Finite Group Databasebeta

← #26109541

Subgroups — #26109541

Order 12032.

414 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252099C211yesmaximal, characteristic
2#20243525C211yesmaximal, characteristic, complemented
3C2×#15688873C211yesmaximal, characteristic, complemented
4#20242066C211yesmaximal, characteristic
5#20242063C211yesmaximal, characteristic
6#20241262C211yesmaximal, characteristic, complemented
7C47×#1292C211yesmaximal, characteristic, complemented
8C2×#13094032—21no—
9#15689673C2211yescharacteristic
10#15689200—21no—
11#15689200C412yes—
12#15689188—21no—
13#15689188C412yes—
14#15689147C2211yescharacteristic
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16#15689091—21no—
17C2×C4×C47⋊C8C2211yescharacteristic
18#15688887—21no—
19#15688887—21no—
20#15688873—21no—
21#15688873—21no—
22#15688870—21no—
23#15688868—21no—
24#15688866—21no—
25#15688866—21no—
26C47×#423C2211yescharacteristic
27C47×#423C2211yescharacteristic
28C2×C4⋊C8×C47C2211yescharacteristic
29C2×#10625350—21no—
30#13094032—42no—
31#13094032—42no—
32C2×#10625329—21no—
33C2×#10625324—22no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8D411yescharacteristic
36C22×C47⋊C8D411yescharacteristic
37C2×C8×D47—22no—
38#13094024D411yescharacteristic
39#13094021D411yescharacteristic
40#13094012—21no—
41#13093998—21no—
42#13093998Q812yes—
43C4×D188—21no—
44C4×D188C2×C412yes—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)—21no—
49(C2×D47)⋊C8—22no—
50(C2×D47)⋊C8—22no—
51C4×C47⋊C8C2×C411yescharacteristic
52C4×C47⋊C8C2×C411yescharacteristic
53C4×C47⋊C8D411yescharacteristic
54C4×C47⋊C8D411yescharacteristic
55C2×M16×C47—21no—
56C2×M16×C47—21no—
57C22×C8×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47C2×C412yes—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×Q811yescharacteristic
67C22×C4×C47#4111yescharacteristic
68C2×C22⋊D47—22no—
69#10625350—41no—
70#10625350—41no—
71#10625350—41no—
72#10625350—41no—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188C4⋊C412yes—
76C2×C4×D47—22no—
77C2×Dic94—21no—
78M16×C47—41no—
79M16×C47—41no—
80M16×C47—41no—
81M16×C47—41no—
82C2×C8×C47—22no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C2×C8×C47—81no—
86C42×C47C22×C411yescharacteristic
87C42×C47C2×D411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47#4111yescharacteristic
90(C2×D47)⋊C4—22no—
91(C2×D47)⋊C4—22no—
92C4⋊Dic47—21no—
93C4⋊Dic47C4⋊C412yes—
94#10625329—41no—
95#10625329—41no—
96C2×C47⋊C8—21no—
97C2×C47⋊C8—21no—
98C2×C47⋊C8—21no—
99C2×C47⋊C8—21no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—81no—
103C2×C47⋊C8C22⋊C411yescharacteristic
104C2×C47⋊C8C22⋊C411yescharacteristic
105C2×C47⋊C8C22⋊C411yescharacteristic
106C2×C47⋊C8C22⋊C411yescharacteristic
107#10625324—44no—
108C8×D47—44no—
109C23×C47C22⋊Q811yescharacteristic
110C22×D47—22no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—41no—
114C2×C4×C47C2×C22⋊C411yescharacteristic
115C2×C4×C47C2×C4⋊C411yescharacteristic
116C2×C4×C47#11711yescharacteristic
117C2×C4×C47#12111yescharacteristic
118C2×C4×C47#12111yescharacteristic
119C2×C4×C47C22⋊Q811yescharacteristic
120C2×C4×C47#13511yescharacteristic
121C2×C4×C47#13511yescharacteristic
122C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
124C22⋊D47—42no—
125C22⋊D47—42no—
126C2×Dic47—22no—
127D188—42no—
128D188—41no—
129D188—41no—
130D188—41no—
131D188—41no—
132C4×D47—42no—
133C4×D47—42no—
134Dic94—41no—
135Dic94—41no—
136C8×C47—42no—
137C8×C47—81no—
138C8×C47—81no—
139C47⋊C8—21no—
140C47⋊C8—21no—
141C47⋊C8—21no—
142C47⋊C8—21no—
143C47⋊C8—81no—
144C47⋊C8—81no—
145#8686—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47#38011yescharacteristic
149C22×C47#47811yescharacteristic
150C22×C47#47811yescharacteristic
151C2×D47—42no—
152C2×D47—42no—
153C2×D47—42no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47#41611yescharacteristic
158C4×C47#41811yescharacteristic
159C4×C47#46311yescharacteristic
160C4×C47#46311yescharacteristic
161C4×C47#47111yescharacteristic
162C4×C47#47111yescharacteristic
163Dic47—42no—
164#2931—471no—
165#1837—471no—
166C2×#334—471no—
167#1547—471no—
168#1547—471no—
169#1292—471no—
170#1292D4711yescharacteristic, complemented
171C2×C47—21no—
172C2×C47—21no—
173C2×C47#180411yescharacteristic
174C2×C47#181311yescharacteristic
175C2×C47#181511yescharacteristic
176D47—82no—
177C2×#131—941no—
178#532—471no—
179#436—472no—
180#436—941no—
181C8×D4—472no—
182C8×D4—941no—
183#423—471no—
184#423C2×D4711yescharacteristic
185#423C2×D4711yescharacteristic
186C2×C4⋊C8—471no—
187C2×C4⋊C8C2×D4711yescharacteristic
188#408—941no—
189C2×C4×C8—471no—
190#341—941no—
191#341—941no—
192#334—941no—
193#334—941no—
194#332—941no—
195#332—941no—
196#331—941no—
197#331—941no—
198C47#868611yescharacteristic, complemented, Sylow
199C2×#41—941no—
200#131—1882no—
201#131—1882no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—942no—
205C2×M16—941no—
206C22×C8—21no—
207C22×C8—471no—
208C22×C8—471no—
209C22×C8—942no—
210C22×C8—941no—
211C4⋊Q8—471no—
212#127—471no—
213#124—941no—
214#121—472no—
215#121—941no—
216C4×D4—472no—
217C4×D4—941no—
218C2×C42C22×D4711yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—941no—
225C4⋊C8—941no—
226C4⋊C8—941no—
227C4⋊C8—941no—
228C4⋊C8C4×D4712yes—
229C22⋊C8—942no—
230C22⋊C8—942no—
231C4×C8—471no—
232C4×C8—471no—
233C4×C8—471no—
234C4×C8—471no—
235#41—1881no—
236#41—1881no—
237#41—1881no—
238#41—1881no—
239C2×Q8—941no—
240C2×D4—472no—
241C2×D4—942no—
242C2×D4—941no—
243C2×D4—941no—
244C22×C4—942no—
245C22×C4#1062535011yescharacteristic
246C22×C4D4×D4711yescharacteristic
247C22×C4Q8×D4711yescharacteristic
248M16—41no—
249M16—41no—
250M16—41no—
251M16—41no—
252M16—1884no—
253M16—1881no—
254M16—1881no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—81no—
259C2×C8—471no—
260C2×C8—471no—
261C2×C8—471no—
262C2×C8—471no—
263C2×C8—941no—
264C2×C8—941no—
265C2×C8—941no—
266C2×C8—941no—
267C2×C8—3761no—
268C2×C8—1884no—
269C2×C8—1881no—
270C2×C8—1881no—
271C4⋊C4—472no—
272C4⋊C4—941no—
273C22⋊C4—942no—
274C22⋊C4—942no—
275C42C2×C4×D4711yescharacteristic
276C42#1062535011yescharacteristic
277C42D4×D4711yescharacteristic
278C42D4×D4711yescharacteristic
279C23—942no—
280C23#1309400211yescharacteristic
281Q8—1881no—
282Q8—1881no—
283D4—1882no—
284D4—1882no—
285D4—1882no—
286D4—1881no—
287D4—1881no—
288D4—1881no—
289D4—1881no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—942no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4C22⋊C4×D4711yescharacteristic
297C2×C4C4⋊C4×D4711yescharacteristic
298C2×C4#1309398111yescharacteristic
299C2×C4#1309399711yescharacteristic
300C2×C4#1309399711yescharacteristic
301C2×C4#1309400211yescharacteristic
302C2×C4#1309404511yescharacteristic
303C2×C4#1309404711yescharacteristic
304C2×C4#1309404811yescharacteristic
305C2×C4#1309405111yescharacteristic
306C8—42no—
307C8—81no—
308C8—81no—
309C8—941no—
310C8—941no—
311C8—941no—
312C8—941no—
313C8—3761no—
314C8—3761no—
315C22—21no—
316C22—21no—
317C22—1882no—
318C22—1882no—
319C22—1882no—
320C22#1568898211yescharacteristic
321C22#1568942211yescharacteristic
322C22#1568942611yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—1882no—
327C4#1568911811yescharacteristic
328C4#1568912611yescharacteristic
329C4#1568933711yescharacteristic
330C4#1568934111yescharacteristic
331C4#1568938511yescharacteristic
332C4#1568938911yescharacteristic
333C2—21no—
334C2—21no—
335C2—3762no—
336C2#2024339711yescharacteristic
337C2#2024343711yescharacteristic
338C2#2024344611yescharacteristic
subgroupG/H
centerC22#15688982
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1292C2
socleC22×C47#380
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1292D47
hypocenterC47#8686
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #20243525C2semidirect
3. C2×#15688873C2semidirect
6. #20241262C2semidirect
7. C47×#1292C2semidirect
170. #1292D47semidirect
198. C47#8686semidirect

Maximal subgroups

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

subgroupindex
#202520992
#202435252
C2×#156888732
#202420662
#202420632
#202412622
C47×#12922
#868647