Finite Group Databasebeta

← #26133238

Subgroups — #26133238

Order 12032.

476 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#20251999C211yesmaximal, characteristic, complemented
2#20245116C211yesmaximal, characteristic, complemented
3C2×#15688930C211yesmaximal, characteristic, complemented
4C2×#15688907C211yesmaximal, characteristic, complemented
5#20241501C211yesmaximal, characteristic, complemented
6#20241481C211yesmaximal, characteristic, complemented
7C47×#3067C211yesmaximal, characteristic
8#15689890—21no—
9C2×#13093964C2211yescharacteristic
10C22×C47⋊C16C2211yescharacteristic
11#15689172—22no—
12#15689172—21no—
13C2×#13093960—22no—
14C2×#13093960C2211yescharacteristic
15C2×#13093960C2211yescharacteristic
16#15688930—22no—
17#15688930—22no—
18#15688907—22no—
19C47×#575—21no—
20C2×#135×C47C2211yescharacteristic
21C2×#131×C47C2211yescharacteristic
22C47×#473C412yes—
23C47×#472C412yes—
24C47×#471—22no—
25C2×#108×C47C2211yescharacteristic
26C47×#413—21no—
27#13094031—42no—
28#13094031—42no—
29C2×#10625329—24no—
30C2×#10625329—21no—
31C2×#10625329—21no—
32#13093964—21no—
33#13093964—21no—
34C2×C47⋊C16—21no—
35C2×C47⋊C16—21no—
36C2×C47⋊C16—21no—
37C2×C47⋊C16—21no—
38#13093960—24no—
39#13093960—21no—
40#13093960—21no—
41#13093960D412yes—
42#13093960D412yes—
43C2×#41×C47—21no—
44C2×#41×C47D411yescharacteristic
45#137×C47—22no—
46#137×C47—41no—
47AGL(1,ℤ/8)×C47—22no—
48AGL(1,ℤ/8)×C47—41no—
49#135×C47—21no—
50#135×C47—21no—
51#135×C47—21no—
52#135×C47—21no—
53#135×C47—42no—
54C2×Q16×C47C2×C411yescharacteristic
55C2×SD16×C47—21no—
56C2×D8×C47C2×C411yescharacteristic
57#131×C47—21no—
58#131×C47—21no—
59#131×C47C2×C412yes—
60#131×C47D412yes—
61C2×M16×C47—21no—
62C2×M16×C47—21no—
63C2×M16×C47D411yescharacteristic
64C22×C8×C47—21no—
65C22×C8×C47C2311yescharacteristic
66#108×C47—21no—
67#108×C47C2×C412yes—
68#101×C47—22no—
69#100×C47—22no—
70#41×C47—21no—
71#41×C47—21no—
72#41×C47—41no—
73#41×C47—41no—
74#41×C47—41no—
75#41×C47—41no—
76#41×C47C22⋊C412yes—
77#41×C47D812yes—
78C2×Q8×C47—21no—
79C2×Q8×C47SD1611yescharacteristic
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—41no—
83C2×D4×C47SD1611yescharacteristic
84C22×C4×C47—21no—
85C22×C4×C47—41no—
86C22×C4×C47C2×D411yescharacteristic
87Q16×C47—21no—
88Q16×C47—21no—
89Q16×C47—42no—
90SD16×C47—42no—
91SD16×C47—42no—
92SD16×C47—41no—
93SD16×C47—41no—
94D8×C47—21no—
95D8×C47—21no—
96D8×C47—42no—
97M16×C47—21no—
98M16×C47—22no—
99M16×C47—22no—
100M16×C47—22no—
101M16×C47—41no—
102M16×C47C22⋊C412yes—
103C2×C8×C47—21no—
104C2×C8×C47—22no—
105C2×C8×C47—22no—
106C2×C8×C47—41no—
107C2×C8×C47C22×C411yescharacteristic
108C2×C8×C47C2×D411yescharacteristic
109C2×C8×C47C2×D411yescharacteristic
110C2×C8×C47C2×D411yescharacteristic
111C2×C8×C47#4111yescharacteristic
112C2×C8×C47#4111yescharacteristic
113#10625329—22no—
114#10625329—22no—
115#10625329—44no—
116#10625329—41no—
117#10625329—41no—
118C2×C47⋊C8—44no—
119C2×C47⋊C8—41no—
120C2×C47⋊C8—41no—
121C47⋊C16—41no—
122C47⋊C16—41no—
123C23×C47—21no—
124C23×C47—41no—
125C23×C47#13711yescharacteristic
126Q8×C47—21no—
127Q8×C47—41no—
128Q8×C47—41no—
129Q8×C47#10212yes—
130D4×C47—22no—
131D4×C47—22no—
132D4×C47—21no—
133D4×C47—41no—
134D4×C47—41no—
135D4×C47—81no—
136D4×C47—81no—
137D4×C47#10212yes—
138C2×C4×C47—21no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—41no—
142C2×C4×C47—81no—
143C2×C4×C47—81no—
144C2×C4×C47C2×C22⋊C411yescharacteristic
145C2×C4×C47#12011yescharacteristic
146C2×C4×C47#12311yescharacteristic
147C2×C4×C47C2×D811yescharacteristic
148C2×C4×C47C2×SD1611yescharacteristic
149C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
150C8×C47—22no—
151C8×C47—42no—
152C8×C47C4×D411yescharacteristic
153C8×C47C4×D411yescharacteristic
154C8×C47#12111yescharacteristic
155C8×C47#12111yescharacteristic
156C47⋊C8—42no—
157C47⋊C8—42no—
158#12995—471nomaximal, Sylow
159C22×C47—22no—
160C22×C47—41no—
161C22×C47—81no—
162C22×C47—81no—
163C22×C47—161no—
164C22×C47#41711yescharacteristic
165C22×C47#44811yescharacteristic
166C22×C47#48011yescharacteristic
167C4×C47—22no—
168C4×C47—81no—
169C4×C47#38611yescharacteristic
170C4×C47C2×#10211yescharacteristic
171C4×C47#44711yescharacteristic
172C4×C47#48111yescharacteristic
173#3067D4711yescharacteristic
174#2916—471no—
175#2143—471no—
176C2×#359—471no—
177C2×#350—471no—
178#1388—471no—
179#1381—471no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47—21no—
183C2×C47—42no—
184C2×C47—161no—
185C2×C47#187411yescharacteristic
186#575—21no—
187C2×#135C2×D4711yescharacteristic
188#568—941no—
189C2×#131C2×D4711yescharacteristic
190C2×M32—471no—
191C22×C16—471no—
192#473Dic4712yes—
193#472Dic4712yes—
194#471—22no—
195#430—942no—
196#430—941no—
197C2×#108—471no—
198C2×#108—471no—
199C2×#108—942no—
200C2×#108C2×D4711yescharacteristic
201#413—21no—
202#359—942no—
203#359—942no—
204#350—942no—
205C47#1299511yescharacteristic, complemented, Sylow
206C2×#41—21no—
207C2×#41C22⋊D4711yescharacteristic
208#137—22no—
209#137—41no—
210AGL(1,ℤ/8)—22no—
211AGL(1,ℤ/8)—41no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—42no—
217C2×Q16C2×Dic4711yescharacteristic
218C2×SD16—21no—
219C2×D8C2×Dic4711yescharacteristic
220#131—21no—
221#131—21no—
222#131—1882no—
223#131—1882no—
224#131C2×Dic4712yes—
225#131C22⋊D4712yes—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—944no—
229C2×M16—941no—
230C2×M16—941no—
231C2×M16C22⋊D4711yescharacteristic
232C22×C8—21no—
233C22×C8C22×D4711yescharacteristic
234M32—941no—
235M32—941no—
236C2×C16—941no—
237C2×C16—941no—
238C2×C16—941no—
239C2×C16—941no—
240#108—21no—
241#108—472no—
242#108—472no—
243#108—944no—
244#108—941no—
245#108—941no—
246#108C2×Dic4712yes—
247#101—22no—
248#100—22no—
249#41—21no—
250#41—21no—
251#41—41no—
252#41—41no—
253#41—41no—
254#41—41no—
255#41#1062533412yes—
256#41C22⋊Dic4712yes—
257C2×Q8—21no—
258C2×Q8#1062533511yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4#1062533511yescharacteristic
263C22×C4—21no—
264C22×C4—41no—
265C22×C4C2×C22⋊D4711yescharacteristic
266Q16—21no—
267Q16—21no—
268Q16—42no—
269SD16—42no—
270SD16—42no—
271SD16—41no—
272SD16—41no—
273D8—21no—
274D8—21no—
275D8—42no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—41no—
281M16—942no—
282M16—942no—
283M16—1884no—
284M16—1881no—
285M16—1881no—
286M16C22⋊Dic4712yes—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—1884no—
292C2×C8—1881no—
293C2×C8—1881no—
294C2×C8D4×D4711yescharacteristic
295C2×C8D4×D4711yescharacteristic
296C2×C8#1062535211yescharacteristic
297C2×C8#1062535211yescharacteristic
298C2×C8C22×Dic4711yescharacteristic
299C2×C8C2×C22⋊D4711yescharacteristic
300C16—1881no—
301C16—1881no—
302C23—21no—
303C23—41no—
304C23#1309405411yescharacteristic
305Q8—21no—
306Q8—41no—
307Q8—41no—
308Q8#1309394512yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316D4#1309394512yes—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×C22⋊Dic4711yescharacteristic
324C2×C4#1309399211yescharacteristic
325C2×C4#1309400511yescharacteristic
326C2×C4C2×#1062533411yescharacteristic
327C2×C4C2×#1062533511yescharacteristic
328C2×C4#1309404911yescharacteristic
329C8—22no—
330C8—42no—
331C8—1882no—
332C8—1882no—
333C8D4×Dic4711yescharacteristic
334C8D4×Dic4711yescharacteristic
335C8#1309399611yescharacteristic
336C8#1309399611yescharacteristic
337C22—22no—
338C22—41no—
339C22—81no—
340C22—81no—
341C22—161no—
342C22#1568912411yescharacteristic
343C22#1568926111yescharacteristic
344C22#1568943411yescharacteristic
345C4—22no—
346C4—81no—
347C4#1568900311yescharacteristic
348C4C2×#1309394511yescharacteristic
349C4#1568925611yescharacteristic
350C4#1568943911yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—161no—
356C2#2024370611yescharacteristic
subgroupG/H
centerC4#15689256
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
Fitting, prime radicalC47×#3067C2
socleC2×C47#1874
cosocleC22×C8×C47C23
hypercenter#3067D47
hypocenterC47#12995
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #20251999C2semidirect
2. #20245116C2semidirect
3. C2×#15688930C2semidirect
4. C2×#15688907C2semidirect
5. #20241501C2semidirect
6. #20241481C2semidirect
205. C47#12995semidirect

Maximal subgroups

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

subgroupindex
#202519992
#202451162
C2×#156889302
C2×#156889072
#202415012
#202414812
C47×#30672
#1299547