Finite Group Databasebeta

← #26118851

Subgroups — #26118851

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689410C211yesmaximal, characteristic
2C2×#15689325C211yesmaximal, characteristic
3#20244182C211yesmaximal, characteristic
4C2×#15688872C211yesmaximal, characteristic, complemented
5C2×#15688869C211yesmaximal, characteristic, complemented
6#20241223C211yesmaximal, characteristic, complemented
7C2×C47×#498C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×C4×D188C2211yescharacteristic
10#15689410—22no—
11#15689410—22no—
12#15689325—22no—
13#15689325—22no—
14C2×#13093955—21no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4⋊(C47⋊C8)C2211yescharacteristic
17C2×C4⋊(C47⋊C8)C2211yescharacteristic
18C2×#13093948—21no—
19C2×#13093944—21no—
20C2×(C2×D47)⋊C8—22no—
21#15688872—22no—
22#15688869—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C2×C4⋊Q8×C47C2211yescharacteristic
25C47×#498—22no—
26C47×#498—22no—
27C2×#106×C47—22no—
28C2×C4×C8×C47C2211yescharacteristic
29C22×D188D411yescharacteristic
30C22×C4×D47—21no—
31C2×#10625336—24no—
32C2×#10625336—42no—
33C2×#10625336—42no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8—21no—
37C4×D188—21no—
38C4×D188—21no—
39C4×D188—21no—
40C4×D188—21no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×(C2×D47)⋊C4—21no—
43#13093955—24no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)—21no—
48C4⋊(C47⋊C8)D412yes—
49C4⋊(C47⋊C8)D412yes—
50#13093948—24no—
51#13093944—24no—
52(C2×D47)⋊C8—44no—
53C22×Q8×C47—21no—
54C22×Q8×C47—21no—
55C22×C8×C47D412yes—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47—21no—
61C4⋊Q8×C47—21no—
62C4⋊Q8×C47D412yes—
63C4⋊Q8×C47D412yes—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C42×C47C2311yescharacteristic
69#106×C47—28no—
70C4×C8×C47—21no—
71C4×C8×C47—21no—
72C23×D47—21no—
73C2×Q8×C47—22no—
74C2×Q8×C47—22no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C2×Q8×C47—41no—
79C2×Q8×C47—41no—
80C2×Q8×C47—41no—
81C2×Q8×C47—82no—
82C22×C4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47#4111yescharacteristic
87C22×Dic47—21no—
88C2×D188—21no—
89C2×D188—21no—
90C2×D188—21no—
91C2×D188—21no—
92C2×D188—21no—
93C2×D188SD1612yes—
94C2×C4×D47—41no—
95C2×C4×D47—41no—
96C2×C4×D47—42no—
97C2×C4×D47—42no—
98C2×C8×C47—22no—
99C2×C8×C47—22no—
100C2×C8×C47SD1614yescomplemented
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—41no—
109C42×C47C2×D411yescharacteristic
110C42×C47C2×D411yescharacteristic
111C42×C47C2×D411yescharacteristic
112C42×C47C2×D411yescharacteristic
113#10625336—48no—
114(C2×D47)⋊C4—41no—
115(C2×D47)⋊C4—41no—
116C4⋊Dic47—21no—
117C4⋊Dic47—21no—
118C2×C47⋊C8—22no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—41no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—42no—
126C23×C47#12111yescharacteristic
127C22×D47—22no—
128C22×D47—41no—
129C22×D47—42no—
130C22×D47—41no—
131C22×D47—42no—
132Q8×C47—42no—
133Q8×C47—42no—
134Q8×C47—82no—
135Q8×C47—82no—
136Q8×C47—81no—
137Q8×C47—81no—
138C2×C4×C47—22no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—81no—
142C2×C4×C47—81no—
143C2×C4×C47—82no—
144C2×C4×C47#12011yescharacteristic
145C2×C4×C47#12011yescharacteristic
146C2×C4×C47#12011yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47#12111yescharacteristic
149C2×C4×C47#12711yescharacteristic
150C2×C4×C47C2×SD1611yescharacteristic
151C2×C4×C47C2×SD1612yes—
152C2×C4×C47#13511yescharacteristic
153C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C47#13711yescharacteristic
155C2×C4×C47#13712yes—
156C2×Dic47—41no—
157C2×Dic47—42no—
158D188—22no—
159D188—22no—
160D188—41no—
161D188—41no—
162C4×D47—82no—
163C4×D47—81no—
164C4×D47—81no—
165C8×C47—24no—
166C47⋊C8—42no—
167C47⋊C8—42no—
168C47⋊C8—81no—
169#10271—471nomaximal, Sylow
170C22×C47#39211yescharacteristic
171C22×C47#46112yes—
172C22×C47#46511yescharacteristic
173C22×C47#46811yescharacteristic
174C22×C47#47112yes—
175C2×D47—44no—
176C2×D47—42no—
177C2×D47—82no—
178C2×D47—81no—
179C2×D47—81no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—82no—
183C4×C47—82no—
184C4×C47#44811yescharacteristic
185C4×C47#44812yes—
186C4×C47#44911yescharacteristic
187C4×C47#45011yescharacteristic
188C4×C47#45211yescharacteristic
189C4×C47#49211yescharacteristic
190C4×C47#49711yescharacteristic
191Dic47—81no—
192C2×#498D4711yescharacteristic, complemented
193C2×#476—471no—
194C2×#461—471no—
195#1954—471no—
196C2×#333—471no—
197C2×#332—471no—
198#1277—471no—
199C2×C47#167112yes—
200C2×C47#169112yes—
201C2×C47#200011yescharacteristic
202C2×C47#201311yescharacteristic
203C2×C47#201911yescharacteristic
204D47—82no—
205C22×SD16—941no—
206C2×C4⋊Q8C2×D4711yescharacteristic
207C2×C4⋊Q8C2×D4711yescharacteristic
208C2×C4×D4—471no—
209#498—22no—
210#498—22no—
211#476—942no—
212#476—942no—
213#461—942no—
214#461—942no—
215C2×#106—22no—
216C2×#106—941no—
217C2×C4⋊C8—471no—
218C2×C4⋊C8—471no—
219C2×C4⋊C8—471no—
220C2×#103—941no—
221C2×#102—941no—
222C2×C22⋊C8—942no—
223C2×C4×C8C2×D4711yescharacteristic
224#333—942no—
225#332—942no—
226C47#1027111yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—471no—
230C23×C4—941no—
231C2×SD16—944no—
232C2×SD16—1882no—
233C2×SD16—1882no—
234C22×C8—941no—
235C22×C8—941no—
236C22×C8—941no—
237C22×C8C22⋊D4712yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D4712yes—
245C4⋊Q8C22⋊D4712yes—
246C4×D4—941no—
247C4×D4—941no—
248C4×D4—941no—
249C4×D4—941no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—471no—
255C2×C22⋊C4—941no—
256C2×C42C22×D4711yescharacteristic
257#106—28no—
258#106—944no—
259C4⋊C8—472no—
260C4⋊C8—472no—
261C4⋊C8—941no—
262C4⋊C8—941no—
263C4⋊C8—941no—
264C4⋊C8—941no—
265#103—944no—
266#102—944no—
267C22⋊C8—1884no—
268C4×C8—21no—
269C4×C8—21no—
270C24—941no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×D4—472no—
281C2×D4—941no—
282C2×D4—941no—
283C2×D4—941no—
284C2×D4—941no—
285C2×D4—941no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—941no—
289C22×C4—1881no—
290C22×C4—1881no—
291C22×C4—1882no—
292C22×C4—1882no—
293C22×C4D4×D4711yescharacteristic
294C22×C4#1062535211yescharacteristic
295C22×C4C2×C22⋊D4711yescharacteristic
296SD16—1888no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—942no—
300C2×C8—942no—
301C2×C8—1881no—
302C2×C8—1882no—
303C2×C8—1881no—
304C2×C8—1881no—
305C2×C8—1881no—
306C2×C8—1881no—
307C2×C8#1062533614yescomplemented
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—941no—
317C4⋊C4—941no—
318C22⋊C4—1881no—
319C22⋊C4—1881no—
320C42D4×D4711yescharacteristic
321C42D4×D4711yescharacteristic
322C42C2×C22⋊D4711yescharacteristic
323C42C2×C22⋊D4711yescharacteristic
324C23—942no—
325C23—1881no—
326C23—1882no—
327C23—1881no—
328C23—1882no—
329C23#1309399511yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334Q8—81no—
335Q8—81no—
336D4—942no—
337D4—942no—
338D4—1881no—
339D4—1881no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—3761no—
347C2×C4—3761no—
348C2×C4—3762no—
349C2×C4—1881no—
350C2×C4—1882no—
351C2×C4#1309399211yescharacteristic
352C2×C4#1309399211yescharacteristic
353C2×C4#1309399311yescharacteristic
354C2×C4#1309399611yescharacteristic
355C2×C4#1309399611yescharacteristic
356C2×C4#1309402011yescharacteristic
357C2×C4SD16×D4711yescharacteristic
358C2×C4C2×#1062533612yes—
359C2×C4#1309404511yescharacteristic
360C2×C4#1309405111yescharacteristic
361C2×C4#1309405311yescharacteristic
362C2×C4#1309405512yes—
363C8—24no—
364C8—3761no—
365C8—1882no—
366C8—1882no—
367C22—3761no—
368C22—3762no—
369C22—3761no—
370C22—1884no—
371C22—1882no—
372C22#1568903111yescharacteristic
373C22#1568932512yes—
374C22#1568934911yescharacteristic
375C22#1568936611yescharacteristic
376C22#1568938312yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—82no—
381C4—3761no—
382C4#1568926011yescharacteristic
383C4#1568926212yes—
384C4#1568926711yescharacteristic
385C4#1568927211yescharacteristic
386C4#1568928211yescharacteristic
387C4#1568949711yescharacteristic
388C4#1568951511yescharacteristic
389C2—3762no—
390C2#2024276012yes—
391C2#2024288412yes—
392C2#2024438811yescharacteristic
393C2#2024447111yescharacteristic
394C2#2024450011yescharacteristic
subgroupG/H
centerC23#13093995
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC2×C47×#498C2
socleC23×C47#121
prime radicalC22×D188D4
hypercenterC2×#498D47
hypocenterC47#10271
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
4. C2×#15688872C2semidirect
5. C2×#15688869C2semidirect
6. #20241223C2semidirect
7. C2×C47×#498C2semidirect
100. C2×C8×C47SD16semidirect
192. C2×#498D47semidirect
226. C47#10271semidirect
307. C2×C8#10625336semidirect

Maximal subgroups

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

subgroupindex
C2×#156894102
C2×#156893252
#202441822
C2×#156888722
C2×#156888692
#202412232
C2×C47×#4982
#1027147