Finite Group Databasebeta

← C2×#26025614 #36725758

Subgroups — C2×#26025614 #36725758

Order 22784.

2614 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×C4×Dic178C211yesmaximal, characteristic
2#26025614C2116yesmaximal, direct product factor
3C2×#19151537C212yesmaximal
4C2×#19151518C212yesmaximal
5C2×#19151505C212yesmaximal
6C2×#19151494C212yesmaximal
7C2×#19151482C214yesmaximal
8C2×#19151464C211yesmaximal, characteristic
9C22×C4×Q8×C89C211yesmaximal, characteristic
10C23×Dic178C2211yescharacteristic
11C2×C4×Dic178—28no—
12C2×C4×Dic178C2218yes—
13C22×C4⋊Dic89C2211yescharacteristic
14C22×C4⋊Dic89C2212yes—
15C22×#13083553C2212yes—
16C22×#13083553C2214yes—
17C22×C4×Dic89C2212yes—
18#19151537C22116yes—
19#19151518C22116yes—
20#19151505C22116yes—
21#19151494C22116yes—
22#19151482C22132yes—
23#19151464C2218yes—
24C2×#15667919C2212yes—
25C2×#15667919C2214yes—
26C2×#15667919C2214yes—
27C2×#15667919C2214yes—
28C2×#15667919C2212yes—
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
36C22×Dic178—28no—
37C22×Dic178—24no—
38C22×Dic178C2314yes—
39C23×Dic89C2314yes—
40C23×Dic89C2312yes—
41C23×Dic89C2312yes—
42C4×Dic178—232no—
43C2×C4⋊Dic89—24no—
44C2×C4⋊Dic89—28no—
45C2×C4⋊Dic89C2318yes—
46C2×C4⋊Dic89C2314yes—
47C2×#13083553—216no—
48C2×#13083553—28no—
49C2×#13083553C2318yes—
50C2×#13083553C23116yes—
51C2×C4×Dic89—28no—
52C2×C4×Dic89C2318yes—
53#15667919C2318yes—
54#15667919C2318yes—
55#15667919C23116yes—
56#15667919C23116yes—
57#15667919C23116yes—
58C22×Q8×C89—24no—
59C22×Q8×C89—24no—
60C22×Q8×C89Q818yescomplemented
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×C89C2314yes—
71C2×C4⋊C4×C89C2318yes—
72C2×C42×C89—28no—
73C2×C42×C89D418yes—
74C2×C42×C89C2318yes—
75C2×C42×C89C2314yes—
76C2×Dic178—232no—
77C2×Dic178—28no—
78C2×Dic178—216no—
79C22×Dic89—24no—
80C22×Dic89—24no—
81C22×Dic89—24no—
82C22×Dic89—28no—
83C22×Dic89—28no—
84C22×Dic89—28no—
85C22×Dic89—28no—
86C22×Dic89#4118yes—
87C22×Dic89C2418yes—
88C22×Dic89C2414yes—
89C22×Dic89C2414yes—
90C4⋊Dic89—28no—
91C4⋊Dic89—216no—
92#13083553—216no—
93#13083553—232no—
94C4×Dic89—216no—
95C24×C89C2411yescharacteristic
96C2×Q8×C89—232no—
97C2×Q8×C89—28no—
98C2×Q8×C89—28no—
99C2×Q8×C89C2×Q8116yescomplemented
100C22×C4×C89—24no—
101C22×C4×C89—24no—
102C22×C4×C89—24no—
103C22×C4×C89—24no—
104C22×C4×C89—24no—
105C22×C4×C89—28no—
106C22×C4×C89C2×D414yes—
107C22×C4×C89C2×D414yes—
108C22×C4×C89C2×D414yes—
109C22×C4×C89C2×Q818yes—
110C22×C4×C89C2×Q814yes—
111C22×C4×C89#4114yes—
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×Q812yes—
123C23×C89C2×#4114yes—
124C23×C89C2×#4112yes—
125C23×C892−1+414yes—
126C23×C89C2511yescharacteristic
127Q8×C89—232no—
128C2×C4×C89—216no—
129C2×C4×C89—28no—
130C2×C4×C89—28no—
131C2×C4×C89—24no—
132C2×C4×C89—24no—
133C2×C4×C89—24no—
134C2×C4×C89—24no—
135C2×C4×C89—24no—
136C2×C4×C89—28no—
137C2×C4×C89—48no—
138C2×C4×C89C22⋊Q8116yes—
139C2×C4×C89C22×D414yes—
140C2×C4×C89C22×D414yes—
141C2×C4×C89C22×D414yes—
142C2×C4×C89C22×Q818yes—
143C2×C4×C89C22×Q814yes—
144C2×C4×C89C2×#4114yes—
145C2×Dic89—216no—
146C2×Dic89—24no—
147C2×Dic89—24no—
148C2×Dic89—24no—
149C2×Dic89—28no—
150C2×Dic89—28no—
151C2×Dic89—28no—
152C2×Dic89—28no—
153C2×Dic89—48no—
154C2×Dic89—416no—
155C2×Dic89C2×#4118yes—
156Dic178—232no—
157C22×C89C2×C22⋊Q814yes—
158C22×C89#52514yes—
159C22×C89#53614yes—
160C22×C89#55618yes—
161C22×C89#55718yes—
162C22×C89C23×D411yescharacteristic
163C22×C89C23×Q811yescharacteristic
164C22×C89C22×#4111yescharacteristic
165C22×C89C22×#4112yes—
166C22×C89C2×2−1+412yes—
167C4×C89—28no—
168C4×C89—24no—
169C4×C89—24no—
170C4×C89—44no—
171C4×C89C2×C22⋊Q818yes—
172Dic89—28no—
173Dic89—44no—
174Dic89—48no—
175C2×#2581—891nomaximal, Sylow
176C2×C89#258118yescomplemented
177C2×C89C22×C22⋊Q811yescharacteristic
178C2×C89C2×#52511yescharacteristic
179C2×C89C2×#53611yescharacteristic
180C2×C89C2×#55612yes—
181C2×C89C2×#55712yes—
182C22×C4×Q8—891no—
183C22×C4×Q8D8911yescharacteristic
184#2581—8916no—
185C2×#400—892no—
186C2×#395—892no—
187C2×#391—892no—
188C2×#389—892no—
189C2×#387—894no—
190C2×#382—891no—
191C89C2×#258111yescharacteristic, complemented, Sylow
192C23×Q8—891no—
193C23×Q8C2×D8911yescharacteristic
194C2×C4×Q8—28no—
195C2×C4×Q8—898no—
196C2×C4×Q8—1788no—
197C2×C4×Q8C2×D8918yes—
198C22×C4⋊C4—894no—
199C22×C4⋊C4—891no—
200C22×C4⋊C4—892no—
201C22×C4⋊C4—892no—
202C22×C4⋊C4C2×D8912yes—
203C22×C4⋊C4C2×D8911yescharacteristic
204C22×C42—892no—
205C22×C42C2×D8912yes—
206C22×C42C2×D8911yescharacteristic
207#400—8916no—
208#395—8916no—
209#391—8916no—
210#389—8916no—
211#387—8932no—
212#382—898no—
213C2×#95—892no—
214C2×#95—894no—
215C2×#95—894no—
216C2×#95—894no—
217C2×#95—892no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—894no—
221C22×Q8—1788no—
222C22×Q8—1784no—
223C22×Q8Dic17818yescomplemented
224C22×Q8C22×D8914yes—
225C23×C4—892no—
226C23×C4—894no—
227C23×C4—892no—
228C23×C4C22×D8912yes—
229C23×C4C22×D8911yescharacteristic
230C23×C4C22×D8911yescharacteristic
231C23×C4C22×D8912yes—
232C23×C4C22×D8911yescharacteristic
233C4×Q8—232no—
234C4×Q8—17832no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—898no—
238C2×C4⋊C4—894no—
239C2×C4⋊C4—898no—
240C2×C4⋊C4—8916no—
241C2×C4⋊C4—17816no—
242C2×C4⋊C4—1788no—
243C2×C4⋊C4—1784no—
244C2×C4⋊C4—1788no—
245C2×C4⋊C4C22×D8914yes—
246C2×C4⋊C4C22×D8918yes—
247C2×C42—28no—
248C2×C42—898no—
249C2×C42—1788no—
250C2×C42C22⋊D8918yes—
251C2×C42C22×D8918yes—
252C2×C42C22×D8914yes—
253#95—898no—
254#95—898no—
255#95—8916no—
256#95—8916no—
257#95—8916no—
258C24C23×D8911yescharacteristic
259C2×Q8—232no—
260C2×Q8—28no—
261C2×Q8—28no—
262C2×Q8—17816no—
263C2×Q8—17832no—
264C2×Q8—1788no—
265C2×Q8C2×Dic178116yescomplemented
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—898no—
273C22×C4—894no—
274C22×C4—898no—
275C22×C4—894no—
276C22×C4—1784no—
277C22×C4—1788no—
278C22×C4—1788no—
279C22×C4—1788no—
280C22×C4—1788no—
281C22×C4—1784no—
282C22×C4—1784no—
283C22×C4C2×C22⋊D8914yes—
284C22×C4C2×C22⋊D8914yes—
285C22×C4C2×C22⋊D8914yes—
286C22×C4C2×Dic17818yes—
287C22×C4C2×Dic17814yes—
288C22×C4#1308357414yes—
289C22×C4C23×D8914yes—
290C22×C4C23×D8912yes—
291C22×C4C23×D8912yes—
292C22×C4C23×D8914yes—
293C22×C4C23×D8912yes—
294C4⋊C4—216no—
295C4⋊C4—28no—
296C4⋊C4—17832no—
297C4⋊C4—17816no—
298C4⋊C4—1788no—
299C4⋊C4—17816no—
300C42—216no—
301C42—17816no—
302C42C2×C22⋊D89116yes—
303C23C22×C22⋊D8912yes—
304C23C22×Dic17812yes—
305C23#41×D8914yes—
306C23C2×#1308357412yes—
307C23#1566806014yes—
308C23C24×D8911yescharacteristic
309Q8—232no—
310Q8—17832no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—216no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—48no—
321C2×C4—898no—
322C2×C4—3568no—
323C2×C4—35616no—
324C2×C4—1784no—
325C2×C4—1788no—
326C2×C4—1788no—
327C2×C4—1788no—
328C2×C4—1788no—
329C2×C4—1784no—
330C2×C4—1784no—
331C2×C4—17816no—
332C2×C4#15667991116yes—
333C2×C4C22×C22⋊D8914yes—
334C2×C4C22×C22⋊D8914yes—
335C2×C4C22×C22⋊D8914yes—
336C2×C4C22×Dic17818yes—
337C2×C4C22×Dic17814yes—
338C2×C4C2×#1308357414yes—
339C22C2×#1566799114yes—
340C22#1915210514yes—
341C22#1915217214yes—
342C22#1915232018yes—
343C22#1915232418yes—
344C22C23×C22⋊D8911yescharacteristic
345C22C23×Dic17811yescharacteristic
346C22C2×#41×D8912yes—
347C22C22×#1308357411yescharacteristic
348C22C2×#1566806012yes—
349C4—28no—
350C4—24no—
351C4—24no—
352C4—44no—
353C4—3568no—
354C4—3564no—
355C4—1788no—
356C4C2×#1566799118yes—
357C2#2602561418yesdirect product factor
358C2C22×#1566799111yescharacteristic
359C2C2×#1915210511yescharacteristic
360C2C2×#1915217211yescharacteristic
361C2C2×#1915232012yes—
362C2C2×#1915232412yes—
subgroupG/H
centerC24C23×D89
commutator, cosocleC23×C89C25
FrattiniC23C24×D89
FittingC22×C4×Q8×C89C2
socle, prime radicalC24×C89C24
hypercenterC22×C4×Q8D89
hypocenterC89C2×#2581
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #26025614C2direct (N × H)
60. C22×Q8×C89Q8semidirect
99. C2×Q8×C89C2×Q8semidirect
176. C2×C89#2581semidirect
191. C89C2×#2581semidirect
223. C22×Q8Dic178semidirect
265. C2×Q8C2×Dic178semidirect
357. C2#26025614direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Dic1782
#260256142
C2×#191515372
C2×#191515182
C2×#191515052
C2×#191514942
C2×#191514822
C2×#191514642
C22×C4×Q8×C892
C2×#258189