yescomplemented
16C2×#14519761C412yescomplemented
17C2×#14519761C2212yes—
18#16840235C2214yes—
19#16839734—24no—
20C2×#14519692C2212yes—
21C2×#14519691C2214yes—
22C2×#14519691C2214yescomplemented
23#99×D61—24no—
24#16839623C2214yescomplemented
25#16839528C2214yescomplemented
26C61×#518C2211yescharacteristic
27C2×#99×C61C2212yes—
28C61×#342C2214yescomplemented
29#14519822—44no—
30C2×#11758222C2×C412yes—
31C2×#11758224—24no—
32C2×#11758224—44no—
33C2×#11758224D414yes—
34C22×Dic122C2×C411yescharacteristic, complemented
35C22×C22⋊D61C2×C412yes—
36C22×C22⋊D61C2×C411yescharacteristic, complemented
37C22×C22⋊D61C2311yescharacteristic
38C22×C22⋊D61C2312yes—
39C2×D4×D61D414yes—
40C23×Dic61C2×C411yescharacteristic, complemented
41C23×Dic61C2311yescharacteristic
42C22×C4×D61C2×C411yescharacteristic, complemented
43#14519777—22no—
44#14519765—22no—
45#14519761—24no—
46#14519761—24no—
47#14519761—22no—
48#14519761—22no—
49#14519761C2×C414yescomplemented
50#14519761C2×C414yescomplemented
51#14519761C2×C414yescomplemented
52#14519757—24no—
53#14519757—22no—
54#14519752—22no—
55#14519751—24no—
56#14519751—24no—
57#14519747—22no—
58#14519747—22no—
59#14519747C2×C414yescomplemented
60C4×D244—24no—
61C4×C22⋊D61—44no—
62#14519737—44no—
63#14519733—24no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C22⋊Dic61C2314yes—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C22⋊C4×D61—24no—
70#14519692—24no—
71#14519691—28no—
72#14519691—44no—
73#14519684—44no—
74C22×D4×C61C2311yescharacteristic
75C4×D4×C61—24no—
76#117×C61C2×C412yescomplemented
77C2×C22⋊C4×C61C2×C412yescomplemented
78C2×C22⋊C4×C61C2314yes—
79C2×C22⋊C4×C61C2312yes—
80#99×C61—24no—
81C24×C61C22×C411yescharacteristic
82C24×C61C2411yescharacteristic
83C23×D61C22⋊C412yes—
84C23×D61C22×C411yescharacteristic
85C23×D61C2×D412yes—
86C2×D4×C61—24no—
87C2×D4×C61C2×D414yes—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—24no—
91C22×C4×C61C22×C411yescharacteristic
92C2×C22⋊D61—24no—
93C2×C22⋊D61—24no—
94C2×C22⋊D61—22no—
95C2×C22⋊D61—22no—
96C2×C22⋊D61—24no—
97C2×C22⋊D61—42no—
98C2×C22⋊D61—44no—
99C2×C22⋊D61C22⋊C414yes—
100C2×C22⋊D61C22×C414yes—
101C2×C22⋊D61C2×D414yes—
102C22×Dic61—22no—
103C22×Dic61—22no—
104C22×Dic61—24no—
105C22×Dic61—24no—
106C22×Dic61—42no—
107C22×Dic61C22⋊C412yes—
108C22×Dic61C22×C411yescharacteristic
109C22×Dic61C22×C411yescharacteristic
110C22×Dic61C22×C411yescharacteristic
111C22×Dic61C2×D412yes—
112#11758224—48no—
113#11758224—416no—
114D4×D61—48no—
115#11758222—44no—
116C2×D244—24no—
117C2×D244C22⋊C412yes—
118C2×C4×D61—22no—
119C2×C4×D61—42no—
120C2×C4×D61—44no—
121C2×C4×D61C22⋊C414yes—
122C2×Dic122—22no—
123C2×Dic122—42no—
124C2×Dic122C22⋊C412yes—
125C4⋊C4×C61—22no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—22no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—24no—
130C22⋊C4×C61C22×C414yes—
131C42×C61—22no—
132C22⋊Dic61—24no—
133C22⋊Dic61—24no—
134#11758198—24no—
135#11758198—24no—
136#11758198—24no—
137#11758198—24no—
138#11758198—44no—
139C4⋊Dic61—24no—
140C4⋊Dic61—24no—
141C4⋊Dic61—24no—
142#11758196—44no—
143C4×Dic61—44no—
144C23×C61—22no—
145C23×C61—81no—
146C23×C61—81no—
147C23×C61C2×C22⋊C412yes—
148C23×C61C23×C411yescharacteristic
149C23×C61C22×D412yes—
150C23×C61H(5,𝔽2)14yes—
151C22×D61—24no—
152C22×D61—21no—
153C22×D61—22no—
154C22×D61—42no—
155C22×D61—81no—
156C22×D61—82no—
157C22×D61—82no—
158C22×D61C2×C22⋊C414yes—
159D4×C61—28no—
160C2×C4×C61—22no—
161C2×C4×C61—24no—
162C2×C4×C61—44no—
163C2×C4×C61—82no—
164C2×C4×C61—82no—
165C2×C4×C61—84no—
166C2×C4×C61C2×C22⋊C412yes—
167C22⋊D61—24no—
168C22⋊D61—24no—
169C22⋊D61—28no—
170C22⋊D61—44no—
171C22⋊D61—44no—
172C22⋊D61—44no—
173C2×Dic61—24no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—22no—
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—42no—
181C2×Dic61—44no—
182C2×Dic61—44no—
183C2×Dic61—44no—
184C2×Dic61—84no—
185C2×Dic61C2×C22⋊C414yes—
186D244—42no—
187D244—84no—
188C4×D61—44no—
189C4×D61—44no—
190C4×D61—164no—
191Dic122—42no—
192Dic122—44no—
193#21777—611nomaximal, Sylow
194C22×C61—22no—
195C22×C61—44no—
196C22×C61—42no—
197C22×C61—41no—
198C22×C61—42no—
199C22×C61—82no—
200C22×C61C22×C22⋊C411yescharacteristic
201C22×C61#51812yes—
202C22×C61#53414yes—
203C2×D61—24no—
204C2×D61—44no—
205C2×D61—42no—
206C2×D61—41no—
207C2×D61—84no—
208C2×D61—82no—
209C2×D61—162no—
210C4×C61—22no—
211C4×C61—44no—
212C4×C61—84no—
213Dic61—24no—
214Dic61—41no—
215Dic61—41no—
216Dic61—41no—
217Dic61—84no—
218#3485—611no—
219#2883—612no—
220#1913—614no—
221#1890—614no—
222#1828—611no—
223#1828D6111yescharacteristic, complemented
224#1782—612no—
225C2×C61—24no—
226C2×C61—21no—
227C2×C61—22no—
228C2×C61—42no—
229C2×C61#230811yescharacteristic
230D61—44no—
231D61—81no—
232C2×H(5,𝔽2)—611no—
233C22×#41—611no—
234#547—1224no—
235#546—1224no—
236#536—612no—
237#534—612no—
238#534—612no—
239#525—612no—
240C2×#123—612no—
241C2×#123—612no—
242#518—614no—
243#518C2×D6111yescharacteristic
244#410—1224no—
245C2×#99—612no—
246C2×#99—614no—
247C2×#99—614no—
248C2×#99—1224no—
249C2×#99C2×D6112yes—
250#381—614no—
251#342—614no—
252#342C2×D6114yescomplemented
253C61#2177711yescharacteristic, complemented, Sylow
254H(5,𝔽2)—2444no—
255C2×#41—614no—
256C2×#41—612no—
257C2×#41—1224no—
258C2×#41—2444no—
259C22×Q8—611no—
260C22×D4—612no—
261C22×D4—612no—
262C22×D4—614no—
263C22×D4—611no—
264C22×D4—611no—
265C22×D4C22×D6111yescharacteristic
266C23×C4—611no—
267C23×C4—611no—
268C23×C4—611no—
269C4⋊Q8—1222no—
270#124—1222no—
271#123—614no—
272#123—614no—
273#123—614no—
274#123—1222no—
275#123—1222no—
276#123—1224no—
277#123—1224no—
278C22⋊Q8—1224no—
279C22⋊Q8—1222no—
280#121—1222no—
281#121—1224no—
282#121—1224no—
283#120—614no—
284#120—1222no—
285#120—1222no—
286C4×D4—24no—
287C4×D4—1224no—
288C4×D4—2444no—
289C4×D4—2444no—
290#117—1224no—
291#117C4×D6112yescomplemented
292C2×C4⋊C4—1222no—
293C2×C4⋊C4—1222no—
294C2×C22⋊C4—614no—
295C2×C22⋊C4—1224no—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1222no—
298C2×C22⋊C4C4×D6112yescomplemented
299C2×C22⋊C4C22×D6114yes—
300C2×C22⋊C4C22×D6112yes—
301#99—24no—
302#99—1228no—
303#99—1224no—
304#99—2444no—
305#95—2444no—
306C24—612no—
307C24—612no—
308C24—611no—
309C24C2×C4×D6111yescharacteristic
310C24C23×D6111yescharacteristic
311#41—2444no—
312#41—2448no—
313#41—24416no—
314C2×Q8—612no—
315C2×Q8—1222no—
316C2×Q8—2442no—
317C2×D4—24no—
318C2×D4—614no—
319C2×D4—614no—
320C2×D4—614no—
321C2×D4—612no—
322C2×D4—1224no—
323C2×D4—1224no—
324C2×D4—1224no—
325C2×D4—1222no—
326C2×D4—1224no—
327C2×D4—1222no—
328C2×D4—2442no—
329C2×D4—2448no—
330C2×D4—2444no—
331C2×D4D4×D6114yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—612no—
336C22×C4—612no—
337C22×C4—614no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—1222no—
342C22×C4—1224no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—1224no—
346C22×C4—2442no—
347C22×C4—2444no—
348C22×C4—2442no—
349C22×C4C2×C4×D6111yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—1224no—
352C4⋊C4—1224no—
353C4⋊C4—1224no—
354C4⋊C4—2444no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—1224no—
360C22⋊C4—1224no—
361C22⋊C4—1224no—
362C22⋊C4—1224no—
363C22⋊C4—1224no—
364C22⋊C4—1224no—
365C22⋊C4—2444no—
366C22⋊C4C2×C4×D6114yes—
367C42—22no—
368C42—2444no—
369C23—22no—
370C23—81no—
371C23—81no—
372C23—614no—
373C23—1224no—
374C23—1222no—
375C23—1221no—
376C23—4882no—
377C23—4882no—
378C23—4881no—
379C23—2442no—
380C23C22⋊C4×D6112yes—
381C23C22×C4×D6111yescharacteristic
382C23C2×D4×D6112yes—
383C23#1451982314yes—
384Q8—2442no—
385Q8—2444no—
386D4—28no—
387D4—1224no—
388D4—1224no—
389D4—1228no—
390D4—4884no—
391D4—2442no—
392D4—2444no—
393D4—2444no—
394D4—2444no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—614no—
402C2×C4—1224no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—1222no—
407C2×C4—1222no—
408C2×C4—1222no—
409C2×C4—9764no—
410C2×C4—4884no—
411C2×C4—2444no—
412C2×C4—2444no—
413C2×C4—2442no—
414C2×C4—2444no—
415C2×C4—2444no—
416C2×C4—2444no—
417C2×C4C22⋊C4×D6112yes—
418C22—22no—
419C22—44no—
420C22—42no—
421C22—41no—
422C22—42no—
423C22—82no—
424C22—1224no—
425C22—9762no—
426C22—4882no—
427C22—4884no—
428C22—2442no—
429C22—2444no—
430C22—2441no—
431C22C2×C22⋊C4×D6111yescharacteristic
432C22#1684023612yes—
433C22#1684032314yes—
434C4—22no—
435C4—44no—
436C4—84no—
437C4—1224no—
438C4—4884no—
439C4—2441no—
440C4—2441no—
441C4—2441no—
442C2—21no—
443C2—24no—
444C2—22no—
445C2—42no—
446C2—4881no—
447C2—2444no—
448C2#2144839611yescharacteristic
subgroupG/H
centerC2#21448396
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1828C2
socleC2×C61#2308
prime radical#21459464C2
cosocleC24×C61C24
hypercenter#1828D61
hypocenterC61#21777
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ #1828

Complements

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

NH ≅ G/Ntype
2. #21454277C2semidirect
3. #21446360C2semidirect
4. #21446231C2semidirect
5. #21445927C2semidirect
6. #21445735C2semidirect
7. C61×#1828C2semidirect
12. #16840338C4semidirect
13. #16840324C4semidirect
15. #16840274C4semidirect
16. C2×#14519761C4semidirect
22. C2×#14519691C22semidirect
24. #16839623C22semidirect
25. #16839528C22semidirect
28. C61×#342C22semidirect
34. C22×Dic122C2×C4semidirect
36. C22×C22⋊D61C2×C4semidirect
40. C23×Dic61C2×C4semidirect
42. C22×C4×D61C2×C4semidirect
49. #14519761C2×C4semidirect
50. #14519761C2×C4semidirect
51. #14519761C2×C4semidirect
59. #14519747C2×C4semidirect
76. #117×C61C2×C4semidirect
77. C2×C22⋊C4×C61C2×C4semidirect
223. #1828D61semidirect
252. #342C2×D61semidirect
253. C61#21777semidirect
291. #117C4×D61semidirect
298. C2×C22⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214594642
#214542772
#214463602
#214462312
#214459272
#214457352
C61×#18282
#2177761