Finite Group Databasebeta

← #26110151

Subgroups — #26110151

Order 12032.

454 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252428C211yesmaximal, characteristic
2#20244182C211yesmaximal, characteristic
3C4×#13093940C211yesmaximal, characteristic
4#20242141C211yesmaximal, characteristic, complemented
5#20242140C211yesmaximal, characteristic, complemented
6#20241225C211yesmaximal, characteristic, complemented
7C47×#1278C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9#15689202—21no—
10#15689202—21no—
11#15689200—21no—
12#15689200—21no—
13#15689176C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093940C411yescharacteristic
16C2×#13093940C411yescharacteristic
17#15689085—22no—
18C2×(C2×D47)⋊C8—22no—
19C4×#10625329C2211yescharacteristic
20#15688891C411yescharacteristic
21#15688891C411yescharacteristic
22#15688876—21no—
23#15688875—21no—
24#15688874—21no—
25#15688873—21no—
26C47×#431C2211yescharacteristic
27C4×M16×C47C2211yescharacteristic
28C2×C4×C8×C47C2211yescharacteristic
29C22×D188C2×C411yescharacteristic
30C22×C4×D47D412yes—
31C2×#10625329C2×C411yescharacteristic
32C2×#10625329C2×C411yescharacteristic
33C2×#10625324—22no—
34C2×#10625324—22no—
35C22×C47⋊C8—21no—
36C4×D188—21no—
37C4×D188—21no—
38C4×D188—21no—
39C4×D188—21no—
40C2×C4⋊Dic47C2×C411yescharacteristic
41C2×(C2×D47)⋊C4Q812yes—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)D411yescharacteristic
45C4⋊(C47⋊C8)D411yescharacteristic
46#13093940—22no—
47#13093940—22no—
48C22⋊(C47⋊C8)—21no—
49(C2×D47)⋊C8—22no—
50(C2×D47)⋊C8—22no—
51(C2×D47)⋊C8—44no—
52#13093935—21no—
53C4×C47⋊C8—21no—
54C4×C47⋊C8—21no—
55C2×M16×C47C2×C411yescharacteristic
56C2×M16×C47C2×C411yescharacteristic
57C22×C8×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47D411yescharacteristic
60C4⋊C8×C47D411yescharacteristic
61C22⋊C8×C47—21no—
62#97×C47—21no—
63C4×C8×C47—21no—
64C4×C8×C47—21no—
65C4×C8×C47—21no—
66C4×C8×C47—21no—
67C23×D47C4⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×Q811yescharacteristic
71C22×Dic47C4⋊C412yes—
72C2×D188—21no—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188—21no—
78C2×C4×D47—22no—
79C2×C4×D47—22no—
80C2×C4×D47—42no—
81C2×C4×D47—42no—
82M16×C47—21no—
83M16×C47—21no—
84M16×C47—21no—
85M16×C47M1612yescomplemented
86C2×C8×C47—21no—
87C2×C8×C47—22no—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47—21no—
91C2×C8×C47—41no—
92C2×C8×C47—41no—
93C42×C47C2×D411yescharacteristic
94C42×C47C2×D411yescharacteristic
95C42×C47#4111yescharacteristic
96C42×C47#4111yescharacteristic
97(C2×D47)⋊C4—22no—
98(C2×D47)⋊C4—22no—
99C4⋊Dic47—21no—
100C4⋊Dic47—21no—
101#10625329—21no—
102#10625329—21no—
103#10625329—21no—
104#10625329M1612yescomplemented
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—42no—
110C2×C47⋊C8—41no—
111#10625324—82no—
112#10625324—82no—
113C23×C47C2×C4⋊C411yescharacteristic
114C22×D47—22no—
115C22×D47—22no—
116C22×D47—22no—
117C22×D47—42no—
118C22×D47—42no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—41no—
122C2×C4×C47C4×D411yescharacteristic
123C2×C4×C47C4×D411yescharacteristic
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47C22⋊Q811yescharacteristic
126C2×C4×C47C22⋊Q811yescharacteristic
127C2×C4×C47#12311yescharacteristic
128C2×C4×C47C2×M1611yescharacteristic
129C2×C4×C47C2×M1611yescharacteristic
130C2×C4×C47#13111yescharacteristic
131C2×C4×C47#13111yescharacteristic
132C2×Dic47—22no—
133C2×Dic47—42no—
134D188—41no—
135D188—41no—
136D188—41no—
137D188—41no—
138C4×D47—42no—
139C4×D47—42no—
140C4×D47—82no—
141C8×C47—22no—
142C8×C47—42no—
143C8×C47—41no—
144C8×C47—41no—
145C8×C47—41no—
146C47⋊C8—22no—
147C47⋊C8—41no—
148C47⋊C8—41no—
149C47⋊C8—41no—
150C47⋊C8—81no—
151#8837—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47#38511yescharacteristic
155C22×C47#42311yescharacteristic
156C22×C47#42411yescharacteristic
157C2×D47—44no—
158C2×D47—42no—
159C2×D47—42no—
160C2×D47—42no—
161C2×D47—82no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47#43511yescharacteristic
166C4×C47#43511yescharacteristic
167C4×C47#43611yescharacteristic
168C4×C47#43611yescharacteristic
169C4×C47#45311yescharacteristic
170C4×C47#45511yescharacteristic
171Dic47—42no—
172#2975—471no—
173#1954—471no—
174C4×#100—471no—
175#1565—471no—
176#1565—471no—
177#1278—471no—
178#1278D4711yescharacteristic, complemented
179C2×C47—21no—
180C2×C47—21no—
181C2×C47#185111yescharacteristic
182C2×C47#185611yescharacteristic
183C2×C47#185811yescharacteristic
184D47—82no—
185C2×C4×D4—471no—
186#436—941no—
187#436—941no—
188#436—941no—
189#436—941no—
190#431—471no—
191#431C2×D4711yescharacteristic
192C2×C4⋊C8—471no—
193C2×#100—471no—
194C2×#100—471no—
195#407—942no—
196C2×C22⋊C8—942no—
197C4×M16—471no—
198C4×M16C2×D4711yescharacteristic
199C2×C4×C8C2×D4711yescharacteristic
200#343—471no—
201#343—471no—
202#335—941no—
203#335—941no—
204#334—941no—
205#334—941no—
206C47#883711yescharacteristic, complemented, Sylow
207C22×D4—471no—
208C23×C4—472no—
209C2×M16—471no—
210C2×M16—471no—
211C2×M16—942no—
212C2×M16—942no—
213C2×M16C4×D4711yescharacteristic
214C2×M16C4×D4711yescharacteristic
215C22×C8—21no—
216C22×C8—941no—
217C4×D4—941no—
218C4×D4—941no—
219C4×D4—941no—
220C4×D4—941no—
221C2×C4⋊C4—471no—
222C2×C22⋊C4—472no—
223C2×C42C22×D4711yescharacteristic
224C4⋊C8—471no—
225C4⋊C8—471no—
226C4⋊C8—941no—
227C4⋊C8—941no—
228C4⋊C8D18811yescharacteristic
229C4⋊C8D18811yescharacteristic
230#100—942no—
231#100—942no—
232C22⋊C8—21no—
233C22⋊C8—942no—
234C22⋊C8—942no—
235C22⋊C8—941no—
236C22⋊C8—1884no—
237#97—21no—
238#97—941no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—941no—
244C4×C8—941no—
245C24—472no—
246C2×D4—941no—
247C2×D4—941no—
248C2×D4—941no—
249C2×D4—941no—
250C2×D4—941no—
251C2×D4—941no—
252C22×C4—472no—
253C22×C4—942no—
254C22×C4—942no—
255C22×C4—1882no—
256C22×C4—1882no—
257C22×C4C2×C4×D4711yescharacteristic
258C22×C4D4×D4711yescharacteristic
259C22×C4Q8×D4711yescharacteristic
260M16—21no—
261M16—21no—
262M16—21no—
263M16—472no—
264M16—941no—
265M16—941no—
266M16—941no—
267M16—3762no—
268M16—3762no—
269M16#1062532412yescomplemented
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—941no—
278C2×C8—941no—
279C2×C8—941no—
280C2×C8—941no—
281C2×C8—1882no—
282C2×C8—1881no—
283C4⋊C4—941no—
284C4⋊C4—941no—
285C22⋊C4—942no—
286C22⋊C4—942no—
287C42C2×D18811yescharacteristic
288C42#1062535011yescharacteristic
289C42D4×D4711yescharacteristic
290C42#1062535211yescharacteristic
291C23—942no—
292C23—942no—
293C23—942no—
294C23—1882no—
295C23—1882no—
296C23C4⋊C4×D4711yescharacteristic
297D4—1881no—
298D4—1881no—
299D4—1881no—
300D4—1881no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—942no—
305C2×C4—3762no—
306C2×C4—1882no—
307C2×C4—1882no—
308C2×C4—1882no—
309C2×C4(C22⋊D47)⋊C411yescharacteristic
310C2×C4C4×D18811yescharacteristic
311C2×C4#1309399411yescharacteristic
312C2×C4#1309400211yescharacteristic
313C2×C4#1309400411yescharacteristic
314C2×C4#1309400611yescharacteristic
315C2×C4M16×D4711yescharacteristic
316C2×C4C2×#1062532411yescharacteristic
317C2×C4#1309403011yescharacteristic
318C2×C4#1309403211yescharacteristic
319C8—22no—
320C8—42no—
321C8—41no—
322C8—41no—
323C8—41no—
324C8—942no—
325C8—3761no—
326C8—1881no—
327C8—1881no—
328C8—1881no—
329C22—21no—
330C22—21no—
331C22—3762no—
332C22—1884no—
333C22—1882no—
334C22—1882no—
335C22—1882no—
336C22#1568899711yescharacteristic
337C22#1568914511yescharacteristic
338C22#1568914811yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—1882no—
343C4#1568919111yescharacteristic
344C4#1568919611yescharacteristic
345C4#1568919811yescharacteristic
346C4#1568920211yescharacteristic
347C4#1568929211yescharacteristic
348C4#1568930211yescharacteristic
349C2—21no—
350C2—21no—
351C2—3762no—
352C2#2024357211yescharacteristic
353C2#2024359911yescharacteristic
354C2#2024360911yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1278C2
socleC22×C47#385
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1278D47
hypocenterC47#8837
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
4. #20242141C2semidirect
5. #20242140C2semidirect
6. #20241225C2semidirect
7. C47×#1278C2semidirect
85. M16×C47M16semidirect
104. #10625329M16semidirect
178. #1278D47semidirect
206. C47#8837semidirect
269. M16#10625324semidirect

Maximal subgroups

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

subgroupindex
#202524282
#202441822
C4×#130939402
#202421412
#202421402
#202412252
C47×#12782
#883747