Finite Group Databasebeta

← #26132164

Subgroups — #26132164

Order 12032.

848 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688922C211yesmaximal, characteristic, complemented
2C2×#15688920C212yesmaximal, complemented
3#20243792C212yesmaximal, complemented
4#20243782C211yesmaximal, characteristic, complemented
5C2×C47×#555C211yesmaximal, characteristic
6#15689601C2211yescharacteristic
7#15689132—21no—
8#15689128—22no—
9C2×#13093943C2211yescharacteristic
10C2×#13093943C2212yes—
11#15688922—22no—
12#15688920—24no—
13#15688883—21no—
14#15688883—22no—
15C47×#555—22no—
16C47×#555—24no—
17C47×#555C414yescomplemented
18C2×C4⋊Q8×C47C2211yescharacteristic
19C2×#124×C47—21no—
20C2×#124×C47C2212yes—
21C2×#121×C47—22no—
22C2×C4×D4×C47—21no—
23C2×#117×C47—22no—
24C2×#10625329—22no—
25C2×#10625329—21no—
26Q8×Dic47—42no—
27#13093983—22no—
28#13093983—21no—
29#13093948—42no—
30#13093945—44no—
31#13093943—22no—
32#13093943—22no—
33#13093943—22no—
34C22×Q8×C47D411yescharacteristic
35C22×Q8×C47C2311yescharacteristic
36C22×D4×C47—21no—
37C22×D4×C47—21no—
38C22×D4×C47D412yes—
39C23×C4×C47—22no—
40C4⋊Q8×C47—22no—
41C4⋊Q8×C47C2×C412yes—
42C4⋊Q8×C47D412yes—
43#124×C47—24no—
44#124×C47—24no—
45#124×C47—22no—
46#124×C47—22no—
47#124×C47C2×C414yes—
48#124×C47D414yes—
49#121×C47—28no—
50#121×C47—28no—
51C4×D4×C47—24no—
52C4×D4×C47—24no—
53#117×C47—28no—
54#117×C47—28no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—22no—
57C2×C22⋊C4×C47—22no—
58C2×C22⋊C4×C47—22no—
59C2×C22⋊C4×C47—22no—
60C2×C42×C47—21no—
61C2×C42×C47D412yes—
62C2×C42×C47D411yescharacteristic
63C24×C47—22no—
64C2×Q8×C47—21no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47C22⋊C412yes—
70C2×Q8×C47C22×C411yescharacteristic
71C2×Q8×C47C2×D411yescharacteristic
72C2×Q8×C47C2×D412yes—
73C2×D4×C47—24no—
74C2×D4×C47—24no—
75C2×D4×C47—22no—
76C2×D4×C47—44no—
77C2×D4×C47—42no—
78C2×D4×C47—42no—
79C2×D4×C47—81no—
80C2×D4×C47—81no—
81C2×D4×C47C22⋊C414yes—
82C22×C4×C47—21no—
83C22×C4×C47—28no—
84C22×C4×C47—22no—
85C22×C4×C47—21no—
86C22×C4×C47—48no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D412yes—
89C4⋊C4×C47—24no—
90C4⋊C4×C47—24no—
91C4⋊C4×C47—24no—
92C22⋊C4×C47—28no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—24no—
96C22⋊C4×C47—24no—
97C42×C47—24no—
98C42×C47—22no—
99C42×C47—21no—
100C42×C47C22⋊C414yes—
101C42×C47C22⋊C412yes—
102C22⋊Dic47—41no—
103C4⋊Dic47—24no—
104C4⋊Dic47—22no—
105C4×Dic47—42no—
106C4×Dic47—41no—
107#10625329—42no—
108#10625329—41no—
109#10625329—41no—
110#10625329—42no—
111C2×C47⋊C8—42no—
112C2×C47⋊C8—41no—
113C23×C47—24no—
114C23×C47—44no—
115C23×C47—44no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47#12011yescharacteristic
119Q8×C47—41no—
120Q8×C47—41no—
121Q8×C47—42no—
122Q8×C47—42no—
123Q8×C47—81no—
124D4×C47—44no—
125D4×C47—48no—
126D4×C47—82no—
127D4×C47—84no—
128C2×C4×C47—24no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—21no—
132C2×C4×C47—22no—
133C2×C4×C47—21no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—416no—
137C2×C4×C47—44no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47C2×C22⋊C412yes—
142C2×C4×C47#12011yescharacteristic
143C2×C4×C47#12012yes—
144C2×Dic47—42no—
145C47⋊C8—82no—
146C47⋊C8—81no—
147#12776—471nomaximal, Sylow
148C22×C47—22no—
149C22×C47—44no—
150C22×C47—44no—
151C22×C47—44no—
152C22×C47—84no—
153C22×C47—84no—
154C22×C47#37911yescharacteristic
155C22×C47B(4,𝔽2)11yescharacteristic
156C22×C47#45811yescharacteristic
157C4×C47—24no—
158C4×C47—22no—
159C4×C47—21no—
160C4×C47—22no—
161C4×C47—21no—
162C4×C47—44no—
163C4×C47—42no—
164Dic47—82no—
165C2×#555D4711yescharacteristic
166C2×#356—471no—
167C2×#355—472no—
168#1888—472no—
169#1886—471no—
170C2×C47—21no—
171C2×C47—21no—
172C2×C47—84no—
173C2×C47#179011yescharacteristic
174C2×C47#220412yes—
175#555—22no—
176#555—24no—
177#555Dic4714yescomplemented
178C2×C4⋊Q8C2×D4711yescharacteristic
179C2×#124—21no—
180C2×#124C2×D4712yes—
181C2×#121—22no—
182#519—471no—
183C2×C4×D4—21no—
184C2×#117—22no—
185#419—941no—
186#418—942no—
187C2×#101—471no—
188C2×#101—472no—
189#356—942no—
190#355—944no—
191#339—941no—
192#339—942no—
193C47#1277611yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D4711yescharacteristic
195C22×Q8C22×D4711yescharacteristic
196C22×D4—21no—
197C22×D4—21no—
198C22×D4C22⋊D4712yes—
199C23×C4—22no—
200C2×M16—942no—
201C2×M16—941no—
202C4⋊Q8—22no—
203C4⋊Q8C2×Dic4712yes—
204C4⋊Q8C22⋊D4712yes—
205#124—24no—
206#124—24no—
207#124—22no—
208#124—22no—
209#124C2×Dic4714yes—
210#124C22⋊D4714yes—
211#121—28no—
212#121—28no—
213C4×Q8—1882no—
214C4×D4—24no—
215C4×D4—24no—
216#117—28no—
217#117—28no—
218#117—942no—
219#117—941no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42—21no—
226C2×C42C22⋊D4712yes—
227C2×C42C22⋊D4711yescharacteristic
228#103—1882no—
229#102—1884no—
230#101—942no—
231#101—942no—
232#101—942no—
233C24—22no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8C22⋊Dic4712yes—
240C2×Q8C22×Dic4711yescharacteristic
241C2×Q8C2×C22⋊D4711yescharacteristic
242C2×Q8C2×C22⋊D4712yes—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—81no—
250C2×D4—81no—
251C2×D4C22⋊Dic4714yes—
252C22×C4—21no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—48no—
257C22×C4C2×C22⋊D4711yescharacteristic
258C22×C4C2×C22⋊D4712yes—
259M16—1882no—
260M16—1881no—
261M16—1881no—
262M16—1882no—
263C2×C8—1882no—
264C2×C8—1881no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—944no—
269C4⋊C4—942no—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—1881no—
276C42—24no—
277C42—22no—
278C42—21no—
279C42—1882no—
280C42—1881no—
281C42C22⋊Dic4714yes—
282C42C22⋊Dic4712yes—
283C23—24no—
284C23—44no—
285C23—44no—
286C23—42no—
287C23—42no—
288C23#1309399311yescharacteristic
289Q8—41no—
290Q8—41no—
291Q8—42no—
292Q8—42no—
293Q8—81no—
294D4—44no—
295D4—48no—
296D4—82no—
297D4—84no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—416no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—1882no—
311C2×C4C2×C22⋊Dic4711yescharacteristic
312C2×C4C2×C22⋊Dic4712yes—
313C2×C4#1309399311yescharacteristic
314C2×C4#1309399312yes—
315C8—3762no—
316C8—3761no—
317C22—22no—
318C22—44no—
319C22—44no—
320C22—44no—
321C22—84no—
322C22—84no—
323C22#1568897911yescharacteristic
324C22#1568931011yescharacteristic
325C22#1568931311yescharacteristic
326C4—24no—
327C4—22no—
328C4—21no—
329C4—22no—
330C4—21no—
331C4—44no—
332C4—42no—
333C4—3762no—
334C2—21no—
335C2—21no—
336C2—84no—
337C2#2024333611yescharacteristic
338C2#2024541412yes—
subgroupG/H
centerC22#15689310
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
Fitting, prime radicalC2×C47×#555C2
socleC22×C47B(4,𝔽2)
cosocleC22×Q8×C47C23
hypercenterC2×#555D47
hypocenterC47#12776
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#15688922C2semidirect
2. C2×#15688920C2semidirect
3. #20243792C2semidirect
4. #20243782C2semidirect
17. C47×#555C4semidirect
177. #555Dic47semidirect
193. C47#12776semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
C2×#156889202
#202437922
#202437822
C2×C47×#5552
#1277647