Finite Group Databasebeta

← #26106830

Subgroups — #26106830

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×#15689191C211yesmaximal, characteristic
2C4×#13093938C211yesmaximal, characteristic
3#20243166C211yesmaximal, characteristic, complemented
4#20243162C211yesmaximal, characteristic, complemented
5#20241704C211yesmaximal, characteristic
6#20241235C211yesmaximal, characteristic, complemented
7C47×#1282C211yesmaximal, characteristic, complemented
8C22×#10625324C411yescharacteristic, complemented
9C23×C47⋊C8C2211yescharacteristic
10C2×C4×C22⋊D47C2211yescharacteristic
11#15689191—21no—
12#15689191—21no—
13#15689191—21no—
14#15689191—21no—
15C2×Dic47⋊C8C411yescharacteristic, complemented
16C2×#13093938C412yes—
17C2×C22⋊(C47⋊C8)C2211yescharacteristic
18C2×(C2×D47)⋊C8C411yescharacteristic, complemented
19C2×#13093935—21no—
20C2×#13093934C411yescharacteristic, complemented
21C2×C8×Dic47—21no—
22C4×C22⋊Dic47C2211yescharacteristic
23#15688889C412yes—
24#15688878—21no—
25#15688877—21no—
26#15688848C412yescomplemented
27#15688846C412yescomplemented
28C2×C22⋊C8×C47C2211yescharacteristic
29C2×C22⋊C8×C47C2211yescharacteristic
30C4×C22⋊C4×C47C2211yescharacteristic
31C22×C22⋊D47C2×C411yescharacteristic
32C22×C4×D47C2×C411yescharacteristic
33C2×#10625324—21no—
34C2×#10625324—21no—
35C2×#10625324—42no—
36C22×C47⋊C8—21no—
37C22×C47⋊C8—42no—
38C22×C47⋊C8C2×C411yescharacteristic
39C22×C47⋊C8C2×C411yescharacteristic
40C22×C47⋊C8D411yescharacteristic
41C22×C47⋊C8D411yescharacteristic
42C22×C47⋊C8D411yescharacteristic
43C22×C47⋊C8D411yescharacteristic
44C4×C22⋊D47—21no—
45C4×C22⋊D47—21no—
46C4×C22⋊D47—21no—
47C4×C22⋊D47—21no—
48C2×Dic47⋊C4C2×C411yescharacteristic
49C2×C22⋊Dic47C2×C411yescharacteristic
50C2×C22⋊Dic47C2×C412yes—
51C2×(C2×D47)⋊C4C2×C411yescharacteristic
52C2×C4×Dic47—21no—
53C2×C4×Dic47C2×C411yescharacteristic
54Dic47⋊C8—21no—
55Dic47⋊C8—21no—
56#13093938—24no—
57C22⋊(C47⋊C8)—21no—
58C22⋊(C47⋊C8)C2×C412yes—
59(C2×D47)⋊C8—21no—
60(C2×D47)⋊C8—21no—
61#13093935—42no—
62#13093934—21no—
63#13093934—21no—
64C8×Dic47—42no—
65C47⋊#95—21no—
66C23×C4×C47C2311yescharacteristic
67C22×C8×C47—21no—
68C22×C8×C47C2×C411yescharacteristic
69C22×C8×C47C2×C411yescharacteristic
70C2×C22⋊C4×C47C2×C412yes—
71C2×C42×C47—21no—
72C22⋊C8×C47—21no—
73C22⋊C8×C47—21no—
74C22⋊C8×C47—21no—
75C22⋊C8×C47C2×C412yes—
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—21no—
101C2×C8×C47—21no—
102C2×C8×C47—22no—
103C2×C8×C47—41no—
104C2×C8×C47—41no—
105C2×C8×C47—41no—
106C2×C8×C47—41no—
107C22⋊C4×C47—22no—
108C22⋊C4×C47—22no—
109C42×C47—41no—
110C42×C47—41no—
111C22⋊Dic47—21no—
112C22⋊Dic47—21no—
113C22⋊Dic47—22no—
114C22⋊Dic47M1614yescomplemented
115(C2×D47)⋊C4—21no—
116(C2×D47)⋊C4—21no—
117Dic47⋊C4—21no—
118Dic47⋊C4—21no—
119C4×Dic47—21no—
120C4×Dic47—21no—
121C4×Dic47—41no—
122C4×Dic47—41no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C2×C47⋊C8—41no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8C22⋊C411yescharacteristic
133C2×C47⋊C8C22⋊C411yescharacteristic
134C2×C47⋊C8C22⋊C411yescharacteristic
135C2×C47⋊C8C22⋊C411yescharacteristic
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—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—42no—
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—24no—
174C2×Dic47—42no—
175C2×Dic47—42no—
176C2×Dic47—41no—
177C2×Dic47—41no—
178C4×D47—42no—
179C4×D47—41no—
180C4×D47—41no—
181C8×C47—42no—
182C8×C47—41no—
183C8×C47—41no—
184C47⋊C8—21no—
185C47⋊C8—21no—
186C47⋊C8—21no—
187C47⋊C8—42no—
188C47⋊C8#9912yescomplemented
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—82no—
212C4×C47C2×#9911yescharacteristic
213C4×C47#41011yescharacteristic
214Dic47—44no—
215Dic47—41no—
216Dic47—41no—
217Dic47—41no—
218C2×#435—471no—
219C4×#99—471no—
220#1752—471no—
221C8×C22⋊C4—471no—
222#1459—471no—
223#1282—471no—
224#1282D4711yescharacteristic, complemented
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×C8—471no—
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⋊C8C2×D4711yescharacteristic
245C2×C22⋊C8C2×D4711yescharacteristic
246C2×#97—471no—
247C2×#97—941no—
248C2×C4×C8—941no—
249C4×C22⋊C4—471no—
250C4×C22⋊C4C2×D4711yescharacteristic
251#342—472no—
252#336—941no—
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—471no—
265C22×C8—471no—
266C22×C8—471no—
267C22×C8—471no—
268C22×C8—471no—
269C22×C8—471no—
270C22×C8—941no—
271C22×C8—1882no—
272C22×C8C4×D4711yescharacteristic
273C22×C8C4×D4711yescharacteristic
274C4×D4—941no—
275C4×D4—941no—
276C4×D4—941no—
277C4×D4—941no—
278C2×C4⋊C4—471no—
279C2×C22⋊C4—472no—
280C2×C22⋊C4—471no—
281C2×C22⋊C4—471no—
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—21no—
291C22⋊C8—21no—
292C22⋊C8—472no—
293C22⋊C8—941no—
294C22⋊C8—941no—
295C22⋊C8—941no—
296C22⋊C8C4×D4712yes—
297#97—941no—
298#97—941no—
299#97—1882no—
300C4×C8—1882no—
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—942no—
316C22×C4—941no—
317C22×C4—941no—
318C22×C4—1881no—
319C22×C4—1881no—
320C22×C4C2×C4×D4711yescharacteristic
321C22×C4C2×C4×D4711yescharacteristic
322C22×C4D4×D4711yescharacteristic
323C22×C4D4×D4711yescharacteristic
324C22×C4#1062535211yescharacteristic
325C22×C4#1062535411yescharacteristic
326M16—1882no—
327M16—1882no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—471no—
336C2×C8—471no—
337C2×C8—471no—
338C2×C8—471no—
339C2×C8—942no—
340C2×C8—942no—
341C2×C8—942no—
342C2×C8—942no—
343C2×C8—1881no—
344C2×C8—1881no—
345C2×C8—1881no—
346C2×C8—1881no—
347C2×C8—1881no—
348C4⋊C4—941no—
349C4⋊C4—941no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—474no—
353C22⋊C4—941no—
354C22⋊C4—941no—
355C22⋊C4—942no—
356C22⋊C4—941no—
357C22⋊C4—941no—
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
372C23(C22⋊D47)⋊C411yescharacteristic
373C23#1309398611yescharacteristic
374C23M16×D4712yes—
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—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—941no—
391C2×C4—941no—
392C2×C4—941no—
393C2×C4—941no—
394C2×C4—941no—
395C2×C4—944no—
396C2×C4—1882no—
397C2×C4—1882no—
398C2×C4—1882no—
399C2×C4—1881no—
400C2×C4—1881no—
401C2×C4—1881no—
402C2×C4—1881no—
403C2×C4C22⋊C4×D4711yescharacteristic
404C2×C4#1309398211yescharacteristic
405C2×C4(C22⋊D47)⋊C411yescharacteristic
406C2×C4#1309398611yescharacteristic
407C8—42no—
408C8—41no—
409C8—41no—
410C8—472no—
411C8—941no—
412C8—941no—
413C8—941no—
414C8—1882no—
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—
428C22#1568897011yescharacteristic
429C22#1568907511yescharacteristic
430C22#1568907711yescharacteristic
431C22#1568919312yes—
432C22#1568919812yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—82no—
437C4—1884no—
438C4—1881no—
439C4—1881no—
440C4—1881no—
441C4#99×D4711yescharacteristic
442C4#1568909511yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—3761no—
448C2#2024317011yescharacteristic
449C2#2024317511yescharacteristic
450C2#2024318711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC23×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1282C2
socleC22×C47C4×C22⋊C4
prime radicalC22×C22⋊D47C2×C4
cosocleC23×C4×C47C23
hypercenter#1282D47
hypocenterC47#8161
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
3. #20243166C2semidirect
4. #20243162C2semidirect
6. #20241235C2semidirect
7. C47×#1282C2semidirect
8. C22×#10625324C4semidirect
15. C2×Dic47⋊C8C4semidirect
18. C2×(C2×D47)⋊C8C4semidirect
20. C2×#13093934C4semidirect
26. #15688848C4semidirect
27. #15688846C4semidirect
114. C22⋊Dic47M16semidirect
188. C47⋊C8#99semidirect
224. #1282D47semidirect
256. C47#8161semidirect

Maximal subgroups

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

subgroupindex
C2×#156891912
C4×#130939382
#202431662
#202431622
#202417042
#202412352
C47×#12822
#816147