Finite Group Databasebeta

← #26190179

Subgroups — #26190179

Order 12032.

1282 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1282 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689704C211yesmaximal, characteristic, complemented
2#20254136C212yesmaximal, complemented
3#20242390C214yesmaximal, complemented
4#20242322C214yesmaximal, complemented
5#20241955C212yesmaximal, complemented
6C2×#15688849C211yesmaximal, characteristic, complemented
7C47×#2618C211yesmaximal, characteristic, complemented
8C22×#10625350—21no—
9#15689704—24no—
10#15689704—22no—
11#15689704C2214yes—
12C2×#13094004—22no—
13C2×#13093999—22no—
14C2×#13093998—21no—
15C2×#13093998C2212yes—
16#15689436C2214yescomplemented
17#15689434—22no—
18#15689357—24no—
19#15689257C2214yescomplemented
20#15689256—22no—
21C2×C22⋊(C47⋊C8)C2212yes—
22#15688856C2218yescomplemented
23#15688849C2214yescomplemented
24C2×#123×C47—21no—
25C2×#121×C47C2211yescharacteristic
26C22×C4⋊C4×C47C2211yescharacteristic
27C47×#398C2212yescomplemented
28C47×#396—22no—
29C47×#394C2212yescomplemented
30C47×#386—21no—
31C47×#380C2211yescharacteristic
32C2×#10625350—24no—
33C2×#10625350—24no—
34C2×#10625350—44no—
35C2×#10625350—44no—
36C22×Dic94—21no—
37C22×D188—21no—
38C22×D188—21no—
39C22×D188D412yes—
40C22×C22⋊D47—22no—
41C22×C4×D47—22no—
42C2×#10625334—24no—
43C22×C47⋊C8—22no—
44#13094004—28no—
45#13094004—28no—
46#13093999—28no—
47#13093999—28no—
48#13093998—24no—
49#13093998—24no—
50#13093998—22no—
51#13093998—22no—
52#13093998D414yes—
53#13093998C2314yes—
54C2×C4⋊Dic47—21no—
55C2×C4⋊Dic47D412yes—
56C2×(C2×D47)⋊C4—22no—
57C2×(C2×D47)⋊C4—22no—
58#13093957—24no—
59#13093945—24no—
60#13093944—24no—
61C22⋊(C47⋊C8)—22no—
62C22⋊(C47⋊C8)C2314yes—
63C22×D4×C47—21no—
64C22×D4×C47C2311yescharacteristic
65C23×C4×C47D411yescharacteristic
66C23×C4×C47D411yescharacteristic
67C23×C4×C47C2311yescharacteristic
68#123×C47—44no—
69#121×C47—22no—
70#121×C47—22no—
71C2×C4⋊C4×C47—22no—
72C2×C4⋊C4×C47—22no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47D412yes—
76C2×C4⋊C4×C47C2311yescharacteristic
77C2×C4⋊C4×C47C2312yes—
78C2×C22⋊C4×C47—21no—
79C2×C22⋊C4×C47—21no—
80C2×C22⋊C4×C47—22no—
81C2×C22⋊C4×C47—22no—
82#95×C47—22no—
83#95×C47C2312yes—
84C24×C47—21no—
85C24×C47C2×D411yescharacteristic
86C23×D47—22no—
87C2×D4×C47—22no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—42no—
91C2×D4×C47—42no—
92C2×D4×C47—81no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—22no—
99C22×C4×C47—21no—
100C22×C4×C47—81no—
101C22×C4×C47—81no—
102C22×C4×C47D814yescomplemented
103C22×C4×C47C2×D412yes—
104C22×C4×C47C2×D412yes—
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D412yes—
107C22×C4×C47C2×D412yes—
108C22×C4×C47C2411yescharacteristic
109C2×C22⋊D47—24no—
110C2×C22⋊D47—24no—
111C2×C22⋊D47—44no—
112C2×C22⋊D47—44no—
113C22×Dic47—22no—
114#10625350—48no—
115#10625350—416no—
116#10625350—48no—
117C2×D188—24no—
118C2×D188—24no—
119C2×D188—22no—
120C2×D188—44no—
121C2×D188—82no—
122C2×D188—82no—
123C2×D188C2×D414yes—
124C2×C4×D47—28no—
125C2×C4×D47—48no—
126C2×Dic94—24no—
127C2×Dic94—44no—
128C4⋊C4×C47—24no—
129C4⋊C4×C47—24no—
130C4⋊C4×C47—22no—
131C4⋊C4×C47—44no—
132C4⋊C4×C47—81no—
133C22⋊C4×C47—42no—
134C22⋊C4×C47—42no—
135C22⋊C4×C47—44no—
136C22⋊C4×C47—82no—
137#10625334—84no—
138(C2×D47)⋊C4—28no—
139(C2×D47)⋊C4—24no—
140(C2×D47)⋊C4—24no—
141C4⋊Dic47—24no—
142C4⋊Dic47—22no—
143C4⋊Dic47C2×D414yes—
144C2×C47⋊C8—24no—
145C2×C47⋊C8—82no—
146C23×C47—41no—
147C23×C47—41no—
148C23×C47—42no—
149C23×C47—42no—
150C23×C47—41no—
151C23×C47—41no—
152C23×C47—41no—
153C23×C47#12011yescharacteristic
154C23×C47#12011yescharacteristic
155C23×C47C2×D812yes—
156C23×C47AGL(1,ℤ/8)12yes—
157C23×C47C22×D411yescharacteristic
158C22×D47—24no—
159C22×D47—44no—
160C22×D47—44no—
161C22×D47—44no—
162D4×C47—44no—
163D4×C47—82no—
164D4×C47—82no—
165C2×C4×C47—22no—
166C2×C4×C47—22no—
167C2×C4×C47—24no—
168C2×C4×C47—24no—
169C2×C4×C47—22no—
170C2×C4×C47—42no—
171C2×C4×C47—41no—
172C2×C4×C47—42no—
173C2×C4×C47—41no—
174C2×C4×C47—42no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47—42no—
178C2×C4×C47—41no—
179C2×C4×C47—42no—
180C2×C4×C47—41no—
181C2×C4×C47—41no—
182C2×C4×C47—81no—
183C2×C4×C47#12012yes—
184C2×C4×C47C2×D814yes—
185C2×C4×C47C22×D412yes—
186C22⋊D47—44no—
187C22⋊D47—44no—
188C22⋊D47—48no—
189C2×Dic47—24no—
190C2×Dic47—44no—
191D188—44no—
192D188—48no—
193D188—82no—
194D188—84no—
195C4×D47—416no—
196Dic94—48no—
197C47⋊C8—44no—
198#18292—471nomaximal, Sylow
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—21no—
204C22×C47—21no—
205C22×C47—42no—
206C22×C47—42no—
207C22×C47—82no—
208C22×C47—82no—
209C22×C47—81no—
210C22×C47—81no—
211C22×C47—81no—
212C22×C47#44712yes—
213C22×C47#44712yes—
214C22×C47C2×#12011yescharacteristic
215C22×C47C22×D811yescharacteristic
216C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
217C2×D47—44no—
218C2×D47—44no—
219C2×D47—44no—
220C2×D47—88no—
221C4×C47—24no—
222C4×C47—22no—
223C4×C47—22no—
224C4×C47—42no—
225C4×C47—82no—
226C4×C47—81no—
227Dic47—44no—
228C2×#537—471no—
229#3186—472no—
230#2618D4711yescharacteristic, complemented
231#1610—474no—
232#1598—474no—
233#1519—472no—
234C2×#324—471no—
235C2×C47—21no—
236C2×C47—22no—
237C2×C47—21no—
238C2×C47—22no—
239C2×C47—161no—
240C2×C47C2×#44711yescharacteristic
241C2×C47#302011yescharacteristic
242C2×C47#302611yescharacteristic
243D47—84no—
244C22×#41—941no—
245#537—474no—
246#537—944no—
247#537—942no—
248C2×#123—21no—
249C2×C22⋊Q8—942no—
250C2×#121—472no—
251C2×#121—942no—
252C2×#121—941no—
253C2×#121C2×D4711yescharacteristic
254C22×C4⋊C4C2×D4711yescharacteristic
255#480—474no—
256#480—942no—
257#466—944no—
258#447—474no—
259#447—942no—
260C2×C22⋊C8—472no—
261#398C2×D4712yescomplemented
262#396—22no—
263#394C2×D4712yescomplemented
264#386—21no—
265#380C2×D4711yescharacteristic
266#327—478no—
267#324—474no—
268C47#1829211yescharacteristic, complemented, Sylow
269C2×#41—944no—
270C2×#41—944no—
271C2×#41—1884no—
272C2×#41—1884no—
273C22×Q8—941no—
274C22×D4—21no—
275C22×D4—472no—
276C22×D4—941no—
277C22×D4—941no—
278C22×D4—942no—
279C22×D4C22×D4711yescharacteristic
280C23×C4—942no—
281C23×C4C22⋊D4711yescharacteristic
282C23×C4C22⋊D4711yescharacteristic
283C23×C4C22×D4711yescharacteristic
284C2×D8—944no—
285C22×C8—942no—
286#123—44no—
287C22⋊Q8—948no—
288C22⋊Q8—948no—
289#121—22no—
290#121—22no—
291#121—474no—
292#121—474no—
293#121—948no—
294#121—948no—
295#121—944no—
296#121—942no—
297#121—942no—
298#121—944no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—472no—
304C2×C4⋊C4—941no—
305C2×C4⋊C4C22⋊D4712yes—
306C2×C4⋊C4C22×D4711yescharacteristic
307C2×C4⋊C4C22×D4712yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—942no—
313C2×C22⋊C4—942no—
314#107—944no—
315#102—944no—
316#102—944no—
317C22⋊C8—474no—
318C22⋊C8—942no—
319#95—22no—
320#95C22×D4712yes—
321C24—21no—
322C24—942no—
323C24C2×C22⋊D4711yescharacteristic
324#41—18816no—
325#41—1888no—
326#41—1888no—
327C2×Q8—944no—
328C2×Q8—1884no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C2×D4—474no—
336C2×D4—944no—
337C2×D4—944no—
338C2×D4—944no—
339C2×D4—944no—
340C2×D4—942no—
341C2×D4—3762no—
342C2×D4—3762no—
343C2×D4—1884no—
344C2×D4—1884no—
345C2×D4—1884no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—948no—
356C22×C4—942no—
357C22×C4—1888no—
358C22×C4#1062533414yescomplemented
359C22×C4D4×D4712yes—
360C22×C4D4×D4712yes—
361C22×C4C2×C22⋊D4712yes—
362C22×C4C2×C22⋊D4712yes—
363C22×C4C2×C22⋊D4711yescharacteristic
364C22×C4C23×D4711yescharacteristic
365D8—3764no—
366C2×C8—944no—
367C2×C8—3762no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—44no—
372C4⋊C4—81no—
373C4⋊C4—474no—
374C4⋊C4—944no—
375C4⋊C4—942no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—82no—
380C22⋊C4—948no—
381C22⋊C4—944no—
382C22⋊C4—944no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—944no—
391C23—1884no—
392C23—1884no—
393C23—1884no—
394C23#1309399211yescharacteristic
395C23#1309399211yescharacteristic
396C23C2×#1062533412yes—
397C23#1309404912yes—
398C23C22×C22⋊D4711yescharacteristic
399Q8—1888no—
400D4—44no—
401D4—82no—
402D4—82no—
403D4—3762no—
404D4—3764no—
405D4—1884no—
406D4—1888no—
407D4—1888no—
408D4—1884no—
409D4—1884no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—81no—
428C2×C4—944no—
429C2×C4—18816no—
430C2×C4—1884no—
431C2×C4#1309399212yes—
432C2×C4C2×#1062533414yes—
433C2×C4C2×D4×D4712yes—
434C8—1884no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—3768no—
449C22—1884no—
450C22—1884no—
451C22—1884no—
452C22#1568925612yes—
453C22#1568925612yes—
454C22C2×#1309399211yescharacteristic
455C22C22×#1062533411yescharacteristic
456C22C2×#1309404911yescharacteristic
457C4—22no—
458C4—22no—
459C4—24no—
460C4—42no—
461C4—82no—
462C4—81no—
463C4—1884no—
464C2—21no—
465C2—21no—
466C2—22no—
467C2—22no—
468C2—161no—
469C2—3764no—
470C2C2×#1568925611yescharacteristic
471C2#2025275111yescharacteristic
472C2#2025278411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2618C2
socleC22×C47C2×#120
hypercenter#2618D47
hypocenterC47#18292
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#15689704C2semidirect
2. #20254136C2semidirect
3. #20242390C2semidirect
4. #20242322C2semidirect
5. #20241955C2semidirect
6. C2×#15688849C2semidirect
7. C47×#2618C2semidirect
16. #15689436C22semidirect
19. #15689257C22semidirect
22. #15688856C22semidirect
23. #15688849C22semidirect
27. C47×#398C22semidirect
29. C47×#394C22semidirect
102. C22×C4×C47D8semidirect
230. #2618D47semidirect
261. #398C2×D47semidirect
263. #394C2×D47semidirect
268. C47#18292semidirect
358. C22×C4#10625334semidirect

Maximal subgroups

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

subgroupindex
C2×#156897042
#202541362
#202423902
#202423222
#202419552
C2×#156888492
C47×#26182
#1829247