Finite Group Databasebeta

← #29834280

Subgroups — #29834280

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840192C212yesmaximal, complemented
2#21456247C211yesmaximal, characteristic, complemented
3#21447881C212yesmaximal, complemented
4#21447874C212yesmaximal, complemented
5#21447869C211yesmaximal, characteristic, complemented
6#21447864C211yesmaximal, characteristic, complemented
7#21447859C211yesmaximal, characteristic, complemented
8#21447854C211yesmaximal, characteristic, complemented
9C2×C4⋊#11758185C211yesmaximal, characteristic, complemented
10#21447524C212yesmaximal, complemented
11C4×#135×C61C211yesmaximal, characteristic, complemented
12C2×#14519799—21no—
13C2×#14519799C2212yes—
14#16840312—21no—
15#16840192—28no—
16C2×#14519723C2212yes—
17C2×#14519721C2212yes—
18C2×#14519720C2212yes—
19C2×#14519718C2212yes—
20C22×#11758185C2212yes—
21#16840146—21no—
22#16840146C2212yes—
23#16840142C2211yescharacteristic
24#16840139C2211yescharacteristic
25#16840135—21no—
26C4⋊#11758185C2211yescharacteristic
27C4⋊#11758185C2211yescharacteristic
28C4⋊#11758185C2212yes—
29#16839565C2212yes—
30#16839565C2212yes—
31#16839562C2212yes—
32#16839562C2212yes—
33C2×#135×C61C2211yescharacteristic
34C4×#41×C61—21no—
35C4×Q16×C61C2211yescharacteristic
36C4×Q16×C61C2211yescharacteristic
37C4×SD16×C61—21no—
38C4×SD16×C61—21no—
39C4×D8×C61C2211yescharacteristic
40C4×D8×C61C2211yescharacteristic
41C61×#427C2211yescharacteristic
42C61×#416—21no—
43C2×C4×C8×C61C2211yescharacteristic
44C2×#11758222—22no—
45#14519799—24no—
46#14519799—24no—
47#14519799—44no—
48C2×Dic244—21no—
49C2×Dic244C2312yes—
50C2×#11758190—22no—
51C2×#11758190—22no—
52C2×D488—21no—
53C2×D488C2312yes—
54#14519777—21no—
55#14519774—21no—
56#14519765—22no—
57#14519751—41no—
58#14519751—41no—
59C4×D244—41no—
60C4×D244—41no—
61#14519723—24no—
62#14519721—24no—
63#14519720—24no—
64#14519718—24no—
65C2×#11758185—24no—
66C2×#11758185C2312yes—
67C2×#11758185C2312yes—
68C2×#41×C61—21no—
69#135×C61—22no—
70#135×C61D414yescomplemented
71C2×Q16×C61C2311yescharacteristic
72C2×SD16×C61—21no—
73C2×D8×C61C2311yescharacteristic
74C22×C8×C61D411yescharacteristic
75C22×C8×C61C2311yescharacteristic
76C4×Q8×C61—21no—
77C4×Q8×C61—21no—
78C4×D4×C61—21no—
79C4×D4×C61—21no—
80C4×D4×C61—41no—
81C4×D4×C61—41no—
82#117×C61—21no—
83#117×C61—41no—
84C2×C42×C61—41no—
85C2×C42×C61D411yescharacteristic
86#107×C61C2311yescharacteristic
87#107×C61C2311yescharacteristic
88#106×C61—21no—
89#103×C61—21no—
90#103×C61—21no—
91#102×C61—21no—
92#102×C61—21no—
93C4×C8×C61D411yescharacteristic
94C4×C8×C61D411yescharacteristic
95C4×C8×C61C2311yescharacteristic
96C4×C8×C61C2311yescharacteristic
97#41×C61—24no—
98#41×C61—42no—
99C2×Q8×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61D811yescharacteristic
106C22×C4×C61D811yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C2×C22⋊D61—42no—
109#11758222—44no—
110#11758222—44no—
111C2×D244—22no—
112C2×D244—41no—
113C2×D244—41no—
114C2×C4×D61—42no—
115C2×Dic122—22no—
116Q16×C61—21no—
117Q16×C61C2×D412yes—
118SD16×C61—24no—
119D8×C61—21no—
120D8×C61C2×D412yes—
121C2×C8×C61—22no—
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D412yes—
126C2×C8×C61#4112yes—
127C2×C8×C61C2411yescharacteristic
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C22⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61—41no—
136C42×C61—41no—
137C42×C61—41no—
138C42×C61D811yescharacteristic
139C42×C61D811yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142#11758198—42no—
143#11758198—42no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146Dic244—22no—
147Dic244—22no—
148Dic244—42no—
149D488—22no—
150D488—22no—
151D488—42no—
152#11758190—44no—
153#11758190—44no—
154#11758185—22no—
155#11758185—22no—
156C23×C61—41no—
157C23×C61C2×D811yescharacteristic
158C22×D61—42no—
159Q8×C61—22no—
160Q8×C61—41no—
161D4×C61—22no—
162D4×C61—44no—
163D4×C61—41no—
164C2×C4×C61—22no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—44no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—82no—
171C2×C4×C61—82no—
172C2×C4×C61#12112yes—
173C2×C4×C61C2×D811yescharacteristic
174C2×C4×C61C2×D811yescharacteristic
175C2×C4×C61C2×D811yescharacteristic
176C2×C4×C61C2×D811yescharacteristic
177C2×C4×C61C2×D811yescharacteristic
178C2×C4×C61AGL(1,ℤ/8)12yes—
179C2×C4×C61C22×D411yescharacteristic
180C22⋊D61—84no—
181C2×Dic61—42no—
182D244—42no—
183D244—42no—
184D244—82no—
185D244—82no—
186C4×D61—84no—
187Dic122—42no—
188Dic122—42no—
189C8×C61—21no—
190C8×C61—21no—
191C8×C61#12112yes—
192C8×C61C22×D411yescharacteristic
193C8×C61C2×#4111yescharacteristic
194#33579—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—42no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#45912yes—
201C22×C61C22×D811yescharacteristic
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—82no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—42no—
209C4×C61—81no—
210C4×C61—81no—
211C4×C61#45912yes—
212C4×C61#50812yescomplemented
213C4×C61C2×#12111yescharacteristic
214C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
215Dic61—82no—
216C2×#508—612no—
217#3147—611no—
218C4×#135D6111yescharacteristic, complemented
219#2211—612no—
220#2210—612no—
221#2209—611no—
222#2208—611no—
223#2207—611no—
224#2206—611no—
225C2×C4⋊C16—611no—
226#2138—612no—
227C2×C61—22no—
228C2×C61—82no—
229C2×C61C2×#45911yescharacteristic
230C2×C61C2×#50811yescharacteristic
231C2×C61#341411yescharacteristic
232D61—162no—
233C2×#135—612no—
234C2×#135—1221no—
235C2×#135C2×D6111yescharacteristic
236#532—1221no—
237C4×#41—21no—
238#508—1228no—
239C2×Q32—612no—
240C2×SD32—612no—
241C2×SD32—612no—
242C2×D16—612no—
243C22×C16—612no—
244#495—612no—
245#495—1221no—
246#494—611no—
247#493—611no—
248#492—1221no—
249C4×Q16C2×D6111yescharacteristic
250C4×Q16C2×D6111yescharacteristic
251C4×SD16—21no—
252C4×SD16—21no—
253C4×D8C2×D6111yescharacteristic
254C4×D8C2×D6111yescharacteristic
255#427C2×D6111yescharacteristic
256#416—21no—
257C2×C4×C8C2×D6111yescharacteristic
258C4⋊C16—611no—
259C4⋊C16—611no—
260C4⋊C16—612no—
261#358—612no—
262#358—612no—
263#357—612no—
264#357—612no—
265C61#3357911yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C2×#41—1222no—
268#135—22no—
269#135—1224no—
270#135—1224no—
271#135—2444no—
272#135D24414yescomplemented
273C2×Q16—612no—
274C2×Q16—1221no—
275C2×Q16C22×D6111yescharacteristic
276C2×SD16—21no—
277C2×SD16—1222no—
278C2×SD16—1222no—
279C2×D8—612no—
280C2×D8—1221no—
281C2×D8C22×D6111yescharacteristic
282C22×C8C22⋊D6111yescharacteristic
283C22×C8C22×D6111yescharacteristic
284C4⋊Q8—1221no—
285#127—1221no—
286#124—1222no—
287#121—2441no—
288#121—2441no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—41no—
294C4×D4—41no—
295C4×D4—2441no—
296C4×D4—2441no—
297#117—21no—
298#117—41no—
299C2×C42—41no—
300C2×C42C22⋊D6111yescharacteristic
301Q32—1224no—
302SD32—1224no—
303SD32—1224no—
304D16—1224no—
305C2×C16—612no—
306C2×C16—612no—
307C2×C16—1224no—
308#107C22×D6111yescharacteristic
309#107C22×D6111yescharacteristic
310#106—21no—
311#103—21no—
312#103—21no—
313#102—21no—
314#102—21no—
315C4×C8C22⋊D6111yescharacteristic
316C4×C8C22⋊D6111yescharacteristic
317C4×C8C22×D6111yescharacteristic
318C4×C8C22×D6111yescharacteristic
319#41—24no—
320#41—42no—
321#41—2444no—
322#41—2444no—
323C2×Q8—21no—
324C2×Q8—1222no—
325C2×D4—21no—
326C2×D4—41no—
327C2×D4—1222no—
328C2×D4—2442no—
329C2×D4—2441no—
330C2×D4—2441no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—2442no—
335C22×C4#1175819911yescharacteristic
336C22×C4#1175819911yescharacteristic
337C22×C4C2×C22⋊D6111yescharacteristic
338Q16—21no—
339Q16—1222no—
340Q16—1222no—
341Q16—2442no—
342Q16C2×D24412yes—
343SD16—24no—
344SD16—2444no—
345SD16—2444no—
346D8—21no—
347D8—1222no—
348D8—1222no—
349D8—2442no—
350D8C2×D24412yes—
351C2×C8—22no—
352C2×C8C2×D24412yes—
353C2×C8#1175822212yes—
354C2×C8C2×C22⋊D6111yescharacteristic
355C2×C8C2×C22⋊D6111yescharacteristic
356C2×C8C2×C22⋊D6111yescharacteristic
357C2×C8C23×D6111yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—2441no—
363C4⋊C4—2441no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—2442no—
368C22⋊C4—2442no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42#1175819911yescharacteristic
373C42#1175819911yescharacteristic
374C42C2×C22⋊D6111yescharacteristic
375C42C2×C22⋊D6111yescharacteristic
376C16—1222no—
377C16—1222no—
378C23—41no—
379C23—2442no—
380C23C2×#1175819911yescharacteristic
381Q8—22no—
382Q8—41no—
383Q8—2442no—
384Q8—2442no—
385D4—22no—
386D4—44no—
387D4—41no—
388D4—4882no—
389D4—4884no—
390D4—4882no—
391D4—2442no—
392D4—2442no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—4884no—
402C2×C4—2442no—
403C2×C4#1451975112yes—
404C2×C4C2×#1175819911yescharacteristic
405C2×C4C2×#1175819911yescharacteristic
406C2×C4C2×#1175819911yescharacteristic
407C2×C4C2×#1175819911yescharacteristic
408C2×C4C2×#1175819911yescharacteristic
409C2×C4#1451980212yes—
410C2×C4C22×C22⋊D6111yescharacteristic
411C8—21no—
412C8—21no—
413C8#1451975112yes—
414C8C22×D24411yescharacteristic
415C8C2×#1175822211yescharacteristic
416C22—21no—
417C22—21no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—4882no—
422C22—4882no—
423C22—4882no—
424C22#1683995712yes—
425C22C22×#1175819911yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—81no—
431C4—81no—
432C4—4882no—
433C4#1683995712yes—
434C4#1684019212yescomplemented
435C4C2×#1451975111yescharacteristic
436C4C2×#1451980211yescharacteristic
437C2—22no—
438C2—82no—
439C2—9762no—
440C2C2×#1683995711yescharacteristic
441C2C2×#1684019211yescharacteristic
442C2#2145895311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC4×#135×C61C2
socleC22×C61C22×D8
hypercenterC4×#135D61
hypocenterC61#33579
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16840192C2semidirect
2. #21456247C2semidirect
3. #21447881C2semidirect
4. #21447874C2semidirect
5. #21447869C2semidirect
6. #21447864C2semidirect
7. #21447859C2semidirect
8. #21447854C2semidirect
9. C2×C4⋊#11758185C2semidirect
10. #21447524C2semidirect
11. C4×#135×C61C2semidirect
70. #135×C61D4semidirect
212. C4×C61#508semidirect
218. C4×#135D61semidirect
265. C61#33579semidirect
272. #135D244semidirect
434. C4#16840192semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
#214562472
#214478812
#214478742
#214478692
#214478642
#214478592
#214478542
C2×C4⋊#117581852
#214475242
C4×#135×C612
#3357961