Finite Group Databasebeta

← #26199412

Subgroups — #26199412

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254705C212yesmaximal
2#20253902C211yesmaximal, characteristic, complemented
3#20252101C211yesmaximal, characteristic
4#20242888C214yesmaximal, complemented
5#20242788C212yesmaximal, complemented
6#20242778C212yesmaximal, complemented
7#20242076C212yesmaximal, complemented
8C2×C47×#335C211yesmaximal, characteristic, complemented
9C2×#13094030—21no—
10#15689680C2212yes—
11#15689673C2211yescharacteristic
12#15689528C2211yescharacteristic
13#15689528C2211yescharacteristic
14#15689525—21no—
15#15689525—21no—
16#15689521C2212yes—
17#15689414C2214yes—
18#15689409—22no—
19#15689352—24no—
20#15689329C2214yes—
21#15689324—22no—
22#15689202—21no—
23#15689202C2212yes—
24C8×D188—21no—
25C8×D188C2212yes—
26#15689161—21no—
27#15689161—21no—
28#15689147C2211yescharacteristic
29#15689089—21no—
30#15688871C2214yes—
31#15688870C2214yes—
32C2×C4⋊C8×C47C2212yes—
33C2×C4×C8×C47C2211yescharacteristic
34C47×#335C2212yes—
35C47×#335C2212yes—
36C2×#10625350—21no—
37C2×#10625325—24no—
38#13094030—44no—
39C2×#10625329—21no—
40C2×#10625324—22no—
41C2×C8×D47—22no—
42#13094024D411yescharacteristic
43#13094024C2312yes—
44#13094024C2312yes—
45#13094021D411yescharacteristic
46#13094016—22no—
47#13094012—21no—
48#13094003—22no—
49#13094003—22no—
50#13093998—21no—
51#13093998D412yes—
52C4×Dic94—22no—
53C4×Dic94—22no—
54C4×D188—21no—
55C4×D188C2312yes—
56#13093983—22no—
57#13093958—21no—
58#13093958—21no—
59#13093958—21no—
60#13093958—21no—
61#13093956—22no—
62#13093956—22no—
63#13093956—24no—
64C4⋊(C47⋊C8)—21no—
65C4⋊(C47⋊C8)C2312yes—
66#13093949—24no—
67#13093946—24no—
68(C2×D47)⋊C8—22no—
69(C2×D47)⋊C8—22no—
70C22×C8×C47—22no—
71C22×C8×C47D411yescharacteristic
72C22×C8×C47D411yescharacteristic
73C2×C42×C47C2311yescharacteristic
74C4⋊C8×C47—22no—
75C4⋊C8×C47C2314yes—
76C4×C8×C47D411yescharacteristic
77C4×C8×C47D411yescharacteristic
78C4×C8×C47C2311yescharacteristic
79C4×C8×C47C2311yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C2×C22⋊D47—22no—
84#10625350—42no—
85#10625350—42no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188C2×D412yes—
89C2×C4×D47—22no—
90C2×Dic94—21no—
91C2×Dic94—22no—
92C2×Dic94—22no—
93C2×C8×C47—22no—
94C2×C8×C47—24no—
95C2×C8×C47—82no—
96C2×C8×C47SD1614yescomplemented
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47#4111yescharacteristic
104C42×C47C2411yescharacteristic
105C22⋊Dic47—42no—
106(C2×D47)⋊C4—22no—
107(C2×D47)⋊C4—22no—
108C4⋊Dic47—21no—
109C4⋊Dic47—22no—
110C4⋊Dic47—22no—
111C4⋊Dic47—22no—
112C4⋊Dic47—22no—
113C4⋊Dic47C2×D412yes—
114Dic47⋊C4—42no—
115Dic47⋊C4—42no—
116C4×Dic47—42no—
117#10625329—42no—
118C2×C47⋊C8—22no—
119#10625325—84no—
120#10625324—44no—
121C8×D47—44no—
122C23×C47#12111yescharacteristic
123C22×D47—22no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—41no—
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47#12111yescharacteristic
130C2×C4×C47C2×SD1612yes—
131C2×C4×C47#13712yes—
132C2×C4×C47C22×D411yescharacteristic
133C2×C4×C47C22×D411yescharacteristic
134C2×C4×C47C2×#4111yescharacteristic
135C22⋊D47—44no—
136C2×Dic47—22no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139D188—22no—
140D188—22no—
141D188—42no—
142D188—42no—
143C4×D47—42no—
144C4×D47—42no—
145Dic94—42no—
146Dic94—42no—
147Dic94—42no—
148Dic94—82no—
149C8×C47—21no—
150C8×C47—21no—
151C8×C47—44no—
152C8×C47C2×SD1612yes—
153C8×C47C2×SD1612yes—
154C47⋊C8—42no—
155#19195—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47#46112yes—
159C22×C47C2×#12111yescharacteristic
160C2×D47—42no—
161C2×D47—42no—
162C2×D47—42no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47#46112yes—
167C4×C47C22×SD1611yescharacteristic
168C4×C47C2×#13711yescharacteristic
169C4×C47#57611yescharacteristic
170C4×C47#57811yescharacteristic
171Dic47—42no—
172Dic47—82no—
173Dic47—82no—
174#3246—472no—
175#3160—471no—
176#2931—471no—
177#1691—474no—
178#1675—472no—
179#1673—472no—
180C2×#335D4711yescharacteristic, complemented
181#1550—472no—
182C2×C47—22no—
183C2×C47C2×#46111yescharacteristic
184C2×C47#306111yescharacteristic
185C2×C47#306711yescharacteristic
186D47—82no—
187C2×#131—941no—
188#533—472no—
189#532—471no—
190#500—471no—
191#500—471no—
192#499—941no—
193#499—941no—
194#498—472no—
195#476—474no—
196#475—942no—
197#465—944no—
198#461—474no—
199#460—942no—
200#436—472no—
201#436—941no—
202C8×D4—472no—
203C8×D4—941no—
204#427—941no—
205#427—941no—
206#423—471no—
207C2×C4⋊C8C2×D4712yes—
208#408—941no—
209C2×C4×C8C2×D4711yescharacteristic
210#335C2×D4712yes—
211#335C2×D4712yes—
212#333—474no—
213#332—474no—
214C47#1919511yescharacteristic, complemented, Sylow
215C2×#41—941no—
216C2×SD16—944no—
217#131—1884no—
218C2×M16—941no—
219C2×M16—942no—
220C22×C8—22no—
221C22×C8—942no—
222C22×C8D18811yescharacteristic
223C22×C8D18811yescharacteristic
224C4⋊Q8—471no—
225C4⋊Q8—472no—
226C4⋊Q8—472no—
227#127—471no—
228#125—942no—
229#124—941no—
230C22⋊Q8—942no—
231C22⋊Q8—942no—
232#121—472no—
233#121—941no—
234C4×Q8—942no—
235C4×Q8—942no—
236C4×D4—472no—
237C4×D4—941no—
238#117—942no—
239C2×C42C22×D4711yescharacteristic
240#107—941no—
241#107—941no—
242#107—941no—
243#107—941no—
244#106—944no—
245#106—942no—
246#106—942no—
247C4⋊C8—22no—
248C4⋊C8—472no—
249C4⋊C8—941no—
250C4⋊C8C22×D4714yes—
251#103—944no—
252#102—944no—
253C22⋊C8—942no—
254C22⋊C8—942no—
255C4×C8D18811yescharacteristic
256C4×C8D18811yescharacteristic
257C4×C8C22×D4711yescharacteristic
258C4×C8C22×D4711yescharacteristic
259#41—1882no—
260#41—1882no—
261C2×Q8—941no—
262C2×Q8—942no—
263C2×Q8—942no—
264C2×D4—472no—
265C2×D4—942no—
266C2×D4—941no—
267C2×D4—941no—
268C22×C4—942no—
269C22×C4C2×D18811yescharacteristic
270C22×C4D4×D4711yescharacteristic
271C22×C4#1062535411yescharacteristic
272SD16—3764no—
273M16—1882no—
274M16—1884no—
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—82no—
278C2×C8—942no—
279C2×C8—1884no—
280C2×C8#1062532514yescomplemented
281C2×C8C2×D18811yescharacteristic
282C2×C8C2×D18811yescharacteristic
283C2×C8C2×D18811yescharacteristic
284C2×C8C2×D18811yescharacteristic
285C4⋊C4—472no—
286C4⋊C4—941no—
287C4⋊C4—942no—
288C4⋊C4—942no—
289C4⋊C4—942no—
290C4⋊C4—942no—
291C4⋊C4—1882no—
292C4⋊C4—1882no—
293C22⋊C4—942no—
294C22⋊C4—942no—
295C22⋊C4—1882no—
296C42—1882no—
297C42C2×D18811yescharacteristic
298C42D4×D4711yescharacteristic
299C42#1062535411yescharacteristic
300C42C23×D4711yescharacteristic
301C23—942no—
302C23#1309399911yescharacteristic
303Q8—3762no—
304Q8—1882no—
305Q8—1882no—
306Q8—1882no—
307D4—942no—
308D4—942no—
309D4—1882no—
310D4—1884no—
311D4—1882no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—942no—
316C2×C4—1882no—
317C2×C4—1882no—
318C2×C4—1882no—
319C2×C4—1882no—
320C2×C4#1309399911yescharacteristic
321C2×C4#1309399911yescharacteristic
322C2×C4#1309399911yescharacteristic
323C2×C4C2×#1062532512yes—
324C2×C4#1309405712yes—
325C2×C4C2×D4×D4711yescharacteristic
326C2×C4C22×D18811yescharacteristic
327C2×C4C2×#1062535411yescharacteristic
328C8—21no—
329C8—21no—
330C8—44no—
331C8—1882no—
332C8C2×#1062532512yes—
333C8C2×#1062532512yes—
334C22—21no—
335C22—21no—
336C22—1882no—
337C22—1882no—
338C22—1882no—
339C22#1568932912yes—
340C22C2×#1309399911yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—3762no—
345C4—3762no—
346C4—1882no—
347C4#1568932912yes—
348C4C22×#1062532511yescharacteristic
349C4C2×#1309405711yescharacteristic
350C4#1568992411yescharacteristic
351C4#1568993611yescharacteristic
352C2—22no—
353C2—3762no—
354C2C2×#1568932911yescharacteristic
355C2#2025306011yescharacteristic
356C2#2025310911yescharacteristic
subgroupG/H
centerC22C2×#13093999
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#335C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenterC2×#335D47
hypocenterC47#19195
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #20253902C2semidirect
4. #20242888C2semidirect
5. #20242788C2semidirect
6. #20242778C2semidirect
7. #20242076C2semidirect
8. C2×C47×#335C2semidirect
96. C2×C8×C47SD16semidirect
180. C2×#335D47semidirect
214. C47#19195semidirect
280. C2×C8#10625325semidirect

Maximal subgroups

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

subgroupindex
#202547052
#202539022
#202521012
#202428882
#202427882
#202427782
#202420762
C2×C47×#3352
#1919547