Finite Group Databasebeta

← #26199403

Subgroups — #26199403

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#20254705C212yesmaximal
2#20253902C211yesmaximal, characteristic, complemented
3#20252101C211yesmaximal, characteristic
4#20242912C212yesmaximal, complemented
5#20242902C212yesmaximal, complemented
6#20242764C214yesmaximal, complemented
7#20242076C212yesmaximal, complemented
8C2×C47×#334C211yesmaximal, 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×#334C2212yes—
35C47×#334C2212yes—
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—22no—
125C2×C4×C47—41no—
126C2×C4×C47#12111yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47C2×SD1612yes—
130C2×C4×C47#13712yes—
131C2×C4×C47C22×D411yescharacteristic
132C2×C4×C47C22×D411yescharacteristic
133C2×C4×C47C2×#4111yescharacteristic
134C22⋊D47—44no—
135C2×Dic47—22no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138D188—24no—
139D188—42no—
140D188—42no—
141C4×D47—42no—
142C4×D47—42no—
143Dic94—42no—
144Dic94—42no—
145Dic94—42no—
146Dic94—82no—
147C8×C47—21no—
148C8×C47—21no—
149C8×C47—44no—
150C8×C47C2×SD1612yes—
151C8×C47C2×SD1612yes—
152C47⋊C8—42no—
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#46112yes—
165C4×C47C22×SD1611yescharacteristic
166C4×C47C2×#13711yescharacteristic
167C4×C47#57712yes—
168Dic47—42no—
169Dic47—82no—
170Dic47—82no—
171#3246—472no—
172#3160—471no—
173#2931—471no—
174#1695—472no—
175#1693—472no—
176#1671—474no—
177C2×#334D4711yescharacteristic, complemented
178#1550—472no—
179C2×C47—22no—
180C2×C47C2×#46111yescharacteristic
181C2×C47#306011yescharacteristic
182C2×C47#306711yescharacteristic
183D47—82no—
184C2×#131—941no—
185#533—472no—
186#532—471no—
187#500—471no—
188#500—471no—
189#499—941no—
190#499—941no—
191#498—472no—
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—941no—
202#427—941no—
203#423—471no—
204C2×C4⋊C8C2×D4712yes—
205#408—941no—
206C2×C4×C8C2×D4711yescharacteristic
207#334C2×D4712yes—
208#334C2×D4712yes—
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—22no—
218C22×C8—942no—
219C22×C8D18811yescharacteristic
220C22×C8D18811yescharacteristic
221C4⋊Q8—471no—
222C4⋊Q8—472no—
223C4⋊Q8—472no—
224#127—471no—
225#125—942no—
226#124—941no—
227C22⋊Q8—942no—
228C22⋊Q8—942no—
229#121—472no—
230#121—941no—
231C4×Q8—942no—
232C4×Q8—942no—
233C4×D4—472no—
234C4×D4—941no—
235#117—942no—
236C2×C42C22×D4711yescharacteristic
237#107—941no—
238#107—941no—
239#107—941no—
240#107—941no—
241#106—944no—
242#106—942no—
243#106—942no—
244C4⋊C8—22no—
245C4⋊C8—472no—
246C4⋊C8—941no—
247C4⋊C8C22×D4714yes—
248#103—944no—
249#102—944no—
250C22⋊C8—942no—
251C22⋊C8—942no—
252C4×C8D18811yescharacteristic
253C4×C8D18811yescharacteristic
254C4×C8C22×D4711yescharacteristic
255C4×C8C22×D4711yescharacteristic
256#41—1882no—
257#41—1882no—
258C2×Q8—941no—
259C2×Q8—942no—
260C2×Q8—942no—
261C2×D4—472no—
262C2×D4—942no—
263C2×D4—941no—
264C2×D4—941no—
265C22×C4—942no—
266C22×C4C2×D18811yescharacteristic
267C22×C4D4×D4711yescharacteristic
268C22×C4#1062535411yescharacteristic
269SD16—3764no—
270M16—1882no—
271M16—1884no—
272C2×C8—22no—
273C2×C8—24no—
274C2×C8—82no—
275C2×C8—942no—
276C2×C8—1884no—
277C2×C8#1062532514yescomplemented
278C2×C8C2×D18811yescharacteristic
279C2×C8C2×D18811yescharacteristic
280C2×C8C2×D18811yescharacteristic
281C2×C8C2×D18811yescharacteristic
282C4⋊C4—472no—
283C4⋊C4—941no—
284C4⋊C4—942no—
285C4⋊C4—942no—
286C4⋊C4—942no—
287C4⋊C4—942no—
288C4⋊C4—1882no—
289C4⋊C4—1882no—
290C22⋊C4—942no—
291C22⋊C4—942no—
292C22⋊C4—1882no—
293C42—1882no—
294C42C2×D18811yescharacteristic
295C42D4×D4711yescharacteristic
296C42#1062535411yescharacteristic
297C42C23×D4711yescharacteristic
298C23—942no—
299C23#1309399911yescharacteristic
300Q8—3762no—
301Q8—1882no—
302Q8—1882no—
303Q8—1882no—
304D4—944no—
305D4—1882no—
306D4—1884no—
307D4—1882no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—942no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4—1882no—
314C2×C4—1882no—
315C2×C4#1309399911yescharacteristic
316C2×C4#1309399911yescharacteristic
317C2×C4#1309399911yescharacteristic
318C2×C4C2×#1062532512yes—
319C2×C4#1309405712yes—
320C2×C4C2×D4×D4711yescharacteristic
321C2×C4C22×D18811yescharacteristic
322C2×C4C2×#1062535411yescharacteristic
323C8—21no—
324C8—21no—
325C8—44no—
326C8—1882no—
327C8C2×#1062532512yes—
328C8C2×#1062532512yes—
329C22—21no—
330C22—21no—
331C22—1882no—
332C22—1882no—
333C22—1882no—
334C22#1568932912yes—
335C22C2×#1309399911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—3762no—
340C4—3762no—
341C4—1882no—
342C4#1568932912yes—
343C4C22×#1062532511yescharacteristic
344C4C2×#1309405711yescharacteristic
345C4#1568992912yes—
346C2—22no—
347C2—3762no—
348C2C2×#1568932911yescharacteristic
349C2#2025305111yescharacteristic
350C2#2025310911yescharacteristic
subgroupG/H
centerC22C2×#13093999
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#334C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenterC2×#334D47
hypocenterC47#19194
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #20253902C2semidirect
4. #20242912C2semidirect
5. #20242902C2semidirect
6. #20242764C2semidirect
7. #20242076C2semidirect
8. C2×C47×#334C2semidirect
96. C2×C8×C47SD16semidirect
177. C2×#334D47semidirect
211. C47#19194semidirect
277. 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
#202429122
#202429022
#202427642
#202420762
C2×C47×#3342
#1919447