Finite Group Databasebeta

← #26190186

Subgroups — #26190186

Order 12032.

1220 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1220 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
2#20254141C212yesmaximal, complemented
3#20242424C214yesmaximal, complemented
4#20242356C214yesmaximal, complemented
5#20241955C212yesmaximal, complemented
6C2×#15688849C211yesmaximal, characteristic, complemented
7C47×#2627C211yesmaximal, 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#15689451—22no—
17#15689441C2214yescomplemented
18#15689354—24no—
19#15689273—22no—
20#15689262C2214yescomplemented
21C2×C22⋊(C47⋊C8)C2212yes—
22#15688856C2218yescomplemented
23#15688849C2214yescomplemented
24C2×C22⋊Q8×C47C2211yescharacteristic
25C22×C4⋊C4×C47C2211yescharacteristic
26C47×#400C2212yescomplemented
27C47×#398—22no—
28C47×#395C2212yescomplemented
29C47×#385—21no—
30C47×#381—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×#10625336—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#13093955—24no—
59#13093948—24no—
60#13093944—24no—
61C22⋊(C47⋊C8)—22no—
62C22⋊(C47⋊C8)C2314yes—
63C22×Q8×C47C2311yescharacteristic
64C23×C4×C47D411yescharacteristic
65C23×C4×C47D411yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C22⋊Q8×C47—22no—
68C22⋊Q8×C47—22no—
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47—22no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47D412yes—
75C2×C4⋊C4×C47C2311yescharacteristic
76C2×C4⋊C4×C47C2312yes—
77C2×C22⋊C4×C47—21no—
78C2×C22⋊C4×C47—21no—
79#95×C47—22no—
80#95×C47—22no—
81#95×C47—22no—
82#95×C47C2312yes—
83C24×C47C2×D411yescharacteristic
84C23×D47—22no—
85C2×Q8×C47—22no—
86C2×Q8×C47—81no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—22no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—81no—
96C22×C4×C47—81no—
97C22×C4×C47SD1614yescomplemented
98C22×C4×C47C2×D412yes—
99C22×C4×C47C2×D412yes—
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2×D412yes—
102C22×C4×C47C2×D412yes—
103C22×C4×C47C2411yescharacteristic
104C2×C22⋊D47—24no—
105C2×C22⋊D47—24no—
106C2×C22⋊D47—44no—
107C2×C22⋊D47—44no—
108C22×Dic47—22no—
109#10625350—48no—
110#10625350—416no—
111#10625350—48no—
112C2×D188—24no—
113C2×D188—24no—
114C2×D188—22no—
115C2×D188—44no—
116C2×D188—82no—
117C2×D188—82no—
118C2×D188C2×D414yes—
119C2×C4×D47—28no—
120C2×C4×D47—48no—
121C2×Dic94—24no—
122C2×Dic94—44no—
123C4⋊C4×C47—24no—
124C4⋊C4×C47—24no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—42no—
127C4⋊C4×C47—81no—
128C4⋊C4×C47—82no—
129C22⋊C4×C47—41no—
130C22⋊C4×C47—41no—
131C22⋊C4×C47—42no—
132#10625336—84no—
133(C2×D47)⋊C4—28no—
134(C2×D47)⋊C4—24no—
135(C2×D47)⋊C4—24no—
136C4⋊Dic47—24no—
137C4⋊Dic47—22no—
138C4⋊Dic47C2×D414yes—
139C2×C47⋊C8—24no—
140C2×C47⋊C8—82no—
141C23×C47—41no—
142C23×C47—41no—
143C23×C47#12011yescharacteristic
144C23×C47#12011yescharacteristic
145C23×C47C2×SD1612yes—
146C23×C47AGL(1,ℤ/8)12yes—
147C23×C47C22×D411yescharacteristic
148C22×D47—24no—
149C22×D47—44no—
150C22×D47—44no—
151C22×D47—44no—
152Q8×C47—44no—
153C2×C4×C47—22no—
154C2×C4×C47—22no—
155C2×C4×C47—24no—
156C2×C4×C47—24no—
157C2×C4×C47—22no—
158C2×C4×C47—42no—
159C2×C4×C47—41no—
160C2×C4×C47—42no—
161C2×C4×C47—41no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—42no—
165C2×C4×C47—42no—
166C2×C4×C47—42no—
167C2×C4×C47—42no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—42no—
171C2×C4×C47—81no—
172C2×C4×C47#12012yes—
173C2×C4×C47C2×SD1614yes—
174C2×C4×C47C22×D412yes—
175C22⋊D47—44no—
176C22⋊D47—44no—
177C22⋊D47—48no—
178C2×Dic47—24no—
179C2×Dic47—44no—
180D188—44no—
181D188—48no—
182D188—82no—
183D188—84no—
184C4×D47—416no—
185Dic94—48no—
186C47⋊C8—44no—
187#18293—471nomaximal, Sylow
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—42no—
195C22×C47—42no—
196C22×C47#45012yes—
197C22×C47#45012yes—
198C22×C47C2×#12011yescharacteristic
199C22×C47C22×SD1611yescharacteristic
200C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
201C2×D47—44no—
202C2×D47—44no—
203C2×D47—44no—
204C2×D47—88no—
205C4×C47—24no—
206C4×C47—22no—
207C4×C47—22no—
208C4×C47—42no—
209C4×C47—82no—
210C4×C47—81no—
211C4×C47—81no—
212Dic47—44no—
213C2×#537—471no—
214#3187—472no—
215#2627D4711yescharacteristic, complemented
216#1616—474no—
217#1604—474no—
218#1519—472no—
219C2×#324—471no—
220C2×C47—21no—
221C2×C47—22no—
222C2×C47—21no—
223C2×C47—22no—
224C2×C47C2×#45011yescharacteristic
225C2×C47#302011yescharacteristic
226C2×C47#302611yescharacteristic
227D47—84no—
228C22×#41—941no—
229#537—474no—
230#537—944no—
231#537—942no—
232C2×C22⋊Q8—942no—
233C2×C22⋊Q8C2×D4711yescharacteristic
234C2×#121—472no—
235C2×#121—942no—
236C2×#121—941no—
237C22×C4⋊C4C2×D4711yescharacteristic
238#483—942no—
239#481—474no—
240#465—944no—
241#450—942no—
242#448—474no—
243C2×C22⋊C8—472no—
244#400C2×D4712yescomplemented
245#398—22no—
246#395C2×D4712yescomplemented
247#385—21no—
248#381—21no—
249#380C2×D4711yescharacteristic
250#327—478no—
251#324—474no—
252C47#1829311yescharacteristic, complemented, Sylow
253C2×#41—944no—
254C2×#41—944no—
255C2×#41—1884no—
256C2×#41—1884no—
257C22×Q8—941no—
258C22×Q8C22×D4711yescharacteristic
259C22×D4—472no—
260C22×D4—941no—
261C22×D4—941no—
262C22×D4—942no—
263C23×C4—942no—
264C23×C4C22⋊D4711yescharacteristic
265C23×C4C22⋊D4711yescharacteristic
266C23×C4C22×D4711yescharacteristic
267C2×SD16—944no—
268C22×C8—942no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—948no—
272C22⋊Q8—948no—
273#121—474no—
274#121—474no—
275#121—948no—
276#121—948no—
277#121—944no—
278#121—942no—
279#121—942no—
280#121—944no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—472no—
287C2×C4⋊C4—941no—
288C2×C4⋊C4C22⋊D4712yes—
289C2×C4⋊C4C22×D4711yescharacteristic
290C2×C4⋊C4C22×D4712yes—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—942no—
294C2×C22⋊C4—942no—
295#106—944no—
296#103—944no—
297#102—944no—
298C22⋊C8—474no—
299C22⋊C8—942no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95C22×D4712yes—
304C24—942no—
305C24C2×C22⋊D4711yescharacteristic
306#41—18816no—
307#41—1888no—
308#41—1888no—
309C2×Q8—22no—
310C2×Q8—81no—
311C2×Q8—944no—
312C2×Q8—1884no—
313C2×D4—474no—
314C2×D4—944no—
315C2×D4—944no—
316C2×D4—944no—
317C2×D4—944no—
318C2×D4—942no—
319C2×D4—3762no—
320C2×D4—3762no—
321C2×D4—1884no—
322C2×D4—1884no—
323C2×D4—1884no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—948no—
335C22×C4—942no—
336C22×C4—1888no—
337C22×C4#1062533614yescomplemented
338C22×C4D4×D4712yes—
339C22×C4D4×D4712yes—
340C22×C4C2×C22⋊D4712yes—
341C22×C4C2×C22⋊D4712yes—
342C22×C4C2×C22⋊D4711yescharacteristic
343C22×C4C23×D4711yescharacteristic
344SD16—3764no—
345C2×C8—944no—
346C2×C8—3762no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—474no—
354C4⋊C4—944no—
355C4⋊C4—942no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—948no—
360C22⋊C4—944no—
361C22⋊C4—944no—
362C23—41no—
363C23—41no—
364C23—944no—
365C23—1884no—
366C23—1884no—
367C23—1884no—
368C23#1309399211yescharacteristic
369C23#1309399211yescharacteristic
370C23C2×#1062533612yes—
371C23#1309404912yes—
372C23C22×C22⋊D4711yescharacteristic
373Q8—44no—
374Q8—1888no—
375D4—3762no—
376D4—3764no—
377D4—1884no—
378D4—1888no—
379D4—1888no—
380D4—1884no—
381D4—1884no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—944no—
402C2×C4—18816no—
403C2×C4—1884no—
404C2×C4#1309399212yes—
405C2×C4C2×#1062533614yes—
406C2×C4C2×D4×D4712yes—
407C8—1884no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—3768no—
417C22—1884no—
418C22—1884no—
419C22—1884no—
420C22#1568927312yes—
421C22#1568927312yes—
422C22C2×#1309399211yescharacteristic
423C22C22×#1062533611yescharacteristic
424C22C2×#1309404911yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—82no—
430C4—81no—
431C4—81no—
432C4—1884no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—3764no—
438C2C2×#1568927311yescharacteristic
439C2#2025275111yescharacteristic
440C2#2025278411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2627C2
socleC22×C47C2×#120
prime radicalC2×#15689704C2
hypercenter#2627D47
hypocenterC47#18293
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
2. #20254141C2semidirect
3. #20242424C2semidirect
4. #20242356C2semidirect
5. #20241955C2semidirect
6. C2×#15688849C2semidirect
7. C47×#2627C2semidirect
17. #15689441C22semidirect
20. #15689262C22semidirect
22. #15688856C22semidirect
23. #15688849C22semidirect
26. C47×#400C22semidirect
28. C47×#395C22semidirect
97. C22×C4×C47SD16semidirect
215. #2627D47semidirect
244. #400C2×D47semidirect
246. #395C2×D47semidirect
252. C47#18293semidirect
337. C22×C4#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156897042
#202541412
#202424242
#202423562
#202419552
C2×#156888492
C47×#26272
#1829347