Finite Group Databasebeta

← #26110757

Subgroups — #26110757

Order 12032.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252131C211yesmaximal, characteristic
2#20244164C211yesmaximal, characteristic, complemented
3#20242162C211yesmaximal, characteristic, complemented
4#20242038C211yesmaximal, characteristic
5#20241840C211yesmaximal, characteristic
6#20241225C211yesmaximal, characteristic, complemented
7C47×#1277C211yesmaximal, characteristic, complemented
8C2×#13094030—21no—
9#15689673C2211yescharacteristic
10#15689200—21no—
11#15689193—22no—
12#15689188—21no—
13#15689176C2211yescharacteristic
14C2×#13093960—21no—
15#15689147C2211yescharacteristic
16#15689089—21no—
17C4×#10625329C2211yescharacteristic
18#15688888—21no—
19#15688875—21no—
20#15688875—21no—
21#15688870C411yescharacteristic
22#15688870C411yescharacteristic
23#15688866C411yescharacteristic
24#15688866C411yescharacteristic
25#15688852—22no—
26C2×C4⋊C8×C47C2211yescharacteristic
27C2×C4⋊C8×C47C2211yescharacteristic
28C2×C4×C8×C47C2211yescharacteristic
29C2×#10625350—21no—
30#13094030—42no—
31#13094030—42no—
32C2×#10625329—21no—
33C2×#10625329D411yescharacteristic
34C2×#10625329D411yescharacteristic
35C2×#10625324—22no—
36C2×C8×D47—22no—
37#13094024C2×C411yescharacteristic
38#13094021C2×C411yescharacteristic
39#13094012—21no—
40#13093998—21no—
41#13093998Q812yes—
42C4×D188—21no—
43C4×D188D412yes—
44#13093960—42no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)C2×C411yescharacteristic
48C4⋊(C47⋊C8)C2×C411yescharacteristic
49C22⋊(C47⋊C8)—21no—
50(C2×D47)⋊C8—22no—
51(C2×D47)⋊C8—22no—
52#13093935—21no—
53C4×C47⋊C8—21no—
54C4×C47⋊C8—21no—
55C22×C8×C47—21no—
56C22×C8×C47—21no—
57C22×C8×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47C2×C411yescharacteristic
62C4⋊C8×C47C2×C411yescharacteristic
63C4⋊C8×C47D411yescharacteristic
64C4⋊C8×C47D411yescharacteristic
65C4×C8×C47—21no—
66C4×C8×C47—21no—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×Q811yescharacteristic
69C22×C4×C47#4111yescharacteristic
70C2×C22⋊D47—22no—
71#10625350—41no—
72#10625350—41no—
73#10625350—41no—
74#10625350—41no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188C4⋊C412yes—
78C2×C4×D47—22no—
79C2×Dic94—21no—
80C2×C8×C47—22no—
81C2×C8×C47—21no—
82C2×C8×C47—21no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—41no—
86C2×C8×C47—41no—
87C2×C8×C47—41no—
88C2×C8×C47—41no—
89C2×C8×C47—41no—
90C2×C8×C47—41no—
91C42×C47C22×C411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47C2×D411yescharacteristic
94C42×C47#4111yescharacteristic
95(C2×D47)⋊C4—22no—
96(C2×D47)⋊C4—22no—
97C4⋊Dic47—21no—
98C4⋊Dic47C4⋊C412yes—
99#10625329—21no—
100#10625329—21no—
101#10625329—21no—
102#10625329—41no—
103#10625329—41no—
104#10625329D812yes—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—41no—
110#10625324—44no—
111C8×D47—44no—
112C23×C47C22⋊Q811yescharacteristic
113C22×D47—22no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—41no—
117C2×C4×C47C2×C4⋊C411yescharacteristic
118C2×C4×C47C4×D411yescharacteristic
119C2×C4×C47C4×D411yescharacteristic
120C2×C4×C47#12011yescharacteristic
121C2×C4×C47C22⋊Q811yescharacteristic
122C2×C4×C47#12311yescharacteristic
123C2×C4×C47C2×D811yescharacteristic
124C2×C4×C47#13511yescharacteristic
125C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C47#13711yescharacteristic
127C22⋊D47—42no—
128C22⋊D47—42no—
129C2×Dic47—22no—
130D188—42no—
131D188—41no—
132D188—41no—
133D188—41no—
134D188—41no—
135C4×D47—42no—
136C4×D47—42no—
137Dic94—41no—
138Dic94—41no—
139C8×C47—42no—
140C8×C47—41no—
141C8×C47—41no—
142C8×C47—41no—
143C8×C47—41no—
144C47⋊C8—22no—
145C47⋊C8—41no—
146C47⋊C8—41no—
147C47⋊C8—41no—
148C47⋊C8—81no—
149#8930—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47#38511yescharacteristic
153C22×C47#47411yescharacteristic
154C22×C47#47811yescharacteristic
155C2×D47—42no—
156C2×D47—42no—
157C2×D47—42no—
158C4×C47—21no—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47C4×D811yescharacteristic
162C4×C47#44011yescharacteristic
163C4×C47#44311yescharacteristic
164C4×C47#44411yescharacteristic
165C4×C47#44711yescharacteristic
166C4×C47#45211yescharacteristic
167Dic47—42no—
168#2935—471no—
169#1950—471no—
170#1571—471no—
171#1540—471no—
172#1492—471no—
173#1278—471no—
174#1277D4711yescharacteristic, complemented
175C2×C47—21no—
176C2×C47—21no—
177C2×C47#185811yescharacteristic
178C2×C47#186211yescharacteristic
179C2×C47#186611yescharacteristic
180D47—82no—
181C2×#131—941no—
182#532—471no—
183#436—941no—
184#435—942no—
185C8×D4—941no—
186#431—471no—
187C2×#108—941no—
188#423—471no—
189C2×C4⋊C8C2×D4711yescharacteristic
190C2×C4⋊C8C2×D4711yescharacteristic
191#408—941no—
192C4×M16—471no—
193C2×C4×C8C2×D4711yescharacteristic
194#341—941no—
195#335—941no—
196#335—941no—
197#332—471no—
198#332—471no—
199#331—471no—
200#331—471no—
201D4⋊C8—942no—
202C47#893011yescharacteristic, complemented, Sylow
203C2×#41—941no—
204#131—1882no—
205#131—1882no—
206C2×M16—471no—
207C2×M16—471no—
208C2×M16—942no—
209C2×M16—941no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—942no—
214C4⋊Q8—471no—
215#127—471no—
216#124—941no—
217#121—472no—
218#121—941no—
219C4×D4—472no—
220C4×D4—941no—
221C2×C42C22×D4711yescharacteristic
222#108—1882no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—471no—
226C4⋊C8—471no—
227C4⋊C8—941no—
228C4⋊C8—941no—
229C4⋊C8C4×D4711yescharacteristic
230C4⋊C8C4×D4711yescharacteristic
231C4⋊C8D18811yescharacteristic
232C4⋊C8D18811yescharacteristic
233C22⋊C8—942no—
234C22⋊C8—942no—
235C22⋊C8—941no—
236#97—941no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—941no—
240C4×C8—941no—
241#41—1881no—
242#41—1881no—
243#41—1881no—
244#41—1881no—
245C2×Q8—941no—
246C2×D4—472no—
247C2×D4—942no—
248C2×D4—941no—
249C2×D4—941no—
250C22×C4—942no—
251C22×C4#1062535011yescharacteristic
252C22×C4D4×D4711yescharacteristic
253C22×C4Q8×D4711yescharacteristic
254M16—472no—
255M16—941no—
256M16—941no—
257M16—941no—
258M16—1884no—
259M16—1881no—
260M16—1881no—
261C2×C8—22no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—941no—
273C2×C8—941no—
274C2×C8—941no—
275C2×C8—941no—
276C2×C8—1884no—
277C2×C8—1881no—
278C4⋊C4—472no—
279C4⋊C4—941no—
280C22⋊C4—942no—
281C22⋊C4—942no—
282C42C2×C4×D4711yescharacteristic
283C42C2×D18811yescharacteristic
284C42D4×D4711yescharacteristic
285C42#1062535211yescharacteristic
286C23—942no—
287C23#1309400211yescharacteristic
288Q8—1881no—
289Q8—1881no—
290D4—1882no—
291D4—1882no—
292D4—1882no—
293D4—1881no—
294D4—1881no—
295D4—1881no—
296D4—1881no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—942no—
301C2×C4—1882no—
302C2×C4—1882no—
303C2×C4C4⋊C4×D4711yescharacteristic
304C2×C4(C22⋊D47)⋊C411yescharacteristic
305C2×C4C4×D18811yescharacteristic
306C2×C4#1309399411yescharacteristic
307C2×C4#1309400411yescharacteristic
308C2×C4#1309400611yescharacteristic
309C2×C4D8×D4711yescharacteristic
310C2×C4#1309404511yescharacteristic
311C2×C4#1309405111yescharacteristic
312C2×C4#1309405611yescharacteristic
313C8—42no—
314C8—41no—
315C8—41no—
316C8—41no—
317C8—41no—
318C8—942no—
319C8—3761no—
320C8—1881no—
321C8—1881no—
322C8—1881no—
323C22—21no—
324C22—21no—
325C22—1882no—
326C22—1882no—
327C22—1882no—
328C22#1568899711yescharacteristic
329C22#1568940211yescharacteristic
330C22#1568942211yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—1882no—
335C4#1568920311yescharacteristic
336C4#1568922611yescharacteristic
337C4#1568924111yescharacteristic
338C4#1568924611yescharacteristic
339C4#1568925811yescharacteristic
340C4#1568928811yescharacteristic
341C2—21no—
342C2—21no—
343C2—3762no—
344C2#2024360911yescharacteristic
345C2#2024363611yescharacteristic
346C2#2024366611yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1277C2
socleC22×C47#385
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1277D47
hypocenterC47#8930
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #20244164C2semidirect
3. #20242162C2semidirect
6. #20241225C2semidirect
7. C47×#1277C2semidirect
174. #1277D47semidirect
202. C47#8930semidirect

Maximal subgroups

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

subgroupindex
#202521312
#202441642
#202421622
#202420382
#202418402
#202412252
C47×#12772
#893047