Finite Group Databasebeta

← C2×#24906917 #35386708

Subgroups — C2×#24906917 #35386708

Order 21248.

726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128053C211yesmaximal, characteristic, complemented
2C2×#19127988C211yesmaximal, characteristic
3C2×#19127982C211yesmaximal, characteristic
4C2×#19127766C211yesmaximal, characteristic
5#24906917C218yesmaximal, direct product factor
6C2×#19127453C211yesmaximal, characteristic, complemented
7C2×#19127448C211yesmaximal, characteristic, complemented
8C2×#333×C83C211yesmaximal, characteristic, complemented
9C22×C8×D83—21no—
10C2×#14581792C2211yescharacteristic
11C2×C4×D332C2211yescharacteristic
12#19128053—22no—
13#19128053C2214yescomplemented
14#19127988—22no—
15#19127988C2214yes—
16#19127982—22no—
17#19127982C2214yes—
18#19127766—22no—
19#19127766C2214yes—
20C2×#14581726—21no—
21C2×#14581723—21no—
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581715—21no—
24C2×#14581715—21no—
25C2×#14581714—21no—
26C2×#14581712—21no—
27C2×(C2×D83)⋊C8—21no—
28C2×C4×C83⋊C8C2211yescharacteristic
29#19127453C2214yescomplemented
30#19127448C2214yescomplemented
31C2×C4⋊Q8×C83C2211yescharacteristic
32C2×C4⋊C8×C83C2211yescharacteristic
33C2×C4⋊C8×C83C2211yescharacteristic
34#333×C83C2214yescomplemented
35C22×Dic166—21no—
36C22×D332D411yescharacteristic
37C22×C4×D83—21no—
38C22×C83⋊C8—21no—
39C22×C83⋊C8D411yescharacteristic
40C22×C83⋊C8D411yescharacteristic
41C2×C8×D83—24no—
42C2×C8×D83—44no—
43#14581792—21no—
44#14581792—21no—
45#14581792—21no—
46#14581792C2312yes—
47C4×D332—21no—
48C4×D332—21no—
49C4×D332D412yes—
50C4×D332C2312yes—
51C2×C4⋊Dic83—21no—
52C2×C4⋊Dic83—21no—
53C2×C4⋊Dic83D411yescharacteristic
54C2×(C2×D83)⋊C4—21no—
55#14581726—24no—
56#14581723—24no—
57C4⋊(C83⋊C8)—21no—
58C4⋊(C83⋊C8)C2312yes—
59#14581715—24no—
60#14581715—24no—
61#14581714—24no—
62#14581712—24no—
63(C2×D83)⋊C8—24no—
64C4×C83⋊C8D412yes—
65C4×C83⋊C8C2312yes—
66C22×Q8×C83—21no—
67C22×C8×C83—21no—
68C22×C8×C83—21no—
69C4⋊Q8×C83—21no—
70C4⋊Q8×C83—21no—
71C4⋊Q8×C83—21no—
72C4⋊Q8×C83C2312yes—
73C2×C4⋊C4×C83—21no—
74C2×C4⋊C4×C83—21no—
75C2×C42×C83C2311yescharacteristic
76C4⋊C8×C83—21no—
77C4⋊C8×C83—21no—
78C4⋊C8×C83C2312yes—
79C4⋊C8×C83C2312yes—
80C23×D83—21no—
81C2×Dic166—22no—
82C2×Dic166—42no—
83C2×Dic166—42no—
84C2×Dic166—41no—
85C2×D332—21no—
86C2×D332—22no—
87C2×D332Q1614yes—
88C2×D332C2×D412yes—
89C22×Dic83—21no—
90C22×Dic83—41no—
91C2×C4×D83—22no—
92C2×C4×D83—22no—
93C2×C4×D83—42no—
94C2×C4×D83—42no—
95C2×C83⋊C8—22no—
96C2×C83⋊C8—22no—
97C2×C83⋊C8—22no—
98C2×C83⋊C8—42no—
99C2×C83⋊C8C2×D412yes—
100C2×C83⋊C8C2×D412yes—
101C8×D83—48no—
102C4⋊Dic83—21no—
103C4⋊Dic83—22no—
104C4⋊Dic83—22no—
105C4⋊Dic83—41no—
106C4⋊Dic83—41no—
107C4⋊Dic83C2×D412yes—
108(C2×D83)⋊C4—22no—
109(C2×D83)⋊C4—22no—
110C2×Q8×C83—22no—
111C2×Q8×C83—41no—
112C2×Q8×C83—82no—
113C22×C4×C83—41no—
114C22×C4×C83C2×D411yescharacteristic
115C22×C4×C83C2×D411yescharacteristic
116C22×C4×C83#4111yescharacteristic
117C2×C8×C83—22no—
118C2×C8×C83—22no—
119C2×C8×C83—42no—
120C2×C8×C83—42no—
121C4⋊C4×C83—22no—
122C4⋊C4×C83—22no—
123C4⋊C4×C83—41no—
124C4⋊C4×C83—41no—
125C42×C83C2×D411yescharacteristic
126C42×C83C2×D411yescharacteristic
127C42×C83#4111yescharacteristic
128C42×C83C2411yescharacteristic
129C23×C83#12111yescharacteristic
130C22×D83—22no—
131C22×D83—24no—
132C22×D83—42no—
133C22×D83—42no—
134Q8×C83—82no—
135Q8×C83—82no—
136C2×C4×C83—22no—
137C2×C4×C83—42no—
138C2×C4×C83—82no—
139C2×C4×C83#12111yescharacteristic
140C2×C4×C83#12111yescharacteristic
141C2×C4×C83#12111yescharacteristic
142C2×C4×C83C2×Q1612yes—
143C2×C4×C83#13512yes—
144C2×C4×C83#13512yes—
145C2×C4×C83AGL(1,ℤ/8)12yes—
146C2×C4×C83C22×D411yescharacteristic
147C2×C4×C83C22×D411yescharacteristic
148C2×C4×C83C2×#4111yescharacteristic
149C2×Dic83—22no—
150C2×Dic83—42no—
151C2×Dic83—42no—
152C2×Dic83—82no—
153D332—22no—
154D332—24no—
155D332C2×Q1614yes—
156C4×D83—42no—
157C4×D83—42no—
158C4×D83—44no—
159Dic166—42no—
160Dic166—42no—
161Dic166—82no—
162C8×C83—42no—
163C8×C83—42no—
164C83⋊C8—22no—
165C83⋊C8—22no—
166C83⋊C8—42no—
167C22×C83#46312yes—
168C22×C83#46712yes—
169C22×C83#47112yes—
170C22×C83C2×#12111yescharacteristic
171C2×D83—24no—
172C2×D83—48no—
173C2×D83—42no—
174C2×D83—42no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—82no—
178C4×C83#46312yes—
179C4×C83#46712yes—
180C4×C83C22×Q1611yescharacteristic
181C4×C83C2×#13511yescharacteristic
182C4×C83C2×#13511yescharacteristic
183C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
184Dic83—42no—
185Dic83—82no—
186C2×#1683—831nomaximal, Sylow
187C2×C83#168314yescomplemented
188C2×C83C2×#46311yescharacteristic
189C2×C83C2×#46711yescharacteristic
190C2×C83C2×#47111yescharacteristic
191D83—44no—
192C2×#487—831no—
193C2×#476—831no—
194C2×#474—831no—
195C2×C8×D4—831no—
196#1683—838no—
197C2×#335—831no—
198C2×#333D8311yescharacteristic, complemented
199C2×#332—831no—
200C83C2×#168311yescharacteristic, complemented, Sylow
201C23×C8—1661no—
202C2×C4⋊Q8—831no—
203C2×C4⋊Q8C2×D8311yescharacteristic
204C2×C4×D4—831no—
205#487—834no—
206#487—1662no—
207#476—834no—
208#476—1662no—
209#474—834no—
210#474—1662no—
211C8×D4—834no—
212C8×D4—1662no—
213C2×#107—1661no—
214C2×#106—1661no—
215C2×C4⋊C8—831no—
216C2×C4⋊C8C2×D8311yescharacteristic
217C2×C4⋊C8C2×D8311yescharacteristic
218C2×#103—1661no—
219C2×#103—1661no—
220C2×#102—1661no—
221C2×#102—1661no—
222C2×C22⋊C8—1661no—
223C2×C4×C8—831no—
224#335—834no—
225#333C2×D8314yescomplemented
226#332—834no—
227C22×Q8—21no—
228C22×Q8—1661no—
229C22×D4—831no—
230C23×C4—1661no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—3324no—
234C22×C8—831no—
235C22×C8—831no—
236C22×C8—1664no—
237C22×C8—1661no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—832no—
242C4⋊Q8—1661no—
243C4⋊Q8—1661no—
244C4⋊Q8—1661no—
245C4⋊Q8C22×D8312yes—
246C4×D4—832no—
247C4×D4—832no—
248C4×D4—1661no—
249C4×D4—1661no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—831no—
253C2×C4⋊C4—1661no—
254C2×C4⋊C4—1661no—
255C2×C22⋊C4—1661no—
256C2×C42C22×D8311yescharacteristic
257#107—1664no—
258#106—1664no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—832no—
262C4⋊C8—1661no—
263C4⋊C8C22×D8312yes—
264C4⋊C8C22×D8312yes—
265#103—1664no—
266#103—1664no—
267#102—1664no—
268#102—1664no—
269C22⋊C8—1664no—
270C4×C8—832no—
271C4×C8—832no—
272C24—1661no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×Q8—3321no—
277C2×Q8—3322no—
278C2×Q8—3322no—
279C2×Q8—1662no—
280C2×D4—834no—
281C2×D4—832no—
282C2×D4—1661no—
283C2×D4—1662no—
284C22×C4—41no—
285C22×C4—3322no—
286C22×C4—3322no—
287C22×C4—3321no—
288C22×C4—1661no—
289C22×C4—1662no—
290C22×C4—1662no—
291C22×C4D4×D8311yescharacteristic
292C22×C4D4×D8311yescharacteristic
293C22×C4#1305059011yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—3328no—
299C2×C8—3322no—
300C2×C8—832no—
301C2×C8—832no—
302C2×C8—1662no—
303C2×C8—1662no—
304C2×C8—1662no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—3321no—
310C4⋊C4—3321no—
311C4⋊C4—832no—
312C4⋊C4—1662no—
313C4⋊C4—1662no—
314C4⋊C4—1661no—
315C22⋊C4—1662no—
316C22⋊C4—1662no—
317C42D4×D8311yescharacteristic
318C42D4×D8311yescharacteristic
319C42#1305059011yescharacteristic
320C42C23×D8311yescharacteristic
321C23—3322no—
322C23—3322no—
323C23—1662no—
324C23—1664no—
325C23#1458176511yescharacteristic
326Q8—82no—
327Q8—82no—
328Q8—3322no—
329Q8—3322no—
330Q8—6642no—
331D4—834no—
332D4—1664no—
333D4—1662no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—3322no—
338C2×C4—3324no—
339C2×C4—3322no—
340C2×C4—3322no—
341C2×C4—3322no—
342C2×C4—6642no—
343C2×C4—1662no—
344C2×C4#1458176511yescharacteristic
345C2×C4#1458176511yescharacteristic
346C2×C4#1458176511yescharacteristic
347C2×C4Q16×D8312yes—
348C2×C4#1458181312yes—
349C2×C4#1458181512yes—
350C2×C4#1458181912yes—
351C2×C4C2×D4×D8311yescharacteristic
352C2×C4C2×D4×D8311yescharacteristic
353C2×C4C2×#1305059011yescharacteristic
354C8—42no—
355C8—42no—
356C8—3322no—
357C8—1662no—
358C8—1662no—
359C22—3322no—
360C22—3328no—
361C22—3322no—
362C22—1664no—
363C22#1912791912yes—
364C22#1912794112yes—
365C22#1912796312yes—
366C22C2×#1458176511yescharacteristic
367C4—21no—
368C4—21no—
369C4—82no—
370C4—3322no—
371C4—6642no—
372C4#1912791912yes—
373C4#1912794112yes—
374C4C2×Q16×D8311yescharacteristic
375C4C2×#1458181311yescharacteristic
376C4C2×#1458181511yescharacteristic
377C4C2×#1458181911yescharacteristic
378C2—3324no—
379C2#2490691714yesdirect product factor
380C2C2×#1912791911yescharacteristic
381C2C2×#1912794111yescharacteristic
382C2C2×#1912796311yescharacteristic
subgroupG/H
centerC23#14581765
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC2×#333×C83C2
socleC23×C83#121
prime radicalC22×D332D4
hypercenterC2×#333D83
hypocenterC83C2×#1683
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
1. C2×#19128053C2semidirect
5. #24906917C2direct (N × H)
6. C2×#19127453C2semidirect
7. C2×#19127448C2semidirect
8. C2×#333×C83C2semidirect
13. #19128053C22semidirect
29. #19127453C22semidirect
30. #19127448C22semidirect
34. #333×C83C22semidirect
187. C2×C83#1683semidirect
198. C2×#333D83semidirect
200. C83C2×#1683semidirect
225. #333C2×D83semidirect
379. C2#24906917direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191280532
C2×#191279882
C2×#191279822
C2×#191277662
#249069172
C2×#191274532
C2×#191274482
C2×#333×C832
C2×#168383