Finite Group Databasebeta

← #26106831

Subgroups — #26106831

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689194C211yesmaximal, characteristic
2C4×#13093938C211yesmaximal, characteristic
3#20243164C211yesmaximal, characteristic, complemented
4#20241704C211yesmaximal, characteristic
5#20241236C211yesmaximal, characteristic, complemented
6#20241234C211yesmaximal, characteristic, complemented
7C8×C22⋊C4×C47C211yesmaximal, characteristic, complemented
8C22×#10625324C411yescharacteristic, complemented
9C2×C4×C22⋊D47C2211yescharacteristic
10#15689194—21no—
11#15689194—21no—
12#15689194—21no—
13#15689194—21no—
14C2×Dic47⋊C8C411yescharacteristic, complemented
15C2×#13093938C412yes—
16C2×C22⋊(C47⋊C8)C2211yescharacteristic
17C2×C22⋊(C47⋊C8)C2211yescharacteristic
18C2×(C2×D47)⋊C8C411yescharacteristic, complemented
19C2×#13093934—21no—
20C2×#13093934C411yescharacteristic, complemented
21C4×C22⋊Dic47C2211yescharacteristic
22#15688889C412yes—
23#15688877—21no—
24#15688848C412yescomplemented
25#15688846C412yescomplemented
26C23×C8×C47C2211yescharacteristic
27C2×C22⋊C8×C47C2211yescharacteristic
28C2×C4×C8×C47—21no—
29C4×C22⋊C4×C47C2211yescharacteristic
30C47×#336—21no—
31C22×C22⋊D47C2×C411yescharacteristic
32C22×C4×D47C2×C411yescharacteristic
33C2×#10625324—21no—
34C2×#10625324—21no—
35C2×#10625324—42no—
36C22×C47⋊C8—21no—
37C22×C47⋊C8C2×C411yescharacteristic
38C22×C47⋊C8C2×C411yescharacteristic
39C4×C22⋊D47—21no—
40C4×C22⋊D47—21no—
41C4×C22⋊D47—21no—
42C4×C22⋊D47—21no—
43C2×Dic47⋊C4C2×C411yescharacteristic
44C2×C22⋊Dic47C2×C412yes—
45C2×C22⋊Dic47C2×C411yescharacteristic
46C2×(C2×D47)⋊C4C2×C411yescharacteristic
47C2×C4×Dic47—21no—
48C2×C4×Dic47C2×C411yescharacteristic
49Dic47⋊C8—21no—
50Dic47⋊C8—21no—
51#13093938—24no—
52C22⋊(C47⋊C8)—21no—
53C22⋊(C47⋊C8)—21no—
54C22⋊(C47⋊C8)—21no—
55C22⋊(C47⋊C8)C2×C412yes—
56(C2×D47)⋊C8—21no—
57(C2×D47)⋊C8—21no—
58#13093934—21no—
59#13093934—21no—
60#13093934—42no—
61C47⋊#95—21no—
62C23×C4×C47C2311yescharacteristic
63C22×C8×C47—21no—
64C22×C8×C47—42no—
65C22×C8×C47C2×C411yescharacteristic
66C22×C8×C47C2×C411yescharacteristic
67C22×C8×C47D411yescharacteristic
68C22×C8×C47D411yescharacteristic
69C22×C8×C47D411yescharacteristic
70C22×C8×C47D411yescharacteristic
71C2×C22⋊C4×C47C2×C412yes—
72C2×C42×C47—21no—
73C22⋊C8×C47—21no—
74C22⋊C8×C47C2×C412yes—
75C4×C8×C47—42no—
76#95×C47—21no—
77C24×C47C22×C411yescharacteristic
78C23×D47C4211yescharacteristic
79C22×C4×C47—22no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47#4111yescharacteristic
87C22×C4×C47#4111yescharacteristic
88C2×C22⋊D47—21no—
89C2×C22⋊D47—21no—
90C2×C22⋊D47—22no—
91C2×C22⋊D47—22no—
92C22×Dic47—22no—
93C22×Dic47C4211yescharacteristic
94C22×Dic47C4211yescharacteristic
95C22×Dic47C4211yescharacteristic
96C2×C4×D47—21no—
97C2×C4×D47—21no—
98C2×C4×D47—41no—
99C2×C4×D47—41no—
100C2×C8×C47—22no—
101C2×C8×C47—22no—
102C2×C8×C47—22no—
103C2×C8×C47—22no—
104C2×C8×C47—41no—
105C2×C8×C47—41no—
106C2×C8×C47—41no—
107C2×C8×C47—41no—
108C2×C8×C47—41no—
109C2×C8×C47C22⋊C411yescharacteristic
110C2×C8×C47C22⋊C411yescharacteristic
111C2×C8×C47C22⋊C411yescharacteristic
112C2×C8×C47C22⋊C411yescharacteristic
113C22⋊C4×C47—22no—
114C22⋊C4×C47M1614yescomplemented
115C42×C47—41no—
116C42×C47—41no—
117C22⋊Dic47—21no—
118C22⋊Dic47—21no—
119C22⋊Dic47—22no—
120C22⋊Dic47—22no—
121(C2×D47)⋊C4—21no—
122(C2×D47)⋊C4—21no—
123Dic47⋊C4—21no—
124Dic47⋊C4—21no—
125C4×Dic47—21no—
126C4×Dic47—21no—
127C4×Dic47—41no—
128C4×Dic47—41no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—21no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136#10625324—42no—
137#10625324—42no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47C2×C4211yescharacteristic
141C23×C47C4×D411yescharacteristic
142C23×C47C4×D411yescharacteristic
143C23×C47C2×M1612yes—
144C23×C47#13112yes—
145C22×D47—21no—
146C22×D47—21no—
147C22×D47—21no—
148C22×D47—41no—
149C22×D47—41no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—24no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47C2×C22⋊C411yescharacteristic
163C2×C4×C47#11711yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47C4×D411yescharacteristic
166C22⋊D47—42no—
167C22⋊D47—42no—
168C2×Dic47—21no—
169C2×Dic47—21no—
170C2×Dic47—21no—
171C2×Dic47—21no—
172C2×Dic47—21no—
173C2×Dic47—41no—
174C2×Dic47—41no—
175C2×Dic47—42no—
176C2×Dic47—42no—
177C2×Dic47—42no—
178C4×D47—42no—
179C4×D47—41no—
180C4×D47—41no—
181C8×C47—21no—
182C8×C47—21no—
183C8×C47—21no—
184C8×C47—42no—
185C8×C47#9912yescomplemented
186C47⋊C8—42no—
187C47⋊C8—41no—
188C47⋊C8—41no—
189#8161—471nomaximal, Sylow
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—42no—
197C22×C47—42no—
198C22×C47C4×C22⋊C411yescharacteristic
199C22×C47C4×M1611yescharacteristic
200C22×C47#40511yescharacteristic
201C22×C47#43512yes—
202C22×C47#43612yes—
203C2×D47—41no—
204C2×D47—41no—
205C2×D47—41no—
206C2×D47—81no—
207C2×D47—81no—
208C4×C47—21no—
209C4×C47—21no—
210C4×C47—21no—
211C4×C47—44no—
212C4×C47C2×#9911yescharacteristic
213C4×C47#41011yescharacteristic
214Dic47—41no—
215Dic47—41no—
216Dic47—41no—
217Dic47—82no—
218C2×#435—471no—
219C4×#99—471no—
220#1752—471no—
221C8×C22⋊C4D4711yescharacteristic, complemented
222#1459—471no—
223#1282—471no—
224#1282—471no—
225C2×C47—22no—
226C2×C47—22no—
227C2×C47—21no—
228C2×C47—21no—
229C2×C47#175311yescharacteristic
230C2×C47C4×#9911yescharacteristic
231C2×C47#175711yescharacteristic
232D47—81no—
233C22×M16—471no—
234C23×C8C2×D4711yescharacteristic
235C2×C4×D4—471no—
236#435—941no—
237#435—941no—
238#435—941no—
239#435—941no—
240C2×C4⋊C8—471no—
241C2×#99—472no—
242C2×C22⋊C8—471no—
243C2×C22⋊C8—471no—
244C2×C22⋊C8—471no—
245C2×C22⋊C8C2×D4711yescharacteristic
246C2×#97—471no—
247C2×#97—941no—
248C2×C4×C8—21no—
249C4×C22⋊C4—471no—
250C4×C22⋊C4C2×D4711yescharacteristic
251#342—472no—
252#336—21no—
253#336—941no—
254#324—472no—
255#323—472no—
256C47#816111yescharacteristic, complemented, Sylow
257C22×D4—471no—
258C23×C4—471no—
259C23×C4C22×D4711yescharacteristic
260C2×M16—941no—
261C2×M16—941no—
262C2×M16—1882no—
263C22×C8—21no—
264C22×C8—42no—
265C22×C8—471no—
266C22×C8—471no—
267C22×C8—941no—
268C22×C8C4×D4711yescharacteristic
269C22×C8C4×D4711yescharacteristic
270C22×C8D18811yescharacteristic
271C22×C8D18811yescharacteristic
272C22×C8C22⋊D4711yescharacteristic
273C22×C8C22⋊D4711yescharacteristic
274C4×D4—941no—
275C4×D4—941no—
276C4×D4—941no—
277C4×D4—941no—
278C2×C4⋊C4—471no—
279C2×C22⋊C4—471no—
280C2×C22⋊C4—471no—
281C2×C22⋊C4—472no—
282C2×C22⋊C4C4×D4712yes—
283C2×C42—21no—
284C2×C42—471no—
285C2×C42—941no—
286C4⋊C8—941no—
287C4⋊C8—941no—
288#99—944no—
289C22⋊C8—21no—
290C22⋊C8—472no—
291C22⋊C8—941no—
292C22⋊C8—941no—
293C22⋊C8—941no—
294C22⋊C8—941no—
295C22⋊C8—941no—
296C22⋊C8C4×D4712yes—
297#97—941no—
298#97—941no—
299#97—1882no—
300C4×C8—42no—
301#95—21no—
302#95—941no—
303C24—471no—
304C24C2×C4×D4711yescharacteristic
305C2×D4—941no—
306C2×D4—941no—
307C2×D4—942no—
308C2×D4—942no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—471no—
313C22×C4—471no—
314C22×C4—471no—
315C22×C4—941no—
316C22×C4—941no—
317C22×C4—942no—
318C22×C4—1881no—
319C22×C4—1881no—
320C22×C4C2×C4×D4711yescharacteristic
321C22×C4C2×C4×D4711yescharacteristic
322C22×C4C2×D18811yescharacteristic
323C22×C4#1062535011yescharacteristic
324C22×C4#1062535011yescharacteristic
325C22×C4C2×C22⋊D4711yescharacteristic
326M16—1882no—
327M16—1882no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—941no—
338C2×C8—941no—
339C2×C8—942no—
340C2×C8—1881no—
341C2×C8—1881no—
342C2×C8—1881no—
343C2×C8—1881no—
344C2×C8(C2×D47)⋊C411yescharacteristic
345C2×C8(C2×D47)⋊C411yescharacteristic
346C2×C8(C2×D47)⋊C411yescharacteristic
347C2×C8(C2×D47)⋊C411yescharacteristic
348C4⋊C4—941no—
349C4⋊C4—941no—
350C22⋊C4—22no—
351C22⋊C4—941no—
352C22⋊C4—941no—
353C22⋊C4—941no—
354C22⋊C4—941no—
355C22⋊C4—942no—
356C22⋊C4—942no—
357C22⋊C4#1062532414yescomplemented
358C42—41no—
359C42—41no—
360C42—941no—
361C42—941no—
362C42—1881no—
363C42—1881no—
364C23—41no—
365C23—41no—
366C23—941no—
367C23—941no—
368C23—941no—
369C23—1881no—
370C23—1881no—
371C23C42×D4711yescharacteristic
372C23C4×C22⋊D4711yescharacteristic
373C23C4×D18811yescharacteristic
374C23C2×#1062532412yes—
375C23#1309403012yes—
376D4—1882no—
377D4—1882no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—941no—
391C2×C4—941no—
392C2×C4—941no—
393C2×C4—941no—
394C2×C4—941no—
395C2×C4—1882no—
396C2×C4—1882no—
397C2×C4—1882no—
398C2×C4—1882no—
399C2×C4—1881no—
400C2×C4—1881no—
401C2×C4—1881no—
402C2×C4—1881no—
403C2×C4C2×(C2×D47)⋊C411yescharacteristic
404C2×C4#1309398111yescharacteristic
405C2×C4C4×C22⋊D4711yescharacteristic
406C2×C4C4×D18811yescharacteristic
407C8—21no—
408C8—21no—
409C8—21no—
410C8—42no—
411C8—1882no—
412C8—1881no—
413C8—1881no—
414C8#1309393812yescomplemented
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—3761no—
424C22—3761no—
425C22—1881no—
426C22—1881no—
427C22—1881no—
428C22C4×(C2×D47)⋊C411yescharacteristic
429C22C4×#1062532411yescharacteristic
430C22#1568907711yescharacteristic
431C22#1568919412yes—
432C22#1568920212yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—3762no—
438C4—1881no—
439C4—1881no—
440C4—1881no—
441C4C2×#1309393811yescharacteristic
442C4#1568909611yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—3761no—
448C2#2024317111yescharacteristic
449C2C4×#1309393811yescharacteristic
450C2#2024318911yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D47)⋊C4
commutatorC23×C47C2×C42
FrattiniC23×C4C22×D47
FittingC8×C22⋊C4×C47C2
socleC22×C47C4×C22⋊C4
prime radicalC22×C22⋊D47C2×C4
cosocleC23×C4×C47C23
hypercenterC8×C22⋊C4D47
hypocenterC47#8161
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20243164C2semidirect
5. #20241236C2semidirect
6. #20241234C2semidirect
7. C8×C22⋊C4×C47C2semidirect
8. C22×#10625324C4semidirect
14. C2×Dic47⋊C8C4semidirect
18. C2×(C2×D47)⋊C8C4semidirect
20. C2×#13093934C4semidirect
24. #15688848C4semidirect
25. #15688846C4semidirect
114. C22⋊C4×C47M16semidirect
185. C8×C47#99semidirect
221. C8×C22⋊C4D47semidirect
256. C47#8161semidirect
357. C22⋊C4#10625324semidirect
414. C8#13093938semidirect

Maximal subgroups

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

subgroupindex
C2×#156891942
C4×#130939382
#202431642
#202417042
#202412362
#202412342
C8×C22⋊C4×C472
#816147