Finite Group Databasebeta

← #26112965

Subgroups — #26112965

Order 12032.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252106C211yesmaximal, characteristic, complemented
2#20244213C211yesmaximal, characteristic, complemented
3#20242056C211yesmaximal, characteristic, complemented
4#20241948C211yesmaximal, characteristic, complemented
5#20241247C211yesmaximal, characteristic, complemented
6#20241218C211yesmaximal, characteristic, complemented
7C47×#3205C211yesmaximal, characteristic
8C2×#13094031—21no—
9#15689195—22no—
10#15689195—22no—
11#15689176C2211yescharacteristic
12#15689147C2211yescharacteristic
13C2×#13093945—21no—
14C2×#13093945—21no—
15#15689090—21no—
16C2×#13093935C2211yescharacteristic
17#15688989C2211yescharacteristic
18#15688867—22no—
19#15688857—22no—
20C47×#532C2211yescharacteristic
21C2×#127×C47C2211yescharacteristic
22C47×#493C412yescomplemented
23C47×#486C412yescomplemented
24C47×#459—22no—
25C47×#447—22no—
26C47×#431C2211yescharacteristic
27#13094031—42no—
28#13094031—42no—
29C2×#10625329—22no—
30C2×#10625329—21no—
31C22×C47⋊C8—22no—
32C22×C47⋊C8D411yescharacteristic
33C22×C47⋊C8D411yescharacteristic
34C2×C4⋊Dic47—21no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)D412yes—
38#13093945—42no—
39#13093945—42no—
40C22⋊(C47⋊C8)—24no—
41C22⋊(C47⋊C8)—21no—
42#13093935—21no—
43#13093935—21no—
44C4×C47⋊C8—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C2×#41×C47—21no—
48C22×D4×C47—21no—
49C22×D4×C47—21no—
50C22×D4×C47—21no—
51C2×D8×C47—24no—
52C4⋊Q8×C47C2×C411yescharacteristic
53#127×C47—21no—
54#127×C47—21no—
55#127×C47—21no—
56#127×C47C2×C412yes—
57#127×C47C2×C411yescharacteristic
58#124×C47—21no—
59#121×C47—21no—
60#121×C47D412yes—
61C4×D4×C47—21no—
62C4×D4×C47D412yes—
63C2×C42×C47C2311yescharacteristic
64C4⋊C8×C47—21no—
65#102×C47—24no—
66C22⋊C8×C47—21no—
67C4×C8×C47C2×C412yes—
68C24×C47—41no—
69#41×C47—41no—
70#41×C47—41no—
71#41×C47—41no—
72#41×C47—41no—
73C2×Q8×C47—21no—
74C2×D4×C47—22no—
75C2×D4×C47—22no—
76C2×D4×C47—21no—
77C2×D4×C47—22no—
78C2×D4×C47—21no—
79C2×D4×C47—22no—
80C2×D4×C47—41no—
81C2×D4×C47—41no—
82C2×D4×C47—41no—
83C2×D4×C47—41no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—82no—
88C2×D4×C47—82no—
89C2×D4×C47C22⋊C412yes—
90C22×C4×C47—22no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×Dic47—41no—
95D8×C47—84no—
96C2×C8×C47—22no—
97C4⋊C4×C47—21no—
98C4⋊C4×C47C22⋊C412yes—
99C22⋊C4×C47—22no—
100C22⋊C4×C47—22no—
101C42×C47C22×C411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47C2×D411yescharacteristic
104C42×C47#4111yescharacteristic
105C4⋊Dic47—41no—
106C4⋊Dic47—41no—
107#10625329—44no—
108#10625329—41no—
109#10625329—41no—
110C2×C47⋊C8—21no—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—22no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116C2×C47⋊C8—44no—
117C2×C47⋊C8—41no—
118C2×C47⋊C8D812yescomplemented
119C23×C47—22no—
120C23×C47—42no—
121C23×C47—82no—
122C23×C47—82no—
123C23×C47—81no—
124C23×C47—81no—
125C23×C47#12111yescharacteristic
126Q8×C47—41no—
127Q8×C47—41no—
128D4×C47—42no—
129D4×C47—44no—
130D4×C47—42no—
131D4×C47—42no—
132D4×C47—41no—
133D4×C47—42no—
134D4×C47—41no—
135D4×C47—42no—
136D4×C47—81no—
137D4×C47—82no—
138D4×C47—82no—
139D4×C47—82no—
140D4×C47—81no—
141D4×C47—81no—
142C2×C4×C47—22no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47C2×C22⋊C411yescharacteristic
148C2×C4×C47C4×D411yescharacteristic
149C2×C4×C47C4×D411yescharacteristic
150C2×C4×C47#12011yescharacteristic
151C2×C4×C47#12111yescharacteristic
152C2×C4×C47#12311yescharacteristic
153C2×C4×C47C2×D811yescharacteristic
154C2×C4×C47#13511yescharacteristic
155C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
157C2×Dic47—81no—
158C2×Dic47—81no—
159C2×Dic47—81no—
160C8×C47—42no—
161C47⋊C8—22no—
162C47⋊C8—42no—
163C47⋊C8—41no—
164C47⋊C8—81no—
165#9293—471nomaximal, Sylow
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—82no—
172C22×C47—82no—
173C22×C47—82no—
174C22×C47—162no—
175C22×C47—162no—
176C22×C47#38611yescharacteristic
177C22×C47#45911yescharacteristic
178C22×C47#46311yescharacteristic
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—42no—
184C4×C47C4×D811yescharacteristic
185C4×C47#44211yescharacteristic
186C4×C47#44711yescharacteristic
187C4×C47#44911yescharacteristic
188Dic47—161no—
189#3205D4711yescharacteristic
190#2932—471no—
191#1961—471no—
192#1545—471no—
193#1517—471no—
194#1286—471no—
195#1275—471no—
196C2×C47—21no—
197C2×C47—21no—
198C2×C47—82no—
199C2×C47—162no—
200C2×C47#187911yescharacteristic
201C2×C47#190311yescharacteristic
202C2×C47#191011yescharacteristic
203C2×#131—941no—
204#532C2×D4711yescharacteristic
205C2×#127C2×D4711yescharacteristic
206#493Dic4712yescomplemented
207#486Dic4712yescomplemented
208#459—22no—
209#447—22no—
210#435—942no—
211#435—942no—
212#431—471no—
213#431C2×D4711yescharacteristic
214#423—471no—
215C2×#102—941no—
216C2×#102—941no—
217#408—941no—
218C2×#97—471no—
219#382—471no—
220#331—942no—
221#327—942no—
222C47#929311yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—21no—
227C2×D8—24no—
228#131—1882no—
229#131—1882no—
230C2×M16—942no—
231C2×M16—941no—
232C22×C8—471no—
233C22×C8—471no—
234C22×C8—942no—
235C4⋊Q8C2×Dic4711yescharacteristic
236#127—21no—
237#127—21no—
238#127—21no—
239#127C2×Dic4712yes—
240#127C2×Dic4711yescharacteristic
241#124—21no—
242#121—21no—
243#121C22⋊D4712yes—
244C4×D4—21no—
245C4×D4C22⋊D4712yes—
246C2×C4⋊C4—941no—
247C2×C42C22×D4711yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—472no—
250C4⋊C8—941no—
251C4⋊C8—941no—
252#102—24no—
253#102—1882no—
254#102—1882no—
255C22⋊C8—21no—
256C22⋊C8—944no—
257C22⋊C8—941no—
258#97—941no—
259#97—941no—
260C4×C8—941no—
261C4×C8C2×Dic4712yes—
262#95—941no—
263#95—941no—
264C24—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4C22⋊Dic4712yes—
286C22×C4—22no—
287C22×C4—1881no—
288C22×C4D4×D4711yescharacteristic
289C22×C4#1062535211yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291D8—84no—
292M16—1884no—
293M16—1881no—
294M16—1881no—
295C2×C8—22no—
296C2×C8—472no—
297C2×C8—941no—
298C2×C8—941no—
299C2×C8—941no—
300C2×C8—942no—
301C2×C8—941no—
302C2×C8—941no—
303C2×C8—1884no—
304C2×C8—1881no—
305C4⋊C4—21no—
306C4⋊C4—1881no—
307C4⋊C4—1881no—
308C4⋊C4C22⋊Dic4712yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42D4×D4711yescharacteristic
312C42#1062535211yescharacteristic
313C42C22×Dic4711yescharacteristic
314C42C2×C22⋊D4711yescharacteristic
315C23—22no—
316C23—42no—
317C23—82no—
318C23—82no—
319C23—81no—
320C23—81no—
321C23#1309399611yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—44no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—42no—
330D4—41no—
331D4—42no—
332D4—81no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—81no—
337D4—81no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—3761no—
344C2×C4—3761no—
345C2×C4—3761no—
346C2×C4C2×C22⋊Dic4711yescharacteristic
347C2×C4D4×Dic4711yescharacteristic
348C2×C4D4×Dic4711yescharacteristic
349C2×C4#1309399211yescharacteristic
350C2×C4#1309399611yescharacteristic
351C2×C4#1309400511yescharacteristic
352C2×C4D8×D4711yescharacteristic
353C2×C4#1309404311yescharacteristic
354C2×C4#1309404811yescharacteristic
355C2×C4#1309404811yescharacteristic
356C8—42no—
357C8—942no—
358C8—3761no—
359C8—1882no—
360C8—1881no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—162no—
370C22—162no—
371C22#1568900311yescharacteristic
372C22#1568931611yescharacteristic
373C22#1568933611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—7521no—
380C4D8×Dic4711yescharacteristic
381C4#1568923511yescharacteristic
382C4#1568925511yescharacteristic
383C4#1568926511yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—162no—
388C2#2024373811yescharacteristic
389C2#2024387011yescharacteristic
390C2#2024391611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3205C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3205D47
hypocenterC47#9293
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #20252106C2semidirect
2. #20244213C2semidirect
3. #20242056C2semidirect
4. #20241948C2semidirect
5. #20241247C2semidirect
6. #20241218C2semidirect
22. C47×#493C4semidirect
23. C47×#486C4semidirect
118. C2×C47⋊C8D8semidirect
206. #493Dic47semidirect
207. #486Dic47semidirect
222. C47#9293semidirect

Maximal subgroups

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

subgroupindex
#202521062
#202442132
#202420562
#202419482
#202412472
#202412182
C47×#32052
#929347