Finite Group Databasebeta

← #36559538

Subgroups — #36559538

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
2C4×#15667971C211yesmaximal, characteristic, complemented
3#26021203C212yesmaximal, complemented
4#26021178C212yesmaximal, complemented
5#26020572C214yesmaximal, complemented
6#26020078C214yesmaximal, complemented
7C2×#432×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—
18C2×#15667971C2212yes—
19C2×#15667934C2218yes—
20C4×C4⋊Dic89C2212yes—
21C4×C4⋊Dic89C2212yes—
22C4×C22⋊Dic89—21no—
23C4×C22⋊Dic89—21no—
24#19151444—21no—
25C42×Dic89—21no—
26#19151361C2218yes—
27C22×C42×C89C2211yescharacteristic
28#432×C89—24no—
29C2×C4⋊C8×C89C2212yes—
30C2×C22⋊C8×C89C2212yes—
31C2×#97×C89C2212yes—
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—
40#15668012—22no—
41#15668012C2×C414yes—
42#15668009—22no—
43#15668009C2×C414yes—
44#15668000—28no—
45#15667986—28no—
46#15667986—24no—
47#15667986C2×C418yes—
48C4×D356—28no—
49C4×D356—24no—
50C4×D356C2×C418yes—
51#15667971—28no—
52C2×C4⋊Dic89—21no—
53C2×C4⋊Dic89C2312yes—
54C2×C4⋊Dic89C2314yes—
55C2×C22⋊Dic89—22no—
56C2×#13083551—22no—
57C2×#13083551—22no—
58C2×C4×Dic89—22no—
59C2×C4×Dic89—22no—
60C2×C4×Dic89—22no—
61#15667934—216no—
62#15667919—22no—
63#15667919—22no—
64C23×C4×C89D411yescharacteristic
65C23×C4×C89D411yescharacteristic
66C23×C4×C89C2311yescharacteristic
67C22×C8×C89C2314yes—
68C2×C42×C89C2×C414yes—
69C2×C42×C89D411yescharacteristic
70C2×C42×C89D411yescharacteristic
71C2×C42×C89D414yes—
72C2×C42×C89C2311yescharacteristic
73C2×C42×C89C2311yescharacteristic
74C4⋊C8×C89—24no—
75C22⋊C8×C89—24no—
76#97×C89—24no—
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—
91C22×Dic89—24no—
92C2×C4×D89—28no—
93C2×C4×D89—48no—
94C4⋊Dic89—24no—
95C4⋊Dic89—22no—
96C4⋊Dic89—24no—
97C4⋊Dic89C22×C414yes—
98C4⋊Dic89#4118yescomplemented
99C22⋊Dic89—28no—
100#13083551—28no—
101#13083551—28no—
102C4×Dic89—28no—
103C4×Dic89—44no—
104C4×Dic89—44no—
105C24×C89C2×D411yescharacteristic
106C22×C4×C89—24no—
107C22×C4×C89—24no—
108C22×C4×C89—42no—
109C22×C4×C89C22⋊C414yes—
110C22×C4×C89C22⋊C414yes—
111C22×C4×C89C22×C412yes—
112C22×C4×C89C2×D411yescharacteristic
113C22×C4×C89C2×D411yescharacteristic
114C22×C4×C89C2×D411yescharacteristic
115C22×C4×C89C2×D411yescharacteristic
116C22×C4×C89C2×D412yes—
117C22×C4×C89C2×D411yescharacteristic
118C22×C4×C89C2411yescharacteristic
119C2×C8×C89—24no—
120C2×C8×C89—28no—
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—44no—
144C2×C4×C89C2×C22⋊C412yes—
145C2×C4×C89C2×C22⋊C412yes—
146C2×C4×C89C2×C22⋊C412yes—
147C2×C4×C89C2×C22⋊C412yes—
148C2×C4×C89C2×C22⋊C412yes—
149C2×C4×C89AGL(1,ℤ/8)14yes—
150C2×C4×C89C23×C411yescharacteristic
151C2×C4×C89C22×D411yescharacteristic
152C2×C4×C89C2×#4114yes—
153C22⋊D89—416no—
154C2×Dic89—24no—
155C2×Dic89—28no—
156C2×Dic89—44no—
157C2×Dic89—44no—
158C2×Dic89—44no—
159D356—28no—
160D356—48no—
161D356—44no—
162D356—48no—
163C4×D89—416no—
164Dic178—48no—
165C8×C89—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#41814yes—
183C4×C89C4×#4112yes—
184C4×C89#53212yes—
185Dic89—48no—
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×#432D8911yescharacteristic, complemented
196C4×#117—891no—
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×#117—892no—
213C22×C42C2×D8911yescharacteristic
214#432—24no—
215C2×C4⋊C8C2×D8912yes—
216C2×#102—898no—
217C2×C22⋊C8C2×D8912yes—
218C2×#97C2×D8912yes—
219C4×C4⋊C4—892no—
220C4×C4⋊C4—892no—
221C4×C22⋊C4—1781no—
222C4×C22⋊C4—1781no—
223#376—1781no—
224C43—1781no—
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×C4D35611yescharacteristic
235C23×C4D35611yescharacteristic
236C23×C4C22×D8911yescharacteristic
237C22×C8C22×D8914yes—
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—1788no—
250C2×C4⋊C4—894no—
251C2×C4⋊C4—892no—
252C2×C4⋊C4—1781no—
253C2×C22⋊C4—1782no—
254C2×C22⋊C4—1782no—
255C2×C22⋊C4—1782no—
256C2×C42—1782no—
257C2×C42—1782no—
258C2×C42—1782no—
259C2×C42C4×D8914yes—
260C2×C42D35611yescharacteristic
261C2×C42D35611yescharacteristic
262C2×C42C22⋊D8914yes—
263C2×C42C22×D8911yescharacteristic
264C2×C42C22×D8911yescharacteristic
265C4⋊C8—24no—
266#102—17816no—
267C22⋊C8—24no—
268#97—24no—
269#95—1782no—
270#95—1782no—
271C24—1782no—
272C24C2×D35611yescharacteristic
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—42no—
288C22×C4—3568no—
289C22×C4—1784no—
290C22×C4—1782no—
291C22×C4—1788no—
292C22×C4#1308355114yes—
293C22×C4#1308355114yes—
294C22×C4C2×C4×D8912yes—
295C22×C4C2×C22⋊D8912yes—
296C22×C4C2×D35611yescharacteristic
297C22×C4C2×D35611yescharacteristic
298C22×C4C2×D35611yescharacteristic
299C22×C4C2×D35611yescharacteristic
300C22×C4C2×D35611yescharacteristic
301C22×C4C23×D8911yescharacteristic
302C2×C8—24no—
303C2×C8—28no—
304C4⋊C4—894no—
305C4⋊C4—898no—
306C4⋊C4—1782no—
307C4⋊C4—1784no—
308C4⋊C4—1784no—
309C22⋊C4—1788no—
310C22⋊C4—1788no—
311C22⋊C4—1788no—
312C42—3564no—
313C42—3564no—
314C42—1788no—
315C42#1308355114yes—
316C42#1308355114yes—
317C42C2×C4×D8912yes—
318C42C2×C4×D8912yes—
319C42C2×C22⋊D8912yes—
320C42C2×C22⋊D8912yes—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—3564no—
325C23—3564no—
326C23—3564no—
327C23—1784no—
328C23C2×#1308355112yes—
329C23C22×D35611yescharacteristic
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—44no—
344C2×C4—35616no—
345C2×C4—3564no—
346C2×C4—3564no—
347C2×C4—3564no—
348C2×C4—1784no—
349C2×C4—1788no—
350C2×C4C2×#1308355112yes—
351C2×C4C2×#1308355112yes—
352C2×C4C2×#1308355112yes—
353C2×C4C2×#1308355112yes—
354C2×C4C2×#1308355112yes—
355C2×C4#1566804014yes—
356C2×C4C22×C4×D8911yescharacteristic
357C2×C4C22×C22⋊D8911yescharacteristic
358C2×C4#41×D8914yes—
359C8—44no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3564no—
365C22—3564no—
366C22—3564no—
367C22—7128no—
368C22C22×#1308355111yescharacteristic
369C22#1915211012yes—
370C22C2×#1566804012yes—
371C22#1915239812yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—3568no—
377C4—3564no—
378C4#1915160414yes—
379C4#1915206612yes—
380C4#1915214412yes—
381C2—42no—
382C2—7124no—
383C2#2602274011yescharacteristic
384C2C2×#1915160411yescharacteristic
385C2#2602877211yescharacteristic
386C2#2603040414yes—
subgroupG/H
centerC23C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC2×#432×C89C2
socleC23×C89C22×D4
prime radicalC2×#19152148C2
cosocleC22×C4×C89C24
hypercenterC2×#432D89
hypocenterC89#21393
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
2. C4×#15667971C2semidirect
3. #26021203C2semidirect
4. #26021178C2semidirect
5. #26020572C2semidirect
6. #26020078C2semidirect
7. C2×#432×C89C2semidirect
10. #19152148C4semidirect
98. C4⋊Dic89#41semidirect
195. C2×#432D89semidirect
201. C89#21393semidirect

Maximal subgroups

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

subgroupindex
C2×#191521482
C4×#156679712
#260212032
#260211782
#260205722
#260200782
C2×#432×C892
#2139389