Finite Group Databasebeta

← #26112058

Subgroups — #26112058

Order 12032.

692 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 147 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#20251719C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic, complemented
3C2×#15688856C212yesmaximal
4#20241267C212yesmaximal, complemented
5C2×C47×#382C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C2211yescharacteristic
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689025C412yescomplemented
10#15689021C412yescomplemented
11#15689014—22no—
12#15689002—21no—
13#15688983C2211yescharacteristic
14#15688878—22no—
15#15688856—24no—
16C22×C4⋊C4×C47—21no—
17C22×C42×C47C2211yescharacteristic
18C47×#382—24no—
19C2×#95×C47—21no—
20C2×#95×C47C2212yes—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23C22×C47⋊C8—22no—
24#13094007—42no—
25#13094007—42no—
26#13093998—22no—
27#13093998—22no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C412yes—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—21no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—21no—
35C22⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C23×C4×C47—22no—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C2×C4⋊C4×C47—24no—
44C2×C4⋊C4×C47—42no—
45C2×C4⋊C4×C47—42no—
46C2×C42×C47—22no—
47C2×C42×C47—22no—
48C2×C42×C47C2×C412yes—
49C2×C42×C47D412yes—
50#95×C47—24no—
51#95×C47—22no—
52#95×C47D414yes—
53C24×C47C2×D411yescharacteristic
54C23×D47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—24no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—44no—
65C22×C4×C47—44no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22⋊C412yes—
70C22×C4×C47D812yes—
71C22×C4×C47SD1612yes—
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D412yes—
75C22×C4×C47#4112yes—
76C2×C22⋊D47—41no—
77C2×C22⋊D47—41no—
78C2×C22⋊D47—41no—
79C2×C22⋊D47—41no—
80C2×C22⋊D47—41no—
81C2×C22⋊D47—41no—
82C22×Dic47—22no—
83C22×Dic47—21no—
84C2×D188—22no—
85C2×D188—42no—
86C4⋊C4×C47—48no—
87C42×C47—24no—
88C42×C47—24no—
89C22⋊Dic47—41no—
90C22⋊Dic47—41no—
91(C2×D47)⋊C4—44no—
92(C2×D47)⋊C4—44no—
93(C2×D47)⋊C4—42no—
94C4⋊Dic47—22no—
95C4⋊Dic47—42no—
96Dic47⋊C4—44no—
97C2×C47⋊C8—44no—
98C2×C47⋊C8—42no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47C2×C22⋊C411yescharacteristic
102C23×C47#12011yescharacteristic
103C23×C47#12311yescharacteristic
104C23×C47C2×D811yescharacteristic
105C23×C47C2×SD1611yescharacteristic
106C23×C47AGL(1,ℤ/8)11yescharacteristic
107C23×C47#13711yescharacteristic
108C22×D47—41no—
109C22×D47—42no—
110C22×D47—41no—
111C22×D47—41no—
112C22×D47—42no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—44no—
129C2×C4×C47—44no—
130C2×C4×C47—44no—
131C2×C4×C47—84no—
132C2×C4×C47—84no—
133C2×C4×C47#10214yes—
134C2×C4×C47C4×D412yes—
135C2×C4×C47#12112yes—
136C22⋊D47—81no—
137C22⋊D47—81no—
138C22⋊D47—81no—
139C22⋊D47—81no—
140C2×Dic47—41no—
141C2×Dic47—42no—
142C2×Dic47—41no—
143C2×Dic47—42no—
144C2×Dic47—41no—
145C2×Dic47—42no—
146D188—82no—
147C47⋊C8—82no—
148#9156—471nomaximal, Sylow
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—22no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47#38611yescharacteristic
158C22×C47#41012yes—
159C22×C47C2×#10211yescharacteristic
160C22×C47#41711yescharacteristic
161C22×C47#42112yes—
162C22×C47#44711yescharacteristic
163C22×C47#44811yescharacteristic
164C22×C47#48011yescharacteristic
165C22×C47#48111yescharacteristic
166C2×D47—81no—
167C2×D47—81no—
168C2×D47—81no—
169C2×D47—82no—
170C2×D47—82no—
171C4×C47—24no—
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—42no—
175C4×C47—84no—
176Dic47—81no—
177Dic47—82no—
178#2854—471no—
179C2×#382D4711yescharacteristic, complemented
180#1817—471no—
181C2×#327—472no—
182#1294—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47#151212yes—
188C2×C47#151712yes—
189C2×C47#187411yescharacteristic
190C2×C47#189711yescharacteristic
191C2×C47#189911yescharacteristic
192D47—161no—
193C2×#123—941no—
194C2×#121—471no—
195C22×C4⋊C4—21no—
196C22×C42C2×D4711yescharacteristic
197C2×C22⋊C8—472no—
198#390—472no—
199#389—472no—
200#388—942no—
201#386—941no—
202#382—24no—
203#380—471no—
204C2×#95—21no—
205C2×#95C2×D4712yes—
206#336—942no—
207#327—944no—
208C47#915611yescharacteristic, complemented, Sylow
209C22×D4—471no—
210C22×D4—941no—
211C23×C4—22no—
212C23×C4C22⋊D4711yescharacteristic
213C23×C4C22⋊D4711yescharacteristic
214C23×C4C22×D4711yescharacteristic
215C22×C8—942no—
216#123—1882no—
217#123—1882no—
218#121—942no—
219#121—942no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—471no—
224C2×C4⋊C4—472no—
225C2×C4⋊C4—942no—
226C2×C22⋊C4—941no—
227C2×C22⋊C4—942no—
228C2×C22⋊C4—942no—
229C2×C22⋊C4—941no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C4×D4712yes—
233C2×C42D18812yes—
234C22⋊C8—942no—
235C22⋊C8—942no—
236#95—24no—
237#95—22no—
238#95—942no—
239#95—941no—
240#95C22⋊D4714yes—
241C24—941no—
242C24C2×C22⋊D4711yescharacteristic
243C2×D4—942no—
244C2×D4—1882no—
245C2×D4—1881no—
246C2×D4—1881no—
247C2×D4—1881no—
248C2×D4—1881no—
249C2×D4—1881no—
250C2×D4—1881no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—942no—
265C22×C4—941no—
266C22×C4(C2×D47)⋊C412yes—
267C22×C4(C2×D47)⋊C412yes—
268C22×C4#1062533412yes—
269C22×C4#1062533612yes—
270C22×C4C2×C4×D4711yescharacteristic
271C22×C4C2×D18811yescharacteristic
272C22×C4#1062535012yes—
273C22×C4C2×C22⋊D4712yes—
274C2×C8—1884no—
275C2×C8—1882no—
276C4⋊C4—48no—
277C4⋊C4—942no—
278C4⋊C4—1882no—
279C4⋊C4—1884no—
280C22⋊C4—1884no—
281C22⋊C4—1881no—
282C22⋊C4—1881no—
283C22⋊C4—1884no—
284C22⋊C4—1882no—
285C42—24no—
286C42—24no—
287C23—22no—
288C23—22no—
289C23—1881no—
290C23—1882no—
291C23—1881no—
292C23—1881no—
293C23—1882no—
294C23C2×(C2×D47)⋊C411yescharacteristic
295C23#1309399311yescharacteristic
296C23#1309400711yescharacteristic
297C23C2×#1062533411yescharacteristic
298C23C2×#1062533611yescharacteristic
299C23#1309405011yescharacteristic
300C23#1309405511yescharacteristic
301D4—3762no—
302D4—3761no—
303D4—3761no—
304D4—3761no—
305D4—3761no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—1881no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4—1882no—
332C2×C4#1309394414yes—
333C2×C4C4×C22⋊D4712yes—
334C2×C4#1309399812yes—
335C8—3762no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—3762no—
345C22—3762no—
346C22—3761no—
347C22—3761no—
348C22—3761no—
349C22#1568900211yescharacteristic
350C22#1568909612yes—
351C22C2×#1309394411yescharacteristic
352C22#1568912211yescharacteristic
353C22#1568914112yes—
354C22#1568925711yescharacteristic
355C22#1568926211yescharacteristic
356C22#1568943611yescharacteristic
357C22#1568944111yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—84no—
363C4—3762no—
364C4—3761no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—7521no—
370C2#2024192412yes—
371C2#2024194712yes—
372C2#2024370511yescharacteristic
373C2#2024383611yescharacteristic
374C2#2024384611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#382C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenterC2×#382D47
hypocenterC47#9156
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. #20243456C2semidirect
4. #20241267C2semidirect
5. C2×C47×#382C2semidirect
9. #15689025C4semidirect
10. #15689021C4semidirect
179. C2×#382D47semidirect
208. C47#9156semidirect

Maximal subgroups

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

subgroupindex
#202517192
#202434562
C2×#156888562
#202412672
C2×C47×#3822
#915647