Finite Group Databasebeta

← #26116796

Subgroups — #26116796

Order 12032.

650 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#20254145C211yesmaximal, characteristic
2C2×#15689453C211yesmaximal, characteristic, complemented
3#20244373C211yesmaximal, characteristic, complemented
4#20243474C211yesmaximal, characteristic, complemented
5#20241956C211yesmaximal, characteristic
6#20241236C211yesmaximal, characteristic, complemented
7C47×#2501C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689453—22no—
10#15689453—22no—
11#15689453—21no—
12#15689439—21no—
13#15689350—22no—
14#15689274—21no—
15#15689261—21no—
16C2×#13093955—21no—
17C2×#13093947—21no—
18C2×C22⋊(C47⋊C8)C2211yescharacteristic
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20#15688983C2211yescharacteristic
21#15688889—22no—
22#15688885—22no—
23#15688857—22no—
24C2×#123×C47—22no—
25C2×#121×C47C2211yescharacteristic
26C22×C4⋊C4×C47C2211yescharacteristic
27C47×#399—22no—
28C47×#390—21no—
29C47×#384—21no—
30C4×C22⋊C4×C47C2211yescharacteristic
31C22×Dic94D411yescharacteristic
32C2×#10625335—22no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8D412yes—
35#13094003—21no—
36#13094003—21no—
37#13094003—21no—
38#13094003—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47D411yescharacteristic
41C2×Dic47⋊C4—21no—
42C2×C22⋊Dic47—21no—
43C2×C22⋊Dic47—21no—
44#13093955—22no—
45#13093955—24no—
46#13093947—22no—
47#13093947—24no—
48#13093945—22no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)D412yes—
53C47⋊#95—21no—
54C22×D4×C47—21no—
55C22×D4×C47D411yescharacteristic
56C23×C4×C47—21no—
57C23×C4×C47C2311yescharacteristic
58#123×C47—44no—
59#123×C47—44no—
60#121×C47—21no—
61#121×C47—21no—
62#121×C47—21no—
63#121×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—24no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47D412yes—
71C2×C4⋊C4×C47D411yescharacteristic
72C2×C22⋊C4×C47—24no—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47D412yes—
75C2×C42×C47—21no—
76#95×C47—21no—
77C24×C47—21no—
78C24×C47C2×D411yescharacteristic
79C2×D4×C47—21no—
80C2×D4×C47—21no—
81C2×D4×C47—42no—
82C2×D4×C47—41no—
83C2×D4×C47—41no—
84C2×D4×C47—42no—
85C2×D4×C47—42no—
86C22×C4×C47—22no—
87C22×C4×C47—22no—
88C22×C4×C47—21no—
89C22×C4×C47—42no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—42no—
93C22×C4×C47—42no—
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47#4111yescharacteristic
100C22×Dic47—21no—
101C22×Dic47—21no—
102C22×Dic47—41no—
103C2×Dic94—21no—
104C2×Dic94—21no—
105C2×Dic94—42no—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—48no—
111C4⋊C4×C47—41no—
112C4⋊C4×C47—41no—
113C22⋊C4×C47—22no—
114C22⋊C4×C47—48no—
115C22⋊C4×C47—41no—
116C22⋊C4×C47—41no—
117C22⋊C4×C47SD1614yescomplemented
118C42×C47—42no—
119C22⋊Dic47—41no—
120C22⋊Dic47—41no—
121C22⋊Dic47—42no—
122#10625335—82no—
123C4⋊Dic47—21no—
124C4⋊Dic47—21no—
125C4⋊Dic47—42no—
126Dic47⋊C4—41no—
127Dic47⋊C4—41no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—42no—
131C2×C47⋊C8—42no—
132C23×C47—41no—
133C23×C47—41no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47—41no—
137C23×C47—42no—
138C23×C47#12011yescharacteristic
139C23×C47#12111yescharacteristic
140C23×C47#12711yescharacteristic
141C23×C47C2×SD1612yes—
142C23×C47AGL(1,ℤ/8)12yes—
143D4×C47—42no—
144D4×C47—81no—
145D4×C47—82no—
146D4×C47—81no—
147D4×C47—81no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—24no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—42no—
156C2×C4×C47—42no—
157C2×C4×C47—41no—
158C2×C4×C47—42no—
159C2×C4×C47—44no—
160C2×C4×C47—42no—
161C2×C4×C47—82no—
162C2×C4×C47—81no—
163C2×C4×C47—81no—
164C2×C4×C47#12011yescharacteristic
165C2×C4×C47#12011yescharacteristic
166C2×C4×C47#12111yescharacteristic
167C2×C4×C47#12111yescharacteristic
168C2×Dic47—41no—
169C2×Dic47—41no—
170C2×Dic47—81no—
171C2×Dic47—81no—
172C2×Dic47—81no—
173C2×Dic47—81no—
174C2×Dic47—81no—
175Dic94—42no—
176Dic94—42no—
177C47⋊C8—42no—
178C47⋊C8—42no—
179#9960—471nomaximal, Sylow
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—42no—
187C22×C47—42no—
188C22×C47—84no—
189C22×C47—81no—
190C22×C47—81no—
191C22×C47—81no—
192C22×C47#39211yescharacteristic
193C22×C47#45012yes—
194C22×C47#46012yes—
195C22×C47#49211yescharacteristic
196C22×C47#49611yescharacteristic
197C4×C47—21no—
198C4×C47—21no—
199C4×C47—21no—
200C4×C47—44no—
201C4×C47—82no—
202C4×C47—81no—
203C4×C47#45812yes—
204Dic47—81no—
205Dic47—81no—
206Dic47—161no—
207#3187—471no—
208C2×#483—471no—
209#2501D4711yescharacteristic
210#1996—471no—
211#1823—471no—
212#1519—471no—
213#1282—471no—
214C2×C47—21no—
215C2×C47—22no—
216C2×C47—22no—
217C2×C47—21no—
218C2×C47—161no—
219C2×C47#200911yescharacteristic
220C2×C47#201411yescharacteristic
221C2×C47#202211yescharacteristic
222C2×#123—22no—
223C2×C22⋊Q8—471no—
224C2×#121C2×D4711yescharacteristic
225C22×C4⋊C4C2×D4711yescharacteristic
226#483—942no—
227#483—942no—
228#483—941no—
229#481—941no—
230#465—942no—
231#450—941no—
232#448—941no—
233C2×#106—941no—
234C2×#103—941no—
235C2×C22⋊C8—471no—
236C2×C22⋊C8—471no—
237#399—22no—
238#390—21no—
239#384—21no—
240#380—471no—
241C4×C22⋊C4C2×D4711yescharacteristic
242#342—942no—
243#340—942no—
244#327—942no—
245C47#996011yescharacteristic, complemented, Sylow
246C22×Q8—471no—
247C22×D4—21no—
248C22×D4C22⋊D4711yescharacteristic
249C23×C4—21no—
250C23×C4C22×D4711yescharacteristic
251C2×SD16—942no—
252C22×C8—472no—
253C22×C8—941no—
254#123—44no—
255#123—44no—
256C22⋊Q8—941no—
257C22⋊Q8—941no—
258C22⋊Q8—941no—
259C22⋊Q8—941no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—471no—
271C2×C4⋊C4—941no—
272C2×C4⋊C4—941no—
273C2×C4⋊C4C22⋊D4712yes—
274C2×C4⋊C4C22⋊D4711yescharacteristic
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—941no—
278C2×C22⋊C4—941no—
279C2×C22⋊C4C22⋊D4712yes—
280C2×C42—21no—
281#106—944no—
282#106—942no—
283#103—944no—
284#103—942no—
285#102—942no—
286C22⋊C8—472no—
287C22⋊C8—941no—
288C22⋊C8—941no—
289C22⋊C8—941no—
290#95—21no—
291#95—941no—
292C24—21no—
293C24C2×C22⋊D4711yescharacteristic
294C2×Q8—941no—
295C2×Q8—941no—
296C2×Q8—1882no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—941no—
313C22×C4—941no—
314C22×C4—1881no—
315C22×C4D4×D4711yescharacteristic
316C22×C4D4×D4711yescharacteristic
317C22×C4D4×D4711yescharacteristic
318C22×C4#1062535211yescharacteristic
319C22×C4C2×C22⋊D4711yescharacteristic
320C22×C4C2×C22⋊D4711yescharacteristic
321SD16—3762no—
322C2×C8—942no—
323C2×C8—942no—
324C2×C8—1882no—
325C2×C8—1882no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—48no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—941no—
334C4⋊C4—941no—
335C4⋊C4—1882no—
336C4⋊C4—1881no—
337C4⋊C4—1881no—
338C22⋊C4—22no—
339C22⋊C4—48no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—1882no—
343C22⋊C4—1881no—
344C22⋊C4—1881no—
345C22⋊C4#1062533514yescomplemented
346C42—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23#1309399311yescharacteristic
354C23#1309399511yescharacteristic
355C23#1309402011yescharacteristic
356C23C2×#1062533512yes—
357C23#1309405012yes—
358Q8—1882no—
359Q8—1882no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—82no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—3761no—
382C2×C4—3761no—
383C2×C4—3761no—
384C2×C4—3761no—
385C2×C4—3761no—
386C2×C4—1881no—
387C2×C4—1881no—
388C2×C4#1309399211yescharacteristic
389C2×C4#1309399211yescharacteristic
390C2×C4#1309399611yescharacteristic
391C2×C4#1309399611yescharacteristic
392C8—1882no—
393C8—1882no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1568903111yescharacteristic
407C22#1568927412yes—
408C22#1568932012yes—
409C22#1568949511yescharacteristic
410C22#1568951011yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—44no—
415C4—82no—
416C4—81no—
417C4—3761no—
418C4—3761no—
419C4—7521no—
420C4#1568931212yes—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—22no—
425C2—161no—
426C2#2024444811yescharacteristic
427C2#2024447611yescharacteristic
428C2#2024451611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2501C2
socleC22×C47#392
prime radicalC2×#121×C47C22
hypercenter#2501D47
hypocenterC47#9960
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. C2×#15689453C2semidirect
3. #20244373C2semidirect
4. #20243474C2semidirect
6. #20241236C2semidirect
117. C22⋊C4×C47SD16semidirect
245. C47#9960semidirect
345. C22⋊C4#10625335semidirect

Maximal subgroups

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

subgroupindex
#202541452
C2×#156894532
#202443732
#202434742
#202419562
#202412362
C47×#25012
#996047