Finite Group Databasebeta

← #29576122

Subgroups — #29576122

Order 15616.

536 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455879C211yesmaximal, characteristic
2C4×#14519784C211yesmaximal, characteristic, complemented
3#21444410C211yesmaximal, characteristic, complemented
4#21444269C211yesmaximal, characteristic, complemented
5#21443708C211yesmaximal, characteristic, complemented
6#21443657C211yesmaximal, characteristic, complemented
7C61×#1489C211yesmaximal, characteristic, complemented
8C2×#14519784—21no—
9#16840312C2211yescharacteristic
10#16840086—21no—
11#16840057C411yescharacteristic
12#16840052C411yescharacteristic
13#16839962C411yescharacteristic
14#16839957C411yescharacteristic
15#16839907—21no—
16C4×#11758201—21no—
17C4×#11758199—21no—
18#16839815C2211yescharacteristic
19#16839718—22no—
20#16839718—21no—
21C4×#11758194—21no—
22C4×#11758194C2211yescharacteristic
23C2×C4×#92418—21no—
24C2×C4×#92418C2211yescharacteristic
25#16839501—22no—
26#16839489—21no—
27#16839489—21no—
28C4×#41×C61C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C4×M16×C61C2211yescharacteristic
31C61×Q8⋊C8—21no—
32C61×D4⋊C8—21no—
33C2×#11758222—21no—
34C2×#11758201—21no—
35C2×#11758199—21no—
36#14519784—24no—
37#14519784—24no—
38C2×#11758194—22no—
39C2×#11758194—21no—
40C22×#92418—22no—
41C22×#92418—21no—
42#14519777C2×C411yescharacteristic
43#14519774C2×C411yescharacteristic
44#14519765—21no—
45#14519751—21no—
46#14519751—21no—
47C4×D244—21no—
48C4×D244—21no—
49#14519710—21no—
50#14519708—21no—
51C4⋊#92418C2×C411yescharacteristic
52C4⋊#92418C2×C411yescharacteristic
53#14519701—21no—
54#14519698—21no—
55#14519697—21no—
56#14519697—21no—
57C22⋊#92418—21no—
58#14519688—22no—
59#14519688D412yes—
60C4×#92418—22no—
61C4×#92418—21no—
62C4×#92418—21no—
63C4×#92418—21no—
64C4×#92418D411yescharacteristic
65C4×#92418D411yescharacteristic
66C2×#41×C61D411yescharacteristic
67C2×M16×C61—21no—
68C2×M16×C61—21no—
69C4×Q8×C61C2×C411yescharacteristic
70C4×Q8×C61D411yescharacteristic
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61C2×C411yescharacteristic
74C4×D4×C61D411yescharacteristic
75#117×C61—21no—
76#117×C61D411yescharacteristic
77C2×C42×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61—21no—
81#97×C61—21no—
82C4×C8×C61—21no—
83#41×C61—22no—
84#41×C61—22no—
85C2×Q8×C61C22⋊C411yescharacteristic
86C2×D4×C61—21no—
87C2×D4×C61C22⋊C411yescharacteristic
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—41no—
95#11758222—81no—
96#11758222—81no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×C4×D61—41no—
101C2×Dic122—21no—
102M16×C61—22no—
103M16×C61—41no—
104M16×C61—81no—
105C2×C8×C61—41no—
106C2×C8×C61—41no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61C22⋊C411yescharacteristic
110C4⋊C4×C61C22⋊C411yescharacteristic
111C22⋊C4×C61—21no—
112C22⋊C4×C61—21no—
113C22⋊C4×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61C22×C411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121#11758201—42no—
122#11758199—42no—
123#11758198—41no—
124#11758198—41no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127#11758194—28no—
128#11758194—22no—
129#11758194—22no—
130C2×#92418—28no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—21no—
134C2×#92418—22no—
135C2×#92418—21no—
136C2×#92418—21no—
137C2×#92418—21no—
138C23×C61—21no—
139C23×C61#12311yescharacteristic
140C22×D61—41no—
141Q8×C61—21no—
142Q8×C61#10412yes—
143D4×C61—24no—
144D4×C61—21no—
145D4×C61#10412yes—
146C2×C4×C61—24no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61C2×C22⋊C411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C22⋊D61—82no—
167C2×Dic61—41no—
168D244—41no—
169D244—41no—
170D244—41no—
171D244—41no—
172D244—81no—
173C4×D61—81no—
174C4×D61—81no—
175Dic122—81no—
176C8×C61—42no—
177C8×C61—81no—
178#92418—22no—
179#92418—24no—
180#92418—22no—
181#92418—41no—
182#92418—41no—
183#9211—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—22no—
187C22×C61—21no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61#38611yescharacteristic
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—22no—
197C4×C61—21no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61C2×#10411yescharacteristic
201C4×C61C2×#10411yescharacteristic
202C4×C61#44311yescharacteristic
203C4×C61#44411yescharacteristic
204C4×C61#47111yescharacteristic
205C4×C61#47211yescharacteristic
206Dic61—81no—
207#3106—611no—
208C4×#131—611no—
209#1525—611no—
210#1492—611no—
211#1489D6111yescharacteristic, complemented
212#1299—611no—
213#1278—611no—
214C2×C61—41no—
215C2×C61—42no—
216C2×C61#187611yescharacteristic
217C2×C61#189611yescharacteristic
218C2×C61#190711yescharacteristic
219D61—161no—
220C2×#131—1221no—
221#532—611no—
222C4×#41C2×D6111yescharacteristic
223#482—1221no—
224#477—611no—
225#476—611no—
226#460—611no—
227#459—611no—
228#449—1221no—
229C4×SD16—1221no—
230C4×D8—1221no—
231#431—611no—
232#423C2×D6111yescharacteristic
233#405—1222no—
234#405—1221no—
235C4×M16—611no—
236C4×M16—1221no—
237C4×M16C2×D6111yescharacteristic
238C2×C4×C8—611no—
239C2×C4×C8—1221no—
240#329—1222no—
241Q8⋊C8—21no—
242D4⋊C8—21no—
243D4⋊C8—1221no—
244D4⋊C8—1221no—
245C61#921111yescharacteristic, complemented, Sylow
246C2×#41—1221no—
247C2×#41D24411yescharacteristic
248C2×SD16—1221no—
249C2×D8—1221no—
250#131—1224no—
251#131—1224no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—1222no—
255C2×M16—1221no—
256C22×C8—1222no—
257C22×C8—1221no—
258C4⋊Q8—611no—
259#127—611no—
260#124—1221no—
261#121—1221no—
262#121—1221no—
263C4×Q8C4×D6111yescharacteristic
264C4×Q8C22⋊D6111yescharacteristic
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—1221no—
268C4×D4—1221no—
269C4×D4C4×D6111yescharacteristic
270C4×D4C22⋊D6111yescharacteristic
271#117—21no—
272#117D24411yescharacteristic
273C2×C42—21no—
274C2×C42C22×D6111yescharacteristic
275#107—1221no—
276#106—1221no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—611no—
280C4⋊C8—611no—
281#103—1221no—
282#102—1221no—
283#102—1221no—
284#102—1221no—
285C22⋊C8—1221no—
286#97—21no—
287#97—612no—
288#97—1222no—
289C4×C8—21no—
290C4×C8—611no—
291C4×C8—611no—
292C4×C8—1222no—
293C4×C8—1221no—
294C4×C8—1221no—
295C4×C8—1221no—
296#41—22no—
297#41—22no—
298#41—4881no—
299#41—4881no—
300C2×Q8—1221no—
301C2×Q8#1175819811yescharacteristic
302C2×D4—21no—
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—1221no—
306C2×D4—2441no—
307C2×D4#1175819811yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—2441no—
312C22×C4C2×D24411yescharacteristic
313C22×C4#1175822411yescharacteristic
314C22×C4#1175822411yescharacteristic
315SD16—2442no—
316D8—2442no—
317M16—22no—
318M16—41no—
319M16—81no—
320M16—1228no—
321M16—1222no—
322M16—1222no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—1228no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—1222no—
331C2×C8—1221no—
332C2×C8—1221no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4#1175819811yescharacteristic
338C4⋊C4#1175819811yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42C2×C4×D6111yescharacteristic
348C42D4×D6111yescharacteristic
349C42D4×D6111yescharacteristic
350C42C2×C22⋊D6111yescharacteristic
351C23—21no—
352C23—2441no—
353C23#1451976111yescharacteristic
354Q8—21no—
355Q8—4881no—
356Q8#1451970312yes—
357D4—24no—
358D4—21no—
359D4—4881no—
360D4—4882no—
361D4—2441no—
362D4—2441no—
363D4—2441no—
364D4—2441no—
365D4#1451970312yes—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—4881no—
381C2×C4—4881no—
382C2×C4—2441no—
383C2×C4C2×#1175819811yescharacteristic
384C2×C4#1451973711yescharacteristic
385C2×C4#1451973711yescharacteristic
386C2×C4#1451974711yescharacteristic
387C2×C4#1451974911yescharacteristic
388C2×C4#1451974911yescharacteristic
389C8—42no—
390C8—81no—
391C8—1222no—
392C8—1222no—
393C8—1224no—
394C8—2441no—
395C8—2441no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—4881no—
403C22—4881no—
404C22—4881no—
405C22#1683963911yescharacteristic
406C4—21no—
407C4—21no—
408C4—22no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—4881no—
413C4C2×#1451970311yescharacteristic
414C4C2×#1451970311yescharacteristic
415C4#1683987611yescharacteristic
416C4#1683988111yescharacteristic
417C4#1684002711yescharacteristic
418C4#1684003011yescharacteristic
419C2—41no—
420C2—42no—
421C2—9761no—
422C2#2144614411yescharacteristic
423C2#2144626011yescharacteristic
424C2#2144632511yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1489C2
socleC22×C61#386
prime radical#21455879C2
cosocleC2×C42×C61C23
hypercenter#1489D61
hypocenterC61#9211
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1489

Complements

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

NH ≅ G/Ntype
2. C4×#14519784C2semidirect
3. #21444410C2semidirect
4. #21444269C2semidirect
5. #21443708C2semidirect
6. #21443657C2semidirect
7. C61×#1489C2semidirect
211. #1489D61semidirect
245. C61#9211semidirect

Maximal subgroups

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

subgroupindex
#214558792
C4×#145197842
#214444102
#214442692
#214437082
#214436572
C61×#14892
#921161