Finite Group Databasebeta

← #26118873

Subgroups — #26118873

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689502C211yesmaximal, characteristic, complemented
2C2×#15689404C211yesmaximal, characteristic
3C2×#15689400C211yesmaximal, characteristic
4#20244181C211yesmaximal, characteristic
5C2×#15688870C211yesmaximal, characteristic, complemented
6#20241222C211yesmaximal, characteristic, complemented
7C2×C47×#332C211yesmaximal, characteristic, complemented
8C22×#10625337—22no—
9C2×#13094024C2211yescharacteristic
10C2×C4×D188C2211yescharacteristic
11#15689502—22no—
12#15689502—22no—
13#15689404—22no—
14#15689404—22no—
15#15689400—22no—
16#15689400—22no—
17C2×#13093958—21no—
18C2×#13093957—21no—
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093946—21no—
21C2×#13093944—21no—
22C2×(C2×D47)⋊C8—22no—
23C2×C4×C47⋊C8C2211yescharacteristic
24#15688870—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×#10625337—28no—
33C2×#10625337—44no—
34C2×#10625337—44no—
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#13093958—24no—
50#13093957—24no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)D412yes—
53#13093946—24no—
54#13093944—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—42no—
76C2×Q8×C47—42no—
77C22×C4×C47—41no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—21no—
82C22×Dic47—41no—
83C2×D188—21no—
84C2×D188—21no—
85C2×D188—21no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188Q1612yescomplemented
89C2×C4×D47—41no—
90C2×C4×D47—41no—
91C2×C4×D47—42no—
92C2×C4×D47—42no—
93C2×Dic94—22no—
94C2×Dic94—42no—
95C2×Dic94—42no—
96C2×Dic94—41no—
97C2×C8×C47—22no—
98C2×C8×C47—41no—
99C2×C8×C47—41no—
100C2×C8×C47—41no—
101C2×C8×C47—42no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C42×C47C2×D411yescharacteristic
107C42×C47C2×D411yescharacteristic
108C42×C47C2×D411yescharacteristic
109C42×C47C2×D411yescharacteristic
110#10625337—416no—
111(C2×D47)⋊C4—41no—
112(C2×D47)⋊C4—41no—
113C4⋊Dic47—22no—
114C4⋊Dic47—21no—
115C4⋊Dic47—21no—
116C4⋊Dic47—22no—
117C4⋊Dic47—41no—
118C4⋊Dic47—41no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—41no—
123C2×C47⋊C8—41no—
124C2×C47⋊C8D814yes—
125C23×C47#12111yescharacteristic
126C22×D47—22no—
127C22×D47—41no—
128C22×D47—42no—
129C22×D47—41no—
130C22×D47—42no—
131Q8×C47—44no—
132Q8×C47—81no—
133Q8×C47—81no—
134C2×C4×C47—22no—
135C2×C4×C47—42no—
136C2×C4×C47—82no—
137C2×C4×C47#12011yescharacteristic
138C2×C4×C47#12011yescharacteristic
139C2×C4×C47#12011yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12711yescharacteristic
143C2×C4×C47C2×D812yes—
144C2×C4×C47C2×Q1611yescharacteristic
145C2×C4×C47#13511yescharacteristic
146C2×C4×C47AGL(1,ℤ/8)12yes—
147C2×C4×C47#13711yescharacteristic
148C2×C4×C47#13711yescharacteristic
149C2×Dic47—42no—
150C2×Dic47—41no—
151C2×Dic47—42no—
152C2×Dic47—82no—
153D188—22no—
154D188—22no—
155D188—41no—
156D188—41no—
157C4×D47—82no—
158C4×D47—81no—
159C4×D47—81no—
160Dic94—44no—
161Dic94—81no—
162Dic94—81no—
163C8×C47—42no—
164C8×C47—81no—
165C47⋊C8—24no—
166C47⋊C8—42no—
167#10274—471nomaximal, Sylow
168C22×C47#39211yescharacteristic
169C22×C47#45912yes—
170C22×C47#46711yescharacteristic
171C22×C47#47011yescharacteristic
172C22×C47#47312yes—
173C2×D47—44no—
174C2×D47—42no—
175C2×D47—82no—
176C2×D47—81no—
177C2×D47—81no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—82no—
181C4×C47#44712yes—
182C4×C47#45111yescharacteristic
183C4×C47#45111yescharacteristic
184C4×C47#45211yescharacteristic
185C4×C47#45211yescharacteristic
186C4×C47#49311yescharacteristic
187C4×C47#49611yescharacteristic
188Dic47—82no—
189Dic47—81no—
190C2×#494—471no—
191C2×#474—471no—
192C2×#474—471no—
193#1954—471no—
194C2×#332—471no—
195C2×#332D4711yescharacteristic, complemented
196#1277—471no—
197C2×C47#166912yes—
198C2×C47#168912yes—
199C2×C47#200111yescharacteristic
200C2×C47#201111yescharacteristic
201C2×C47#202011yescharacteristic
202D47—82no—
203C22×Q16—942no—
204C2×C4⋊Q8—471no—
205C2×C4⋊Q8C2×D4711yescharacteristic
206C2×C4×D4—471no—
207#494—942no—
208#494—942no—
209#474—942no—
210#474—942no—
211#474—942no—
212#474—942no—
213C2×#107—941no—
214C2×#107—941no—
215C2×C4⋊C8—471no—
216C2×C4⋊C8C2×D4711yescharacteristic
217C2×C4⋊C8C2×D4711yescharacteristic
218C2×#102—941no—
219C2×#102—941no—
220C2×C22⋊C8—942no—
221C2×C4×C8—471no—
222#332—22no—
223#332—942no—
224C47#1027411yescharacteristic, complemented, Sylow
225C22×Q8—21no—
226C22×Q8—941no—
227C22×D4—471no—
228C23×C4—941no—
229C2×Q16—948no—
230C2×Q16—1884no—
231C2×Q16—1884no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—472no—
235C22×C8—941no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—472no—
240C4⋊Q8—941no—
241C4⋊Q8—941no—
242C4⋊Q8—941no—
243C4⋊Q8D18812yes—
244C4×D4—941no—
245C4×D4—941no—
246C4×D4—941no—
247C4×D4—941no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—471no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—941no—
253C2×C22⋊C4—941no—
254C2×C42C22×D4711yescharacteristic
255#107—944no—
256#107—944no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—472no—
261C4⋊C8—941no—
262C4⋊C8C22⋊D4712yes—
263#102—944no—
264#102—944no—
265C22⋊C8—1884no—
266C4×C8—941no—
267C4×C8—941no—
268C24—941no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—942no—
274C2×Q8—1882no—
275C2×Q8—1882no—
276C2×Q8—1881no—
277C2×D4—472no—
278C2×D4—941no—
279C2×D4—941no—
280C2×D4—941no—
281C2×D4—941no—
282C2×D4—941no—
283C22×C4—41no—
284C22×C4—941no—
285C22×C4—1881no—
286C22×C4—1881no—
287C22×C4—1881no—
288C22×C4—1882no—
289C22×C4—1882no—
290C22×C4#1062535011yescharacteristic
291C22×C4D4×D4711yescharacteristic
292C22×C4D4×D4711yescharacteristic
293Q16—18816no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—474no—
300C2×C8—942no—
301C2×C8—942no—
302C2×C8—942no—
303C2×C8—1881no—
304C2×C8—1881no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—942no—
310C4⋊C4—941no—
311C4⋊C4—941no—
312C4⋊C4—942no—
313C4⋊C4—1881no—
314C4⋊C4—1881no—
315C22⋊C4—1881no—
316C22⋊C4—1881no—
317C42C2×D18811yescharacteristic
318C42D4×D4711yescharacteristic
319C42D4×D4711yescharacteristic
320C42C2×C22⋊D4711yescharacteristic
321C23—942no—
322C23—1881no—
323C23—1882no—
324C23—1881no—
325C23—1882no—
326C23#1309399711yescharacteristic
327Q8—44no—
328Q8—81no—
329Q8—81no—
330Q8—3761no—
331Q8—3761no—
332Q8—1884no—
333D4—942no—
334D4—942no—
335D4—1881no—
336D4—1881no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—3762no—
341C2×C4—3761no—
342C2×C4—3761no—
343C2×C4—3762no—
344C2×C4—1881no—
345C2×C4—1882no—
346C2×C4—1882no—
347C2×C4#1309399211yescharacteristic
348C2×C4#1309399411yescharacteristic
349C2×C4#1309399411yescharacteristic
350C2×C4#1309399711yescharacteristic
351C2×C4#1309399811yescharacteristic
352C2×C4#1309402011yescharacteristic
353C2×C4D8×D4712yes—
354C2×C4Q16×D4711yescharacteristic
355C2×C4#1309404511yescharacteristic
356C2×C4#1309405112yes—
357C2×C4#1309405311yescharacteristic
358C2×C4#1309405611yescharacteristic
359C8—42no—
360C8—81no—
361C8—944no—
362C8—1882no—
363C22—3761no—
364C22—3762no—
365C22—3761no—
366C22—1884no—
367C22—1882no—
368C22#1568903211yescharacteristic
369C22#1568931712yes—
370C22#1568936311yescharacteristic
371C22#1568938011yescharacteristic
372C22#1568939712yes—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—3761no—
377C4—3762no—
378C4#1568925812yes—
379C4#1568927711yescharacteristic
380C4#1568928011yescharacteristic
381C4#1568928211yescharacteristic
382C4#1568928811yescharacteristic
383C4#1568949811yescharacteristic
384C4#1568950911yescharacteristic
385C2—3762no—
386C2#2024275212yes—
387C2#2024287612yes—
388C2#2024439711yescharacteristic
389C2#2024446011yescharacteristic
390C2#2024450811yescharacteristic
subgroupG/H
centerC23#13093997
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC2×C47×#332C2
socleC23×C47#121
prime radicalC22×D188D4
hypercenterC2×#332D47
hypocenterC47#10274
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×#15689502C2semidirect
5. C2×#15688870C2semidirect
6. #20241222C2semidirect
7. C2×C47×#332C2semidirect
88. C2×D188Q16semidirect
195. C2×#332D47semidirect
224. C47#10274semidirect

Maximal subgroups

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

subgroupindex
C2×#156895022
C2×#156894042
C2×#156894002
#202441812
C2×#156888702
#202412222
C2×C47×#3322
#1027447