Finite Group Databasebeta

← #26112688

Subgroups — #26112688

Order 12032.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689323C211yesmaximal, characteristic
2Q8×#10625329C211yesmaximal, characteristic, complemented
3#20244227C211yesmaximal, characteristic
4#20242055C211yesmaximal, characteristic
5#20241262C211yesmaximal, characteristic, complemented
6#20241237C211yesmaximal, characteristic, complemented
7C47×#1487C211yesmaximal, characteristic, complemented
8C22×#10625336C412yes—
9C2×#13094021C2211yescharacteristic
10#15689323—21no—
11#15689323—21no—
12#15689323—21no—
13#15689323—21no—
14#15689254—22no—
15#15689254—21no—
16Q8×C47⋊C8—21no—
17#15689147—21no—
18#15689147C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093944C412yes—
21C2×#13093940—21no—
22C2×#13093940—21no—
23C4×#10625329—21no—
24C4×#10625329C2211yescharacteristic
25#15688866—22no—
26C2×C4×Q8×C47C2211yescharacteristic
27C47×#431C2211yescharacteristic
28C47×#423C2211yescharacteristic
29C47×Q8⋊C8—21no—
30C47×Q8⋊C8—21no—
31C22×D188—21no—
32C22×D188—21no—
33C22×D188C2×C412yes—
34C2×#10625336—22no—
35C2×#10625336—22no—
36C2×#10625336—44no—
37C2×#10625329—21no—
38C2×#10625329D411yescharacteristic
39C2×#10625329D411yescharacteristic
40C22×C47⋊C8C2×C412yes—
41#13094021—21no—
42#13094021—21no—
43#13094021—21no—
44#13094021—21no—
45C4⋊(C47⋊C8)—22no—
46C4⋊(C47⋊C8)—22no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)D412yes—
51#13093944—22no—
52#13093944—22no—
53#13093940—42no—
54#13093940—42no—
55#13093935—22no—
56#13093935—21no—
57C4×C47⋊C8—22no—
58C4×C47⋊C8—21no—
59C22×Q8×C47C2×C411yescharacteristic
60C2×M16×C47—21no—
61C4×Q8×C47—21no—
62C4×Q8×C47—21no—
63C4×Q8×C47D411yescharacteristic
64C4×Q8×C47D411yescharacteristic
65C4×Q8×C47D411yescharacteristic
66C4×Q8×C47D411yescharacteristic
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47C2×C411yescharacteristic
69C2×C42×C47—21no—
70C2×C42×C47C2311yescharacteristic
71C4⋊C8×C47—21no—
72C4⋊C8×C47—21no—
73C4⋊C8×C47—21no—
74C22⋊C8×C47—21no—
75C4×C8×C47—21no—
76C23×D47—22no—
77C2×Q8×C47—21no—
78C2×Q8×C47—22no—
79C2×Q8×C47—41no—
80C2×Q8×C47C22⋊C411yescharacteristic
81C2×Q8×C47C22⋊C411yescharacteristic
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47#4111yescharacteristic
87C2×D188—22no—
88C2×D188—22no—
89C2×D188—42no—
90C2×D188—42no—
91C2×D188—41no—
92C2×D188—41no—
93C2×D188—41no—
94C2×D188—41no—
95C2×D188—41no—
96C2×D188—41no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188—82no—
100M16×C47—41no—
101M16×C47—41no—
102C2×C8×C47—41no—
103C2×C8×C47—41no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—21no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47C22⋊C411yescharacteristic
110C4⋊C4×C47C22⋊C411yescharacteristic
111C42×C47—21no—
112C42×C47—21no—
113C42×C47—21no—
114C42×C47—21no—
115C42×C47C2×D411yescharacteristic
116C42×C47C2×D411yescharacteristic
117C42×C47C2×D411yescharacteristic
118C42×C47#4111yescharacteristic
119#10625336—44no—
120#10625336—44no—
121#10625329—21no—
122#10625329—21no—
123#10625329—21no—
124#10625329—41no—
125#10625329—41no—
126#10625329SD1612yescomplemented
127C2×C47⋊C8—21no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—42no—
132C23×C47C4×D411yescharacteristic
133C22×D47—42no—
134C22×D47—42no—
135C22×D47—42no—
136C22×D47—82no—
137C22×D47—82no—
138Q8×C47—21no—
139Q8×C47—21no—
140Q8×C47—21no—
141Q8×C47—42no—
142Q8×C47#10012yescomplemented
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—81no—
152C2×C4×C47C2×C22⋊C411yescharacteristic
153C2×C4×C47C4×D411yescharacteristic
154C2×C4×C47#12011yescharacteristic
155C2×C4×C47#12111yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47#12311yescharacteristic
158C2×C4×C47C2×SD1611yescharacteristic
159C2×C4×C47#13511yescharacteristic
160C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C47#13711yescharacteristic
162D188—42no—
163D188—42no—
164D188—82no—
165D188—82no—
166D188—82no—
167D188—81no—
168D188—81no—
169D188—81no—
170D188—81no—
171C8×C47—81no—
172C8×C47—81no—
173C47⋊C8—22no—
174C47⋊C8—42no—
175C47⋊C8—41no—
176C47⋊C8—42no—
177#9254—471nomaximal, Sylow
178C22×C47—21no—
179C22×C47—21no—
180C22×C47#38611yescharacteristic
181C22×C47C4×SD1611yescharacteristic
182C22×C47#44011yescharacteristic
183C2×D47—82no—
184C2×D47—82no—
185C2×D47—82no—
186C2×D47—162no—
187C2×D47—162no—
188C4×C47—21no—
189C4×C47—22no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—41no—
193C4×C47—41no—
194C4×C47—41no—
195C4×C47C2×#10011yescharacteristic
196C4×C47#41311yescharacteristic
197C4×C47#44811yescharacteristic
198C4×C47#44911yescharacteristic
199C4×C47#46011yescharacteristic
200C4×C47#46411yescharacteristic
201C2×#460—471no—
202Q8×M16—471no—
203#1965—471no—
204#1545—471no—
205#1487D4711yescharacteristic, complemented
206#1292—471no—
207#1283—471no—
208C2×C47—21no—
209C2×C47—21no—
210C2×C47#188011yescharacteristic
211C2×C47#190011yescharacteristic
212C2×C47#190311yescharacteristic
213D47—162no—
214C22×SD16—472no—
215C2×#127—471no—
216C2×C4×Q8C2×D4711yescharacteristic
217#460—941no—
218#460—941no—
219#460—941no—
220#460—941no—
221#446—942no—
222#446—941no—
223C8×Q8—941no—
224#431C2×D4711yescharacteristic
225#423—471no—
226#423—941no—
227#423C2×D4711yescharacteristic
228C2×C4⋊C8—471no—
229C2×#102—472no—
230C2×#100—941no—
231C2×#100—941no—
232C4×M16—471no—
233C4×M16—941no—
234#331—942no—
235Q8⋊C8—21no—
236Q8⋊C8—21no—
237C47#925411yescharacteristic, complemented, Sylow
238C22×Q8C4×D4711yescharacteristic
239C22×D4—472no—
240C22×D4—941no—
241C22×D4—941no—
242C2×SD16—942no—
243C2×SD16—942no—
244C2×SD16—1884no—
245C2×M16—21no—
246C2×M16—471no—
247C2×M16—471no—
248C2×M16—941no—
249C22×C8—472no—
250#127—941no—
251#127—941no—
252#127—941no—
253#127—941no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8D18811yescharacteristic
257C4×Q8D18811yescharacteristic
258C4×Q8C22⋊D4711yescharacteristic
259C4×Q8C22⋊D4711yescharacteristic
260C2×C4⋊C4—21no—
261C2×C4⋊C4C4×D4711yescharacteristic
262C2×C42—21no—
263C2×C42C22×D4711yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—472no—
268C4⋊C8—942no—
269C4⋊C8—942no—
270C4⋊C8—941no—
271C4⋊C8—941no—
272C4⋊C8—941no—
273#102—942no—
274#102—942no—
275#100—1882no—
276#100—1882no—
277C22⋊C8—21no—
278#97—942no—
279#97—941no—
280C4×C8—21no—
281C4×C8—942no—
282C4×C8—941no—
283C24—942no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8(C2×D47)⋊C411yescharacteristic
288C2×Q8(C2×D47)⋊C411yescharacteristic
289C2×D4—942no—
290C2×D4—942no—
291C2×D4—3762no—
292C2×D4—1882no—
293C2×D4—1882no—
294C2×D4—1881no—
295C2×D4—1881no—
296C2×D4—1881no—
297C2×D4—1881no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1881no—
301C2×D4—1881no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×C4×D4711yescharacteristic
305C22×C4D4×D4711yescharacteristic
306C22×C4#1062535211yescharacteristic
307SD16—1884no—
308SD16—1884no—
309M16—41no—
310M16—41no—
311M16—472no—
312M16—941no—
313M16—941no—
314M16—941no—
315M16—1881no—
316M16—1881no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—941no—
320C2×C8—942no—
321C2×C8—942no—
322C2×C8—941no—
323C2×C8—1882no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4(C2×D47)⋊C411yescharacteristic
330C4⋊C4(C2×D47)⋊C411yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D18811yescharacteristic
336C42D4×D4711yescharacteristic
337C42#1062535211yescharacteristic
338C42C2×C22⋊D4711yescharacteristic
339C23—3762no—
340C23—3762no—
341C23—1882no—
342C23—1882no—
343C23—1882no—
344C23(C22⋊D47)⋊C411yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#1309394012yescomplemented
350D4—3762no—
351D4—3762no—
352D4—3762no—
353D4—3761no—
354D4—3761no—
355D4—3761no—
356D4—3761no—
357D4—1882no—
358D4—1882no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4C2×(C2×D47)⋊C411yescharacteristic
369C2×C4(C22⋊D47)⋊C411yescharacteristic
370C2×C4#1309399411yescharacteristic
371C2×C4#1309399611yescharacteristic
372C2×C4#1309399611yescharacteristic
373C2×C4#1309400811yescharacteristic
374C2×C4SD16×D4711yescharacteristic
375C2×C4#1309404711yescharacteristic
376C2×C4#1309405111yescharacteristic
377C2×C4#1309405611yescharacteristic
378C8—81no—
379C8—81no—
380C8—942no—
381C8—1881no—
382C8—1882no—
383C8—1882no—
384C22—21no—
385C22—21no—
386C22—3762no—
387C22—3762no—
388C22—3762no—
389C22—7522no—
390C22—7522no—
391C22#1568900011yescharacteristic
392C22#1568920811yescharacteristic
393C22#1568922611yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#1309394011yescharacteristic
402C4#1568910511yescharacteristic
403C4#1568926311yescharacteristic
404C4#1568927111yescharacteristic
405C4#1568932111yescharacteristic
406C4#1568934711yescharacteristic
407C2—21no—
408C2—21no—
409C2—7522no—
410C2#2024374111yescharacteristic
411C2#2024384911yescharacteristic
412C2#2024386611yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1487C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1487D47
hypocenterC47#9254
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. Q8×#10625329C2semidirect
5. #20241262C2semidirect
6. #20241237C2semidirect
7. C47×#1487C2semidirect
126. #10625329SD16semidirect
142. Q8×C47#100semidirect
205. #1487D47semidirect
237. C47#9254semidirect
349. Q8#13093940semidirect

Maximal subgroups

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

subgroupindex
C2×#156893232
Q8×#106253292
#202442272
#202420552
#202412622
#202412372
C47×#14872
#925447