Finite Group Databasebeta

← #26199392

Subgroups — #26199392

Order 12032.

650 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254513C211yesmaximal, characteristic
2C2×#15689526C211yesmaximal, characteristic, complemented
3#20252100C211yesmaximal, characteristic
4#20242909C214yesmaximal, complemented
5#20242785C214yesmaximal, complemented
6#20242134C211yesmaximal, characteristic, complemented
7#20242069C211yesmaximal, characteristic, complemented
8#20242068C211yesmaximal, characteristic, complemented
9C47×#3227C211yesmaximal, characteristic
10C2×#13094031—21no—
11C2×#13094024C2211yescharacteristic
12#15689526—21no—
13#15689526—21no—
14#15689526C2212yes—
15#15689526C2212yes—
16#15689519—21no—
17#15689469—21no—
18#15689439—22no—
19#15689413C2214yes—
20#15689201—21no—
21#15689201C2212yes—
22D4×C47⋊C8—21no—
23D4×C47⋊C8C2212yes—
24#15689176C2211yescharacteristic
25C2×#13093957—21no—
26C2×#13093957—21no—
27#15689147C2211yescharacteristic
28#15689090—21no—
29C2×C4×C47⋊C8C2211yescharacteristic
30#15688875C2212yes—
31#15688873C2212yes—
32#15688872C2214yes—
33#15688871C2214yes—
34C47×#532C2211yescharacteristic
35C2×C4⋊Q8×C47C2211yescharacteristic
36C47×#498—21no—
37C47×#486—21no—
38C47×#481—22no—
39C47×#476C2214yes—
40C47×#431C2211yescharacteristic
41C22×Dic94—21no—
42#13094031—44no—
43C2×#10625329—21no—
44C2×#10625329—22no—
45C22×C47⋊C8—22no—
46C22×C47⋊C8D411yescharacteristic
47C22×C47⋊C8D411yescharacteristic
48#13094024—21no—
49#13094024—21no—
50#13094024—21no—
51#13094024C2312yes—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47—21no—
54#13093957—22no—
55#13093957—22no—
56#13093957—22no—
57#13093957—22no—
58#13093955—24no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)C2312yes—
61C4⋊(C47⋊C8)C2312yes—
62#13093945—24no—
63C22⋊(C47⋊C8)—21no—
64C22⋊(C47⋊C8)—22no—
65C22⋊(C47⋊C8)—22no—
66C4×C47⋊C8—21no—
67C4×C47⋊C8D411yescharacteristic
68C4×C47⋊C8D411yescharacteristic
69C4×C47⋊C8C2311yescharacteristic
70C4×C47⋊C8C2311yescharacteristic
71C2×#41×C47—21no—
72C22×Q8×C47—21no—
73C4⋊Q8×C47—21no—
74C4⋊Q8×C47—21no—
75C4⋊Q8×C47—21no—
76C4⋊Q8×C47D411yescharacteristic
77C4⋊Q8×C47C2312yes—
78#127×C47D411yescharacteristic
79#124×C47—21no—
80#121×C47—21no—
81#121×C47D412yes—
82C4×D4×C47—21no—
83C4×D4×C47C2312yes—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47—21no—
86C2×C42×C47C2311yescharacteristic
87#106×C47—24no—
88C4⋊C8×C47C2312yes—
89#102×C47—24no—
90C22⋊C8×C47—21no—
91C4×C8×C47—21no—
92#41×C47—42no—
93#41×C47—42no—
94C2×Q8×C47—21no—
95C2×Q8×C47—22no—
96C2×Q8×C47—41no—
97C2×Q8×C47—82no—
98C2×D4×C47—22no—
99C2×D4×C47—21no—
100C2×D4×C47—21no—
101C2×D4×C47C2×D412yes—
102C22×C4×C47—22no—
103C22×C4×C47—41no—
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47#4111yescharacteristic
107C22×Dic47—41no—
108C2×Dic94—22no—
109C2×Dic94—41no—
110C2×Dic94—82no—
111C2×C8×C47—22no—
112C4⋊C4×C47—21no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—22no—
115C4⋊C4×C47—41no—
116C4⋊C4×C47—41no—
117C4⋊C4×C47C2×D412yes—
118C22⋊C4×C47—22no—
119C22⋊C4×C47—22no—
120C42×C47C2×D411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123C42×C47C2411yescharacteristic
124C4⋊Dic47—22no—
125C4⋊Dic47—22no—
126C4⋊Dic47—41no—
127C4⋊Dic47—41no—
128#10625329—42no—
129#10625329—44no—
130C2×C47⋊C8—22no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—44no—
134C2×C47⋊C8SD1614yes—
135C2×C47⋊C8C2×D411yescharacteristic
136C2×C47⋊C8C2×D411yescharacteristic
137C2×C47⋊C8C2×D411yescharacteristic
138C2×C47⋊C8C2×D411yescharacteristic
139C23×C47—22no—
140C23×C47#12111yescharacteristic
141Q8×C47—42no—
142Q8×C47—82no—
143Q8×C47—82no—
144D4×C47—24no—
145D4×C47—44no—
146D4×C47—42no—
147D4×C47—42no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47—82no—
155C2×C4×C47#12111yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×C4×C47#12111yescharacteristic
158C2×C4×C47C2×SD1612yes—
159C2×C4×C47#13712yes—
160C2×C4×C47C22×D411yescharacteristic
161C2×C4×C47C22×D411yescharacteristic
162C2×C4×C47C2×#4111yescharacteristic
163C2×Dic47—42no—
164C2×Dic47—82no—
165Dic94—82no—
166Dic94—82no—
167C8×C47—42no—
168C47⋊C8—21no—
169C47⋊C8—21no—
170C47⋊C8—42no—
171C47⋊C8—42no—
172C47⋊C8C2×SD1612yes—
173C47⋊C8C2×SD1612yes—
174#19193—471nomaximal, Sylow
175C22×C47—21no—
176C22×C47—21no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47#46112yes—
181C22×C47C2×#12111yescharacteristic
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—42no—
185C4×C47—82no—
186C4×C47#46112yes—
187C4×C47#47212yes—
188C4×C47C22×SD1611yescharacteristic
189C4×C47C2×#13711yescharacteristic
190C4×C47#57812yes—
191Dic47—82no—
192#3227—471no—
193#3227D4711yescharacteristic
194C2×#500—471no—
195#2931—471no—
196#1695—474no—
197#1675—474no—
198#1564—471no—
199#1548—471no—
200#1548—471no—
201C2×C47—22no—
202C2×C47—82no—
203C2×C47C2×#46111yescharacteristic
204C2×C47#305911yescharacteristic
205C2×C47C2×#47211yescharacteristic
206C2×#131—941no—
207#532C2×D4711yescharacteristic
208C2×C4⋊Q8—471no—
209C2×C4⋊Q8C2×D4711yescharacteristic
210#500—472no—
211#500—472no—
212#500—941no—
213#500—941no—
214#498—21no—
215#498—941no—
216#486—21no—
217#486—941no—
218#481—22no—
219#481—942no—
220#476—474no—
221#476C2×D4714yes—
222#436—472no—
223#436—941no—
224C8×D4—472no—
225C8×D4—941no—
226#431—471no—
227#431C2×D4711yescharacteristic
228C2×#107—941no—
229C2×#107—941no—
230#423—471no—
231#408—941no—
232C2×C4×C8—471no—
233#335—472no—
234#334—472no—
235#333—474no—
236#333—474no—
237C47#1919311yescharacteristic, complemented, Sylow
238C2×#41—21no—
239C22×Q8—21no—
240C22×Q8—941no—
241#131—1884no—
242C2×M16—942no—
243C2×M16—941no—
244C22×C8—471no—
245C22×C8—471no—
246C22×C8—942no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—472no—
251C4⋊Q8—941no—
252C4⋊Q8—941no—
253C4⋊Q8—941no—
254C4⋊Q8C22⋊D4711yescharacteristic
255C4⋊Q8C22×D4712yes—
256#127C22⋊D4711yescharacteristic
257#124—21no—
258#121—21no—
259#121C22⋊D4712yes—
260C4×D4—21no—
261C4×D4C22×D4712yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—941no—
265C2×C4⋊C4—941no—
266C2×C42C22×D4711yescharacteristic
267#107—942no—
268#107—942no—
269#107—942no—
270#107—942no—
271#106—24no—
272#106—944no—
273C4⋊C8—472no—
274C4⋊C8—472no—
275C4⋊C8—941no—
276C4⋊C8C22×D4712yes—
277#102—24no—
278#102—944no—
279C22⋊C8—21no—
280C22⋊C8—942no—
281C22⋊C8—942no—
282C22⋊C8—941no—
283C4×C8—21no—
284C4×C8—471no—
285C4×C8—471no—
286C4×C8—471no—
287C4×C8—471no—
288C4×C8—941no—
289#41—42no—
290#41—42no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8—942no—
296C2×Q8—3762no—
297C2×Q8—1881no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4C2×C22⋊D4712yes—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—1881no—
305C22×C4D4×D4711yescharacteristic
306C22×C4#1062535211yescharacteristic
307C22×C4C2×C22⋊D4711yescharacteristic
308M16—1884no—
309M16—1882no—
310C2×C8—22no—
311C2×C8—474no—
312C2×C8—471no—
313C2×C8—471no—
314C2×C8—471no—
315C2×C8—471no—
316C2×C8—942no—
317C2×C8—942no—
318C2×C8—942no—
319C2×C8—1884no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—942no—
326C4⋊C4—942no—
327C4⋊C4—1881no—
328C4⋊C4—1881no—
329C4⋊C4C2×C22⋊D4712yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C42D4×D4711yescharacteristic
333C42#1062535211yescharacteristic
334C42C2×C22⋊D4711yescharacteristic
335C42C23×D4711yescharacteristic
336C23—22no—
337C23#1309399611yescharacteristic
338Q8—42no—
339Q8—82no—
340Q8—82no—
341Q8—3762no—
342Q8—3762no—
343D4—24no—
344D4—44no—
345D4—42no—
346D4—42no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—3762no—
355C2×C4—1882no—
356C2×C4#1309399611yescharacteristic
357C2×C4#1309399611yescharacteristic
358C2×C4#1309399611yescharacteristic
359C2×C4SD16×D4712yes—
360C2×C4#1309405312yes—
361C2×C4C2×D4×D4711yescharacteristic
362C2×C4C22×C22⋊D4711yescharacteristic
363C2×C4C2×#1062535211yescharacteristic
364C8—42no—
365C8—472no—
366C8—472no—
367C8—941no—
368C8—941no—
369C8—1882no—
370C8—1882no—
371C22—21no—
372C22—21no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22#1568932612yes—
377C22C2×#1309399611yescharacteristic
378C4—21no—
379C4—21no—
380C4—42no—
381C4—82no—
382C4—3762no—
383C4#1568932612yes—
384C4#1568939112yes—
385C4C2×SD16×D4711yescharacteristic
386C4C2×#1309405311yescharacteristic
387C4#1568993812yes—
388C2—22no—
389C2—82no—
390C2C2×#1568932611yescharacteristic
391C2#2025304011yescharacteristic
392C2C2×#1568939111yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3227C2
socleC22×C47C2×#121
prime radicalC47×#532C22
hypercenter#3227D47
hypocenterC47#19193
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. C2×#15689526C2semidirect
4. #20242909C2semidirect
5. #20242785C2semidirect
6. #20242134C2semidirect
7. #20242069C2semidirect
8. #20242068C2semidirect
237. C47#19193semidirect

Maximal subgroups

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

subgroupindex
#202545132
C2×#156895262
#202521002
#202429092
#202427852
#202421342
#202420692
#202420682
C47×#32272
#1919347