Finite Group Databasebeta

← C2×#26025640 #36725661

Subgroups — C2×#26025640 #36725661

Order 22784.

2574 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15668004C211yesmaximal, characteristic
2#26025640C2116yesmaximal, direct product factor
3C2×#19151529C212yesmaximal
4C2×#19151517C212yesmaximal
5C2×#19151496C212yesmaximal
6C2×#19151496C212yesmaximal
7C2×#19151493C214yesmaximal
8C2×C4×C4⋊Dic89C211yesmaximal, characteristic
9C22×C4×Q8×C89C211yesmaximal, characteristic
10C2×#15668004—28no—
11C2×#15668004C2218yes—
12C22×C4⋊Dic89C2211yescharacteristic
13C22×C4⋊Dic89C2211yescharacteristic
14C22×C4⋊Dic89C2212yes—
15C22×#13083553C2214yes—
16C22×#13083553C2212yes—
17C22×#13083553C2212yes—
18C22×#13083553C2214yes—
19C22×C4×Dic89C2212yes—
20#19151529C22116yes—
21#19151517C22116yes—
22#19151496C22116yes—
23#19151496C22116yes—
24#19151493C22132yes—
25C4×C4⋊Dic89C2218yes—
26C2×#15667919C2214yes—
27C2×#15667919C2212yes—
28C2×#15667919C2214yes—
29C23×Q8×C89C2211yescharacteristic
30C2×C4×Q8×C89—28no—
31C2×C4×Q8×C89C2218yes—
32C22×C4⋊C4×C89C2212yes—
33C22×C4⋊C4×C89C2211yescharacteristic
34C22×C42×C89C2212yes—
35C22×C42×C89C2211yescharacteristic
36C23×Dic89C2312yes—
37C23×Dic89C2312yes—
38C23×Dic89C2314yes—
39#15668004—232no—
40C2×C4⋊Dic89—24no—
41C2×C4⋊Dic89—24no—
42C2×C4⋊Dic89Q8116yescomplemented
43C2×C4⋊Dic89C2314yes—
44C2×C4⋊Dic89C2314yes—
45C2×C4⋊Dic89C2318yes—
46C2×#13083553—28no—
47C2×#13083553—28no—
48C2×#13083553—216no—
49C2×#13083553—48no—
50C2×#13083553C2318yes—
51C2×#13083553C23116yes—
52C2×#13083553C2318yes—
53C2×#13083553C23116yes—
54C2×C4×Dic89—44no—
55C2×C4×Dic89C2318yes—
56#15667919C2318yes—
57#15667919C23116yes—
58#15667919C23116yes—
59C22×Q8×C89—24no—
60C22×Q8×C89—28no—
61C22×Q8×C89C2314yes—
62C23×C4×C89C2312yes—
63C23×C4×C89C2311yescharacteristic
64C23×C4×C89C2311yescharacteristic
65C23×C4×C89C2312yes—
66C23×C4×C89C2311yescharacteristic
67C4×Q8×C89—232no—
68C2×C4⋊C4×C89—28no—
69C2×C4⋊C4×C89—24no—
70C2×C4⋊C4×C89C2318yes—
71C2×C4⋊C4×C89C2314yes—
72C2×C42×C89—28no—
73C2×C42×C89D418yes—
74C2×C42×C89C2318yes—
75C2×C42×C89C2314yes—
76C22×Dic89—28no—
77C22×Dic89—28no—
78C22×Dic89—24no—
79C22×Dic89—24no—
80C22×Dic89—28no—
81C22×Dic89—24no—
82C22×Dic89—24no—
83C22×Dic89C2×Q8116yes—
84C22×Dic89C2414yes—
85C22×Dic89C2414yes—
86C22×Dic89C2418yes—
87C4⋊Dic89—28no—
88C4⋊Dic89—28no—
89C4⋊Dic89C2×Q8132yescomplemented
90#13083553—216no—
91#13083553—216no—
92#13083553—232no—
93#13083553—416no—
94C4×Dic89—48no—
95C24×C89C2411yescharacteristic
96C2×Q8×C89—232no—
97C2×Q8×C89—28no—
98C2×Q8×C89—216no—
99C22×C4×C89—24no—
100C22×C4×C89—24no—
101C22×C4×C89—24no—
102C22×C4×C89—22no—
103C22×C4×C89—22no—
104C22×C4×C89—24no—
105C22×C4×C89—24no—
106C22×C4×C89—24no—
107C22×C4×C89C2×D414yes—
108C22×C4×C89C2×D414yes—
109C22×C4×C89C2×D414yes—
110C22×C4×C89C2×Q818yes—
111C22×C4×C89#4118yes—
112C22×C4×C89C2414yes—
113C22×C4×C89C2412yes—
114C22×C4×C89C2412yes—
115C22×C4×C89C2414yes—
116C22×C4×C89C2412yes—
117C4⋊C4×C89—216no—
118C4⋊C4×C89—28no—
119C42×C89—216no—
120C42×C89C2×D4116yes—
121C23×C89C22×D412yes—
122C23×C89C22×Q814yes—
123C23×C89C2×#4114yes—
124C23×C89H(5,𝔽2)12yes—
125C23×C892−1+412yes—
126C23×C89C2511yescharacteristic
127Q8×C89—232no—
128C2×C4×C89—28no—
129C2×C4×C89—216no—
130C2×C4×C89—28no—
131C2×C4×C89—24no—
132C2×C4×C89—24no—
133C2×C4×C89—24no—
134C2×C4×C89—22no—
135C2×C4×C89—22no—
136C2×C4×C89—24no—
137C2×C4×C89—24no—
138C2×C4×C89—24no—
139C2×C4×C89—48no—
140C2×C4×C89C4⋊Q8116yes—
141C2×C4×C89C22×D414yes—
142C2×C4×C89C22×D414yes—
143C2×C4×C89C22×D414yes—
144C2×C4×C89C22×Q818yes—
145C2×C4×C89C2×#4118yes—
146C2×Dic89—232no—
147C2×Dic89—28no—
148C2×Dic89—28no—
149C2×Dic89—24no—
150C2×Dic89—24no—
151C2×Dic89—28no—
152C2×Dic89—24no—
153C2×Dic89—24no—
154C2×Dic89—48no—
155C2×Dic89—48no—
156C2×Dic89C22×Q8116yes—
157C22×C89#52514yes—
158C22×C89C2×C4⋊Q814yes—
159C22×C89#53714yes—
160C22×C89#55418yes—
161C22×C89#55718yes—
162C22×C89C23×D411yescharacteristic
163C22×C89C23×Q812yes—
164C22×C89C22×#4112yes—
165C22×C89C2×H(5,𝔽2)11yescharacteristic
166C22×C89C2×2−1+411yescharacteristic
167C4×C89—24no—
168C4×C89—28no—
169C4×C89—24no—
170C4×C89—44no—
171C4×C89C2×C4⋊Q818yes—
172Dic89—216no—
173Dic89—44no—
174Dic89—44no—
175C2×#2584—891nomaximal, Sylow
176C2×C89#258418yescomplemented
177C2×C89C2×#52511yescharacteristic
178C2×C89C22×C4⋊Q811yescharacteristic
179C2×C89C2×#53711yescharacteristic
180C2×C89C2×#55412yes—
181C2×C89C2×#55712yes—
182C22×#125—891no—
183C22×C4×Q8D8911yescharacteristic
184#2584—8916no—
185C2×#398—892no—
186C2×#395—892no—
187C2×#389—892no—
188C2×#389—892no—
189C2×#389—894no—
190C2×C4×C4⋊C4—891no—
191C89C2×#258411yescharacteristic, complemented, Sylow
192C23×Q8C2×D8911yescharacteristic
193C2×#125—898no—
194C2×#125—1788no—
195C2×C4×Q8—28no—
196C2×C4×Q8C2×D8918yes—
197C22×C4⋊C4—892no—
198C22×C4⋊C4—894no—
199C22×C4⋊C4—892no—
200C22×C4⋊C4—894no—
201C22×C4⋊C4—892no—
202C22×C4⋊C4—891no—
203C22×C4⋊C4—891no—
204C22×C4⋊C4C2×D8911yescharacteristic
205C22×C4⋊C4C2×D8912yes—
206C22×C42—892no—
207C22×C42C2×D8912yes—
208C22×C42C2×D8911yescharacteristic
209#398—8916no—
210#395—8916no—
211#389—8932no—
212#389—8916no—
213#389—8916no—
214C4×C4⋊C4—898no—
215C2×#95—892no—
216C2×#95—894no—
217C2×#95—894no—
218C22×Q8—24no—
219C22×Q8—28no—
220C22×Q8C22×D8914yes—
221C23×C4—894no—
222C23×C4—892no—
223C23×C4—892no—
224C23×C4C22×D8912yes—
225C23×C4C22×D8911yescharacteristic
226C23×C4C22×D8911yescharacteristic
227C23×C4C22×D8912yes—
228C23×C4C22×D8911yescharacteristic
229#125—17832no—
230C4×Q8—232no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—8916no—
234C2×C4⋊C4—898no—
235C2×C4⋊C4—8916no—
236C2×C4⋊C4—898no—
237C2×C4⋊C4—894no—
238C2×C4⋊C4—894no—
239C2×C4⋊C4—898no—
240C2×C4⋊C4—8916no—
241C2×C4⋊C4—3568no—
242C2×C4⋊C4—1788no—
243C2×C4⋊C4—1788no—
244C2×C4⋊C4—1784no—
245C2×C4⋊C4—1784no—
246C2×C4⋊C4—17816no—
247C2×C4⋊C4C22×D8918yes—
248C2×C4⋊C4C22×D8914yes—
249C2×C42—28no—
250C2×C42—898no—
251C2×C42—3564no—
252C2×C42C22⋊D8918yes—
253C2×C42C22×D8918yes—
254C2×C42C22×D8914yes—
255#95—898no—
256#95—8916no—
257#95—8916no—
258C24C23×D8911yescharacteristic
259C2×Q8—232no—
260C2×Q8—28no—
261C2×Q8—216no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—8916no—
271C22×C4—898no—
272C22×C4—894no—
273C22×C4—894no—
274C22×C4—1784no—
275C22×C4—1784no—
276C22×C4—1788no—
277C22×C4—1784no—
278C22×C4—1784no—
279C22×C4—1788no—
280C22×C4—1788no—
281C22×C4C2×C22⋊D8914yes—
282C22×C4C2×C22⋊D8914yes—
283C22×C4C2×C22⋊D8914yes—
284C22×C4Q8×D8918yes—
285C22×C4#1308357418yes—
286C22×C4C23×D8914yes—
287C22×C4C23×D8912yes—
288C22×C4C23×D8912yes—
289C22×C4C23×D8914yes—
290C22×C4C23×D8912yes—
291C4⋊C4—216no—
292C4⋊C4—28no—
293C4⋊C4—8932no—
294C4⋊C4—35616no—
295C4⋊C4—17816no—
296C4⋊C4—17816no—
297C4⋊C4—1788no—
298C4⋊C4—1788no—
299C4⋊C4—17832no—
300C42—216no—
301C42—3568no—
302C42C2×C22⋊D89116yes—
303C23C22×C22⋊D8912yes—
304C23C2×Q8×D8914yes—
305C23C2×#1308357414yes—
306C23#1566805812yes—
307C23#1566805912yes—
308C23C24×D8911yescharacteristic
309Q8—232no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—216no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—48no—
322C2×C4—8916no—
323C2×C4—3568no—
324C2×C4—3568no—
325C2×C4—1784no—
326C2×C4—1784no—
327C2×C4—1788no—
328C2×C4—1784no—
329C2×C4—1784no—
330C2×C4—1788no—
331C2×C4—1788no—
332C2×C4—17832no—
333C2×C4#15668010116yes—
334C2×C4C22×C22⋊D8914yes—
335C2×C4C22×C22⋊D8914yes—
336C2×C4C22×C22⋊D8914yes—
337C2×C4C2×Q8×D8918yes—
338C2×C4C2×#1308357418yes—
339C22#1915210914yes—
340C22C2×#1566801014yes—
341C22#1915217614yes—
342C22#1915230818yes—
343C22#1915232618yes—
344C22C23×C22⋊D8911yescharacteristic
345C22C22×Q8×D8912yes—
346C22C22×#1308357412yes—
347C22C2×#1566805811yescharacteristic
348C22C2×#1566805911yescharacteristic
349C4—24no—
350C4—28no—
351C4—24no—
352C4—44no—
353C4—3564no—
354C4—3564no—
355C4—17816no—
356C4C2×#1566801018yes—
357C2#2602564018yesdirect product factor
358C2C2×#1915210911yescharacteristic
359C2C22×#1566801011yescharacteristic
360C2C2×#1915217611yescharacteristic
361C2C2×#1915230812yes—
362C2C2×#1915232612yes—
subgroupG/H
centerC24C23×D89
commutator, cosocleC23×C89C25
FrattiniC23C24×D89
FittingC22×C4×Q8×C89C2
socle, prime radicalC24×C89C24
hypercenterC22×C4×Q8D89
hypocenterC89C2×#2584
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #26025640C2direct (N × H)
42. C2×C4⋊Dic89Q8semidirect
89. C4⋊Dic89C2×Q8semidirect
176. C2×C89#2584semidirect
191. C89C2×#2584semidirect
357. C2#26025640direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156680042
#260256402
C2×#191515292
C2×#191515172
C2×#191514962
C2×#191514932
C2×C4×C4⋊Dic892
C22×C4×Q8×C892
C2×#258489