Finite Group Databasebeta

← #26112830

Subgroups — #26112830

Order 12032.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254326C211yesmaximal, characteristic
2#20244228C211yesmaximal, characteristic
3#20242078C211yesmaximal, characteristic, complemented
4C2×#15688866C211yesmaximal, characteristic
5#20241285C211yesmaximal, characteristic, complemented
6#20241239C211yesmaximal, characteristic, complemented
7C47×#2962C211yesmaximal, characteristic, complemented
8C2×#13094021C2211yescharacteristic
9#15689497—21no—
10#15689471—21no—
11#15689323—22no—
12#15689273C414yes—
13#15689176C2211yescharacteristic
14#15689147C2211yescharacteristic
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×#13093940—21no—
17C2×#13093940—21no—
18#15688872—21no—
19#15688869—21no—
20#15688866—21no—
21#15688866—21no—
22C47×#533C2211yescharacteristic
23C47×#446—22no—
24C47×#446—21no—
25C8×Q8×C47—21no—
26C47×#424—21no—
27C2×C4⋊C8×C47C2211yescharacteristic
28C47×#405—21no—
29C4×M16×C47C2211yescharacteristic
30C22×D188—21no—
31C22×D188—21no—
32C22×D188C2×C412yes—
33C2×#10625336—24no—
34C2×#10625329—21no—
35C22×C47⋊C8—21no—
36#13094021—21no—
37#13094021—21no—
38#13094021—21no—
39#13094021—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45#13093944—24no—
46#13093940—42no—
47#13093940—42no—
48C22⋊(C47⋊C8)C2×C412yes—
49C4×C47⋊C8—21no—
50C2×M16×C47D411yescharacteristic
51C2×M16×C47D411yescharacteristic
52C22×C8×C47—21no—
53C4⋊Q8×C47D411yescharacteristic
54C4⋊Q8×C47D411yescharacteristic
55#125×C47—21no—
56C22⋊Q8×C47—21no—
57C22⋊Q8×C47C2×C412yes—
58C4×Q8×C47—21no—
59C4×Q8×C47D412yes—
60#117×C47—21no—
61C2×C42×C47C2311yescharacteristic
62C4⋊C8×C47—22no—
63C4⋊C8×C47—22no—
64C4⋊C8×C47—21no—
65C4⋊C8×C47D411yescharacteristic
66C4⋊C8×C47D411yescharacteristic
67#97×C47—22no—
68#97×C47—21no—
69C4×C8×C47—22no—
70C4×C8×C47—21no—
71C23×D47—22no—
72C2×Q8×C47—21no—
73C2×Q8×C47C22⋊C412yes—
74C22×C4×C47C22×C411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47#4111yescharacteristic
77C2×D188—22no—
78C2×D188—22no—
79C2×D188—41no—
80C2×D188—41no—
81C2×D188—41no—
82C2×D188—41no—
83C2×D188—41no—
84C2×D188—41no—
85C2×D188—41no—
86C2×D188—41no—
87C2×D188—82no—
88C2×D188—81no—
89C2×D188—81no—
90M16×C47—21no—
91M16×C47—21no—
92M16×C47—21no—
93M16×C47SD1612yescomplemented
94C2×C8×C47—21no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—21no—
98C2×C8×C47—41no—
99C2×C8×C47—41no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47C22⋊C412yes—
106C22⋊C4×C47—22no—
107C42×C47—22no—
108C42×C47C2×D411yescharacteristic
109C42×C47C2×D411yescharacteristic
110C42×C47C2×D411yescharacteristic
111C42×C47#4111yescharacteristic
112#10625336—84no—
113#10625329—41no—
114#10625329—41no—
115C2×C47⋊C8—22no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118C2×C47⋊C8—81no—
119C23×C47C4×D411yescharacteristic
120C22×D47—42no—
121C22×D47—42no—
122C22×D47—42no—
123C22×D47—82no—
124C22×D47—82no—
125Q8×C47—42no—
126Q8×C47—41no—
127Q8×C47—41no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—41no—
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×C47C2×SD1611yescharacteristic
140C2×C4×C47#13511yescharacteristic
141C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C47#13711yescharacteristic
143D188—44no—
144D188—82no—
145D188—82no—
146D188—81no—
147D188—81no—
148D188—81no—
149D188—81no—
150D188—81no—
151D188—81no—
152C8×C47—22no—
153C8×C47—41no—
154C8×C47—41no—
155C8×C47—41no—
156C47⋊C8—42no—
157C47⋊C8—81no—
158C47⋊C8—81no—
159#9274—471nomaximal, Sylow
160C22×C47—21no—
161C22×C47—21no—
162C22×C47#38611yescharacteristic
163C22×C47C4×SD1611yescharacteristic
164C22×C47#44011yescharacteristic
165C2×D47—82no—
166C2×D47—82no—
167C2×D47—82no—
168C2×D47—162no—
169C2×D47—162no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—42no—
174C4×C47—42no—
175C4×C47#44811yescharacteristic
176C4×C47#44911yescharacteristic
177C4×C47#46011yescharacteristic
178C4×C47#46411yescharacteristic
179C4×C47#47111yescharacteristic
180C4×C47#47211yescharacteristic
181#3208—471no—
182#2962D4711yescharacteristic, complemented
183#1965—471no—
184#1550—471no—
185C2×#331—471no—
186#1301—471no—
187#1283—471no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47#188011yescharacteristic
191C2×C47#190711yescharacteristic
192C2×C47#191411yescharacteristic
193D47—162no—
194#533C2×D4711yescharacteristic
195C2×#127—471no—
196#492—941no—
197#486—941no—
198#460—942no—
199#450—474no—
200#446—22no—
201#446—21no—
202C8×Q8—21no—
203#431—471no—
204#424—21no—
205#423—471no—
206C2×C4⋊C8—471no—
207C2×C4⋊C8C2×D4711yescharacteristic
208C2×#100—941no—
209C2×#100—941no—
210#405—21no—
211C4×M16C2×D4711yescharacteristic
212#333—941no—
213#332—941no—
214#331—941no—
215#331—941no—
216C47#927411yescharacteristic, complemented, Sylow
217C22×D4—472no—
218C22×D4—941no—
219C22×D4—941no—
220C2×SD16—944no—
221C2×M16—941no—
222C2×M16C22⋊D4711yescharacteristic
223C2×M16C22⋊D4711yescharacteristic
224C22×C8—21no—
225C22×C8—941no—
226C4⋊Q8D18811yescharacteristic
227C4⋊Q8D18811yescharacteristic
228#127—941no—
229#127—941no—
230#127—941no—
231#127—941no—
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8C4×D4712yes—
235C4×Q8—21no—
236C4×Q8C22⋊D4712yes—
237#117—21no—
238C2×C42C22×D4711yescharacteristic
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—941no—
243C4⋊C8—941no—
244C4⋊C8—941no—
245C4⋊C8—941no—
246C4⋊C8—941no—
247C4⋊C8C22⋊D4711yescharacteristic
248C4⋊C8C22⋊D4711yescharacteristic
249#102—944no—
250#100—1882no—
251#100—1882no—
252C22⋊C8—472no—
253#97—22no—
254#97—21no—
255C4×C8—22no—
256C4×C8—21no—
257C4×C8—941no—
258C24—942no—
259C2×Q8—21no—
260C2×Q8(C2×D47)⋊C412yes—
261C2×D4—942no—
262C2×D4—942no—
263C2×D4—3761no—
264C2×D4—3761no—
265C2×D4—3762no—
266C2×D4—1881no—
267C2×D4—1881no—
268C2×D4—1881no—
269C2×D4—1881no—
270C2×D4—1881no—
271C2×D4—1881no—
272C2×D4—1881no—
273C2×D4—1881no—
274C22×C4C2×C4×D4711yescharacteristic
275C22×C4#1062535011yescharacteristic
276C22×C4C2×C22⋊D4711yescharacteristic
277SD16—3764no—
278M16—21no—
279M16—21no—
280M16—21no—
281M16—1881no—
282M16—1881no—
283M16#1062533612yescomplemented
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—942no—
291C2×C8—3761no—
292C2×C8—1881no—
293C2×C8—1881no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4(C2×D47)⋊C412yes—
300C22⋊C4—22no—
301C42—22no—
302C42C2×D18811yescharacteristic
303C42#1062535011yescharacteristic
304C42C2×C22⋊D4711yescharacteristic
305C42C2×C22⋊D4711yescharacteristic
306C23—3762no—
307C23—3762no—
308C23—1882no—
309C23—1882no—
310C23—1882no—
311C23C4×C22⋊D4711yescharacteristic
312Q8—42no—
313Q8—41no—
314Q8—41no—
315D4—3762no—
316D4—3762no—
317D4—3761no—
318D4—3761no—
319D4—3761no—
320D4—3761no—
321D4—3761no—
322D4—3761no—
323D4—1884no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4C2×(C2×D47)⋊C411yescharacteristic
330C2×C4C4×C22⋊D4711yescharacteristic
331C2×C4#1309399311yescharacteristic
332C2×C4#1309399811yescharacteristic
333C2×C4#1309399811yescharacteristic
334C2×C4#1309400711yescharacteristic
335C2×C4C2×#1062533611yescharacteristic
336C2×C4#1309404411yescharacteristic
337C2×C4#1309404911yescharacteristic
338C2×C4#1309405511yescharacteristic
339C8—22no—
340C8—41no—
341C8—41no—
342C8—41no—
343C8—3761no—
344C8—3761no—
345C8—1882no—
346C22—21no—
347C22—21no—
348C22—3762no—
349C22—3762no—
350C22—3762no—
351C22—7522no—
352C22—7522no—
353C22#1568900211yescharacteristic
354C22C4×#1062533611yescharacteristic
355C22#1568922111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4#1568926211yescharacteristic
362C4#1568926811yescharacteristic
363C4#1568932311yescharacteristic
364C4#1568934511yescharacteristic
365C4#1568938611yescharacteristic
366C4#1568939311yescharacteristic
367C2—21no—
368C2—21no—
369C2—7522no—
370C2#2024374311yescharacteristic
371C2#2024389611yescharacteristic
372C2#2024393511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2962C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#2962D47
hypocenterC47#9274
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #20242078C2semidirect
5. #20241285C2semidirect
6. #20241239C2semidirect
7. C47×#2962C2semidirect
93. M16×C47SD16semidirect
182. #2962D47semidirect
216. C47#9274semidirect
283. M16#10625336semidirect

Maximal subgroups

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

subgroupindex
#202543262
#202442282
#202420782
C2×#156888662
#202412852
#202412392
C47×#29622
#927447