Finite Group Databasebeta

← #26118855

Subgroups — #26118855

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×#15689519C211yesmaximal, characteristic, complemented
2C2×#15689414C211yesmaximal, characteristic
3C2×#15689325C211yesmaximal, characteristic
4#20244181C211yesmaximal, characteristic
5C2×#15688871C211yesmaximal, characteristic, complemented
6#20241222C211yesmaximal, characteristic, complemented
7C2×C47×#332C211yesmaximal, characteristic, complemented
8C22×#10625336—21no—
9C2×#13094024C2211yescharacteristic
10C2×C4×D188C2211yescharacteristic
11#15689519—22no—
12#15689519—22no—
13#15689414—22no—
14#15689414—22no—
15#15689325—22no—
16#15689325—22no—
17C2×#13093956—21no—
18C2×#13093955—22no—
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093948—21no—
21C2×#13093946—21no—
22C2×(C2×D47)⋊C8—22no—
23C2×C4×C47⋊C8C2211yescharacteristic
24#15688871—22no—
25C2×C4⋊Q8×C47C2211yescharacteristic
26C2×C4⋊C8×C47C2211yescharacteristic
27C2×C4⋊C8×C47C2211yescharacteristic
28C47×#332—22no—
29C22×Dic94—21no—
30C22×D188D411yescharacteristic
31C22×C4×D47—21no—
32C2×#10625336—24no—
33C2×#10625336—42no—
34C2×#10625336—42no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8D412yes—
37#13094024—21no—
38#13094024—21no—
39#13094024—21no—
40#13094024D412yes—
41C4×D188—21no—
42C4×D188—21no—
43C4×D188—21no—
44C4×D188—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47D411yescharacteristic
48C2×(C2×D47)⋊C4—21no—
49#13093956—24no—
50#13093955—28no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)D412yes—
53#13093948—24no—
54#13093946—24no—
55(C2×D47)⋊C8—44no—
56C4×C47⋊C8—21no—
57C4×C47⋊C8—21no—
58C22×Q8×C47—21no—
59C22×C8×C47—21no—
60C22×C8×C47—21no—
61C4⋊Q8×C47—21no—
62C4⋊Q8×C47—21no—
63C4⋊Q8×C47—21no—
64C4⋊Q8×C47D412yes—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C42×C47C2311yescharacteristic
68C4⋊C8×C47—21no—
69C4⋊C8×C47—21no—
70C4⋊C8×C47—21no—
71C4⋊C8×C47D412yes—
72C23×D47—21no—
73C2×Q8×C47—22no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C2×Q8×C47—41no—
79C22×C4×C47—41no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×Dic47—21no—
84C22×Dic47—41no—
85C2×D188—21no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188—21no—
89C2×D188—21no—
90C2×D188SD1612yes—
91C2×C4×D47—41no—
92C2×C4×D47—41no—
93C2×C4×D47—42no—
94C2×C4×D47—42no—
95C2×Dic94—22no—
96C2×Dic94—41no—
97C2×Dic94—82no—
98C2×C8×C47—22no—
99C2×C8×C47—41no—
100C2×C8×C47—41no—
101C2×C8×C47—41no—
102C2×C8×C47—42no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C42×C47C2×D411yescharacteristic
108C42×C47C2×D411yescharacteristic
109C42×C47C2×D411yescharacteristic
110C42×C47C2×D411yescharacteristic
111#10625336—48no—
112(C2×D47)⋊C4—41no—
113(C2×D47)⋊C4—41no—
114C4⋊Dic47—22no—
115C4⋊Dic47—21no—
116C4⋊Dic47—22no—
117C4⋊Dic47—21no—
118C4⋊Dic47—41no—
119C4⋊Dic47—41no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8SD1614yescomplemented
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—81no—
135Q8×C47—81no—
136C2×C4×C47—22no—
137C2×C4×C47—42no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47#12011yescharacteristic
141C2×C4×C47#12011yescharacteristic
142C2×C4×C47#12011yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12711yescharacteristic
146C2×C4×C47C2×SD1611yescharacteristic
147C2×C4×C47C2×SD1612yes—
148C2×C4×C47#13511yescharacteristic
149C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C47#13711yescharacteristic
151C2×C4×C47#13712yes—
152C2×Dic47—42no—
153C2×Dic47—41no—
154C2×Dic47—42no—
155C2×Dic47—82no—
156D188—22no—
157D188—22no—
158D188—41no—
159D188—41no—
160C4×D47—82no—
161C4×D47—81no—
162C4×D47—81no—
163Dic94—82no—
164Dic94—82no—
165C8×C47—42no—
166C8×C47—81no—
167C47⋊C8—24no—
168C47⋊C8—42no—
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#44811yescharacteristic
184C4×C47#44812yes—
185C4×C47#44911yescharacteristic
186C4×C47#45011yescharacteristic
187C4×C47#45211yescharacteristic
188C4×C47#49211yescharacteristic
189C4×C47#49711yescharacteristic
190Dic47—81no—
191Dic47—82no—
192C2×#498—471no—
193C2×#476—471no—
194C2×#461—471no—
195#1954—471no—
196C2×#333—471no—
197C2×#332D4711yescharacteristic, complemented
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⋊Q8—471no—
207C2×C4⋊Q8C2×D4711yescharacteristic
208C2×C4×D4—471no—
209#498—942no—
210#498—942no—
211#476—942no—
212#476—942no—
213#461—942no—
214#461—942no—
215C2×#106—942no—
216C2×#106—941no—
217C2×C4⋊C8—471no—
218C2×C4⋊C8C2×D4711yescharacteristic
219C2×C4⋊C8C2×D4711yescharacteristic
220C2×#103—941no—
221C2×#102—941no—
222C2×C22⋊C8—942no—
223C2×C4×C8—471no—
224#333—942no—
225#332—22no—
226C47#1027111yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—941no—
229C22×D4—471no—
230C23×C4—941no—
231C2×SD16—944no—
232C2×SD16—1882no—
233C2×SD16—1882no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—472no—
237C22×C8—941no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—472no—
242C4⋊Q8—941no—
243C4⋊Q8—941no—
244C4⋊Q8—941no—
245C4⋊Q8D18812yes—
246C4×D4—941no—
247C4×D4—941no—
248C4×D4—941no—
249C4×D4—941no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—471no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4—941no—
255C2×C22⋊C4—941no—
256C2×C42C22×D4711yescharacteristic
257#106—948no—
258#106—944no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—472no—
263C4⋊C8—941no—
264C4⋊C8C22⋊D4712yes—
265#103—944no—
266#102—944no—
267C22⋊C8—1884no—
268C4×C8—941no—
269C4×C8—941no—
270C24—941no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—942no—
278C2×Q8—3762no—
279C2×Q8—1881no—
280C2×D4—472no—
281C2×D4—941no—
282C2×D4—941no—
283C2×D4—941no—
284C2×D4—941no—
285C2×D4—941no—
286C22×C4—41no—
287C22×C4—941no—
288C22×C4—1881no—
289C22×C4—1881no—
290C22×C4—1881no—
291C22×C4—1882no—
292C22×C4—1882no—
293C22×C4#1062535011yescharacteristic
294C22×C4D4×D4711yescharacteristic
295C22×C4D4×D4711yescharacteristic
296SD16—1888no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—474no—
303C2×C8—942no—
304C2×C8—942no—
305C2×C8—942no—
306C2×C8—1881no—
307C2×C8—1881no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—942no—
313C4⋊C4—941no—
314C4⋊C4—941no—
315C4⋊C4—942no—
316C4⋊C4—1881no—
317C4⋊C4—1881no—
318C22⋊C4—1881no—
319C22⋊C4—1881no—
320C42C2×D18811yescharacteristic
321C42D4×D4711yescharacteristic
322C42D4×D4711yescharacteristic
323C42C2×C22⋊D4711yescharacteristic
324C23—942no—
325C23—1881no—
326C23—1882no—
327C23—1881no—
328C23—1882no—
329C23#1309399711yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334Q8—3762no—
335Q8—3762no—
336D4—942no—
337D4—942no—
338D4—1881no—
339D4—1881no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—3762no—
345C2×C4—3761no—
346C2×C4—3761no—
347C2×C4—3762no—
348C2×C4—1881no—
349C2×C4—1882no—
350C2×C4—1882no—
351C2×C4#1309399211yescharacteristic
352C2×C4#1309399411yescharacteristic
353C2×C4#1309399411yescharacteristic
354C2×C4#1309399711yescharacteristic
355C2×C4#1309399811yescharacteristic
356C2×C4#1309402011yescharacteristic
357C2×C4SD16×D4712yes—
358C2×C4SD16×D4711yescharacteristic
359C2×C4#1309404311yescharacteristic
360C2×C4#1309404811yescharacteristic
361C2×C4#1309405311yescharacteristic
362C2×C4#1309405612yes—
363C8—42no—
364C8—81no—
365C8—944no—
366C8—1882no—
367C22—3761no—
368C22—3762no—
369C22—3761no—
370C22—1884no—
371C22—1882no—
372C22#1568903211yescharacteristic
373C22#1568932712yes—
374C22#1568935111yescharacteristic
375C22#1568936811yescharacteristic
376C22#1568938912yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—3762no—
381C4—3761no—
382C4#1568926011yescharacteristic
383C4#1568926312yes—
384C4#1568926511yescharacteristic
385C4#1568927511yescharacteristic
386C4#1568928611yescharacteristic
387C4#1568949411yescharacteristic
388C4#1568951611yescharacteristic
389C2—3762no—
390C2#2024276212yes—
391C2#2024288612yes—
392C2#2024439211yescharacteristic
393C2#2024447211yescharacteristic
394C2#2024450111yescharacteristic
subgroupG/H
centerC23#13093997
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC2×C47×#332C2
socleC23×C47#121
prime radicalC22×D188D4
hypercenterC2×#332D47
hypocenterC47#10271
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. C2×#15689519C2semidirect
5. C2×#15688871C2semidirect
6. #20241222C2semidirect
7. C2×C47×#332C2semidirect
125. C2×C47⋊C8SD16semidirect
197. C2×#332D47semidirect
226. C47#10271semidirect

Maximal subgroups

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

subgroupindex
C2×#156895192
C2×#156894142
C2×#156893252
#202441812
C2×#156888712
#202412222
C2×C47×#3322
#1027147