Finite Group Databasebeta

← #26185767

Subgroups — #26185767

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252497C212yesmaximal, complemented
2C4×#13094046C211yesmaximal, characteristic
3#20242211C212yesmaximal, complemented
4#20242206C212yesmaximal, complemented
5#20242187C214yesmaximal, complemented
6#20241773C212yesmaximal, complemented
7#20241671C211yesmaximal, characteristic, complemented
8C2×C47×#334C211yesmaximal, characteristic, complemented
9C2×#13094046C411yescharacteristic
10C4×#10625350C2212yes—
11#15689247C2214yes—
12C4×Dic188C411yescharacteristic
13C4×Dic188C2211yescharacteristic
14C4×#10625325C412yes—
15C4×#10625325C2212yes—
16C4×D376C411yescharacteristic
17C4×D376C2211yescharacteristic
18#15689188C2214yes—
19C8×C22⋊D47—24no—
20#15689176C2211yescharacteristic
21#15689176C2211yescharacteristic
22#15689175—22no—
23#15689161C411yescharacteristic
24C4⋊C8×D47—22no—
25#15689120C412yes—
26#15688853C2214yes—
27#15688850C2214yes—
28#15688844C2212yes—
29C8×C47⋊C8C2212yes—
30C2×C4⋊C8×C47C2212yes—
31C2×C4×C8×C47C2211yescharacteristic
32C47×#334C2212yes—
33C47×#334C2212yes—
34C2×#10625350C2×C412yes—
35#13094046C818yescomplemented
36C2×Dic188C2×C411yescharacteristic
37C2×#10625325C2×C412yes—
38C2×D376C2×C411yescharacteristic
39C2×C8×D47—24no—
40C4×Dic94C2×C412yes—
41C4×Dic94C2312yes—
42C4×D188C2×C412yes—
43C4×D188C2312yes—
44C4×C22⋊D47—22no—
45C4×C22⋊D47—22no—
46#13093983C2×C412yes—
47#13093981—22no—
48C42×D47—22no—
49#13093958C2×C411yescharacteristic
50#13093958C2×C411yescharacteristic
51#13093956C2×C412yes—
52C4⋊(C47⋊C8)—21no—
53C4⋊(C47⋊C8)D412yes—
54Dic47⋊C8—24no—
55#13093949C2×C412yes—
56#13093949C2×C412yes—
57#13093946C2×C412yes—
58#13093946C2×C412yes—
59C22⋊(C47⋊C8)—21no—
60C22⋊(C47⋊C8)D412yes—
61(C2×D47)⋊C8—24no—
62C4×C47⋊C8C2312yes—
63C4×C47⋊C8C2312yes—
64C8×Dic47—24no—
65C22×C8×C47—22no—
66C22×C8×C47C2×C411yescharacteristic
67C22×C8×C47C2×C411yescharacteristic
68C2×C42×C47C2311yescharacteristic
69C4⋊C8×C47—22no—
70C4⋊C8×C47C2314yes—
71C4×C8×C47C2×C411yescharacteristic
72C4×C8×C47C2×C411yescharacteristic
73C4×C8×C47C2311yescharacteristic
74C4×C8×C47C2311yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47#4111yescharacteristic
78C2×C22⋊D47—22no—
79#10625350—24no—
80#10625350C2×C818yescomplemented
81C2×D188C22×C412yes—
82C2×C4×D47—22no—
83C2×C4×D47—22no—
84C2×C4×D47—22no—
85C2×Dic94C22×C412yes—
86C2×C8×C47—22no—
87C2×C8×C47—24no—
88C2×C8×C47—82no—
89C2×C8×C47C2×C814yescomplemented
90C2×C8×C47C22×C411yescharacteristic
91C2×C8×C47C22×C411yescharacteristic
92C2×C8×C47C22×C411yescharacteristic
93C2×C8×C47C22×C411yescharacteristic
94C42×C47C22×C411yescharacteristic
95C42×C47C2×D411yescharacteristic
96C42×C47#4111yescharacteristic
97C42×C47C2411yescharacteristic
98C22⋊Dic47—22no—
99(C2×D47)⋊C4—22no—
100(C2×D47)⋊C4—22no—
101C4⋊Dic47C22×C412yes—
102C4⋊Dic47C22×C412yes—
103Dic47⋊C4—22no—
104Dic47⋊C4—22no—
105C4×Dic47—22no—
106C4×Dic47—22no—
107C4×Dic47—22no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8C2×D414yes—
110Dic188C2×C814yescomplemented
111D376C2×C814yescomplemented
112#10625325C2×C818yescomplemented
113C8×D47—84no—
114C23×C47C4×D411yescharacteristic
115C22×D47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—41no—
118C2×C4×C47C4×D411yescharacteristic
119C2×C4×C47C4×D411yescharacteristic
120C2×C4×C47C4×D411yescharacteristic
121C2×C4×C47C22×C812yes—
122C2×C4×C47#13112yes—
123C2×C4×C47C23×C411yescharacteristic
124C2×C4×C47C22×D411yescharacteristic
125C2×C4×C47C2×#4111yescharacteristic
126C22⋊D47—28no—
127C2×Dic47—22no—
128C2×Dic47—22no—
129C2×Dic47—22no—
130D188—22no—
131D188C22×C814yes—
132C4×D47—28no—
133C4×D47—42no—
134C4×D47—42no—
135C4×D47—82no—
136Dic94—22no—
137Dic94C22×C814yes—
138C8×C47—21no—
139C8×C47—21no—
140C8×C47—44no—
141C8×C47C22×C812yes—
142C8×C47C22×C812yes—
143C47⋊C8—24no—
144C47⋊C8—42no—
145#17858—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47C8×D412yes—
149C22×C47C2×C4×D411yescharacteristic
150C2×D47—24no—
151C2×D47—42no—
152C2×D47—42no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47C8×D412yes—
157C4×C47C23×C811yescharacteristic
158C4×C47C2×#13111yescharacteristic
159C4×C47#57712yes—
160Dic47—24no—
161Dic47—42no—
162Dic47—42no—
163#2988—472no—
164C4×#135—471no—
165#1580—472no—
166#1579—472no—
167C8×SD16—474no—
168C2×#334D4711yescharacteristic, complemented
169#1477—472no—
170#1450—471no—
171C2×C47—22no—
172C2×C47C2×C8×D411yescharacteristic
173C2×C47#295211yescharacteristic
174C2×C47#295711yescharacteristic
175D47—44no—
176C2×#135—471no—
177C4×#41—472no—
178C8×Q8—474no—
179C4×Q16—471no—
180C4×Q16—471no—
181C4×SD16—472no—
182C4×SD16—472no—
183C4×D8—471no—
184C4×D8—471no—
185C8×D4—474no—
186C8×D4—944no—
187#431—471no—
188#431—471no—
189#431—942no—
190#427—471no—
191C2×C4⋊C8—942no—
192C2×C4⋊C8C2×D4712yes—
193#416—472no—
194C2×C4×C8C2×D4711yescharacteristic
195#334C2×D4712yes—
196#334C2×D4712yes—
197Q8⋊C8—474no—
198D4⋊C8—474no—
199#322—472no—
200C82—472no—
201C47#1785811yescharacteristic, complemented, Sylow
202C2×#41—472no—
203#135—478no—
204C2×Q16—471no—
205C2×SD16—472no—
206C2×D8—471no—
207C22×C8—22no—
208C22×C8—944no—
209C22×C8C4×D4711yescharacteristic
210C22×C8C4×D4711yescharacteristic
211C4×Q8—472no—
212C4×Q8—472no—
213C4×D4—472no—
214C4×D4—472no—
215C4×D4—942no—
216C4×D4—942no—
217#117—472no—
218#117—942no—
219C2×C42—942no—
220C2×C42C22×D4711yescharacteristic
221#107—471no—
222#107—471no—
223#106—472no—
224C4⋊C8—22no—
225C4⋊C8—472no—
226C4⋊C8—944no—
227C4⋊C8—941no—
228C4⋊C8C22×D4714yes—
229#103—472no—
230#103—472no—
231#102—472no—
232#102—472no—
233C22⋊C8—472no—
234C22⋊C8—944no—
235C22⋊C8—941no—
236C4×C8—472no—
237C4×C8—472no—
238C4×C8—944no—
239C4×C8C4×D4711yescharacteristic
240C4×C8C4×D4711yescharacteristic
241C4×C8C22×D4711yescharacteristic
242C4×C8C22×D4711yescharacteristic
243#41—478no—
244#41—944no—
245C2×Q8—472no—
246C2×D4—472no—
247C2×D4—942no—
248C22×C4—942no—
249C22×C4—942no—
250C22×C4—942no—
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4D4×D4711yescharacteristic
253C22×C4#1062535411yescharacteristic
254Q16—474no—
255SD16—478no—
256D8—474no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—82no—
260C2×C8—474no—
261C2×C8—942no—
262C2×C8—3764no—
263C2×C8C8×D4714yescomplemented
264C2×C8C2×C4×D4711yescharacteristic
265C2×C8C2×C4×D4711yescharacteristic
266C2×C8C2×C4×D4711yescharacteristic
267C2×C8C2×C4×D4711yescharacteristic
268C4⋊C4—472no—
269C4⋊C4—472no—
270C4⋊C4—942no—
271C4⋊C4—942no—
272C22⋊C4—942no—
273C22⋊C4—942no—
274C22⋊C4—942no—
275C42—942no—
276C42—942no—
277C42—942no—
278C42C2×C4×D4711yescharacteristic
279C42D4×D4711yescharacteristic
280C42#1062535411yescharacteristic
281C42C23×D4711yescharacteristic
282C23—942no—
283C23#1309398611yescharacteristic
284Q8—474no—
285Q8—942no—
286D4—474no—
287D4—948no—
288D4—942no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—942no—
292C2×C4—948no—
293C2×C4—942no—
294C2×C4—942no—
295C2×C4—3762no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4#1309398611yescharacteristic
299C2×C4#1309398611yescharacteristic
300C2×C4#1309398611yescharacteristic
301C2×C4C2×C8×D4712yes—
302C2×C4#1309403212yes—
303C2×C4C22×C4×D4711yescharacteristic
304C2×C4C2×D4×D4711yescharacteristic
305C2×C4C2×#1062535411yescharacteristic
306C8—21no—
307C8—21no—
308C8—44no—
309C8—944no—
310C8—1882no—
311C8C2×C8×D4712yes—
312C8C2×C8×D4712yes—
313C22—21no—
314C22—21no—
315C22—944no—
316C22—1882no—
317C22—1882no—
318C22#1568918812yes—
319C22C2×#1309398611yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—944no—
324C4—1882no—
325C4—1882no—
326C4#1568918812yes—
327C4C22×C8×D4711yescharacteristic
328C4C2×#1309403211yescharacteristic
329C4#1568992912yes—
330C2—22no—
331C2—1884no—
332C2C2×#1568918811yescharacteristic
333C2#2025227111yescharacteristic
334C2#2025230411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC4×C47C23×C8
FrattiniC42C23×D47
FittingC2×C47×#334C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094046C4
cosocleC42×C47C24
hypercenterC2×#334D47
hypocenterC47#17858
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
1. #20252497C2semidirect
3. #20242211C2semidirect
4. #20242206C2semidirect
5. #20242187C2semidirect
6. #20241773C2semidirect
7. #20241671C2semidirect
8. C2×C47×#334C2semidirect
35. #13094046C8semidirect
80. #10625350C2×C8semidirect
89. C2×C8×C47C2×C8semidirect
110. Dic188C2×C8semidirect
111. D376C2×C8semidirect
112. #10625325C2×C8semidirect
168. C2×#334D47semidirect
201. C47#17858semidirect
263. C2×C8C8×D47semidirect

Maximal subgroups

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

subgroupindex
#202524972
C4×#130940462
#202422112
#202422062
#202421872
#202417732
#202416712
C2×C47×#3342
#1785847