Finite Group Databasebeta

← #26234670

Subgroups — #26234670

Order 12032.

1134 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D376C211yesmaximal, characteristic
2C2×#15689205C212yesmaximal
3#20246288C212yesmaximal, complemented
4#20244119C212yesmaximal, complemented
5#20243632C214yesmaximal, complemented
6#20243385C212yesmaximal, complemented
7C8×C4⋊Dic47C211yesmaximal, characteristic, complemented
8C47×#1845C211yesmaximal, characteristic, complemented
9C22×D376C2211yescharacteristic
10C2×C4×D188C2212yes—
11C2×#13093986C2214yes—
12C2×C4⋊C4×D47—22no—
13C4×D376C418yescomplemented
14#15689205—28no—
15C2×#13093958C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×#13093946C2212yes—
18C2×#13093944C2214yes—
19C2×C8×Dic47C2212yes—
20#15689019C2214yes—
21#15688994—24no—
22#15688980—22no—
23C4×C4⋊Dic47C2211yescharacteristic
24#15688885C2214yes—
25#15688877C2212yes—
26C2×#107×C47C2212yes—
27C2×#107×C47C2212yes—
28C2×C4×C8×C47C2211yescharacteristic
29C47×#384C2212yes—
30C22×D188C2312yes—
31C22×C4×D47—22no—
32C22×C4×D47—24no—
33C2×D376C2×C414yes—
34C2×D376D414yes—
35C2×D376Q814yes—
36C22×C47⋊C8C2312yes—
37C4×D188—24no—
38C4×D188C2×C418yescomplemented
39#13093986—28no—
40#13093986—44no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×C4⋊Dic47D411yescharacteristic
43C2×C4⋊Dic47C2312yes—
44C2×Dic47⋊C4—24no—
45C4⋊C4×D47—48no—
46C2×(C2×D47)⋊C4—22no—
47C2×(C2×D47)⋊C4—24no—
48C2×C4×Dic47C2312yes—
49#13093958C2×C414yescomplemented
50C4⋊(C47⋊C8)—22no—
51#13093946C2×C418yescomplemented
52#13093944—28no—
53C8×Dic47—24no—
54C47⋊#95—24no—
55C47⋊#95D412yes—
56C22×C8×C47C2311yescharacteristic
57C22×C8×C47C2311yescharacteristic
58C2×C4⋊C4×C47—22no—
59C2×C4⋊C4×C47—22no—
60C2×C4⋊C4×C47C2314yes—
61C2×C42×C47C2311yescharacteristic
62#107×C47—24no—
63#107×C47—42no—
64C4×C8×C47C2×C414yescomplemented
65C23×D47—22no—
66C22×C4×C47—24no—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2411yescharacteristic
70C22×Dic47—22no—
71C22×Dic47C2×D412yes—
72C22×Dic47C2×D412yes—
73C2×D188—22no—
74C2×D188—22no—
75C2×D188—22no—
76C2×D188C22×C414yes—
77C2×D188C2×D414yes—
78C2×D188C2×Q814yes—
79C2×C4×D47—28no—
80C2×C4×D47—44no—
81C2×C4×D47—44no—
82C2×C4×D47—48no—
83C2×C4×D47—48no—
84C2×C4×D47—84no—
85C2×C8×C47—21no—
86C2×C8×C47—21no—
87C2×C8×C47C22×C412yes—
88C2×C8×C47C22×C412yes—
89C2×C8×C47C2×D412yes—
90C2×C8×C47C2×Q812yes—
91C4⋊C4×C47—24no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47—44no—
94C4⋊C4×C47—82no—
95C42×C47C22×C412yes—
96C42×C47#4112yes—
97(C2×D47)⋊C4—28no—
98(C2×D47)⋊C4—48no—
99C4⋊Dic47—22no—
100C4⋊Dic47—22no—
101C4⋊Dic47D814yescomplemented
102C4⋊Dic47C22×C414yes—
103Dic47⋊C4—48no—
104C4×Dic47—22no—
105C4×Dic47—22no—
106C2×C47⋊C8—22no—
107C2×C47⋊C8—22no—
108C2×C47⋊C8—22no—
109D376C4⋊C4116yescomplemented
110C23×C47C22×D411yescharacteristic
111C22×D47—24no—
112C22×D47—24no—
113C22×D47—24no—
114C22×D47—42no—
115C22×D47—42no—
116C22×D47—42no—
117C22×D47—42no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—44no—
121C2×C4×C47—44no—
122C2×C4×C47—44no—
123C2×C4×C47C4×D412yes—
124C2×C4×C47C4×D412yes—
125C2×C4×C47C2×D812yes—
126C2×C4×C47#13512yes—
127C2×C4×C47C23×C411yescharacteristic
128C2×C4×C47C22×D411yescharacteristic
129C2×C4×C47C22×Q811yescharacteristic
130C2×C4×C47C2×#4111yescharacteristic
131C2×C4×C47H(5,𝔽2)11yescharacteristic
132C2×C4×C472−1+411yescharacteristic
133C2×Dic47—22no—
134C2×Dic47—22no—
135C2×Dic47—24no—
136C2×Dic47—22no—
137C2×Dic47—22no—
138C2×Dic47—22no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47C2×D814yes—
142D188—24no—
143D188—24no—
144D188—24no—
145D188C2×C4⋊C418yescomplemented
146C4×D47—416no—
147C4×D47—816no—
148C8×C47—22no—
149C8×C47C2×C4⋊C414yescomplemented
150C47⋊C8—42no—
151C4⋊C4×D8—471nomaximal, Sylow
152C22×C47C2×C4×D411yescharacteristic
153C22×C47D4211yescharacteristic
154C22×C47D4×Q811yescharacteristic
155C22×C47C22×D811yescharacteristic
156C22×C47C2×#13511yescharacteristic
157C22×C47#57611yescharacteristic
158C22×C47#57811yescharacteristic
159C2×D47—28no—
160C2×D47—44no—
161C2×D47—44no—
162C2×D47—44no—
163C2×D47—44no—
164C2×D47—44no—
165C2×D47—44no—
166C4×C47—22no—
167C4×C47—84no—
168C4×C47C4×D814yescomplemented
169C4×C47C22×C4⋊C411yescharacteristic
170C4×C47#52011yescharacteristic
171C4×C47#54711yescharacteristic
172C4×C47#55411yescharacteristic
173Dic47—24no—
174Dic47—44no—
175Dic47—42no—
176C2×C4×D8—472no—
177C2×C4×D8—471no—
178D4×C4⋊C4—472no—
179#1942—472no—
180#1862—474no—
181#1845D4711yescharacteristic, complemented
182#1802—472no—
183C8×C4⋊C4—471no—
184C2×C47D4×C4⋊C411yescharacteristic
185C2×C47C2×C4×D811yescharacteristic
186C2×C47#295111yescharacteristic
187C2×C47D4×D811yescharacteristic
188C2×C47#328311yescharacteristic
189C2×C47Q8×D811yescharacteristic
190C2×C47#338011yescharacteristic
191D47—48no—
192C22×D8—471no—
193C2×C4×D4—474no—
194C2×C4×D4—472no—
195C22×C4⋊C4—942no—
196C4×D8—478no—
197C4×D8—948no—
198C2×#107—471no—
199C2×#107C2×D4712yes—
200C2×#107C2×D4712yes—
201C2×C4⋊C8—471no—
202C2×#102—472no—
203C2×#102—474no—
204C2×C4×C8—472no—
205C2×C4×C8C2×D4711yescharacteristic
206#389—474no—
207#385—944no—
208#384C2×D4712yes—
209#380—942no—
210C4×C4⋊C4—471no—
211#340—474no—
212#336—472no—
213C47C4⋊C4×D811yescharacteristic, complemented, Sylow
214C22×D4—472no—
215C23×C4—944no—
216C23×C4—942no—
217C2×D8—474no—
218C2×D8—474no—
219C2×D8—474no—
220C22×C8—472no—
221C22×C8C22×D4711yescharacteristic
222C22×C8C22×D4711yescharacteristic
223C4×D4—478no—
224C4×D4—948no—
225C4×D4—944no—
226C4×D4—1884no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—471no—
230C2×C4⋊C4—472no—
231C2×C4⋊C4—471no—
232C2×C4⋊C4—944no—
233C2×C4⋊C4—1888no—
234C2×C4⋊C4C22×D4714yes—
235C2×C22⋊C4—942no—
236C2×C22⋊C4—944no—
237C2×C42—472no—
238C2×C42C22×D4711yescharacteristic
239#107—24no—
240#107—42no—
241#107—474no—
242C4⋊C8—942no—
243#102—478no—
244#102—948no—
245C4×C8—944no—
246C4×C8C4×D4714yescomplemented
247#95—472no—
248#95—944no—
249C24—942no—
250C2×D4—474no—
251C2×D4—474no—
252C2×D4—474no—
253C2×D4—942no—
254C2×D4—942no—
255C2×D4—942no—
256C22×C4—24no—
257C22×C4—472no—
258C22×C4—472no—
259C22×C4—942no—
260C22×C4—948no—
261C22×C4—3764no—
262C22×C4—1888no—
263C22×C4—1888no—
264C22×C4—1884no—
265C22×C4—1884no—
266C22×C4D4×D4711yescharacteristic
267C22×C4D4×D4711yescharacteristic
268C22×C4C23×D4711yescharacteristic
269D8—4716no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—942no—
273C2×C8—942no—
274C2×C8—942no—
275C2×C8C2×C4×D4712yes—
276C2×C8C2×C4×D4712yes—
277C2×C8D4×D4712yes—
278C2×C8Q8×D4712yes—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—44no—
282C4⋊C4—82no—
283C4⋊C4—474no—
284C4⋊C4—474no—
285C4⋊C4—942no—
286C4⋊C4—942no—
287C4⋊C4—1888no—
288C22⋊C4—948no—
289C22⋊C4—1888no—
290C42—942no—
291C42—942no—
292C42C2×C4×D4712yes—
293C42#1062535412yes—
294C23—944no—
295C23—944no—
296C23—944no—
297C23—1882no—
298C23—1882no—
299C23—1882no—
300C23—1882no—
301C23C2×D4×D4711yescharacteristic
302D4—478no—
303D4—944no—
304D4—944no—
305D4—944no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—474no—
312C2×C4—942no—
313C2×C4—942no—
314C2×C4—942no—
315C2×C4—944no—
316C2×C4—942no—
317C2×C4—942no—
318C2×C4—37616no—
319C2×C4—1882no—
320C2×C4—1882no—
321C2×C4—18816no—
322C2×C4#1309398612yes—
323C2×C4#1309398612yes—
324C2×C4D8×D4712yes—
325C2×C4#1309404712yes—
326C2×C4C22×C4×D4711yescharacteristic
327C2×C4C2×D4×D4711yescharacteristic
328C2×C4C2×Q8×D4711yescharacteristic
329C2×C4C2×#1062535411yescharacteristic
330C2×C4#1309406911yescharacteristic
331C2×C4#1309407211yescharacteristic
332C8—22no—
333C8—1882no—
334C8C4⋊C4×D4714yescomplemented
335C22—948no—
336C22—1884no—
337C22—1884no—
338C22—1884no—
339C22—1884no—
340C22—1884no—
341C22—1884no—
342C22C2×#1309398611yescharacteristic
343C22#1568975811yescharacteristic
344C22#1568978811yescharacteristic
345C22C2×D8×D4711yescharacteristic
346C22C2×#1309404711yescharacteristic
347C22#1568992411yescharacteristic
348C22#1568993611yescharacteristic
349C4—22no—
350C4—84no—
351C4—944no—
352C4—1882no—
353C4—1884no—
354C4#1568920514yescomplemented
355C4C2×C4⋊C4×D4711yescharacteristic
356C4#1568960411yescharacteristic
357C4#1568977611yescharacteristic
358C4#1568983411yescharacteristic
359C2—1888no—
360C2#2024628811yescharacteristic
361C2C2×#1568920511yescharacteristic
362C2#2025226211yescharacteristic
363C2#2025506311yescharacteristic
364C2#2025510811yescharacteristic
365C2#2025624011yescharacteristic
366C2#2025625711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1845C2
socleC23×C47C22×D4
prime radicalC22×D376C22
cosocleC22×C4×C47C24
hypercenter#1845D47
hypocenterC47C4⋊C4×D8
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
3. #20246288C2semidirect
4. #20244119C2semidirect
5. #20243632C2semidirect
6. #20243385C2semidirect
7. C8×C4⋊Dic47C2semidirect
8. C47×#1845C2semidirect
13. C4×D376C4semidirect
38. C4×D188C2×C4semidirect
49. #13093958C2×C4semidirect
51. #13093946C2×C4semidirect
64. C4×C8×C47C2×C4semidirect
101. C4⋊Dic47D8semidirect
109. D376C4⋊C4semidirect
145. D188C2×C4⋊C4semidirect
149. C8×C47C2×C4⋊C4semidirect
168. C4×C47C4×D8semidirect
181. #1845D47semidirect
213. C47C4⋊C4×D8semidirect
246. C4×C8C4×D47semidirect
334. C8C4⋊C4×D47semidirect
354. C4#15689205semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3762
C2×#156892052
#202462882
#202441192
#202436322
#202433852
C8×C4⋊Dic472
C47×#18452
C4⋊C4×D847