Finite Group Databasebeta

← #26107956

Subgroups — #26107956

Order 12032.

844 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243478C212yesmaximal, complemented
2#20243376C211yesmaximal, characteristic, complemented
3#20241956C212yesmaximal, complemented
4#20241706C211yesmaximal, characteristic, complemented
5C2×C47×#553C211yesmaximal, characteristic
6C2×#13093939—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688983C2212yes—
9#15688890—21no—
10#15688885—24no—
11#15688857—24no—
12#15688849—21no—
13#15688847—21no—
14C47×#553—24no—
15C47×#553—24no—
16C2×#126×C47—21no—
17C2×#125×C47C411yescharacteristic, complemented
18C2×#123×C47—21no—
19C2×#121×C47—21no—
20C2×#121×C47C411yescharacteristic, complemented
21C2×#121×C47C2211yescharacteristic
22C2×C4×D4×C47—21no—
23C2×C4×D4×C47C411yescharacteristic, complemented
24C2×C4×D4×C47C2211yescharacteristic
25C2×#117×C47C411yescharacteristic, complemented
26C22×C4⋊C4×C47C2211yescharacteristic
27C22×C47⋊C8—22no—
28C2×C4⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30#13093939—42no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C47⋊#95—22no—
34C22×D4×C47—21no—
35C22×D4×C47—21no—
36C22×D4×C47C2×C411yescharacteristic
37C22×D4×C47D411yescharacteristic
38C23×C4×C47—21no—
39C23×C4×C47—21no—
40C23×C4×C47C2×C411yescharacteristic
41C23×C4×C47C2311yescharacteristic
42#126×C47—28no—
43#125×C47—22no—
44#125×C47—22no—
45#123×C47—28no—
46#121×C47—28no—
47#121×C47—22no—
48#121×C47—22no—
49#121×C47—22no—
50#121×C47D414yes—
51C4×D4×C47—28no—
52C4×D4×C47—22no—
53C4×D4×C47—22no—
54C4×D4×C47—22no—
55C4×D4×C47—22no—
56#117×C47—22no—
57#117×C47—22no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47—22no—
62C2×C4⋊C4×C47C2×C411yescharacteristic
63C2×C4⋊C4×C47C2×C411yescharacteristic
64C2×C4⋊C4×C47C2×C411yescharacteristic
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C4⋊C4×C47D414yes—
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69C2×C22⋊C4×C47—21no—
70C2×C22⋊C4×C47—21no—
71C2×C22⋊C4×C47D411yescharacteristic
72C2×C22⋊C4×C47D411yescharacteristic
73C2×C42×C47—21no—
74C2×C42×C47C2×C411yescharacteristic
75C24×C47—21no—
76C24×C47C22⋊C411yescharacteristic
77C24×C47C2×D411yescharacteristic
78C2×D4×C47—22no—
79C2×D4×C47—24no—
80C2×D4×C47—22no—
81C2×D4×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47—24no—
84C2×D4×C47—21no—
85C2×D4×C47—42no—
86C2×D4×C47—42no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—82no—
90C2×D4×C47C22⋊C412yes—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—24no—
95C22×C4×C47—24no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—42no—
99C22×C4×C47—42no—
100C22×C4×C47—82no—
101C22×C4×C47—82no—
102C22×C4×C47C22⋊C411yescharacteristic
103C22×C4×C47C22⋊C411yescharacteristic
104C22×C4×C47C22⋊C411yescharacteristic
105C22×C4×C47C22×C411yescharacteristic
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2×D412yes—
108C22×C4×C47C2×D411yescharacteristic
109C22×Dic47—42no—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—24no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47—24no—
116C4⋊C4×C47—22no—
117C4⋊C4×C47C22⋊C412yes—
118C4⋊C4×C47C22⋊C412yes—
119C4⋊C4×C47C22⋊C412yes—
120C22⋊C4×C47—24no—
121C22⋊C4×C47—24no—
122C22⋊C4×C47—24no—
123C22⋊C4×C47—22no—
124C22⋊C4×C47—24no—
125C22⋊C4×C47—22no—
126C42×C47—24no—
127C42×C47—21no—
128C42×C47—21no—
129C22⋊Dic47—42no—
130C22⋊Dic47—42no—
131C4⋊Dic47—44no—
132C2×C47⋊C8—44no—
133C2×C47⋊C8—42no—
134C23×C47—21no—
135C23×C47—22no—
136C23×C47—21no—
137C23×C47—21no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47—42no—
141C23×C47—42no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47—41no—
145C23×C47—41no—
146C23×C47C2×C22⋊C411yescharacteristic
147C23×C47#12012yes—
148C23×C47#13512yes—
149C23×C47AGL(1,ℤ/8)11yescharacteristic
150C23×C47#13711yescharacteristic
151D4×C47—42no—
152D4×C47—42no—
153D4×C47—44no—
154D4×C47—42no—
155D4×C47—44no—
156D4×C47—42no—
157D4×C47—84no—
158C2×C4×C47—24no—
159C2×C4×C47—22no—
160C2×C4×C47—21no—
161C2×C4×C47—21no—
162C2×C4×C47—22no—
163C2×C4×C47—21no—
164C2×C4×C47—22no—
165C2×C4×C47—22no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—41no—
169C2×C4×C47—42no—
170C2×C4×C47—42no—
171C2×C4×C47—42no—
172C2×C4×C47—42no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—42no—
176C2×C4×C47—41no—
177C2×C4×C47—41no—
178C2×C4×C47—84no—
179C2×C4×C47—84no—
180C2×C4×C47C2×C22⋊C411yescharacteristic
181C2×C4×C47C2×C22⋊C411yescharacteristic
182C2×C4×C47#12012yes—
183C2×Dic47—82no—
184C2×Dic47—82no—
185C2×Dic47—82no—
186C47⋊C8—82no—
187#8378—471nomaximal, Sylow
188C22×C47—22no—
189C22×C47—21no—
190C22×C47—22no—
191C22×C47—21no—
192C22×C47—42no—
193C22×C47—41no—
194C22×C47—41no—
195C22×C47—42no—
196C22×C47—41no—
197C22×C47—41no—
198C22×C47—42no—
199C22×C47—41no—
200C22×C47—42no—
201C22×C47—81no—
202C22×C47—81no—
203C22×C47—82no—
204C22×C47—82no—
205C22×C47#37911yescharacteristic
206C22×C47#41611yescharacteristic
207C22×C47#41711yescharacteristic
208C22×C47#44912yes—
209C22×C47#45212yes—
210C4×C47—22no—
211C4×C47—21no—
212C4×C47—42no—
213C4×C47—42no—
214C4×C47—41no—
215C4×C47—41no—
216C4×C47—41no—
217C4×C47—42no—
218C4×C47#41012yes—
219Dic47—162no—
220C2×#553D4711yescharacteristic
221#1824—472no—
222#1799—471no—
223#1519—472no—
224#1459—471no—
225C2×C47—22no—
226C2×C47—21no—
227C2×C47—21no—
228C2×C47—22no—
229C2×C47—81no—
230C2×C47—82no—
231C2×C47#178211yescharacteristic
232C2×C47#178811yescharacteristic
233C2×C47#179511yescharacteristic
234#553—24no—
235#553—24no—
236C2×#126—21no—
237C2×#125Dic4711yescharacteristic, complemented
238C2×#123—21no—
239C2×#121—21no—
240C2×#121Dic4711yescharacteristic, complemented
241C2×#121C2×D4711yescharacteristic
242C2×C4×D4—21no—
243C2×C4×D4Dic4711yescharacteristic, complemented
244C2×C4×D4C2×D4711yescharacteristic
245C2×#117Dic4711yescharacteristic, complemented
246C22×C4⋊C4C2×D4711yescharacteristic
247C2×#99—941no—
248C2×C22⋊C8—472no—
249#380—472no—
250#342—941no—
251#340—944no—
252#327—944no—
253#324—941no—
254#323—941no—
255C47#837811yescharacteristic, complemented, Sylow
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C2×Dic4711yescharacteristic
259C22×D4C22⋊D4711yescharacteristic
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×Dic4711yescharacteristic
263C23×C4C22×D4711yescharacteristic
264C22×C8—942no—
265#126—28no—
266#125—22no—
267#125—22no—
268#123—28no—
269#121—28no—
270#121—22no—
271#121—22no—
272#121—22no—
273#121C22⋊D4714yes—
274C4×D4—28no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279#117—22no—
280#117—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—942no—
286C2×C4⋊C4C2×Dic4711yescharacteristic
287C2×C4⋊C4C2×Dic4711yescharacteristic
288C2×C4⋊C4C2×Dic4711yescharacteristic
289C2×C4⋊C4C22⋊D4711yescharacteristic
290C2×C4⋊C4C22⋊D4714yes—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—942no—
296C2×C22⋊C4C22⋊D4711yescharacteristic
297C2×C22⋊C4C22⋊D4711yescharacteristic
298C2×C42—21no—
299C2×C42C2×Dic4711yescharacteristic
300#99—1882no—
301C22⋊C8—942no—
302C22⋊C8—942no—
303#95—942no—
304C24—21no—
305C24C22⋊Dic4711yescharacteristic
306C24C2×C22⋊D4711yescharacteristic
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—24no—
313C2×D4—21no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—82no—
319C2×D4C22⋊Dic4712yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—1882no—
332C22×C4C22⋊Dic4711yescharacteristic
333C22×C4C22⋊Dic4711yescharacteristic
334C22×C4C22⋊Dic4711yescharacteristic
335C22×C4C22×Dic4711yescharacteristic
336C22×C4C2×C22⋊D4712yes—
337C22×C4C2×C22⋊D4712yes—
338C22×C4C2×C22⋊D4711yescharacteristic
339C2×C8—1884no—
340C2×C8—1882no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—1884no—
349C4⋊C4C22⋊Dic4712yes—
350C4⋊C4C22⋊Dic4712yes—
351C4⋊C4C22⋊Dic4712yes—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—22no—
356C22⋊C4—24no—
357C22⋊C4—22no—
358C22⋊C4—1882no—
359C22⋊C4—1882no—
360C42—24no—
361C42—21no—
362C42—21no—
363C23—21no—
364C23—22no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23C2×C22⋊Dic4711yescharacteristic
376C23#1309399312yes—
377C23#1309404412yes—
378C23#1309404911yescharacteristic
379C23#1309405411yescharacteristic
380D4—42no—
381D4—42no—
382D4—44no—
383D4—42no—
384D4—44no—
385D4—42no—
386D4—84no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—3762no—
410C2×C4—3762no—
411C2×C4—3762no—
412C2×C4C2×C22⋊Dic4711yescharacteristic
413C2×C4C2×C22⋊Dic4711yescharacteristic
414C2×C4#1309399312yes—
415C8—3762no—
416C22—21no—
417C22—22no—
418C22—21no—
419C22—22no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—42no—
425C22—41no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22—82no—
433C22#1568897911yescharacteristic
434C22#1568911911yescharacteristic
435C22#1568912411yescharacteristic
436C22#1568926812yes—
437C22#1568928512yes—
438C4—22no—
439C4—21no—
440C4—41no—
441C4—41no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—41no—
446C4—7522no—
447C4#1568909712yes—
448C2—22no—
449C2—21no—
450C2—21no—
451C2—22no—
452C2—82no—
453C2—81no—
454C2#2024330611yescharacteristic
455C2#2024332811yescharacteristic
456C2#2024335811yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC2×C47×#553C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenterC2×#553D47
hypocenterC47#8378
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#553

Complements

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

NH ≅ G/Ntype
1. #20243478C2semidirect
2. #20243376C2semidirect
3. #20241956C2semidirect
4. #20241706C2semidirect
17. C2×#125×C47C4semidirect
20. C2×#121×C47C4semidirect
23. C2×C4×D4×C47C4semidirect
25. C2×#117×C47C4semidirect
237. C2×#125Dic47semidirect
240. C2×#121Dic47semidirect
243. C2×C4×D4Dic47semidirect
245. C2×#117Dic47semidirect
255. C47#8378semidirect

Maximal subgroups

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

subgroupindex
#202434782
#202433762
#202419562
#202417062
C2×C47×#5532
#837847