Finite Group Databasebeta

← #26132900

Subgroups — #26132900

Order 12032.

398 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689240C211yesmaximal, characteristic
2C2×#15688953C211yesmaximal, characteristic, complemented
3C2×#15688930C211yesmaximal, characteristic
4#20244137C211yesmaximal, characteristic, complemented
5#20241483C211yesmaximal, characteristic, complemented
6#20241465C211yesmaximal, characteristic, complemented
7C47×#2138C211yesmaximal, characteristic
8C2×#13094031—21no—
9C22×C47⋊C16C2211yescharacteristic
10#15689240—22no—
11#15689240—22no—
12C2×#13093960C2211yescharacteristic
13C2×#13093960C2211yescharacteristic
14#15689150—21no—
15C2×#13093951—21no—
16C2×C8×Dic47C2211yescharacteristic
17#15688953—22no—
18#15688930—22no—
19#15688882—21no—
20C2×#135×C47C2211yescharacteristic
21C22×C16×C47C2211yescharacteristic
22C47×#427C2211yescharacteristic
23C47×#358C411yescharacteristic, complemented
24C47×#358C411yescharacteristic, complemented
25C47×#357C411yescharacteristic, complemented
26C47×#357C411yescharacteristic, complemented
27#13094031—24no—
28#13094031—42no—
29C2×#10625329—21no—
30C2×#10625329—21no—
31C2×#10625329—41no—
32C22×C47⋊C8—41no—
33C22×C47⋊C8D411yescharacteristic
34C2×C4×Dic47D411yescharacteristic
35C2×C47⋊C16—21no—
36C2×C47⋊C16—21no—
37C2×C47⋊C16—21no—
38#13093960—21no—
39#13093960—21no—
40#13093960—21no—
41#13093960D412yes—
42C4⋊(C47⋊C8)—41no—
43C4⋊(C47⋊C8)—41no—
44#13093951—24no—
45C8×Dic47—21no—
46C8×Dic47—21no—
47C2×#41×C47—21no—
48#135×C47—21no—
49#135×C47—21no—
50#135×C47D412yes—
51#135×C47Q812yes—
52C2×Q16×C47C2×C411yescharacteristic
53C2×SD16×C47—21no—
54C2×D8×C47C2×C411yescharacteristic
55C22×C8×C47C2311yescharacteristic
56#117×C47—21no—
57C2×C16×C47—21no—
58C2×C16×C47—21no—
59C2×C16×C47C2×C411yescharacteristic
60C2×C16×C47C2×C411yescharacteristic
61#107×C47C2×C411yescharacteristic
62#107×C47C2×C411yescharacteristic
63#106×C47—21no—
64#41×C47—22no—
65#41×C47—22no—
66#41×C47—41no—
67#41×C47—41no—
68C2×Q8×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—21no—
74Q16×C47—21no—
75Q16×C47C4⋊C412yes—
76SD16×C47—22no—
77SD16×C47—22no—
78D8×C47—21no—
79D8×C47C4⋊C412yes—
80C2×C8×C47C22×C411yescharacteristic
81C2×C8×C47C2×D411yescharacteristic
82C2×C8×C47C2×D411yescharacteristic
83C2×C8×C47C2×Q811yescharacteristic
84C2×C8×C47#4111yescharacteristic
85C2×C8×C47#4111yescharacteristic
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C22⋊C4×C47—41no—
89C42×C47—41no—
90C4×Dic47—21no—
91C4×Dic47—21no—
92#10625329—22no—
93#10625329—44no—
94#10625329—41no—
95#10625329—41no—
96#10625329—41no—
97C2×C47⋊C8—21no—
98C2×C47⋊C8—21no—
99C2×C47⋊C8—44no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8D811yescharacteristic
103C2×C47⋊C8D811yescharacteristic
104C16×C47—21no—
105C16×C47—21no—
106C47⋊C16—41no—
107C23×C47—41no—
108C23×C47#13511yescharacteristic
109Q8×C47—22no—
110Q8×C47—41no—
111D4×C47—22no—
112D4×C47—42no—
113D4×C47—42no—
114D4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47C4×D411yescharacteristic
121C2×C4×C47#12011yescharacteristic
122C2×C4×C47C22⋊Q811yescharacteristic
123C2×C4×C47C2×D811yescharacteristic
124C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C47#13711yescharacteristic
126C2×Dic47—22no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C8×C47C2×C4⋊C411yescharacteristic
130C8×C47C4×D411yescharacteristic
131C8×C47C22⋊Q811yescharacteristic
132C8×C47#12311yescharacteristic
133C47⋊C8—21no—
134C47⋊C8—21no—
135C47⋊C8—42no—
136C47⋊C8—81no—
137#12939—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—42no—
141C22×C47—81no—
142C22×C47—81no—
143C22×C47C4×D811yescharacteristic
144C22×C47#45211yescharacteristic
145C22×C47#47811yescharacteristic
146C4×C47—42no—
147C4×C47—81no—
148C4×C47—81no—
149C4×C47#38511yescharacteristic
150C4×C47#44011yescharacteristic
151C4×C47#44711yescharacteristic
152C4×C47#47411yescharacteristic
153Dic47—42no—
154C2×#443—471no—
155C2×#368—471no—
156C2×#359—471no—
157#2138D4711yescharacteristic
158#1945—471no—
159#1382—471no—
160#1374—471no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—82no—
164C2×C47#186211yescharacteristic
165C2×C47#217811yescharacteristic
166C2×C47#217911yescharacteristic
167C2×#135C2×D4711yescharacteristic
168C2×#131—941no—
169C22×C16—471no—
170C22×C16C2×D4711yescharacteristic
171#443—942no—
172#443—942no—
173C2×#108—471no—
174C2×#108—471no—
175#427C2×D4711yescharacteristic
176#424—941no—
177C2×#104—941no—
178C2×C4×C8—471no—
179#368—942no—
180#359—942no—
181#358Dic4711yescharacteristic, complemented
182#358Dic4711yescharacteristic, complemented
183#357Dic4711yescharacteristic, complemented
184#357Dic4711yescharacteristic, complemented
185#339—941no—
186C47#1293911yescharacteristic, complemented, Sylow
187C2×#41—21no—
188#135—21no—
189#135—21no—
190#135Dic9412yes—
191#135D18812yes—
192C2×Q16C2×Dic4711yescharacteristic
193C2×SD16—21no—
194C2×D8C2×Dic4711yescharacteristic
195#131—944no—
196#131—1882no—
197C2×M16—941no—
198C2×M16—941no—
199C2×M16—1881no—
200C22×C8—471no—
201C22×C8—1881no—
202C22×C8C22×D4711yescharacteristic
203#117—21no—
204C2×C42—471no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—941no—
208C2×C16—941no—
209C2×C16—941no—
210C2×C16C2×Dic4711yescharacteristic
211C2×C16C2×Dic4711yescharacteristic
212#108—472no—
213#108—941no—
214#108—941no—
215#108—941no—
216#107C2×Dic4711yescharacteristic
217#107C2×Dic4711yescharacteristic
218#106—21no—
219C4⋊C8—1881no—
220C4⋊C8—1881no—
221#104—944no—
222C4×C8—941no—
223C4×C8—941no—
224#41—22no—
225#41—22no—
226#41—41no—
227#41—41no—
228C2×Q8—21no—
229C2×D4—21no—
230C2×D4—41no—
231C22×C4—41no—
232C22×C4—941no—
233C22×C4D4×D4711yescharacteristic
234Q16—21no—
235Q16C4⋊Dic4712yes—
236SD16—22no—
237SD16—22no—
238D8—21no—
239D8C4⋊Dic4712yes—
240M16—942no—
241M16—1884no—
242M16—1881no—
243M16—1881no—
244M16—1881no—
245C2×C8—471no—
246C2×C8—471no—
247C2×C8—941no—
248C2×C8—941no—
249C2×C8—1884no—
250C2×C8—1881no—
251C2×C8—1881no—
252C2×C8C2×Dic9411yescharacteristic
253C2×C8C2×D18811yescharacteristic
254C2×C8D4×D4711yescharacteristic
255C2×C8#1062535211yescharacteristic
256C2×C8#1062535211yescharacteristic
257C2×C8C22×Dic4711yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C22⋊C4—41no—
261C42—41no—
262C42—941no—
263C42—941no—
264C16—21no—
265C16—21no—
266C16—1881no—
267C23—41no—
268C23#1309404311yescharacteristic
269Q8—22no—
270Q8—41no—
271D4—22no—
272D4—42no—
273D4—42no—
274D4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—942no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4D4×Dic4711yescharacteristic
284C2×C4#1309399411yescharacteristic
285C2×C4#1309400111yescharacteristic
286C2×C4D8×D4711yescharacteristic
287C2×C4#1309405111yescharacteristic
288C2×C4#1309405311yescharacteristic
289C8—941no—
290C8—941no—
291C8—3761no—
292C8—1882no—
293C8C2×C4⋊Dic4711yescharacteristic
294C8D4×Dic4711yescharacteristic
295C8#1309400111yescharacteristic
296C8#1309400811yescharacteristic
297C22—21no—
298C22—21no—
299C22—42no—
300C22—81no—
301C22—81no—
302C22D8×Dic4711yescharacteristic
303C22#1568928611yescharacteristic
304C22#1568942111yescharacteristic
305C4—42no—
306C4—81no—
307C4—81no—
308C4—1882no—
309C4#1568899811yescharacteristic
310C4#1568922411yescharacteristic
311C4#1568925811yescharacteristic
312C4#1568940111yescharacteristic
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2024363711yescharacteristic
317C2#2024526911yescharacteristic
318C2#2024527411yescharacteristic
subgroupG/H
centerC2×C4D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2138C2
socleC22×C47C4×D8
prime radicalC2×#135×C47C22
cosocleC22×C8×C47C23
hypercenter#2138D47
hypocenterC47#12939
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. C2×#15688953C2semidirect
4. #20244137C2semidirect
5. #20241483C2semidirect
6. #20241465C2semidirect
23. C47×#358C4semidirect
24. C47×#358C4semidirect
25. C47×#357C4semidirect
26. C47×#357C4semidirect
181. #358Dic47semidirect
182. #358Dic47semidirect
183. #357Dic47semidirect
184. #357Dic47semidirect
186. C47#12939semidirect

Maximal subgroups

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

subgroupindex
C2×#156892402
C2×#156889532
C2×#156889302
#202441372
#202414832
#202414652
C47×#21382
#1293947