Finite Group Databasebeta

← #29663077

Subgroups — #29663077

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C211yesmaximal, characteristic
2C2×#16840136C211yesmaximal, characteristic, complemented
3#21454531C211yesmaximal, characteristic, complemented
4#21445286C214yesmaximal, complemented
5#21445162C214yesmaximal, complemented
6#21444565C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#3226C211yesmaximal, characteristic, complemented
10C22×#11758201—21no—
11C22×#11758201—21no—
12C2×#14519784—21no—
13C2×#14519774C2211yescharacteristic
14#16840138—21no—
15#16840136—21no—
16#16840136—21no—
17#16840136C2212yes—
18#16840136C2212yes—
19#16840110—21no—
20#16839957C2214yes—
21#16839895—22no—
22#16839840—21no—
23#16839840C2212yescomplemented
24D4×#92418—21no—
25D4×#92418C2212yescomplemented
26#16839815C2211yescharacteristic
27#16839786C2211yescharacteristic
28#16839729—21no—
29C2×C4×#92418C2211yescharacteristic
30#16839514C2212yescomplemented
31#16839512C2212yescomplemented
32#16839508C2214yescomplemented
33#16839505C2214yes—
34C61×#532C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#500—21no—
37C61×#486—21no—
38C61×#480—22no—
39C61×#474C2214yescomplemented
40C61×#431C2211yescharacteristic
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C2×#11758201—22no—
45C2×#11758201—22no—
46C2×#11758201—22no—
47C2×#11758201—22no—
48C2×#11758201—44no—
49C2×#11758201—44no—
50C2×#11758199—24no—
51#14519784—44no—
52C2×#11758194—21no—
53C2×#11758194—22no—
54C22×#92418—22no—
55C22×#92418D411yescharacteristic
56C22×#92418D411yescharacteristic
57#14519774—21no—
58#14519774—21no—
59#14519774—21no—
60#14519774C2312yes—
61C4⋊#92418—21no—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64#14519697—24no—
65C22⋊#92418—21no—
66C22⋊#92418—22no—
67C22⋊#92418—22no—
68C4×#92418—21no—
69C4×#92418D411yescharacteristic
70C4×#92418D411yescharacteristic
71C4×#92418C2311yescharacteristic
72C4×#92418C2311yescharacteristic
73C2×#41×C61—21no—
74C22×Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D411yescharacteristic
79C4⋊Q8×C61C2312yes—
80#127×C61D411yescharacteristic
81#124×C61—21no—
82#121×C61—21no—
83#121×C61D412yes—
84C4×D4×C61—21no—
85C4×D4×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89#107×C61—24no—
90C4⋊C8×C61C2312yes—
91#102×C61—24no—
92C22⋊C8×C61—21no—
93C4×C8×C61—21no—
94C23×D61—41no—
95#41×C61—42no—
96#41×C61—42no—
97C2×Q8×C61—21no—
98C2×Q8×C61—22no—
99C2×Q8×C61—41no—
100C2×Q8×C61—42no—
101C2×Q8×C61—42no—
102C2×D4×C61—22no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61C2×D412yes—
106C22×C4×C61—22no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—42no—
115C2×D244—42no—
116C2×D244—41no—
117C2×D244—41no—
118C2×D244—41no—
119C2×D244—82no—
120C2×D244—82no—
121C2×C8×C61—22no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61C2×D412yes—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61#4111yescharacteristic
133C42×C61C2411yescharacteristic
134#11758201—44no—
135#11758201—44no—
136#11758201—44no—
137#11758201—44no—
138#11758199—84no—
139#11758194—42no—
140#11758194—44no—
141C2×#92418—22no—
142C2×#92418—22no—
143C2×#92418—22no—
144C2×#92418—44no—
145C2×#92418D814yescomplemented
146C2×#92418C2×D411yescharacteristic
147C2×#92418C2×D411yescharacteristic
148C2×#92418C2×D411yescharacteristic
149C2×#92418C2×D411yescharacteristic
150C23×C61—22no—
151C23×C61#12111yescharacteristic
152C22×D61—42no—
153C22×D61—82no—
154C22×D61—82no—
155C22×D61—82no—
156Q8×C61—42no—
157Q8×C61—42no—
158Q8×C61—42no—
159Q8×C61—82no—
160D4×C61—24no—
161D4×C61—44no—
162D4×C61—42no—
163D4×C61—42no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—82no—
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61C2×D812yes—
175C2×C4×C61AGL(1,ℤ/8)12yes—
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179D244—42no—
180D244—42no—
181D244—44no—
182D244—82no—
183D244—82no—
184D244—82no—
185D244—82no—
186C8×C61—42no—
187#92418—21no—
188#92418—21no—
189#92418—42no—
190#92418—42no—
191#92418C2×D812yes—
192#92418C2×D812yes—
193#19186—611nomaximal, Sylow
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—42no—
197C22×C61—42no—
198C22×C61—42no—
199C22×C61#45912yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—164no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—42no—
208C4×C61—82no—
209C4×C61#45912yes—
210C4×C61#47112yes—
211C4×C61C22×D811yescharacteristic
212C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
213C4×C61#57712yes—
214#3226D6111yescharacteristic, complemented
215#3205—611no—
216C2×#492—611no—
217#2931—611no—
218#1685—614no—
219#1665—614no—
220#1564—611no—
221#1546—611no—
222#1545—611no—
223C2×C61—22no—
224C2×C61—82no—
225C2×C61C2×#45911yescharacteristic
226C2×C61#305711yescharacteristic
227C2×C61C2×#47111yescharacteristic
228D61—162no—
229C22×SD16—1221no—
230C22×SD16—1221no—
231C2×#131—1221no—
232#532C2×D6111yescharacteristic
233C2×C4⋊Q8C2×D6111yescharacteristic
234C2×#127—611no—
235#500—21no—
236#493—1221no—
237#492—612no—
238#492—612no—
239#492—1221no—
240#492—1221no—
241#486—21no—
242#486—1221no—
243#480—22no—
244#474C2×D6114yescomplemented
245#459—614no—
246#447—1222no—
247#436—612no—
248#436—1221no—
249C8×D4—612no—
250C8×D4—1221no—
251#431—611no—
252#431C2×D6111yescharacteristic
253#423—611no—
254#408—1221no—
255C2×C4×C8—611no—
256#335—612no—
257#334—612no—
258#332—614no—
259#331—614no—
260C61#1918611yescharacteristic, complemented, Sylow
261C2×#41—21no—
262C22×Q8—21no—
263C22×D4—1221no—
264C22×D4—1221no—
265C22×D4—1221no—
266C2×SD16—1222no—
267C2×SD16—1222no—
268C2×SD16—1222no—
269C2×SD16—1222no—
270C2×SD16—2444no—
271C2×SD16—2444no—
272C2×D8—1224no—
273#131—2444no—
274C2×M16—1222no—
275C2×M16—1221no—
276C22×C8—611no—
277C22×C8—611no—
278C22×C8—1222no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8C22⋊D6111yescharacteristic
283C4⋊Q8C22×D6112yes—
284#127—612no—
285#127—1221no—
286#127—1221no—
287#127—1221no—
288#127C22⋊D6111yescharacteristic
289#124—21no—
290#121—21no—
291#121C22⋊D6112yes—
292C4×D4—21no—
293C4×D4C22×D6112yes—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C42C22×D6111yescharacteristic
297#107—24no—
298C4⋊C8—612no—
299C4⋊C8—612no—
300C4⋊C8—1221no—
301C4⋊C8C22×D6112yes—
302#102—24no—
303#102—1224no—
304C22⋊C8—21no—
305C22⋊C8—1222no—
306C22⋊C8—1222no—
307C22⋊C8—1221no—
308C4×C8—21no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—1221no—
314C24—2441no—
315#41—42no—
316#41—42no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—1222no—
326C2×D4—1222no—
327C2×D4—1222no—
328C2×D4—4882no—
329C2×D4—4882no—
330C2×D4—2441no—
331C2×D4—2441no—
332C2×D4—2441no—
333C2×D4—2442no—
334C2×D4—2442no—
335C2×D4C2×C22⋊D6112yes—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4D4×D6111yescharacteristic
339C22×C4#1175822411yescharacteristic
340C22×C4C2×C22⋊D6111yescharacteristic
341SD16—2444no—
342SD16—2444no—
343SD16—2444no—
344SD16—2444no—
345D8—4884no—
346M16—2444no—
347M16—2442no—
348C2×C8—22no—
349C2×C8—611no—
350C2×C8—611no—
351C2×C8—614no—
352C2×C8—611no—
353C2×C8—611no—
354C2×C8—1222no—
355C2×C8—1222no—
356C2×C8—1222no—
357C2×C8—2444no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4C2×C22⋊D6112yes—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C42D4×D6111yescharacteristic
367C42#1175822411yescharacteristic
368C42C2×C22⋊D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—22no—
371C23—4882no—
372C23—4882no—
373C23—4882no—
374C23—2442no—
375C23#1451974911yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—42no—
379Q8—82no—
380D4—24no—
381D4—44no—
382D4—42no—
383D4—42no—
384D4—4882no—
385D4—4882no—
386D4—4882no—
387D4—4882no—
388D4—2442no—
389D4—2442no—
390D4—2444no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4#1451974911yescharacteristic
399C2×C4#1451974911yescharacteristic
400C2×C4#1451974911yescharacteristic
401C2×C4D8×D6112yes—
402C2×C4#1451980112yes—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C22×C22⋊D6111yescharacteristic
405C2×C4C2×#1175822411yescharacteristic
406C8—42no—
407C8—612no—
408C8—612no—
409C8—1221no—
410C8—1221no—
411C8—2442no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—9764no—
419C22—4882no—
420C22—4882no—
421C22—4882no—
422C22#1683995512yes—
423C22C2×#1451974911yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—82no—
428C4#1683995512yes—
429C4#1684002712yes—
430C4C2×D8×D6111yescharacteristic
431C4C2×#1451980111yescharacteristic
432C4#1684057012yes—
433C2—22no—
434C2—82no—
435C2—9762no—
436C2C2×#1683995511yescharacteristic
437C2#2145546011yescharacteristic
438C2C2×#1684002711yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3226C2
socleC22×C61C2×#121
prime radical#21456734C2
hypercenter#3226D61
hypocenterC61#19186
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
2. C2×#16840136C2semidirect
3. #21454531C2semidirect
4. #21445286C2semidirect
5. #21445162C2semidirect
6. #21444565C2semidirect
7. #21444491C2semidirect
8. #21444486C2semidirect
9. C61×#3226C2semidirect
23. #16839840C22semidirect
25. D4×#92418C22semidirect
30. #16839514C22semidirect
31. #16839512C22semidirect
32. #16839508C22semidirect
39. C61×#474C22semidirect
145. C2×#92418D8semidirect
214. #3226D61semidirect
244. #474C2×D61semidirect
260. C61#19186semidirect

Maximal subgroups

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

subgroupindex
#214567342
C2×#168401362
#214545312
#214452862
#214451622
#214445652
#214444912
#214444862
C61×#32262
#1918661