Finite Group Databasebeta

← #26371002

Subgroups — #26371002

Order 12032.

534 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#20253924C211yesmaximal, characteristic, complemented
2#20252264C211yesmaximal, characteristic
3#20245554C212yesmaximal, complemented
4#20245543C212yesmaximal, complemented
5#20245533C212yesmaximal, complemented
6#20245528C212yesmaximal, complemented
7#20245176C211yesmaximal, characteristic, complemented
8C2×#15688950C211yesmaximal, characteristic, complemented
9#20245148C211yesmaximal, characteristic, complemented
10#20245095C211yesmaximal, characteristic
11C47×#2139C211yesmaximal, characteristic
12#15689754—21no—
13#15689606—21no—
14C22×C47⋊C16C2211yescharacteristic
15#15689531—22no—
16#15689527C2212yes—
17#15689524C2212yes—
18#15689224—22no—
19Q16×Dic47C2212yes—
20D8×Dic47C2212yes—
21#15689163—21no—
22#15689161C2211yescharacteristic
23C2×#13093958C2211yescharacteristic
24#15689124—21no—
25#15689078C2211yescharacteristic
26#15688951C2212yes—
27#15688950C2212yes—
28#15688950C2212yes—
29#15688949C2212yes—
30Dic47⋊C16C2212yes—
31Dic47⋊C16C2212yes—
32#15688927C2211yescharacteristic
33#15688927C2211yescharacteristic
34#15688924C2211yescharacteristic
35#15688924C2211yescharacteristic
36C2×#135×C47C2211yescharacteristic
37C2×M32×C47C2211yescharacteristic
38C2×#107×C47C2211yescharacteristic
39C47×#358C2212yes—
40C47×#357C2212yes—
41C2×#10625329D411yescharacteristic
42#13094023—22no—
43#13094023—41no—
44#13094015—22no—
45#13094015—41no—
46#13094003—42no—
47Q8×Dic47—22no—
48D4×Dic47—22no—
49D4×Dic47—42no—
50#13093983—21no—
51#13093983—41no—
52#13093983D411yescharacteristic
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47—41no—
55C2×C47⋊C16—22no—
56C2×C47⋊C16C2311yescharacteristic
57C2×C47⋊C16C2311yescharacteristic
58#13093958—21no—
59#13093958C2312yes—
60#13093958C2311yescharacteristic
61#13093958C2311yescharacteristic
62#13093957—22no—
63#13093956—21no—
64#13093955—22no—
65#13093948—22no—
66#13093945—22no—
67#13093934D412yes—
68C8×Dic47C2312yes—
69C2×#41×C47—21no—
70#135×C47—22no—
71#135×C47Q814yescomplemented
72C2×Q16×C47C2311yescharacteristic
73C2×SD16×C47—21no—
74C2×D8×C47C2311yescharacteristic
75C22×C8×C47C2311yescharacteristic
76C2×C4⋊C4×C47—21no—
77M32×C47—22no—
78C2×C16×C47C2312yes—
79#107×C47—21no—
80#107×C47C2312yes—
81#41×C47—24no—
82#41×C47—42no—
83C2×Q8×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47C2×D411yescharacteristic
89C22×Dic47—41no—
90C22×Dic47—42no—
91C2×Dic94—42no—
92Q16×C47—21no—
93Q16×C47C2×Q812yes—
94SD16×C47—24no—
95D8×C47—21no—
96D8×C47C2×Q812yes—
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×Q812yes—
99C2×C8×C47#4112yes—
100C2×C8×C47C2411yescharacteristic
101C4⋊C4×C47—22no—
102C4⋊C4×C47—41no—
103C22⋊Dic47—41no—
104C22⋊Dic47—42no—
105C22⋊Dic47D812yes—
106C4⋊Dic47—22no—
107C4⋊Dic47—21no—
108C4⋊Dic47—21no—
109C4⋊Dic47—41no—
110C4⋊Dic47—42no—
111C4⋊Dic47—42no—
112C4⋊Dic47D811yescharacteristic
113C4⋊Dic47D811yescharacteristic
114Dic47⋊C4—42no—
115Dic47⋊C4—42no—
116C4×Dic47—41no—
117C4×Dic47—42no—
118C4×Dic47C2×D412yes—
119#10625329—22no—
120C2×C47⋊C8C2×D412yes—
121C16×C47—22no—
122C47⋊C16—21no—
123C47⋊C16—21no—
124C23×C47—41no—
125C23×C47C2×D811yescharacteristic
126Q8×C47—22no—
127Q8×C47—41no—
128D4×C47—22no—
129D4×C47—44no—
130D4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—44no—
133C2×C4×C47—42no—
134C2×C4×C47—161no—
135C2×C4×C47C22⋊Q812yes—
136C2×C4×C47C2×D811yescharacteristic
137C2×C4×C47#13712yes—
138C2×C4×C47C22×D411yescharacteristic
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—42no—
142C2×Dic47—162no—
143C2×Dic47—161no—
144C2×Dic47C2×D812yes—
145C2×Dic47C2×D812yes—
146Dic94—162no—
147C8×C47C22⋊Q812yes—
148C8×C47C22×Q811yescharacteristic
149C8×C47C2×#4111yescharacteristic
150C47⋊C8—22no—
151#33634—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—42no—
155C22×C47—81no—
156C22×C47—81no—
157C22×C47#47412yes—
158C22×C47C22×D811yescharacteristic
159C4×C47—42no—
160C4×C47—82no—
161C4×C47#47412yes—
162C4×C47C2×C22⋊Q811yescharacteristic
163C4×C47C2×#13711yescharacteristic
164Dic47—22no—
165Dic47—22no—
166Dic47—81no—
167Dic47—81no—
168Dic47—82no—
169#3164—471no—
170#2951—471no—
171#2229—472no—
172#2228—472no—
173#2226—472no—
174#2225—472no—
175#2159—471no—
176C2×#367—471no—
177#2151—471no—
178#2139D4711yescharacteristic
179#2138—471no—
180C2×C47—22no—
181C2×C47—82no—
182C2×C47C2×#47411yescharacteristic
183C2×C47#341611yescharacteristic
184C2×C47#341611yescharacteristic
185C2×#135C2×D4711yescharacteristic
186#544—941no—
187#520—941no—
188C2×M32C2×D4711yescharacteristic
189C22×C16—471no—
190#501—942no—
191#500—472no—
192#499—472no—
193#440—942no—
194C4×Q16—472no—
195C4×D8—472no—
196#428—941no—
197#427—471no—
198C2×#107—471no—
199C2×#107C2×D4711yescharacteristic
200#417—941no—
201#405—471no—
202#367—472no—
203#367—472no—
204#367—472no—
205#366—472no—
206C4⋊C16—472no—
207C4⋊C16—472no—
208#358—471no—
209#358—471no—
210#358C2×D4712yes—
211#357—471no—
212#357—471no—
213#357C2×D4712yes—
214C47#3363411yescharacteristic, complemented, Sylow
215C2×#41—21no—
216#135—22no—
217#135Dic9414yescomplemented
218C2×Q16C22×D4711yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D4711yescharacteristic
221C2×M16—471no—
222C22×C8C22×D4711yescharacteristic
223C4⋊Q8—942no—
224C4⋊Q8—1881no—
225#125—942no—
226#125—1881no—
227C22⋊Q8—1882no—
228C4×Q8—942no—
229C4×D4—942no—
230C4×D4—1882no—
231#117—471no—
232#117—941no—
233#117—1881no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4—1881no—
237M32—22no—
238C2×C16—471no—
239C2×C16—471no—
240C2×C16—942no—
241C2×C16C22×D4712yes—
242#107—21no—
243#107—471no—
244#107—471no—
245#107—472no—
246#107—942no—
247#107—941no—
248#107C22×D4712yes—
249#106—942no—
250#106—941no—
251#103—942no—
252#102—942no—
253#97—472no—
254C4×C8—472no—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—1882no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—1882no—
264C22×C4—1881no—
265C22×C4D4×D4711yescharacteristic
266Q16—21no—
267Q16C2×Dic9412yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic9412yes—
271M16—942no—
272C2×C8—472no—
273C2×C8C2×Dic9412yes—
274C2×C8D4×D4711yescharacteristic
275C2×C8#1062535212yes—
276C2×C8C23×D4711yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—471no—
280C4⋊C4—471no—
281C4⋊C4—942no—
282C4⋊C4—941no—
283C4⋊C4—941no—
284C4⋊C4—1882no—
285C4⋊C4—1882no—
286C4⋊C4—1882no—
287C4⋊C4—1882no—
288C4⋊C4—1881no—
289C22⋊C4—472no—
290C22⋊C4—1882no—
291C22⋊C4—1881no—
292C42—472no—
293C42—1882no—
294C42—1881no—
295C16—22no—
296C16—941no—
297C16—941no—
298C23—41no—
299C23D8×D4711yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—7522no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—161no—
310C2×C4—472no—
311C2×C4—472no—
312C2×C4—1881no—
313C2×C4—1882no—
314C2×C4—1881no—
315C2×C4—7522no—
316C2×C4—7521no—
317C2×C4#1309400112yes—
318C2×C4D8×D4711yescharacteristic
319C2×C4#1309405312yes—
320C2×C4C2×D4×D4711yescharacteristic
321C8—942no—
322C8#1309400112yes—
323C8C22×Dic9411yescharacteristic
324C8C2×#1062535211yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1568940112yes—
331C22C2×D8×D4711yescharacteristic
332C4—42no—
333C4—82no—
334C4—942no—
335C4—942no—
336C4—3762no—
337C4—3761no—
338C4—3761no—
339C4#1568940112yes—
340C4C2×#1309400111yescharacteristic
341C4C2×#1309405311yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1568940111yescharacteristic
345C2#2025653311yescharacteristic
346C2#2025653411yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2139C2
socleC22×C47C22×D8
prime radicalC2×#135×C47C22
hypercenter#2139D47
hypocenterC47#33634
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. #20253924C2semidirect
3. #20245554C2semidirect
4. #20245543C2semidirect
5. #20245533C2semidirect
6. #20245528C2semidirect
7. #20245176C2semidirect
8. C2×#15688950C2semidirect
9. #20245148C2semidirect
71. #135×C47Q8semidirect
214. C47#33634semidirect
217. #135Dic94semidirect

Maximal subgroups

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

subgroupindex
#202539242
#202522642
#202455542
#202455432
#202455332
#202455282
#202451762
C2×#156889502
#202451482
#202450952
C47×#21392
#3363447