Finite Group Databasebeta

← C2×#26028518 #36761019

Subgroups — C2×#26028518 #36761019

Order 22784.

1806 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
1#26028518C2116yesmaximal, direct product factor
2C2×#19151539C212yesmaximal
3C2×#19151537C211yesmaximal, characteristic
4C2×#19151532C212yesmaximal
5C2×#19151531C212yesmaximal
6C2×#19151529C212yesmaximal
7C2×#19151517C212yesmaximal
8C2×#19151467C211yesmaximal, characteristic
9C2×#19151465C212yesmaximal
10C2×#395×C89C211yesmaximal, characteristic
11C22×C4⋊Dic89C2211yescharacteristic
12C22×C4⋊Dic89C2211yescharacteristic
13C22×#13083553C2212yes—
14C22×#13083553C2212yes—
15C22×#13083553C2212yes—
16C22×#13083553C2212yes—
17C22×#13083553C2212yes—
18C22×C4×Dic89C2211yescharacteristic
19#19151539C22116yes—
20#19151537C2218yes—
21#19151532C22116yes—
22#19151531C22116yes—
23#19151529C22116yes—
24#19151517C22116yes—
25#19151467C2218yes—
26#19151465C22116yes—
27C2×#15667919C2212yes—
28C2×#15667919C2212yes—
29C2×#15667919C2212yes—
30C2×#15667919C2211yescharacteristic
31C2×#15667919C2212yes—
32C2×#15667919C2212yes—
33C2×#15667919C2212yes—
34C2×#15667919C2212yes—
35C23×Q8×C89C2211yescharacteristic
36C22×C4⋊C4×C89C2212yes—
37C22×C4⋊C4×C89C2211yescharacteristic
38#395×C89C2218yes—
39C2×#95×C89C2212yes—
40C2×#95×C89C2211yescharacteristic
41C23×Dic89C2312yes—
42C23×Dic89C2312yes—
43C23×Dic89C2312yes—
44C23×Dic89C2312yes—
45C2×C4⋊Dic89—42no—
46C2×C4⋊Dic89—42no—
47C2×C4⋊Dic89C2314yes—
48C2×C4⋊Dic89C2314yes—
49C2×#13083553—44no—
50C2×#13083553—44no—
51C2×#13083553—44no—
52C2×#13083553—44no—
53C2×#13083553—44no—
54C2×#13083553C2318yes—
55C2×#13083553C2318yes—
56C2×#13083553C2318yes—
57C2×#13083553C2318yes—
58C2×#13083553C2318yes—
59C2×C4×Dic89Q818yescomplemented
60C2×C4×Dic89C2314yes—
61#15667919C2318yes—
62#15667919C2318yes—
63#15667919C2318yes—
64#15667919C2318yes—
65#15667919C2318yes—
66#15667919C2318yes—
67#15667919C2314yes—
68#15667919C2318yes—
69C22×Q8×C89—44no—
70C22×Q8×C89—42no—
71C22×Q8×C89C2314yes—
72C23×C4×C89C2312yes—
73C23×C4×C89C2312yes—
74C23×C4×C89C2311yescharacteristic
75C23×C4×C89C2311yescharacteristic
76C23×C4×C89C2311yescharacteristic
77C2×C4⋊C4×C89—42no—
78C2×C4⋊C4×C89—44no—
79C2×C4⋊C4×C89C2314yes—
80C2×C4⋊C4×C89C2318yes—
81#95×C89C2318yes—
82#95×C89C2314yes—
83C22×Dic89—24no—
84C22×Dic89—24no—
85C22×Dic89—24no—
86C22×Dic89—24no—
87C22×Dic89—24no—
88C22×Dic89—24no—
89C22×Dic89—24no—
90C22×Dic89—24no—
91C22×Dic89—24no—
92C22×Dic89—28no—
93C22×Dic89C2×Q818yes—
94C22×Dic89C2414yes—
95C22×Dic89C2414yes—
96C22×Dic89C2414yes—
97C22×Dic89C2414yes—
98C4⋊Dic89—44no—
99C4⋊Dic89—44no—
100#13083553—48no—
101#13083553—48no—
102#13083553—48no—
103#13083553—48no—
104#13083553—48no—
105C4×Dic89C2×Q8116yescomplemented
106C24×C89C2411yescharacteristic
107C2×Q8×C89—44no—
108C2×Q8×C89—48no—
109C2×Q8×C89—416no—
110C22×C4×C89—22no—
111C22×C4×C89—22no—
112C22×C4×C89—24no—
113C22×C4×C89—22no—
114C22×C4×C89—24no—
115C22×C4×C89—24no—
116C22×C4×C89—22no—
117C22×C4×C89—24no—
118C22×C4×C89—24no—
119C22×C4×C89—24no—
120C22×C4×C89—24no—
121C22×C4×C89—24no—
122C22×C4×C89C2×Q814yes—
123C22×C4×C89C2414yes—
124C22×C4×C89C2414yes—
125C22×C4×C89C2412yes—
126C22×C4×C89C2412yes—
127C22×C4×C89C2412yes—
128C4⋊C4×C89—44no—
129C4⋊C4×C89—48no—
130C23×C89C22×Q812yes—
131C23×C89H(5,𝔽2)14yes—
132C23×C89H(5,𝔽2)12yes—
133C23×C892−1+412yes—
134C23×C892−1+414yes—
135C23×C89C2511yescharacteristic
136Q8×C89—416no—
137C2×C4×C89—28no—
138C2×C4×C89—22no—
139C2×C4×C89—22no—
140C2×C4×C89—22no—
141C2×C4×C89—24no—
142C2×C4×C89—24no—
143C2×C4×C89—24no—
144C2×C4×C89—22no—
145C2×C4×C89—24no—
146C2×C4×C89—24no—
147C2×C4×C89—24no—
148C2×C4×C89—24no—
149C2×C4×C89—24no—
150C2×C4×C89—44no—
151C2×C4×C89—48no—
152C2×C4×C89—44no—
153C2×C4×C89—48no—
154C2×C4×C89C22×Q814yes—
155C2×Dic89—216no—
156C2×Dic89—24no—
157C2×Dic89—24no—
158C2×Dic89—24no—
159C2×Dic89—24no—
160C2×Dic89—24no—
161C2×Dic89—24no—
162C2×Dic89—24no—
163C2×Dic89—24no—
164C2×Dic89—24no—
165C2×Dic89—28no—
166C2×Dic89—48no—
167C2×Dic89—48no—
168C2×Dic89—48no—
169C2×Dic89C22×Q818yes—
170C22×C89#54418yes—
171C22×C89#54414yes—
172C22×C89#56014yes—
173C22×C89#56314yes—
174C22×C89#56318yes—
175C22×C89C23×Q811yescharacteristic
176C22×C89C2×H(5,𝔽2)12yes—
177C22×C89C2×H(5,𝔽2)11yescharacteristic
178C22×C89C2×2−1+411yescharacteristic
179C22×C89C2×2−1+412yes—
180C4×C89—24no—
181C4×C89—42no—
182C4×C89—42no—
183C4×C89—44no—
184C4×C89—44no—
185Dic89—28no—
186Dic89—44no—
187Dic89—44no—
188Dic89—44no—
189C2×#2843—891nomaximal, Sylow
190C2×C89#284318yescomplemented
191C2×C89C2×#54412yes—
192C2×C89C2×#54411yescharacteristic
193C2×C89C2×#56011yescharacteristic
194C2×C89C2×#56311yescharacteristic
195C2×C89C2×#56312yes—
196#2843—8916no—
197C2×#400—892no—
198C2×#400—891no—
199C2×#398—892no—
200C2×#398—892no—
201C2×#398—892no—
202C2×#395—892no—
203C2×#395D8911yescharacteristic
204C2×#384—891no—
205C2×#383—892no—
206C89C2×#284311yescharacteristic, complemented, Sylow
207C23×Q8C2×D8911yescharacteristic
208C22×C4⋊C4—892no—
209C22×C4⋊C4—892no—
210C22×C4⋊C4—892no—
211C22×C4⋊C4—892no—
212C22×C4⋊C4—891no—
213C22×C4⋊C4—892no—
214C22×C4⋊C4—891no—
215C22×C4⋊C4C2×D8912yes—
216C22×C4⋊C4C2×D8911yescharacteristic
217C22×C42—891no—
218#400—898no—
219#400—8916no—
220#398—8916no—
221#398—8916no—
222#398—8916no—
223#395—8916no—
224#395C2×D8918yes—
225#384—898no—
226#383—8916no—
227C2×#95—892no—
228C2×#95—892no—
229C2×#95—892no—
230C2×#95—892no—
231C2×#95—891no—
232C2×#95—892no—
233C2×#95—892no—
234C2×#95—892no—
235C2×#95C2×D8912yes—
236C2×#95C2×D8911yescharacteristic
237C22×Q8—44no—
238C22×Q8—42no—
239C22×Q8C22×D8914yes—
240C23×C4—892no—
241C23×C4—892no—
242C23×C4—892no—
243C23×C4—892no—
244C23×C4C22×D8911yescharacteristic
245C23×C4C22×D8912yes—
246C23×C4C22×D8911yescharacteristic
247C23×C4C22×D8912yes—
248C23×C4C22×D8911yescharacteristic
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—898no—
252C2×C4⋊C4—898no—
253C2×C4⋊C4—894no—
254C2×C4⋊C4—898no—
255C2×C4⋊C4—898no—
256C2×C4⋊C4—898no—
257C2×C4⋊C4—894no—
258C2×C4⋊C4—3564no—
259C2×C4⋊C4—3564no—
260C2×C4⋊C4—3564no—
261C2×C4⋊C4—3564no—
262C2×C4⋊C4—3564no—
263C2×C4⋊C4—3562no—
264C2×C4⋊C4—3562no—
265C2×C4⋊C4C22×D8914yes—
266C2×C4⋊C4C22×D8918yes—
267C2×C42—898no—
268C2×C42—894no—
269#95—898no—
270#95—898no—
271#95—898no—
272#95—898no—
273#95—898no—
274#95—898no—
275#95—894no—
276#95—898no—
277#95C22×D8918yes—
278#95C22×D8914yes—
279C24C23×D8911yescharacteristic
280C2×Q8—44no—
281C2×Q8—48no—
282C2×Q8—416no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—898no—
296C22×C4—894no—
297C22×C4—894no—
298C22×C4—894no—
299C22×C4—894no—
300C22×C4—1784no—
301C22×C4—1784no—
302C22×C4—1784no—
303C22×C4—1784no—
304C22×C4—1784no—
305C22×C4—1784no—
306C22×C4—1788no—
307C22×C4—1784no—
308C22×C4—1784no—
309C22×C4—1784no—
310C22×C4Q8×D8914yes—
311C22×C4C23×D8914yes—
312C22×C4C23×D8914yes—
313C22×C4C23×D8912yes—
314C22×C4C23×D8912yes—
315C22×C4C23×D8912yes—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—3568no—
319C4⋊C4—3568no—
320C4⋊C4—3568no—
321C4⋊C4—3568no—
322C4⋊C4—3568no—
323C4⋊C4—3564no—
324C4⋊C4—3564no—
325C42—8916no—
326C23C2×Q8×D8912yes—
327C23#1566805714yes—
328C23#1566805712yes—
329C23#1566805912yes—
330C23#1566806014yes—
331C23C24×D8911yescharacteristic
332Q8—416no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—898no—
351C2×C4—3568no—
352C2×C4—3568no—
353C2×C4—3568no—
354C2×C4—1784no—
355C2×C4—1784no—
356C2×C4—1784no—
357C2×C4—1784no—
358C2×C4—1784no—
359C2×C4—1784no—
360C2×C4—1788no—
361C2×C4—1784no—
362C2×C4—1784no—
363C2×C4—1784no—
364C2×C4—17816no—
365C2×C4C2×Q8×D8914yes—
366C22#1915222518yes—
367C22#1915222714yes—
368C22#1915233714yes—
369C22#1915234714yes—
370C22#1915234918yes—
371C22C22×Q8×D8911yescharacteristic
372C22C2×#1566805711yescharacteristic
373C22C2×#1566805712yes—
374C22C2×#1566805911yescharacteristic
375C22C2×#1566806012yes—
376C4—24no—
377C4—42no—
378C4—44no—
379C4—42no—
380C4—44no—
381C4—3564no—
382C4—3564no—
383C4—3564no—
384C4—1788no—
385C2#2602851818yesdirect product factor
386C2C2×#1915222512yes—
387C2C2×#1915222711yescharacteristic
388C2C2×#1915233711yescharacteristic
389C2C2×#1915234711yescharacteristic
390C2C2×#1915234912yes—
subgroupG/H
centerC24C23×D89
commutator, cosocleC23×C89C25
FrattiniC23C24×D89
FittingC2×#395×C89C2
socle, prime radicalC24×C89C24
hypercenterC2×#395D89
hypocenterC89C2×#2843
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
1. #26028518C2direct (N × H)
59. C2×C4×Dic89Q8semidirect
105. C4×Dic89C2×Q8semidirect
190. C2×C89#2843semidirect
206. C89C2×#2843semidirect
385. C2#26028518direct (N × H)

Maximal subgroups

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

subgroupindex
#260285182
C2×#191515392
C2×#191515372
C2×#191515322
C2×#191515312
C2×#191515292
C2×#191515172
C2×#191514672
C2×#191514652
C2×#395×C892
C2×#284389