Finite Group Databasebeta

← #29667925

Subgroups — #29667925

Order 15616.

784 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#21456739C211yesmaximal, characteristic
2#21456379C211yesmaximal, characteristic, complemented
3#21456215C211yesmaximal, characteristic, complemented
4#21456195C211yesmaximal, characteristic, complemented
5C2×#16840110C211yesmaximal, characteristic, complemented
6#21445482C212yesmaximal, complemented
7#21445461C212yesmaximal, complemented
8#21445432C212yesmaximal, complemented
9#21445426C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3209C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14#16840158C2211yescharacteristic, complemented
15#16840139C2212yes—
16#16840136C2211yescharacteristic, complemented
17#16840128—22no—
18#16840124—22no—
19#16840114C2212yescomplemented
20#16840112C2211yescharacteristic, complemented
21#16840110—22no—
22#16840110C2212yes—
23#16840110C2212yes—
24#16840110C2212yes—
25#16840110C2212yes—
26#16840110C2211yescharacteristic, complemented
27#16840110C2212yescomplemented
28#16840109C2212yes—
29#16839993—22no—
30#16839958—22no—
31#16839898—22no—
32C4×#11758201—22no—
33#16839766—22no—
34#16839761—22no—
35C2×#14519697—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#494C2212yescomplemented
43C61×#487C2212yescomplemented
44C61×#462—22no—
45C61×#451—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51C2×#11758201—22no—
52C2×D488—24no—
53C2×#11758194D412yes—
54C22×#92418D412yes—
55#14519777C2311yescharacteristic
56#14519774—21no—
57#14519774—21no—
58#14519774—21no—
59#14519774C2312yes—
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519708—22no—
67#14519701—22no—
68#14519700—24no—
69#14519699—24no—
70#14519697—22no—
71#14519697—24no—
72#14519697—24no—
73#14519697—24no—
74#14519697—24no—
75#14519697—24no—
76#14519688D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C22×Q8×C61—21no—
82C2×Q16×C61—24no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2312yes—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94#117×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61D412yes—
99#103×C61—24no—
100C22⋊C8×C61D412yes—
101C4×C8×C61C2312yes—
102C23×D61—41no—
103C2×Q8×C61—22no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—41no—
107C2×Q8×C61—82no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C2×C22⋊D61—41no—
113#11758222—44no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—44no—
121C2×D244—44no—
122C2×D244—41no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244—82no—
126C2×C4×D61—41no—
127C2×Dic122—21no—
128Q16×C61—84no—
129C2×C8×C61C2×D414yes—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61—41no—
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61C2411yescharacteristic
146#11758201—44no—
147#11758198—41no—
148#11758198—41no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151#11758194—24no—
152C2×#92418—42no—
153C2×#92418C2×D412yes—
154C2×#92418C2×D412yes—
155C2×#92418C2×D414yes—
156D488—84no—
157C23×C61#12711yescharacteristic
158C22×D61—41no—
159C22×D61—42no—
160C22×D61—82no—
161C22×D61—82no—
162C22×D61—82no—
163Q8×C61—42no—
164Q8×C61—42no—
165Q8×C61—44no—
166Q8×C61—82no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—82no—
173C2×C4×C61#12711yescharacteristic
174C2×C4×C61#12711yescharacteristic
175C2×C4×C61#12711yescharacteristic
176C2×C4×C61#13514yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C22⋊D61—82no—
181C2×Dic61—41no—
182D244—44no—
183D244—42no—
184D244—42no—
185D244—44no—
186D244—42no—
187D244—44no—
188D244—82no—
189D244—82no—
190D244—82no—
191C4×D61—82no—
192Dic122—42no—
193C8×C61—24no—
194#92418—24no—
195#92418—22no—
196#92418—22no—
197#19627—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61#49512yes—
201C22×C61C2×#12711yescharacteristic
202C2×D61—81no—
203C2×D61—82no—
204C2×D61—82no—
205C2×D61—81no—
206C2×D61—82no—
207C2×D61—81no—
208C2×D61—164no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—81no—
212C4×C61—82no—
213C4×C61—81no—
214C4×C61#49512yes—
215C4×C61C2×#13512yes—
216C4×C61#57512yes—
217C4×C61#57612yes—
218Dic61—81no—
219#3209D6111yescharacteristic, complemented
220#3205—611no—
221#3167—611no—
222#3141—611no—
223#3138—611no—
224C2×#486—611no—
225#1720—612no—
226#1717—612no—
227#1711—612no—
228#1709—612no—
229C8×M16—611no—
230C2×C61—22no—
231C2×C61C2×#49511yescharacteristic
232C2×C61#315311yescharacteristic
233C2×C61#315511yescharacteristic
234D61—161no—
235D61—162no—
236#533C2×D6111yescharacteristic
237#532—611no—
238C2×C4⋊Q8C2×D6111yescharacteristic
239C2×#127—611no—
240#498—611no—
241#494C2×D6112yescomplemented
242#493—612no—
243#492—611no—
244#490—1222no—
245#489—1222no—
246#487—612no—
247#487—611no—
248#487C2×D6112yescomplemented
249#486—612no—
250#486—612no—
251#486—612no—
252#486—612no—
253#486—612no—
254#486—611no—
255#486—612no—
256#486—1222no—
257#465—1222no—
258#462—22no—
259#459—1222no—
260#451—22no—
261#447—1222no—
262C4×SD16—1222no—
263#431C2×D6111yescharacteristic
264#418—1222no—
265#417—1222no—
266C2×#102—1222no—
267C4×M16—611no—
268C2×C4×C8—611no—
269#322—612no—
270C82—612no—
271C61#1962711yescharacteristic, complemented, Sylow
272C2×#41—1221no—
273C22×Q8—21no—
274C22×D4—1221no—
275C22×D4—1221no—
276C22×D4—1221no—
277C2×Q16—24no—
278C2×SD16—1222no—
279C2×D8—1224no—
280C2×M16—612no—
281C22×C8—612no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—611no—
286C4⋊Q8C22×D6112yes—
287C4⋊Q8C22×D6111yescharacteristic
288C4⋊Q8C22×D6111yescharacteristic
289#127—612no—
290#127—611no—
291#127—1221no—
292#127—1221no—
293#127—1221no—
294#125—21no—
295#124—1221no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298#121—1221no—
299#121—1221no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×D4—1221no—
303C4×D4—1221no—
304#117—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C42C22×D6111yescharacteristic
308#106—1222no—
309C4⋊C8C22⋊D6112yes—
310#103—24no—
311#103—1224no—
312#103—1222no—
313#102—1224no—
314#102—1224no—
315#102—1224no—
316#102—1224no—
317#102—1224no—
318#102—1222no—
319#102—1224no—
320C22⋊C8C22⋊D6112yes—
321#97—612no—
322C4×C8—612no—
323C4×C8—611no—
324C4×C8—612no—
325C4×C8—611no—
326C4×C8C22×D6112yes—
327C24—2441no—
328#41—2444no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—21no—
332C2×Q8—41no—
333C2×Q8—82no—
334C2×Q8—1221no—
335C2×D4—1221no—
336C2×D4—1222no—
337C2×D4—1222no—
338C2×D4—1221no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4—4882no—
342C2×D4—2444no—
343C2×D4—2441no—
344C2×D4—2441no—
345C2×D4—2441no—
346C2×D4—2441no—
347C2×D4—2444no—
348C22×C4—41no—
349C22×C4—2441no—
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353Q16—84no—
354SD16—2444no—
355D8—4884no—
356M16—1224no—
357C2×C8—612no—
358C2×C8—612no—
359C2×C8—614no—
360C2×C8—2442no—
361C2×C8C2×C22⋊D6114yes—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C22⋊C4—41no—
375C22⋊C4—2441no—
376C22⋊C4—2441no—
377C42—41no—
378C42D4×D6111yescharacteristic
379C42D4×D6111yescharacteristic
380C42C2×C22⋊D6111yescharacteristic
381C42C23×D6111yescharacteristic
382C23—4882no—
383C23—4882no—
384C23—4882no—
385C23—2441no—
386C23—2442no—
387C23#1451977311yescharacteristic
388Q8—42no—
389Q8—42no—
390Q8—44no—
391Q8—82no—
392Q8—2442no—
393D4—4882no—
394D4—4882no—
395D4—4882no—
396D4—4882no—
397D4—2444no—
398D4—2442no—
399D4—2442no—
400D4—2444no—
401D4—2442no—
402D4—2444no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—4882no—
410C2×C4—2441no—
411C2×C4#1451977311yescharacteristic
412C2×C4#1451977311yescharacteristic
413C2×C4#1451977311yescharacteristic
414C2×C4#1451980014yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C2×D4×D6111yescharacteristic
417C2×C4C22×C22⋊D6111yescharacteristic
418C8—24no—
419C8—1224no—
420C8—1222no—
421C8—1222no—
422C22—21no—
423C22—21no—
424C22—9764no—
425C22—4881no—
426C22—4882no—
427C22—4882no—
428C22—4881no—
429C22—4882no—
430C22—4881no—
431C22#1684014712yes—
432C22C2×#1451977311yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—82no—
437C4—81no—
438C4—4881no—
439C4#1684014712yes—
440C4C2×#1451980012yes—
441C4#1684056112yes—
442C4#1684056512yes—
443C2—22no—
444C2—9762no—
445C2—9761no—
446C2C2×#1684014711yescharacteristic
447C2#2145628611yescharacteristic
448C2#2145630211yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3209C2
socleC22×C61C2×#127
prime radical#21456739C2
hypercenter#3209D61
hypocenterC61#19627
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
2. #21456379C2semidirect
3. #21456215C2semidirect
4. #21456195C2semidirect
5. C2×#16840110C2semidirect
6. #21445482C2semidirect
7. #21445461C2semidirect
8. #21445432C2semidirect
9. #21445426C2semidirect
10. #21444100C2semidirect
11. C61×#3209C2semidirect
14. #16840158C22semidirect
16. #16840136C22semidirect
19. #16840114C22semidirect
20. #16840112C22semidirect
26. #16840110C22semidirect
27. #16840110C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
42. C61×#494C22semidirect
43. C61×#487C22semidirect
219. #3209D61semidirect
241. #494C2×D61semidirect
248. #487C2×D61semidirect
271. C61#19627semidirect

Maximal subgroups

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

subgroupindex
#214567392
#214563792
#214562152
#214561952
C2×#168401102
#214454822
#214454612
#214454322
#214454262
#214441002
C61×#32092
#1962761