Finite Group Databasebeta

← #36481172

Subgroups — #36481172

Order 22784.

1262 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151466C211yesmaximal, characteristic
2C2×C4×C4⋊Dic89C211yesmaximal, characteristic
3#26018516C212yesmaximal
4#26018515C212yesmaximal
5#26018511C212yesmaximal
6#26018508C212yesmaximal
7#26018483C214yesmaximal
8C2×#382×C89C211yesmaximal, characteristic
9C22×C4⋊Dic89C2211yescharacteristic
10C22×C4⋊Dic89C2211yescharacteristic
11C22×#13083553C2214yes—
12C22×C4×Dic89C2212yes—
13#19151466C418yescomplemented
14C4×C4⋊Dic89—24no—
15C2×#15667919C2214yes—
16C2×#15667919C2212yes—
17C2×#15667919C2214yes—
18C2×#15667919C2214yes—
19C2×#15667919C2212yes—
20C2×#15667919C2212yes—
21C2×#15667919C2212yes—
22C22×C4⋊C4×C89C2211yescharacteristic
23C22×C4⋊C4×C89C2211yescharacteristic
24C22×C42×C89C2211yescharacteristic
25#382×C89—24no—
26C2×#95×C89C2212yes—
27C2×#95×C89C2212yes—
28C23×Dic89C2314yes—
29C23×Dic89C2312yes—
30C23×Dic89C2312yes—
31C2×C4⋊Dic89—24no—
32C2×C4⋊Dic89—24no—
33C2×C4⋊Dic89—22no—
34C2×C4⋊Dic89—22no—
35C2×#13083553—28no—
36C2×#13083553—44no—
37C2×#13083553—44no—
38C2×C4×Dic89—24no—
39C2×C4×Dic89—44no—
40#15667919—28no—
41#15667919—24no—
42#15667919—44no—
43#15667919—44no—
44#15667919C2×C418yes—
45#15667919C2×C418yes—
46#15667919C2×C418yes—
47C23×C4×C89C2312yes—
48C23×C4×C89C2312yes—
49C23×C4×C89C2311yescharacteristic
50C23×C4×C89C2311yescharacteristic
51C23×C4×C89C2311yescharacteristic
52C2×C4⋊C4×C89—22no—
53C2×C4⋊C4×C89—22no—
54C2×C4⋊C4×C89—42no—
55C2×C4⋊C4×C89—42no—
56C2×C42×C89C2×C414yes—
57C2×C42×C89D414yes—
58C2×C42×C89Q814yes—
59#95×C89—24no—
60#95×C89—24no—
61C22×Dic89—22no—
62C22×Dic89—22no—
63C22×Dic89—24no—
64C22×Dic89—24no—
65C22×Dic89—22no—
66C22×Dic89—22no—
67C22×Dic89—24no—
68C22×Dic89—24no—
69C22×Dic89—24no—
70C22×Dic89—24no—
71C22×Dic89—24no—
72C22×Dic89—24no—
73C22×Dic89—24no—
74C22×Dic89—24no—
75C22×Dic89C22×C418yes—
76C22×Dic89#4114yes—
77C22×Dic89#4114yes—
78C4⋊Dic89—28no—
79C4⋊Dic89—28no—
80#13083553—416no—
81C4×Dic89—48no—
82C24×C89C2411yescharacteristic
83C22×C4×C89—22no—
84C22×C4×C89—22no—
85C22×C4×C89—22no—
86C22×C4×C89—22no—
87C22×C4×C89—22no—
88C22×C4×C89—22no—
89C22×C4×C89—22no—
90C22×C4×C89—22no—
91C22×C4×C89—24no—
92C22×C4×C89—24no—
93C22×C4×C89—24no—
94C22×C4×C89—24no—
95C22×C4×C89—22no—
96C22×C4×C89—22no—
97C22×C4×C89—24no—
98C22×C4×C89C22×C412yes—
99C22×C4×C89C22×C412yes—
100C22×C4×C89C22×C412yes—
101C22×C4×C89C2×D412yes—
102C22×C4×C89C2×D412yes—
103C22×C4×C89C2×D412yes—
104C22×C4×C89C2×Q812yes—
105C22×C4×C89C2×Q812yes—
106C22×C4×C89C2×Q812yes—
107C4⋊C4×C89—44no—
108C4⋊C4×C89—44no—
109C42×C89C4⋊C4116yescomplemented
110C23×C89C23×C411yescharacteristic
111C23×C89C22×D411yescharacteristic
112C23×C89C22×Q811yescharacteristic
113C23×C89C2×#4112yes—
114C23×C89C2×#4112yes—
115C23×C89H(5,𝔽2)12yes—
116C23×C89H(5,𝔽2)12yes—
117C23×C892−1+412yes—
118C23×C892−1+412yes—
119C2×C4×C89—24no—
120C2×C4×C89—24no—
121C2×C4×C89—24no—
122C2×C4×C89—22no—
123C2×C4×C89—22no—
124C2×C4×C89—24no—
125C2×C4×C89—22no—
126C2×C4×C89—22no—
127C2×C4×C89—22no—
128C2×C4×C89—22no—
129C2×C4×C89—24no—
130C2×C4×C89—24no—
131C2×C4×C89—42no—
132C2×C4×C89—42no—
133C2×C4×C89—42no—
134C2×C4×C89—42no—
135C2×C4×C89—42no—
136C2×C4×C89—42no—
137C2×C4×C89—44no—
138C2×C4×C89—44no—
139C2×C4×C89—48no—
140C2×C4×C89C2×C4⋊C414yes—
141C2×C4×C89C2×C4⋊C414yes—
142C2×C4×C89C2×C4⋊C414yes—
143C2×Dic89—28no—
144C2×Dic89—28no—
145C2×Dic89—28no—
146C2×Dic89—24no—
147C2×Dic89—24no—
148C2×Dic89—28no—
149C2×Dic89—28no—
150C2×Dic89—42no—
151C2×Dic89—42no—
152C2×Dic89—42no—
153C2×Dic89—42no—
154C2×Dic89—44no—
155C2×Dic89—44no—
156C2×Dic89—44no—
157C2×Dic89—44no—
158C2×Dic89—44no—
159C2×Dic89—44no—
160C22×C89C22×C4⋊C411yescharacteristic
161C22×C89C4×#4112yes—
162C22×C89#52012yes—
163C22×C89#52012yes—
164C22×C89#52511yescharacteristic
165C22×C89#52511yescharacteristic
166C22×C89#53312yes—
167C22×C89#53511yescharacteristic
168C22×C89#53511yescharacteristic
169C22×C89#53611yescharacteristic
170C22×C89#53611yescharacteristic
171C22×C89#53814yes—
172C22×C89#53914yes—
173C22×C89#54114yes—
174C22×C89#54214yes—
175C22×C89#54412yes—
176C22×C89#54412yes—
177C4×C89—24no—
178C4×C89—24no—
179C4×C89—24no—
180C4×C89—82no—
181C4×C89—82no—
182Dic89—48no—
183Dic89—44no—
184Dic89—44no—
185#14298—891nomaximal, Sylow
186C2×C89#229611yescharacteristic
187C2×C89#231711yescharacteristic
188C2×C89#231711yescharacteristic
189C2×C89#233414yes—
190C2×C89#263111yescharacteristic
191C2×C89#263211yescharacteristic
192C2×C89#263311yescharacteristic
193C2×C89#263411yescharacteristic
194C2×C89#266614yes—
195C2×#383—891no—
196C2×#382D8911yescharacteristic
197C2×C4×C4⋊C4—891no—
198#1444—892no—
199#1444—892no—
200#1442—892no—
201#1442—892no—
202#1436—894no—
203C89#1429811yescharacteristic, complemented, Sylow
204C22×C4⋊C4—894no—
205C22×C4⋊C4—891no—
206C22×C4⋊C4—891no—
207C22×C4⋊C4C2×D8911yescharacteristic
208C22×C4⋊C4C2×D8911yescharacteristic
209C22×C42—892no—
210C22×C42C2×D8911yescharacteristic
211#383—898no—
212#382—24no—
213C4×C4⋊C4—1784no—
214C2×#95—892no—
215C2×#95—892no—
216C2×#95—892no—
217C2×#95—894no—
218C2×#95—894no—
219C2×#95—894no—
220C2×#95—892no—
221C2×#95C2×D8912yes—
222C2×#95C2×D8912yes—
223C23×C4—892no—
224C23×C4—894no—
225C23×C4—892no—
226C23×C4C22×D8912yes—
227C23×C4C22×D8912yes—
228C23×C4C22×D8911yescharacteristic
229C23×C4C22×D8911yescharacteristic
230C23×C4C22×D8911yescharacteristic
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—3564no—
236C2×C4⋊C4—3564no—
237C2×C4⋊C4—1788no—
238C2×C4⋊C4—1784no—
239C2×C4⋊C4—1784no—
240C2×C4⋊C4—1782no—
241C2×C4⋊C4—1782no—
242C2×C42—3564no—
243C2×C42—1784no—
244C2×C42Dic17814yes—
245C2×C42C4×D8914yes—
246C2×C42C22⋊D8914yes—
247#95—24no—
248#95—24no—
249#95—898no—
250#95—898no—
251#95—898no—
252#95—3564no—
253#95—3564no—
254#95—1788no—
255#95—1784no—
256C24C23×D8911yescharacteristic
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—898no—
273C22×C4—894no—
274C22×C4—894no—
275C22×C4—1782no—
276C22×C4—1782no—
277C22×C4—1784no—
278C22×C4—1784no—
279C22×C4—1782no—
280C22×C4—1782no—
281C22×C4—1784no—
282C22×C4—1784no—
283C22×C4—1784no—
284C22×C4—1784no—
285C22×C4—1784no—
286C22×C4—1784no—
287C22×C4—1784no—
288C22×C4—1784no—
289C22×C4C2×C4×D8912yes—
290C22×C4C2×C4×D8912yes—
291C22×C4C2×C4×D8912yes—
292C22×C4C2×C22⋊D8912yes—
293C22×C4C2×C22⋊D8912yes—
294C22×C4C2×C22⋊D8912yes—
295C22×C4C2×Dic17812yes—
296C22×C4C2×Dic17812yes—
297C22×C4C2×Dic17812yes—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—35616no—
301C4⋊C4—1788no—
302C4⋊C4—1788no—
303C42—3568no—
304C42#13083553116yescomplemented
305C23C22×C4×D8911yescharacteristic
306C23C22×C22⋊D8911yescharacteristic
307C23C22×Dic17811yescharacteristic
308C23#41×D8912yes—
309C23#41×D8912yes—
310C23#1566805712yes—
311C23#1566805812yes—
312C23#1566805912yes—
313C23#1566806012yes—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—3562no—
336C2×C4—3562no—
337C2×C4—3562no—
338C2×C4—3562no—
339C2×C4—3564no—
340C2×C4—3564no—
341C2×C4—3564no—
342C2×C4—3564no—
343C2×C4—3564no—
344C2×C4—3564no—
345C2×C4—1788no—
346C2×C4—1788no—
347C2×C4—1788no—
348C2×C4—1788no—
349C2×C4—1788no—
350C2×C4—1784no—
351C2×C4—1784no—
352C2×C4C2×#1308355314yes—
353C2×C4C2×#1308355314yes—
354C2×C4C2×#1308355314yes—
355C22C22×#1308355311yescharacteristic
356C22#1915206612yes—
357C22#1915207812yes—
358C22#1915207912yes—
359C22#1915210511yescharacteristic
360C22#1915210511yescharacteristic
361C22#1915215112yes—
362C22#1915216211yescharacteristic
363C22#1915216511yescharacteristic
364C22#1915216911yescharacteristic
365C22#1915217211yescharacteristic
366C22#1915218214yes—
367C22#1915219614yes—
368C22#1915220714yes—
369C22#1915221214yes—
370C22#1915222612yes—
371C22#1915222912yes—
372C4—24no—
373C4—24no—
374C4—24no—
375C4—82no—
376C4—82no—
377C4—3568no—
378C4—3564no—
379C4—3564no—
380C2#2602273411yescharacteristic
381C2#2602287211yescharacteristic
382C2#2602287611yescharacteristic
383C2#2602302314yes—
384C2#2602610511yescharacteristic
385C2#2602611311yescharacteristic
386C2#2602612111yescharacteristic
387C2#2602612911yescharacteristic
388C2#2602643314yes—
subgroupG/H
center, FrattiniC24C23×D89
commutatorC23×C89C23×C4
FittingC2×#382×C89C2
socle, prime radical, cosocleC24×C89C24
hypercenterC2×#382D89
hypocenterC89#14298
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
13. #19151466C4semidirect
109. C42×C89C4⋊C4semidirect
203. C89#14298semidirect
304. C42#13083553semidirect

Maximal subgroups

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

subgroupindex
C2×#191514662
C2×C4×C4⋊Dic892
#260185162
#260185152
#260185112
#260185082
#260184832
C2×#382×C892
#1429889