Finite Group Databasebeta

← #36559540

Subgroups — #36559540

Order 22784.

1430 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152148C211yesmaximal, characteristic
2C2×#19151655C211yesmaximal, characteristic, complemented
3#26021204C212yesmaximal, complemented
4#26021179C212yesmaximal, complemented
5#26020574C214yesmaximal, complemented
6#26020080C214yesmaximal, complemented
7C4×#117×C89C211yesmaximal, characteristic, complemented
8C22×#13083574—21no—
9#19152148—24no—
10#19152148C418yescomplemented
11C2×#15668012C2211yescharacteristic
12C2×#15668009C2211yescharacteristic
13C2×#15668000—21no—
14C2×#15667986—21no—
15C2×#15667986C2212yes—
16C2×C4×D356—21no—
17C2×C4×D356C2212yes—
18#19151655—24no—
19C2×C4⋊#10623494C2212yes—
20C2×#15667932C2218yes—
21C2×C22⋊#10623494C2212yes—
22C2×#15667923C2212yes—
23#19151360C2218yes—
24C2×#117×C89C2212yes—
25C22×C42×C89C2211yescharacteristic
26C4×C4⋊C4×C89C2212yes—
27C4×C4⋊C4×C89C2212yes—
28C4×C22⋊C4×C89—21no—
29C4×C22⋊C4×C89—21no—
30#376×C89—21no—
31C43×C89—21no—
32C2×#13083574—28no—
33C2×#13083574—48no—
34C22×Dic178—21no—
35C22×D356—21no—
36C22×D356—21no—
37C22×D356C2312yes—
38C22×C22⋊D89—22no—
39C22×C4×D89—22no—
40C22×#10623494C2314yes—
41#15668012—22no—
42#15668012C2×C414yes—
43#15668009—22no—
44#15668009C2×C414yes—
45#15668000—28no—
46#15667986—28no—
47#15667986—24no—
48#15667986C2×C418yes—
49C4×D356—28no—
50C4×D356—24no—
51C4×D356C2×C418yes—
52C2×C4⋊Dic89—21no—
53C2×C4⋊Dic89C2312yes—
54C2×#13083551—22no—
55C2×#13083551—22no—
56C4⋊#10623494—24no—
57#15667932—216no—
58C22⋊#10623494—24no—
59#15667923—24no—
60C23×C4×C89D411yescharacteristic
61C23×C4×C89D411yescharacteristic
62C23×C4×C89C2311yescharacteristic
63#117×C89—28no—
64C2×C4⋊C4×C89C2314yes—
65C2×C22⋊C4×C89—22no—
66C2×C42×C89—22no—
67C2×C42×C89—22no—
68C2×C42×C89—22no—
69C2×C42×C89C2×C414yes—
70C2×C42×C89D414yes—
71C2×C42×C89D411yescharacteristic
72C2×C42×C89D411yescharacteristic
73C2×C42×C89C2311yescharacteristic
74C2×C42×C89C2311yescharacteristic
75#95×C89—22no—
76#95×C89—22no—
77C23×D89—22no—
78#13083574—432no—
79C2×Dic178—24no—
80C2×Dic178—44no—
81C2×D356—28no—
82C2×D356—24no—
83C2×D356—24no—
84C2×D356—22no—
85C2×D356—44no—
86C2×D356—44no—
87C2×D356C22×C414yes—
88C2×C22⋊D89—28no—
89C2×C22⋊D89—48no—
90C22×Dic89—22no—
91C2×C4×D89—28no—
92C2×C4×D89—48no—
93C2×#10623494—24no—
94C2×#10623494—28no—
95C4⋊Dic89—24no—
96C4⋊Dic89—22no—
97C4⋊Dic89C22×C414yes—
98#13083551—28no—
99#13083551—28no—
100C24×C89C2×D411yescharacteristic
101C22×C4×C89—24no—
102C22×C4×C89—24no—
103C22×C4×C89—24no—
104C22×C4×C89—42no—
105C22×C4×C89C22⋊C414yes—
106C22×C4×C89C22⋊C414yes—
107C22×C4×C89C22×C412yes—
108C22×C4×C89C2×D411yescharacteristic
109C22×C4×C89C2×D411yescharacteristic
110C22×C4×C89C2×D411yescharacteristic
111C22×C4×C89C2×D411yescharacteristic
112C22×C4×C89C2×D412yes—
113C22×C4×C89C2×D411yescharacteristic
114C22×C4×C89C2411yescharacteristic
115C4⋊C4×C89—24no—
116C4⋊C4×C89#4118yescomplemented
117C22⋊C4×C89—28no—
118C42×C89—28no—
119C42×C89—44no—
120C42×C89—44no—
121C42×C89C22⋊C414yes—
122C42×C89C22⋊C414yes—
123C42×C89C22×C412yes—
124C42×C89C22×C412yes—
125C42×C89C2×D412yes—
126C42×C89C2×D412yes—
127C23×C89—22no—
128C23×C89—22no—
129C23×C89—22no—
130C23×C89C2×C22⋊C412yes—
131C23×C89C22×D411yescharacteristic
132C22×D89—24no—
133C22×D89—44no—
134C22×D89—44no—
135C22×D89—44no—
136C2×C4×C89—22no—
137C2×C4×C89—22no—
138C2×C4×C89—22no—
139C2×C4×C89—22no—
140C2×C4×C89—24no—
141C2×C4×C89—24no—
142C2×C4×C89—28no—
143C2×C4×C89—28no—
144C2×C4×C89—44no—
145C2×C4×C89—44no—
146C2×C4×C89—44no—
147C2×C4×C89C2×C22⋊C412yes—
148C2×C4×C89C2×C22⋊C412yes—
149C2×C4×C89C2×C22⋊C412yes—
150C2×C4×C89C2×C22⋊C412yes—
151C2×C4×C89C2×C22⋊C412yes—
152C2×C4×C89AGL(1,ℤ/8)14yes—
153C2×C4×C89C23×C411yescharacteristic
154C2×C4×C89C22×D411yescharacteristic
155C2×C4×C89C2×#4114yes—
156C22⋊D89—416no—
157C2×Dic89—24no—
158C2×Dic89—44no—
159D356—28no—
160D356—48no—
161D356—44no—
162D356—48no—
163C4×D89—416no—
164Dic178—48no—
165#10623494—44no—
166C22×C89—22no—
167C22×C89—22no—
168C22×C89—22no—
169C22×C89—44no—
170C22×C89C22×C22⋊C411yescharacteristic
171C22×C89#52512yes—
172C22×C89C2×AGL(1,ℤ/8)12yes—
173C22×C89#57512yes—
174C2×D89—44no—
175C2×D89—44no—
176C2×D89—44no—
177C2×D89—88no—
178C4×C89—22no—
179C4×C89—22no—
180C4×C89—22no—
181C4×C89—22no—
182C4×C89—48no—
183C4×C89#41814yes—
184C4×C89C4×#4112yes—
185C4×C89#53212yes—
186Dic89—44no—
187#21393—891nomaximal, Sylow
188C2×C89—42no—
189C2×C89#229711yescharacteristic
190C2×C89C2×#41811yescharacteristic
191C2×C89#289611yescharacteristic
192C2×C89#311514yes—
193D89—84no—
194C2×#532—891no—
195C2×#432—891no—
196C4×#117D8911yescharacteristic, complemented
197#1988—892no—
198#1982—892no—
199#1875—894no—
200#1762—894no—
201C89#2139311yescharacteristic, complemented, Sylow
202C22×#41—1781no—
203#532—898no—
204#532—1784no—
205C2×C4⋊Q8—891no—
206C2×#127—891no—
207C2×#124—1781no—
208C2×#121—892no—
209C2×#121—1781no—
210C2×C4×D4—892no—
211C2×C4×D4—1781no—
212C2×#117C2×D8912yes—
213C22×C42C2×D8911yescharacteristic
214#432—1784no—
215C2×C4⋊C8—892no—
216C2×#102—898no—
217C2×C22⋊C8—892no—
218C2×#97—892no—
219C4×C4⋊C4C2×D8912yes—
220C4×C4⋊C4C2×D8912yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223#376—21no—
224C43—21no—
225#340—898no—
226C2×#41—3568no—
227C2×#41—1788no—
228C22×Q8—1781no—
229C22×D4—892no—
230C22×D4—1782no—
231C22×D4—1781no—
232C22×D4—1781no—
233C23×C4—1782no—
234C23×C4C22⋊D8911yescharacteristic
235C23×C4C22⋊D8911yescharacteristic
236C23×C4C22×D8911yescharacteristic
237C22×C8—894no—
238C4⋊Q8—894no—
239C4⋊Q8—1782no—
240#127—894no—
241#127—1782no—
242#124—1788no—
243#121—898no—
244#121—1788no—
245#121—1784no—
246C4×D4—898no—
247C4×D4—1788no—
248C4×D4—1784no—
249#117—28no—
250C2×C4⋊C4—892no—
251C2×C4⋊C4—1781no—
252C2×C4⋊C4C22×D8914yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—1782no—
255C2×C22⋊C4—1782no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42C4×D8914yes—
260C2×C42D35614yes—
261C2×C42C22⋊D8911yescharacteristic
262C2×C42C22⋊D8911yescharacteristic
263C2×C42C22×D8911yescharacteristic
264C2×C42C22×D8911yescharacteristic
265C4⋊C8—1784no—
266#102—17816no—
267C22⋊C8—1784no—
268#97—1784no—
269#95—22no—
270#95—22no—
271C24—1782no—
272C24C2×C22⋊D8911yescharacteristic
273#41—35632no—
274C2×Q8—3564no—
275C2×Q8—1784no—
276C2×D4—894no—
277C2×D4—3564no—
278C2×D4—3568no—
279C2×D4—3564no—
280C2×D4—1788no—
281C2×D4—1782no—
282C2×D4—1788no—
283C2×D4—1784no—
284C2×D4—1784no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—3568no—
290C22×C4—1782no—
291C22×C4—1788no—
292C22×C4#1308355114yes—
293C22×C4#1308355114yes—
294C22×C4C2×C4×D8912yes—
295C22×C4C2×C22⋊D8911yescharacteristic
296C22×C4C2×C22⋊D8911yescharacteristic
297C22×C4C2×C22⋊D8911yescharacteristic
298C22×C4C2×C22⋊D8911yescharacteristic
299C22×C4C2×C22⋊D8911yescharacteristic
300C22×C4C2×D35612yes—
301C22×C4C23×D8911yescharacteristic
302C2×C8—1784no—
303C2×C8—1788no—
304C4⋊C4—24no—
305C4⋊C4—894no—
306C4⋊C4—1782no—
307C4⋊C4—1784no—
308C4⋊C4#1308357418yescomplemented
309C22⋊C4—28no—
310C22⋊C4—1788no—
311C22⋊C4—1788no—
312C42—28no—
313C42—44no—
314C42—44no—
315C42#1308355114yes—
316C42#1308355114yes—
317C42C2×C4×D8912yes—
318C42C2×C4×D8912yes—
319C42C2×D35612yes—
320C42C2×D35612yes—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—3564no—
325C23—3564no—
326C23—3564no—
327C23—1784no—
328C23C2×#1308355112yes—
329C23C22×C22⋊D8911yescharacteristic
330Q8—3568no—
331D4—3568no—
332D4—35616no—
333D4—3564no—
334D4—3568no—
335D4—1788no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—35616no—
348C2×C4—3564no—
349C2×C4—1784no—
350C2×C4C2×#1308355112yes—
351C2×C4C2×#1308355112yes—
352C2×C4C2×#1308355112yes—
353C2×C4C2×#1308355112yes—
354C2×C4C2×#1308355112yes—
355C2×C4#1566803714yes—
356C2×C4C22×C4×D8911yescharacteristic
357C2×C4C22×D35611yescharacteristic
358C2×C4C2×#1308357414yes—
359C8—3564no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3564no—
365C22—3564no—
366C22—3564no—
367C22—7128no—
368C22C22×#1308355111yescharacteristic
369C22#1915210912yes—
370C22C2×#1566803712yes—
371C22#1915239612yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—48no—
377C4—3564no—
378C4#1915160214yes—
379C4C4×#1308357412yes—
380C4#1915214812yes—
381C2—42no—
382C2—7124no—
383C2#2602274211yescharacteristic
384C2C2×#1915160211yescharacteristic
385C2#2602876911yescharacteristic
386C2#2603040314yes—
subgroupG/H
centerC23C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC4×#117×C89C2
socleC23×C89C22×D4
prime radicalC2×#19152148C2
cosocleC22×C4×C89C24
hypercenterC4×#117D89
hypocenterC89#21393
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. C2×#19151655C2semidirect
3. #26021204C2semidirect
4. #26021179C2semidirect
5. #26020574C2semidirect
6. #26020080C2semidirect
7. C4×#117×C89C2semidirect
10. #19152148C4semidirect
116. C4⋊C4×C89#41semidirect
196. C4×#117D89semidirect
201. C89#21393semidirect
308. C4⋊C4#13083574semidirect

Maximal subgroups

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

subgroupindex
C2×#191521482
C2×#191516552
#260212042
#260211792
#260205742
#260200802
C4×#117×C892
#2139389