Finite Group Databasebeta

← #26199398

Subgroups — #26199398

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254700C212yesmaximal
2#20252101C211yesmaximal, characteristic
3#20242908C212yesmaximal, complemented
4#20242898C212yesmaximal, complemented
5#20242760C214yesmaximal, complemented
6C2×#15688874C211yesmaximal, characteristic, complemented
7#20242078C212yesmaximal, complemented
8C47×#3160C211yesmaximal, characteristic, complemented
9C2×#13094030—21no—
10#15689673C2211yescharacteristic
11#15689410C2214yes—
12#15689408—22no—
13#15689354—24no—
14#15689325C2214yes—
15#15689323—22no—
16#15689202—21no—
17#15689202C2212yes—
18C8×D188—21no—
19C8×D188C2212yes—
20#15689147C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2212yes—
22#15689089—21no—
23#15688874C2212yes—
24#15688874C2212yes—
25#15688872C2214yes—
26#15688869C2214yes—
27C47×#533C2212yes—
28C47×#500C2211yescharacteristic
29C47×#500C2211yescharacteristic
30C47×#499—21no—
31C47×#499—21no—
32C47×#498C2212yes—
33C47×#427—21no—
34C47×#427—21no—
35C2×C4×C8×C47C2211yescharacteristic
36C2×#10625350—21no—
37C2×#10625336—24no—
38#13094030—44no—
39C2×#10625329—21no—
40C2×#10625324—22no—
41C22×C47⋊C8—22no—
42C2×C8×D47—22no—
43#13094024D411yescharacteristic
44#13094021D411yescharacteristic
45#13094012—21no—
46#13093998—21no—
47#13093998D412yes—
48C4×D188—21no—
49C4×D188C2312yes—
50#13093955—24no—
51C4⋊(C47⋊C8)—22no—
52C4⋊(C47⋊C8)—21no—
53C4⋊(C47⋊C8)C2314yes—
54C4⋊(C47⋊C8)C2312yes—
55#13093948—24no—
56#13093944—24no—
57(C2×D47)⋊C8—22no—
58(C2×D47)⋊C8—22no—
59C22×C8×C47D411yescharacteristic
60C22×C8×C47D411yescharacteristic
61C4⋊Q8×C47C2312yes—
62C4⋊Q8×C47C2312yes—
63#125×C47—22no—
64C22⋊Q8×C47—22no—
65C22⋊Q8×C47—22no—
66C4×Q8×C47—22no—
67C4×Q8×C47—22no—
68#117×C47—22no—
69C2×C42×C47C2311yescharacteristic
70#107×C47—21no—
71#107×C47—21no—
72#107×C47—21no—
73#107×C47—21no—
74#106×C47—22no—
75#106×C47—22no—
76C4×C8×C47D411yescharacteristic
77C4×C8×C47D411yescharacteristic
78C4×C8×C47C2311yescharacteristic
79C4×C8×C47C2311yescharacteristic
80C2×Q8×C47—22no—
81C2×Q8×C47—22no—
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47#4111yescharacteristic
85C2×C22⋊D47—22no—
86#10625350—42no—
87#10625350—42no—
88C2×D188—21no—
89C2×D188—21no—
90C2×D188C2×D412yes—
91C2×C4×D47—22no—
92C2×Dic94—21no—
93C2×C8×C47—22no—
94C2×C8×C47SD1614yescomplemented
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C4⋊C4×C47—22no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—42no—
104C4⋊C4×C47—42no—
105C22⋊C4×C47—42no—
106C42×C47—42no—
107C42×C47C2×D411yescharacteristic
108C42×C47C2×D411yescharacteristic
109C42×C47#4111yescharacteristic
110C42×C47C2411yescharacteristic
111#10625336—84no—
112(C2×D47)⋊C4—22no—
113(C2×D47)⋊C4—22no—
114C4⋊Dic47—21no—
115C4⋊Dic47C2×D412yes—
116#10625329—42no—
117C2×C47⋊C8—22no—
118C2×C47⋊C8—24no—
119C2×C47⋊C8—82no—
120#10625324—44no—
121C8×D47—44no—
122C23×C47#12111yescharacteristic
123C22×D47—22no—
124Q8×C47—42no—
125Q8×C47—42no—
126Q8×C47—82no—
127C2×C4×C47—22no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47#12111yescharacteristic
133C2×C4×C47#12111yescharacteristic
134C2×C4×C47C2×SD1612yes—
135C2×C4×C47#13712yes—
136C2×C4×C47C22×D411yescharacteristic
137C2×C4×C47C22×D411yescharacteristic
138C2×C4×C47C2×#4111yescharacteristic
139C22⋊D47—44no—
140C2×Dic47—22no—
141D188—24no—
142D188—42no—
143D188—42no—
144C4×D47—42no—
145C4×D47—42no—
146Dic94—42no—
147C8×C47—21no—
148C8×C47—21no—
149C8×C47C2×SD1612yes—
150C8×C47C2×SD1612yes—
151C47⋊C8—42no—
152C47⋊C8—44no—
153#19194—471nomaximal, Sylow
154C22×C47—21no—
155C22×C47—21no—
156C22×C47#46112yes—
157C22×C47C2×#12111yescharacteristic
158C2×D47—42no—
159C2×D47—42no—
160C2×D47—42no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—82no—
165C4×C47—82no—
166C4×C47#46112yes—
167C4×C47C22×SD1611yescharacteristic
168C4×C47C2×#13711yescharacteristic
169C4×C47#57712yes—
170Dic47—42no—
171#3246—472no—
172#3160D4711yescharacteristic, complemented
173#2931—471no—
174#1695—472no—
175#1693—472no—
176#1671—474no—
177C2×#334—471no—
178#1550—472no—
179C2×C47—22no—
180C2×C47C2×#46111yescharacteristic
181C2×C47#306011yescharacteristic
182C2×C47#306711yescharacteristic
183D47—82no—
184C2×#131—941no—
185#533C2×D4712yes—
186#532—471no—
187#500C2×D4711yescharacteristic
188#500C2×D4711yescharacteristic
189#499—21no—
190#499—21no—
191#498C2×D4712yes—
192#476—474no—
193#475—942no—
194#465—944no—
195#461—474no—
196#460—942no—
197#436—472no—
198#436—941no—
199C8×D4—472no—
200C8×D4—941no—
201#427—21no—
202#427—21no—
203#423—471no—
204C2×C4⋊C8—472no—
205#408—941no—
206C2×C4×C8C2×D4711yescharacteristic
207#334—472no—
208#334—472no—
209#333—474no—
210#332—474no—
211C47#1919411yescharacteristic, complemented, Sylow
212C2×#41—941no—
213C2×SD16—944no—
214#131—1884no—
215C2×M16—941no—
216C2×M16—942no—
217C22×C8—942no—
218C22×C8—942no—
219C22×C8C22⋊D4711yescharacteristic
220C22×C8C22⋊D4711yescharacteristic
221C4⋊Q8—471no—
222C4⋊Q8C22×D4712yes—
223C4⋊Q8C22×D4712yes—
224#127—471no—
225#125—22no—
226#124—941no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229#121—472no—
230#121—941no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×D4—472no—
234C4×D4—941no—
235#117—22no—
236C2×C42C22×D4711yescharacteristic
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#106—22no—
242#106—22no—
243#106—944no—
244C4⋊C8—472no—
245C4⋊C8—474no—
246C4⋊C8—942no—
247C4⋊C8—941no—
248#103—944no—
249#102—944no—
250C22⋊C8—942no—
251C22⋊C8—942no—
252C4×C8C22⋊D4711yescharacteristic
253C4×C8C22⋊D4711yescharacteristic
254C4×C8C22×D4711yescharacteristic
255C4×C8C22×D4711yescharacteristic
256#41—1882no—
257#41—1882no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—941no—
261C2×D4—472no—
262C2×D4—942no—
263C2×D4—941no—
264C2×D4—941no—
265C22×C4—942no—
266C22×C4D4×D4711yescharacteristic
267C22×C4#1062535211yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269SD16—3764no—
270M16—1882no—
271M16—1884no—
272C2×C8—22no—
273C2×C8—944no—
274C2×C8—942no—
275C2×C8—3762no—
276C2×C8—1884no—
277C2×C8#1062533614yescomplemented
278C2×C8C2×C22⋊D4711yescharacteristic
279C2×C8C2×C22⋊D4711yescharacteristic
280C2×C8C2×C22⋊D4711yescharacteristic
281C2×C8C2×C22⋊D4711yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—472no—
289C4⋊C4—941no—
290C22⋊C4—42no—
291C22⋊C4—942no—
292C22⋊C4—942no—
293C42—42no—
294C42D4×D4711yescharacteristic
295C42#1062535211yescharacteristic
296C42C2×C22⋊D4711yescharacteristic
297C42C23×D4711yescharacteristic
298C23—942no—
299C23#1309399511yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—82no—
303Q8—1882no—
304D4—944no—
305D4—1882no—
306D4—1884no—
307D4—1882no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—942no—
313C2×C4—1882no—
314C2×C4—1882no—
315C2×C4#1309399511yescharacteristic
316C2×C4#1309399511yescharacteristic
317C2×C4#1309399511yescharacteristic
318C2×C4C2×#1062533612yes—
319C2×C4#1309405512yes—
320C2×C4C2×D4×D4711yescharacteristic
321C2×C4C22×C22⋊D4711yescharacteristic
322C2×C4C2×#1062535211yescharacteristic
323C8—21no—
324C8—21no—
325C8—1882no—
326C8—1884no—
327C8C2×#1062533612yes—
328C8C2×#1062533612yes—
329C22—21no—
330C22—21no—
331C22—1882no—
332C22—1882no—
333C22—1882no—
334C22#1568932512yes—
335C22C2×#1309399511yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—82no—
341C4—1882no—
342C4#1568932512yes—
343C4C22×#1062533611yescharacteristic
344C4C2×#1309405511yescharacteristic
345C4#1568993012yes—
346C2—22no—
347C2—3762no—
348C2C2×#1568932511yescharacteristic
349C2#2025304611yescharacteristic
350C2#2025310411yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3160C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenter#3160D47
hypocenterC47#19194
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
3. #20242908C2semidirect
4. #20242898C2semidirect
5. #20242760C2semidirect
6. C2×#15688874C2semidirect
7. #20242078C2semidirect
8. C47×#3160C2semidirect
94. C2×C8×C47SD16semidirect
172. #3160D47semidirect
211. C47#19194semidirect
277. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202547002
#202521012
#202429082
#202428982
#202427602
C2×#156888742
#202420782
C47×#31602
#1919447