Finite Group Databasebeta

← #26107165

Subgroups — #26107165

Order 12032.

712 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093998C211yesmaximal, characteristic
2C2×#15688858C212yesmaximal
3#20241615C211yesmaximal, characteristic, complemented
4#20241268C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic, complemented
6C2×#13093998C2211yescharacteristic
7C2×C4×D188C411yescharacteristic, complemented
8C2×C4×C22⋊D47—21no—
9#15689025C411yescharacteristic, complemented
10#15689021C411yescharacteristic, complemented
11#15689014—21no—
12#15689002—21no—
13#15688983C2211yescharacteristic
14C4×C4⋊Dic47C411yescharacteristic, complemented
15C4×(C2×D47)⋊C4—21no—
16C2×C47⋊#95—21no—
17C2×C47⋊#95—21no—
18C2×C47⋊#95C2212yes—
19#15688858C418yescomplemented
20C22×C42×C47C2211yescharacteristic
21C2×C22⋊C8×C47C2212yes—
22C2×#97×C47—21no—
23C47×#336—21no—
24C22×D188C2×C411yescharacteristic
25C22×C22⋊D47—21no—
26C23×Dic47—22no—
27C22×C4×D47—21no—
28#13093998—22no—
29#13093998C2×C414yes—
30C4×D188—42no—
31C4×C22⋊D47—44no—
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×C4⋊Dic47C2×C411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×C22⋊Dic47—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C2×(C2×D47)⋊C4—21no—
40C2×(C2×D47)⋊C4—21no—
41C2×C4×Dic47—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—22no—
46C47⋊#95—42no—
47C47⋊#95—42no—
48C47⋊#95C2×C414yescomplemented
49C23×C4×C47D411yescharacteristic
50C23×C4×C47D411yescharacteristic
51C23×C4×C47C2311yescharacteristic
52C22×C8×C47—22no—
53C2×C42×C47—42no—
54C2×C42×C47C2×C411yescharacteristic
55C2×C42×C47C2×C411yescharacteristic
56C2×C42×C47D411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C22⋊C8×C47—22no—
59C22⋊C8×C47C2×C414yescomplemented
60#97×C47—42no—
61C24×C47C2×D411yescharacteristic
62C23×D47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—21no—
66C22×C4×C47—22no—
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C22×C412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4112yes—
80C2×C22⋊D47—24no—
81C2×C22⋊D47—42no—
82C2×C22⋊D47—41no—
83C2×C22⋊D47—41no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—24no—
88C22×Dic47—42no—
89C22×Dic47—42no—
90C22×Dic47—42no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C2×D188—21no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188C4212yes—
98C2×C4×D47—42no—
99C2×C4×D47—42no—
100C2×C4×D47—42no—
101C2×C8×C47—24no—
102C2×C8×C47—42no—
103C2×C8×C47—42no—
104C42×C47—22no—
105C42×C47—22no—
106C42×C47—41no—
107C42×C47—41no—
108C22⋊Dic47—42no—
109(C2×D47)⋊C4—24no—
110(C2×D47)⋊C4—42no—
111(C2×D47)⋊C4—42no—
112(C2×D47)⋊C4—42no—
113C4⋊Dic47—21no—
114C4⋊Dic47—41no—
115C4⋊Dic47—41no—
116C4⋊Dic47C4212yes—
117Dic47⋊C4—42no—
118C4×Dic47—42no—
119C23×C47—21no—
120C23×C47—21no—
121C23×C47—21no—
122C23×C47—21no—
123C23×C47C2×C22⋊C411yescharacteristic
124C23×C47C4×D412yes—
125C23×C47AGL(1,ℤ/8)11yescharacteristic
126C23×C47AGL(1,ℤ/8)12yes—
127C23×C47#13711yescharacteristic
128C22×D47—22no—
129C22×D47—41no—
130C22×D47—41no—
131C22×D47—41no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—41no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—44no—
153C2×C4×C47—42no—
154C2×C4×C47—42no—
155C2×C4×C47#9912yes—
156C2×C4×C47C2×C4211yescharacteristic
157C2×C4×C47#11711yescharacteristic
158C2×C4×C47C4×D412yes—
159C22⋊D47—44no—
160C22⋊D47—44no—
161C2×Dic47—22no—
162C2×Dic47—41no—
163C2×Dic47—41no—
164C2×Dic47—44no—
165C2×Dic47—44no—
166C2×Dic47—41no—
167C2×Dic47—41no—
168C2×Dic47—41no—
169C2×Dic47—41no—
170C2×Dic47—41no—
171C2×Dic47—41no—
172C2×Dic47—44no—
173C2×Dic47—82no—
174C2×Dic47—82no—
175C2×Dic47—82no—
176C2×Dic47—82no—
177D188—42no—
178D188—42no—
179C4×D47—84no—
180C8×C47—44no—
181#8216—471nomaximal, Sylow
182C22×C47—22no—
183C22×C47—22no—
184C22×C47—22no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—22no—
190C22×C47C4×C22⋊C411yescharacteristic
191C22×C47C2×#9911yescharacteristic
192C22×C47#41011yescharacteristic
193C22×C47#41711yescharacteristic
194C22×C47#41811yescharacteristic
195C22×C47#42111yescharacteristic
196C22×C47#42111yescharacteristic
197C22×C47#44012yes—
198C22×C47#44212yes—
199C2×D47—42no—
200C2×D47—42no—
201C2×D47—42no—
202C2×D47—81no—
203C2×D47—81no—
204C2×D47—81no—
205C2×D47—81no—
206C4×C47—22no—
207C4×C47—21no—
208C4×C47—21no—
209C4×C47—22no—
210C4×C47—41no—
211C4×C47—41no—
212C4×C47—41no—
213Dic47—42no—
214Dic47—84no—
215Dic47—81no—
216Dic47—81no—
217C4×#121—471no—
218#1749D4711yescharacteristic, complemented
219C2×#327—472no—
220#1434—471no—
221#1294—472no—
222C2×C47—22no—
223C2×C47—22no—
224C2×C47#151011yescharacteristic
225C2×C47#151211yescharacteristic
226C2×C47#151311yescharacteristic
227C2×C47#151411yescharacteristic
228C2×C47C4×#9911yescharacteristic
229C2×C47#175911yescharacteristic
230C2×C47#176211yescharacteristic
231D47—82no—
232C2×#121—471no—
233C2×C4×D4—471no—
234C2×C4×D4—941no—
235C22×C42C2×D4711yescharacteristic
236C2×C22⋊C8C2×D4712yes—
237C2×#97—21no—
238#390—471no—
239#389—471no—
240#388—941no—
241#386—941no—
242#380—471no—
243C4×C4⋊C4—471no—
244C4×C22⋊C4—941no—
245C2×#95—472no—
246C2×#95—941no—
247C2×#95—941no—
248#336—21no—
249#327—478no—
250C47#821611yescharacteristic, complemented, Sylow
251C22×D4—471no—
252C22×D4—941no—
253C23×C4—942no—
254C23×C4—941no—
255C23×C4D18811yescharacteristic
256C23×C4D18811yescharacteristic
257C23×C4C22×D4711yescharacteristic
258C22×C8—22no—
259#121—474no—
260#121—942no—
261C4×D4—1882no—
262C4×D4—1884no—
263C2×C4⋊C4—471no—
264C2×C4⋊C4—471no—
265C2×C4⋊C4—471no—
266C2×C4⋊C4—941no—
267C2×C22⋊C4—941no—
268C2×C22⋊C4—941no—
269C2×C22⋊C4—941no—
270C2×C22⋊C4—941no—
271C2×C22⋊C4—941no—
272C2×C42—42no—
273C2×C42—941no—
274C2×C42C4×D4711yescharacteristic
275C2×C42C4×D4711yescharacteristic
276C2×C42C22⋊D4711yescharacteristic
277C2×C42C22⋊D4711yescharacteristic
278C22⋊C8—22no—
279C22⋊C8C4×D4714yescomplemented
280#97—42no—
281#95—474no—
282#95—941no—
283#95—941no—
284#95—942no—
285#95—941no—
286#95—1882no—
287#95—1882no—
288C24—941no—
289C24C2×D18811yescharacteristic
290C2×D4—472no—
291C2×D4—944no—
292C2×D4—941no—
293C2×D4—1881no—
294C2×D4—1881no—
295C2×D4—1882no—
296C2×D4—1881no—
297C2×D4—1881no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—941no—
308C22×C4—941no—
309C22×C4—941no—
310C22×C4—944no—
311C22×C4—1882no—
312C22×C4—1882no—
313C22×C4—1882no—
314C22×C4—1882no—
315C22×C4—1882no—
316C22×C4—1882no—
317C22×C4—1882no—
318C22×C4—1882no—
319C22×C4—1882no—
320C22×C4(C2×D47)⋊C411yescharacteristic
321C22×C4(C2×D47)⋊C411yescharacteristic
322C22×C4(C2×D47)⋊C411yescharacteristic
323C22×C4(C2×D47)⋊C411yescharacteristic
324C22×C4C2×C4×D4711yescharacteristic
325C22×C4C2×C4×D4712yes—
326C22×C4#1062535012yes—
327C22×C4C2×C22⋊D4711yescharacteristic
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—472no—
332C4⋊C4—941no—
333C4⋊C4—1881no—
334C4⋊C4—1881no—
335C4⋊C4—1882no—
336C22⋊C4—944no—
337C22⋊C4—1882no—
338C22⋊C4—1882no—
339C22⋊C4—1882no—
340C22⋊C4—1882no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C42—1882no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—942no—
351C23—1881no—
352C23—1881no—
353C23—1881no—
354C23—1881no—
355C23—1881no—
356C23—1881no—
357C23C2×(C2×D47)⋊C411yescharacteristic
358C23C4×D18812yes—
359C23#1309405212yes—
360C23#1309405211yescharacteristic
361C23#1309405711yescharacteristic
362D4—1882no—
363D4—1882no—
364D4—1884no—
365D4—1884no—
366C2×C4—21no—
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—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—942no—
387C2×C4—3764no—
388C2×C4—3762no—
389C2×C4—3762no—
390C2×C4—3762no—
391C2×C4—3762no—
392C2×C4—1884no—
393C2×C4—1884no—
394C2×C4—1884no—
395C2×C4—1881no—
396C2×C4—1881no—
397C2×C4—1881no—
398C2×C4—1881no—
399C2×C4—1881no—
400C2×C4—1881no—
401C2×C4—1881no—
402C2×C4—1881no—
403C2×C4#1309393812yes—
404C2×C4C42×D4711yescharacteristic
405C2×C4#1309398111yescharacteristic
406C2×C4C4×C22⋊D4712yes—
407C8—44no—
408C22—22no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—3761no—
417C22—3761no—
418C22—3761no—
419C22—3761no—
420C22—1882no—
421C22—1882no—
422C22—1882no—
423C22C4×(C2×D47)⋊C411yescharacteristic
424C22C2×#1309393811yescharacteristic
425C22#1568909611yescharacteristic
426C22#1568912511yescharacteristic
427C22#1568912911yescharacteristic
428C22#1568913911yescharacteristic
429C22#1568913911yescharacteristic
430C22#1568922512yes—
431C22#1568923612yes—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—3764no—
440C4—3761no—
441C4—3761no—
442C4—1882no—
443C2—22no—
444C2—22no—
445C2—3762no—
446C2#2024191811yescharacteristic
447C2#2024192711yescharacteristic
448C2#2024193211yescharacteristic
449C2#2024193611yescharacteristic
450C2C4×#1309393811yescharacteristic
451C2#2024320111yescharacteristic
452C2#2024321411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1749D47
hypocenterC47#8216
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #20241615C2semidirect
4. #20241268C2semidirect
5. C47×#1749C2semidirect
7. C2×C4×D188C4semidirect
9. #15689025C4semidirect
10. #15689021C4semidirect
14. C4×C4⋊Dic47C4semidirect
19. #15688858C4semidirect
48. C47⋊#95C2×C4semidirect
59. C22⋊C8×C47C2×C4semidirect
218. #1749D47semidirect
250. C47#8216semidirect
279. C22⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C4×#130939982
C2×#156888582
#202416152
#202412682
C47×#17492
#821647