Finite Group Databasebeta

← C2×#19134141 #25079153

Subgroups — C2×#19134141 #25079153

Order 11008.

726 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×#14583249C211yesmaximal, characteristic, complemented
2C2×#14583176C211yesmaximal, characteristic
3C2×#14583105C211yesmaximal, characteristic
4C2×#14582976C211yesmaximal, characteristic
5#19134141C218yesmaximal, direct product factor
6C2×#14582651C211yesmaximal, characteristic, complemented
7C2×#14582644C211yesmaximal, characteristic, complemented
8C2×C43×#332C211yesmaximal, characteristic, complemented
9C22×#10621873—21no—
10C22×#10621872—21no—
11C2×#13063325C2211yescharacteristic
12C2×C4×D172C2211yescharacteristic
13#14583249—22no—
14#14583249C2214yescomplemented
15#14583176—22no—
16#14583176C2214yes—
17#14583105—22no—
18#14583105C2214yes—
19#14582976—22no—
20#14582976C2214yes—
21C2×#13063258—21no—
22C2×C4⋊(C43⋊C8)C2211yescharacteristic
23C2×#13063250—21no—
24C2×#13063249—21no—
25C2×#13063249—21no—
26C2×#13063245—21no—
27C2×(C2×D43)⋊C8—21no—
28C2×C4×C43⋊C8C2211yescharacteristic
29#14582651C2214yescomplemented
30#14582644C2214yescomplemented
31C2×C4⋊Q8×C43C2211yescharacteristic
32C2×C4⋊C8×C43C2211yescharacteristic
33C2×C4⋊C8×C43C2211yescharacteristic
34C43×#332C2214yescomplemented
35C22×Dic86—21no—
36C22×D172D411yescharacteristic
37C22×C4×D43—21no—
38C2×#10621873—24no—
39C2×#10621873—44no—
40C2×#10621872—24no—
41C2×#10621872—44no—
42C22×C43⋊C8—21no—
43C22×C43⋊C8D411yescharacteristic
44C22×C43⋊C8D411yescharacteristic
45#13063325—21no—
46#13063325—21no—
47#13063325—21no—
48#13063325C2312yes—
49C4×D172—21no—
50C4×D172—21no—
51C4×D172D412yes—
52C4×D172C2312yes—
53C2×C4⋊Dic43—21no—
54C2×C4⋊Dic43—21no—
55C2×C4⋊Dic43D411yescharacteristic
56C2×(C2×D43)⋊C4—21no—
57#13063258—24no—
58C4⋊(C43⋊C8)—21no—
59C4⋊(C43⋊C8)C2312yes—
60#13063250—24no—
61#13063249—24no—
62#13063249—24no—
63#13063245—24no—
64(C2×D43)⋊C8—24no—
65C4×C43⋊C8D412yes—
66C4×C43⋊C8C2312yes—
67C22×Q8×C43—21no—
68C22×C8×C43—21no—
69C22×C8×C43—21no—
70C4⋊Q8×C43—21no—
71C4⋊Q8×C43—21no—
72C4⋊Q8×C43—21no—
73C4⋊Q8×C43C2312yes—
74C2×C4⋊C4×C43—21no—
75C2×C4⋊C4×C43—21no—
76C2×C42×C43C2311yescharacteristic
77C4⋊C8×C43—21no—
78C4⋊C8×C43—21no—
79C4⋊C8×C43C2312yes—
80C4⋊C8×C43C2312yes—
81C23×D43—21no—
82C2×Q8×C43—22no—
83C2×Q8×C43—41no—
84C2×Q8×C43—42no—
85C2×Q8×C43—42no—
86C22×C4×C43—41no—
87C22×C4×C43C2×D411yescharacteristic
88C22×C4×C43C2×D411yescharacteristic
89C22×C4×C43#4111yescharacteristic
90C22×Dic43—21no—
91C22×Dic43—41no—
92C2×D172—21no—
93C2×D172—22no—
94C2×D172—22no—
95C2×D172C2×D412yes—
96C2×C4×D43—22no—
97C2×C4×D43—22no—
98C2×C4×D43—42no—
99C2×C4×D43—42no—
100C2×Dic86—22no—
101C2×Dic86—42no—
102C2×Dic86—42no—
103C2×Dic86—41no—
104C2×C8×C43—22no—
105C2×C8×C43—22no—
106C2×C8×C43—42no—
107C2×C8×C43—42no—
108C4⋊C4×C43—22no—
109C4⋊C4×C43—22no—
110C4⋊C4×C43—41no—
111C4⋊C4×C43—41no—
112C42×C43C2×D411yescharacteristic
113C42×C43C2×D411yescharacteristic
114C42×C43#4111yescharacteristic
115C42×C43C2411yescharacteristic
116(C2×D43)⋊C4—22no—
117(C2×D43)⋊C4—22no—
118C4⋊Dic43—21no—
119C4⋊Dic43—22no—
120C4⋊Dic43—22no—
121C4⋊Dic43—41no—
122C4⋊Dic43—41no—
123C4⋊Dic43C2×D412yes—
124C2×C43⋊C8—22no—
125C2×C43⋊C8—22no—
126C2×C43⋊C8—22no—
127C2×C43⋊C8—42no—
128C2×C43⋊C8C2×D412yes—
129C2×C43⋊C8C2×D412yes—
130#10621873—48no—
131#10621872—48no—
132C23×C43#12111yescharacteristic
133C22×D43—22no—
134C22×D43—42no—
135C22×D43—42no—
136C22×D43—42no—
137Q8×C43—42no—
138Q8×C43—42no—
139Q8×C43—82no—
140C2×C4×C43—22no—
141C2×C4×C43—42no—
142C2×C4×C43—82no—
143C2×C4×C43#12111yescharacteristic
144C2×C4×C43#12111yescharacteristic
145C2×C4×C43#12111yescharacteristic
146C2×C4×C43#13512yes—
147C2×C4×C43AGL(1,ℤ/8)12yes—
148C2×C4×C43AGL(1,ℤ/8)12yes—
149C2×C4×C43#13712yes—
150C2×C4×C43C22×D411yescharacteristic
151C2×C4×C43C22×D411yescharacteristic
152C2×C4×C43C2×#4111yescharacteristic
153C2×Dic43—22no—
154C2×Dic43—42no—
155C2×Dic43—42no—
156C2×Dic43—82no—
157D172—22no—
158D172—22no—
159D172—42no—
160C4×D43—42no—
161C4×D43—42no—
162C4×D43—44no—
163Dic86—42no—
164Dic86—42no—
165Dic86—82no—
166C8×C43—42no—
167C8×C43—42no—
168C43⋊C8—22no—
169C43⋊C8—22no—
170C43⋊C8—42no—
171C2×#1699—431nomaximal, Sylow
172C22×C43#46312yes—
173C22×C43#46812yes—
174C22×C43#47312yes—
175C22×C43C2×#12111yescharacteristic
176C2×D43—42no—
177C2×D43—42no—
178C2×D43—42no—
179C2×D43—84no—
180C4×C43—21no—
181C4×C43—21no—
182C4×C43—82no—
183C4×C43#46312yes—
184C4×C43#46812yes—
185C4×C43C2×#13511yescharacteristic
186C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
187C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
188C4×C43C2×#13711yescharacteristic
189Dic43—42no—
190Dic43—82no—
191C2×#487—431no—
192C2×#474—431no—
193C2×#461—431no—
194C2×#436—431no—
195#1699—438no—
196C2×#335—431no—
197C2×#332—431no—
198C2×#332D4311yescharacteristic, complemented
199C2×C43#169914yescomplemented
200C2×C43C2×#46311yescharacteristic
201C2×C43C2×#46811yescharacteristic
202C2×C43C2×#47311yescharacteristic
203D43—82no—
204C22×SD16—861no—
205C22×M16—861no—
206C2×C4⋊Q8—431no—
207C2×C4⋊Q8C2×D4311yescharacteristic
208C2×C4×D4—431no—
209#487—434no—
210#487—862no—
211#474—434no—
212#474—862no—
213#461—434no—
214#461—862no—
215#436—434no—
216#436—862no—
217C2×#107—861no—
218C2×C4⋊C8—431no—
219C2×C4⋊C8C2×D4311yescharacteristic
220C2×C4⋊C8C2×D4311yescharacteristic
221C2×#103—861no—
222C2×#103—861no—
223C2×#103—861no—
224C2×#102—861no—
225C2×C22⋊C8—861no—
226C2×C4×C8—431no—
227#335—434no—
228#332—434no—
229#332C2×D4314yescomplemented
230C43C2×#169911yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—861no—
233C22×D4—431no—
234C23×C4—861no—
235C2×SD16—864no—
236C2×SD16—1724no—
237C2×M16—864no—
238C2×M16—1724no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—431no—
242C22×C8—431no—
243C22×C8—861no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—432no—
248C4⋊Q8—861no—
249C4⋊Q8—861no—
250C4⋊Q8—861no—
251C4⋊Q8C22×D4312yes—
252C4×D4—432no—
253C4×D4—432no—
254C4×D4—861no—
255C4×D4—861no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—431no—
259C2×C4⋊C4—861no—
260C2×C4⋊C4—861no—
261C2×C22⋊C4—861no—
262C2×C42C22×D4311yescharacteristic
263#107—864no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—432no—
267C4⋊C8—861no—
268C4⋊C8C22×D4312yes—
269C4⋊C8C22×D4312yes—
270#103—864no—
271#103—864no—
272#103—864no—
273#102—864no—
274C22⋊C8—864no—
275C4×C8—432no—
276C4×C8—432no—
277C24—861no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—862no—
283C2×Q8—1722no—
284C2×Q8—1722no—
285C2×Q8—1721no—
286C2×D4—432no—
287C2×D4—861no—
288C2×D4—862no—
289C2×D4—862no—
290C22×C4—41no—
291C22×C4—861no—
292C22×C4—862no—
293C22×C4—862no—
294C22×C4—1721no—
295C22×C4—1722no—
296C22×C4—1722no—
297C22×C4#1062189811yescharacteristic
298C22×C4D4×D4311yescharacteristic
299C22×C4D4×D4311yescharacteristic
300SD16—1728no—
301M16—1728no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—432no—
307C2×C8—432no—
308C2×C8—862no—
309C2×C8—862no—
310C2×C8—862no—
311C2×C8—1722no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—432no—
317C4⋊C4—861no—
318C4⋊C4—862no—
319C4⋊C4—862no—
320C4⋊C4—1721no—
321C4⋊C4—1721no—
322C22⋊C4—862no—
323C22⋊C4—862no—
324C42#1062189811yescharacteristic
325C42D4×D4311yescharacteristic
326C42D4×D4311yescharacteristic
327C42C23×D4311yescharacteristic
328C23—862no—
329C23—1722no—
330C23—1722no—
331C23—1722no—
332C23#1306329811yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—82no—
336Q8—3442no—
337Q8—1722no—
338Q8—1722no—
339D4—862no—
340D4—862no—
341D4—1722no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—862no—
346C2×C4—3442no—
347C2×C4—1722no—
348C2×C4—1722no—
349C2×C4—1724no—
350C2×C4—1722no—
351C2×C4—1722no—
352C2×C4#1306329811yescharacteristic
353C2×C4#1306329811yescharacteristic
354C2×C4#1306329811yescharacteristic
355C2×C4#1306334612yes—
356C2×C4#1306334912yes—
357C2×C4#1306335212yes—
358C2×C4#1306335712yes—
359C2×C4C2×D4×D4311yescharacteristic
360C2×C4C2×D4×D4311yescharacteristic
361C2×C4C2×#1062189811yescharacteristic
362C8—42no—
363C8—42no—
364C8—862no—
365C8—862no—
366C8—1722no—
367C22—3444no—
368C22—1722no—
369C22—1722no—
370C22—1722no—
371C22#1458311312yes—
372C22#1458314812yes—
373C22#1458317312yes—
374C22C2×#1306329811yescharacteristic
375C4—21no—
376C4—21no—
377C4—82no—
378C4—3442no—
379C4—1722no—
380C4#1458311312yes—
381C4#1458314812yes—
382C4C2×#1306334611yescharacteristic
383C4C2×#1306334911yescharacteristic
384C4C2×#1306335211yescharacteristic
385C4C2×#1306335711yescharacteristic
386C2—3442no—
387C2#1913414114yesdirect product factor
388C2C2×#1458311311yescharacteristic
389C2C2×#1458314811yescharacteristic
390C2C2×#1458317311yescharacteristic
subgroupG/H
centerC23#13063298
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC2×C43×#332C2
socleC23×C43#121
prime radicalC22×D172D4
hypercenterC2×#332D43
hypocenterC43C2×#1699
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. C2×#14583249C2semidirect
5. #19134141C2direct (N × H)
6. C2×#14582651C2semidirect
7. C2×#14582644C2semidirect
8. C2×C43×#332C2semidirect
14. #14583249C22semidirect
29. #14582651C22semidirect
30. #14582644C22semidirect
34. C43×#332C22semidirect
198. C2×#332D43semidirect
199. C2×C43#1699semidirect
229. #332C2×D43semidirect
230. C43C2×#1699semidirect
387. C2#19134141direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145832492
C2×#145831762
C2×#145831052
C2×#145829762
#191341412
C2×#145826512
C2×#145826442
C2×C43×#3322
C2×#169943