Finite Group Databasebeta

← #26133328

Subgroups — #26133328

Order 12032.

476 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689553C211yesmaximal, characteristic
2C2×#15688930C211yesmaximal, characteristic, complemented
3C2×#15688929C211yesmaximal, characteristic, complemented
4#20244276C211yesmaximal, characteristic, complemented
5#20241481C211yesmaximal, characteristic, complemented
6#20241466C211yesmaximal, characteristic, complemented
7C2×C47×#443C211yesmaximal, characteristic, complemented
8C2×#13094046C2211yescharacteristic
9#15689553—22no—
10#15689553—22no—
11C2×#13093970C411yescharacteristic, complemented
12C2×#13093968C411yescharacteristic, complemented
13C2×#13093967C411yescharacteristic, complemented
14C2×#13093965C411yescharacteristic, complemented
15C22×C47⋊C16C2211yescharacteristic
16C22×C47⋊C16C2211yescharacteristic
17C2×#13093960C2211yescharacteristic
18C2×#13093952—21no—
19#15689089—21no—
20#15688930—22no—
21#15688929—22no—
22C2×#135×C47C2211yescharacteristic
23C2×#131×C47—21no—
24C47×#443—22no—
25C47×#443—22no—
26C2×#108×C47C2211yescharacteristic
27C2×#104×C47—21no—
28C2×C4×C8×C47C2211yescharacteristic
29C2×#10625350—21no—
30#13094046—21no—
31#13094046—21no—
32#13094046—21no—
33#13094046—21no—
34C2×Dic188C2×C411yescharacteristic
35C2×#10625325—21no—
36C2×D376C2×C411yescharacteristic
37C2×#10625329—21no—
38#13093970—22no—
39#13093968—22no—
40#13093967—22no—
41#13093965—22no—
42C2×C47⋊C16—21no—
43C2×C47⋊C16—21no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16—21no—
46C2×C47⋊C16—21no—
47C2×C47⋊C16C2×C411yescharacteristic
48C2×C47⋊C16C2×C411yescharacteristic
49#13093960—21no—
50#13093960—21no—
51#13093952—42no—
52(C2×D47)⋊C8—41no—
53(C2×D47)⋊C8—41no—
54C2×#41×C47—21no—
55#135×C47—21no—
56#135×C47—21no—
57#135×C47D412yes—
58#135×C47D412yes—
59C2×Q16×C47C2×C411yescharacteristic
60C2×SD16×C47—21no—
61C2×D8×C47C2×C411yescharacteristic
62#131×C47—24no—
63#131×C47—42no—
64C2×M16×C47—21no—
65C2×M16×C47—41no—
66C22×C8×C47—41no—
67C22×C8×C47D411yescharacteristic
68C22×C8×C47C2311yescharacteristic
69C2×C42×C47D411yescharacteristic
70#108×C47—21no—
71#108×C47D412yes—
72#104×C47—24no—
73C4×C8×C47—21no—
74C4×C8×C47—21no—
75#41×C47—22no—
76#41×C47—22no—
77#41×C47—41no—
78#41×C47—41no—
79C2×Q8×C47—21no—
80C2×D4×C47—21no—
81C2×D4×C47—41no—
82C22×C4×C47—21no—
83C22×C4×C47—41no—
84C22×C4×C47C2×D411yescharacteristic
85C2×C22⋊D47—41no—
86#10625350—41no—
87#10625350—41no—
88#10625350—41no—
89#10625350—41no—
90C2×D188—21no—
91C2×C4×D47—41no—
92C2×Dic94—21no—
93Q16×C47—21no—
94Q16×C47C22⋊C412yes—
95SD16×C47—22no—
96SD16×C47—22no—
97D8×C47—21no—
98D8×C47C22⋊C412yes—
99M16×C47—22no—
100M16×C47—44no—
101M16×C47—41no—
102C2×C8×C47—21no—
103C2×C8×C47—21no—
104C2×C8×C47—44no—
105C2×C8×C47—41no—
106C2×C8×C47D811yescharacteristic
107C2×C8×C47D811yescharacteristic
108C2×C8×C47C22×C411yescharacteristic
109C2×C8×C47C2×D411yescharacteristic
110C2×C8×C47C2×D411yescharacteristic
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47#4111yescharacteristic
113C2×C8×C47#4111yescharacteristic
114C42×C47—21no—
115C42×C47—21no—
116#10625329—41no—
117#10625329—41no—
118C2×C47⋊C8—41no—
119Dic188—21no—
120Dic188—21no—
121D376—21no—
122D376—21no—
123#10625325—41no—
124#10625325—41no—
125C47⋊C16—21no—
126C47⋊C16—21no—
127C47⋊C16—41no—
128C23×C47—41no—
129C23×C47#13511yescharacteristic
130C22×D47—41no—
131Q8×C47—22no—
132Q8×C47—41no—
133D4×C47—22no—
134D4×C47—42no—
135D4×C47—42no—
136D4×C47—41no—
137C2×C4×C47—22no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47C4×D411yescharacteristic
144C2×C4×C47#12011yescharacteristic
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47C2×D811yescharacteristic
147C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
149C22⋊D47—81no—
150C22⋊D47—81no—
151C2×Dic47—41no—
152D188—41no—
153D188—41no—
154C4×D47—81no—
155C4×D47—81no—
156Dic94—41no—
157Dic94—41no—
158C8×C47—21no—
159C8×C47—21no—
160C8×C47—42no—
161C8×C47C2×C22⋊C411yescharacteristic
162C8×C47C4×D411yescharacteristic
163C8×C47#12111yescharacteristic
164C8×C47#12311yescharacteristic
165C47⋊C8—81no—
166#13011—471nomaximal, Sylow
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—42no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47C4×D811yescharacteristic
173C22×C47#44911yescharacteristic
174C22×C47#46311yescharacteristic
175C2×D47—81no—
176C2×D47—81no—
177C2×D47—81no—
178C4×C47—42no—
179C4×C47—42no—
180C4×C47#38611yescharacteristic
181C4×C47#44211yescharacteristic
182C4×C47#44711yescharacteristic
183C4×C47#45911yescharacteristic
184Dic47—81no—
185C2×#508—471no—
186C2×#443D4711yescharacteristic, complemented
187C2×#359—471no—
188C2×#359—471no—
189#1974—471no—
190#1381—471no—
191#1374—471no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47—82no—
195C2×C47#187911yescharacteristic
196C2×C47#217811yescharacteristic
197C2×C47#217911yescharacteristic
198D47—161no—
199C2×#135—471no—
200C2×#135C2×D4711yescharacteristic
201C2×#131—21no—
202#508—942no—
203#508—942no—
204C2×Q32—471no—
205C2×SD32—471no—
206C2×SD32—471no—
207C2×D16—471no—
208C22×C16—471no—
209C22×C16—471no—
210#443—22no—
211#443—22no—
212C2×#108—471no—
213C2×#108C2×D4711yescharacteristic
214C2×#104—21no—
215C2×#104—941no—
216#408—941no—
217C2×C4×C8C2×D4711yescharacteristic
218#359—942no—
219#359—942no—
220C47#1301111yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—941no—
223#135—21no—
224#135—21no—
225#135—941no—
226#135—941no—
227#135—941no—
228#135—941no—
229#135D18812yes—
230#135C22⋊D4712yes—
231C2×Q16—471no—
232C2×Q16C4×D4711yescharacteristic
233C2×SD16—21no—
234C2×SD16—941no—
235C2×D8—471no—
236C2×D8C4×D4711yescharacteristic
237#131—24no—
238#131—42no—
239C2×M16—21no—
240C2×M16—41no—
241C2×M16—941no—
242C22×C8—41no—
243C22×C8C22⋊D4711yescharacteristic
244C22×C8C22×D4711yescharacteristic
245C2×C42C22⋊D4711yescharacteristic
246Q32—942no—
247SD32—942no—
248SD32—942no—
249D16—942no—
250C2×C16—471no—
251C2×C16—471no—
252C2×C16—941no—
253C2×C16—941no—
254C2×C16—941no—
255C2×C16—941no—
256C2×C16—941no—
257#108—21no—
258#108—941no—
259#108—941no—
260#108C22⋊D4712yes—
261#104—24no—
262#104—1882no—
263C22⋊C8—1881no—
264C22⋊C8—1881no—
265C4×C8—21no—
266C4×C8—21no—
267#41—22no—
268#41—22no—
269#41—41no—
270#41—41no—
271#41—1881no—
272#41—1881no—
273#41—1881no—
274#41—1881no—
275C2×Q8—21no—
276C2×Q8—941no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—941no—
280C2×D4—1881no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—1881no—
284C22×C4C2×C22⋊D4711yescharacteristic
285Q16—21no—
286Q16—941no—
287Q16—941no—
288Q16(C2×D47)⋊C412yes—
289SD16—22no—
290SD16—22no—
291SD16—1881no—
292SD16—1881no—
293D8—21no—
294D8—941no—
295D8—941no—
296D8(C2×D47)⋊C412yes—
297M16—22no—
298M16—44no—
299M16—41no—
300M16—1881no—
301M16—1881no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—44no—
305C2×C8—41no—
306C2×C8—1881no—
307C2×C8#1062533411yescharacteristic
308C2×C8#1062533411yescharacteristic
309C2×C8C2×C4×D4711yescharacteristic
310C2×C8C2×D18811yescharacteristic
311C2×C8#1062535011yescharacteristic
312C2×C8#1062535011yescharacteristic
313C2×C8C2×C22⋊D4711yescharacteristic
314C2×C8C2×C22⋊D4711yescharacteristic
315C42—21no—
316C42—21no—
317C16—941no—
318C16—941no—
319C16—1881no—
320C23—41no—
321C23—1881no—
322C23#1309404411yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—1881no—
326Q8—1881no—
327D4—22no—
328D4—42no—
329D4—42no—
330D4—41no—
331D4—3761no—
332D4—3761no—
333D4—1881no—
334D4—1881no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—3761no—
342C2×C4—3761no—
343C2×C4—1881no—
344C2×C4C4×C22⋊D4711yescharacteristic
345C2×C4#1309399311yescharacteristic
346C2×C4#1309399811yescharacteristic
347C2×C4C2×#1062533411yescharacteristic
348C2×C4#1309404911yescharacteristic
349C2×C4#1309405011yescharacteristic
350C8—21no—
351C8—21no—
352C8—42no—
353C8—3761no—
354C8C2×(C2×D47)⋊C411yescharacteristic
355C8C4×C22⋊D4711yescharacteristic
356C8#1309399811yescharacteristic
357C8#1309400711yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—3761no—
364C22—3761no—
365C22—3761no—
366C22C4×#1062533411yescharacteristic
367C22#1568926811yescharacteristic
368C22#1568933811yescharacteristic
369C4—42no—
370C4—42no—
371C4—3761no—
372C4#1568900211yescharacteristic
373C4#1568923311yescharacteristic
374C4#1568925711yescharacteristic
375C4#1568931811yescharacteristic
376C2—21no—
377C2—21no—
378C2—82no—
379C2—7521no—
380C2#2024373611yescharacteristic
381C2#2024526711yescharacteristic
382C2#2024527211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#443C2
socleC22×C47C4×D8
prime radicalC2×#15689553C2
cosocleC22×C8×C47C23
hypercenterC2×#443D47
hypocenterC47#13011
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
2. C2×#15688930C2semidirect
3. C2×#15688929C2semidirect
4. #20244276C2semidirect
5. #20241481C2semidirect
6. #20241466C2semidirect
7. C2×C47×#443C2semidirect
11. C2×#13093970C4semidirect
12. C2×#13093968C4semidirect
13. C2×#13093967C4semidirect
14. C2×#13093965C4semidirect
186. C2×#443D47semidirect
220. C47#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#156895532
C2×#156889302
C2×#156889292
#202442762
#202414812
#202414662
C2×C47×#4432
#1301147