Finite Group Databasebeta

← #26111692

Subgroups — #26111692

Order 12032.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252411C211yesmaximal, characteristic
2C2×#15689221C211yesmaximal, characteristic
3#20241783C211yesmaximal, characteristic, complemented
4#20241778C211yesmaximal, characteristic
5#20241232C211yesmaximal, characteristic, complemented
6#20241218C211yesmaximal, characteristic, complemented
7C47×#1461C211yesmaximal, characteristic, complemented
8C22×#10625336C411yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689221—22no—
11#15689221—22no—
12C8×D188—22no—
13#15689188—22no—
14#15689176C2211yescharacteristic
15#15689176C2211yescharacteristic
16C2×#13093957C411yescharacteristic
17C2×#13093948C411yescharacteristic
18C2×#13093944C411yescharacteristic
19C22⋊C8×D47—22no—
20C2×#13093935C2211yescharacteristic
21Q8⋊(C47⋊C8)—22no—
22#15688850—22no—
23C2×C4×Q8×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#431C2211yescharacteristic
26C47×#324—22no—
27C22×D188C2×C411yescharacteristic
28C22×C4×D47—21no—
29C2×#10625336—22no—
30C2×#10625336—22no—
31C2×#10625336C814yescomplemented
32C22×C47⋊C8C2×C411yescharacteristic
33C22×C47⋊C8C2×C411yescharacteristic
34C2×C8×D47—24no—
35C4×D188—21no—
36C4×D188—21no—
37C4×D188D412yes—
38C4×D188D412yes—
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×(C2×D47)⋊C4—21no—
41#13093957—22no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44#13093948—22no—
45#13093944—22no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48(C2×D47)⋊C8—24no—
49#13093935—21no—
50#13093935—21no—
51C4×C47⋊C8—21no—
52C4×C47⋊C8D412yes—
53C22×Q8×C47C2×C411yescharacteristic
54C4×Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×Q8×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C42×C47—21no—
61C2×C42×C47C2311yescharacteristic
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65C22⋊C8×C47D412yes—
66C4×C8×C47—21no—
67C4×C8×C47—21no—
68C23×D47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47C2×C812yescomplemented
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4111yescharacteristic
80C22×Dic47—21no—
81C2×D188—21no—
82C2×D188—21no—
83C2×D188—21no—
84C2×D188C22⋊C412yes—
85C2×D188C2×C812yes—
86C2×D188M1612yes—
87C2×C4×D47—22no—
88C2×C4×D47—22no—
89C2×C4×D47—82no—
90C2×C8×C47—22no—
91C2×C8×C47—41no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C42×C47—41no—
97C42×C47—41no—
98C42×C47C2×D411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47#4111yescharacteristic
102#10625336—28no—
103(C2×D47)⋊C4—22no—
104(C2×D47)⋊C4—22no—
105C4⋊Dic47—21no—
106C4⋊Dic47C22⋊C412yes—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8C2×C812yescomplemented
115C8×D47—84no—
116C23×C47C4×D411yescharacteristic
117C22×D47—22no—
118C22×D47—22no—
119C22×D47—22no—
120C22×D47—81no—
121C22×D47—81no—
122Q8×C47—22no—
123Q8×C47—22no—
124Q8×C47—41no—
125Q8×C47—41no—
126C2×C4×C47—22no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—81no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47#12011yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×C4×C47#12311yescharacteristic
139C2×C4×C47C22×C811yescharacteristic
140C2×C4×C47C2×M1611yescharacteristic
141C2×C4×C47#13111yescharacteristic
142C2×C4×C47#13111yescharacteristic
143C2×Dic47—22no—
144C2×Dic47—81no—
145D188—22no—
146D188—22no—
147D188—22no—
148D188C22⋊C814yes—
149C4×D47—44no—
150C4×D47—82no—
151C8×C47—42no—
152C8×C47—81no—
153C47⋊C8—22no—
154C47⋊C8—42no—
155C47⋊C8—41no—
156C47⋊C8—81no—
157#9087—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47#38611yescharacteristic
161C22×C47C8×D411yescharacteristic
162C22×C47#43511yescharacteristic
163C2×D47—24no—
164C2×D47—42no—
165C2×D47—42no—
166C2×D47—82no—
167C2×D47—82no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—42no—
173C4×C47—81no—
174C4×C47C2×C22⋊C811yescharacteristic
175C4×C47#40811yescharacteristic
176C4×C47C8×D411yescharacteristic
177C4×C47#43511yescharacteristic
178Dic47—42no—
179#2972—471no—
180C2×#440—471no—
181#1479—471no—
182#1478—471no—
183#1461D4711yescharacteristic, complemented
184#1281—471no—
185#1275—471no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47#186911yescharacteristic
189C2×C47#190211yescharacteristic
190C2×C47#190411yescharacteristic
191D47—44no—
192C22×SD16—471no—
193C2×C4×Q8C2×D4711yescharacteristic
194C2×C4×D4—471no—
195#440—942no—
196#440—942no—
197C8×D4—942no—
198C8×D4—942no—
199#431—471no—
200#431—471no—
201#431C2×D4711yescharacteristic
202#431C2×D4711yescharacteristic
203C2×#107—471no—
204C2×#103—471no—
205C2×#102—471no—
206C2×C22⋊C8—942no—
207C2×#97—471no—
208Q8⋊C8—942no—
209D4⋊C8—942no—
210#324—22no—
211C47#908711yescharacteristic, complemented, Sylow
212C22×Q8C4×D4711yescharacteristic
213C22×D4—471no—
214C23×C4—941no—
215C2×SD16—474no—
216C2×SD16—942no—
217C2×SD16—942no—
218C22×C8—471no—
219C22×C8—471no—
220C22×C8—944no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×D4—472no—
226C4×D4—472no—
227C4×D4—941no—
228C4×D4—941no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—471no—
231C2×C4⋊C4C4×D4711yescharacteristic
232C2×C22⋊C4—941no—
233C2×C42—21no—
234C2×C42C22×D4711yescharacteristic
235#107—942no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—941no—
239C4⋊C8—941no—
240#103—942no—
241#102—942no—
242C22⋊C8—21no—
243C22⋊C8—944no—
244C22⋊C8—941no—
245C22⋊C8—941no—
246C22⋊C8C22⋊D4712yes—
247#97—941no—
248#97—941no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—472no—
252C4×C8—941no—
253C24—941no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8C8×D4712yescomplemented
260C2×D4—472no—
261C2×D4—472no—
262C2×D4—472no—
263C2×D4—941no—
264C2×D4—941no—
265C2×D4—941no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—941no—
269C22×C4—942no—
270C22×C4—942no—
271C22×C4—3762no—
272C22×C4C2×C4×D4711yescharacteristic
273C22×C4#1062535011yescharacteristic
274C22×C4C2×C22⋊D4711yescharacteristic
275SD16—948no—
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—472no—
279C2×C8—941no—
280C2×C8—941no—
281C2×C8—941no—
282C2×C8—942no—
283C2×C8—941no—
284C2×C8—941no—
285C2×C8—3764no—
286C2×C8—1881no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—472no—
292C4⋊C4—941no—
293C22⋊C4—942no—
294C22⋊C4—942no—
295C42—41no—
296C42—41no—
297C42D4×D4711yescharacteristic
298C42D4×D4711yescharacteristic
299C42D4×D4711yescharacteristic
300C42#1062535211yescharacteristic
301C23—942no—
302C23—942no—
303C23—942no—
304C23—3761no—
305C23—3761no—
306C23C4×C22⋊D4711yescharacteristic
307Q8—22no—
308Q8—22no—
309Q8—41no—
310Q8—41no—
311D4—474no—
312D4—942no—
313D4—942no—
314D4—942no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—942no—
323C2×C4—3762no—
324C2×C4—3761no—
325C2×C4—1884no—
326C2×C4C22⋊C4×D4711yescharacteristic
327C2×C4(C22⋊D47)⋊C411yescharacteristic
328C2×C4#1309399211yescharacteristic
329C2×C4#1309399511yescharacteristic
330C2×C4#1309399711yescharacteristic
331C2×C4#1309400611yescharacteristic
332C2×C4C2×C8×D4711yescharacteristic
333C2×C4M16×D4711yescharacteristic
334C2×C4#1309403011yescharacteristic
335C2×C4#1309403211yescharacteristic
336C8—42no—
337C8—81no—
338C8—942no—
339C8—3761no—
340C8—1882no—
341C8—1881no—
342C22—21no—
343C22—21no—
344C22—944no—
345C22—3762no—
346C22—3762no—
347C22—1882no—
348C22—1882no—
349C22#1568899911yescharacteristic
350C22C8×C22⋊D4711yescharacteristic
351C22#1568919211yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—81no—
358C4—1882no—
359C4C22⋊C8×D4711yescharacteristic
360C4#1568908811yescharacteristic
361C4(C22⋊D47)⋊C811yescharacteristic
362C4#1568919711yescharacteristic
363C2—21no—
364C2—21no—
365C2—1884no—
366C2#2024367711yescharacteristic
367C2#2024386411yescharacteristic
368C2#2024387811yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#9087
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
3. #20241783C2semidirect
5. #20241232C2semidirect
6. #20241218C2semidirect
7. C47×#1461C2semidirect
31. C2×#10625336C8semidirect
74. C2×Q8×C47C2×C8semidirect
114. C2×C47⋊C8C2×C8semidirect
183. #1461D47semidirect
211. C47#9087semidirect
259. C2×Q8C8×D47semidirect

Maximal subgroups

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

subgroupindex
#202524112
C2×#156892212
#202417832
#202417782
#202412322
#202412182
C47×#14612
#908747