Finite Group Databasebeta

← #36477140

Subgroups — #36477140

Order 22784.

2230 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151466C211yesmaximal, characteristic, complemented
2C2×#19151444C211yesmaximal, characteristic, complemented
3#26018520C212yesmaximal, complemented
4#26018493C212yesmaximal
5#26018489C214yesmaximal, complemented
6#26018475C214yesmaximal, complemented
7C22×C4×D4×C89C211yesmaximal, characteristic
8C22×C22⋊Dic89C2218yes—
9C22×C4×Dic89C2212yes—
10#19151466—24no—
11#19151444—24no—
12C2×#15667919C2212yes—
13C2×#15667919C2212yes—
14C2×#15667919C2212yes—
15C2×#15667919C2212yes—
16C2×#15667919C2212yes—
17C2×#15667919C2218yes—
18C23×D4×C89C2211yescharacteristic
19C24×C4×C89C2212yes—
20C2×C4×D4×C89—24no—
21C2×C4×D4×C89—24no—
22C2×C4×D4×C89C418yescomplemented
23C22×C4⋊C4×C89C2211yescharacteristic
24C22×C22⋊C4×C89C2212yes—
25C22×C42×C89C2211yescharacteristic
26C23×Dic89C2314yes—
27C23×Dic89C2314yes—
28C2×C22⋊Dic89—216no—
29C2×C22⋊Dic89—216no—
30C2×C22⋊Dic89—48no—
31C2×C4×Dic89—28no—
32C2×C4×Dic89—24no—
33#15667919—24no—
34#15667919—24no—
35#15667919—24no—
36#15667919—24no—
37#15667919—24no—
38#15667919—48no—
39C25×C89C2312yes—
40C22×D4×C89—28no—
41C22×D4×C89—22no—
42C22×D4×C89—22no—
43C22×D4×C89C2×C414yes—
44C23×C4×C89—24no—
45C23×C4×C89—24no—
46C23×C4×C89—48no—
47C23×C4×C89C2×C418yes—
48C23×C4×C89C2312yes—
49C23×C4×C89C2311yescharacteristic
50C23×C4×C89C2311yescharacteristic
51C23×C4×C89C2311yescharacteristic
52C4×D4×C89—232no—
53C2×C4⋊C4×C89—22no—
54C2×C4⋊C4×C89—22no—
55C2×C4⋊C4×C89C2×C414yes—
56C2×C22⋊C4×C89—24no—
57C2×C22⋊C4×C89—24no—
58C2×C22⋊C4×C89C2×C418yes—
59C2×C42×C89C2×C414yes—
60C2×C42×C89D414yes—
61C2×C42×C89D414yes—
62C22×Dic89—24no—
63C22×Dic89—24no—
64C22×Dic89—28no—
65C22×Dic89—28no—
66C22×Dic89—24no—
67C22×Dic89—28no—
68C22×Dic89—28no—
69C22×Dic89—24no—
70C22×Dic89—24no—
71C22×Dic89#4118yes—
72C22⋊Dic89—432no—
73C4×Dic89—216no—
74C24×C89—22no—
75C24×C89—22no—
76C24×C89—24no—
77C24×C89—28no—
78C24×C89—28no—
79C24×C89C22×C414yes—
80C24×C89#4118yes—
81C24×C89C2411yescharacteristic
82C2×D4×C89—216no—
83C2×D4×C89—216no—
84C2×D4×C89—216no—
85C2×D4×C89—28no—
86C22×C4×C89—24no—
87C22×C4×C89—24no—
88C22×C4×C89—24no—
89C22×C4×C89—22no—
90C22×C4×C89—22no—
91C22×C4×C89—28no—
92C22×C4×C89—216no—
93C22×C4×C89—416no—
94C22×C4×C89—416no—
95C22×C4×C89—416no—
96C22×C4×C89C22×C414yes—
97C22×C4×C89C22×C412yes—
98C22×C4×C89C22×C412yes—
99C22×C4×C89C22×C412yes—
100C22×C4×C89C2×D412yes—
101C22×C4×C89C2×D412yes—
102C22×C4×C89C2×D412yes—
103C22×C4×C89C2×D412yes—
104C22×C4×C89C2×D412yes—
105C22×C4×C89C2×D412yes—
106C4⋊C4×C89—28no—
107C22⋊C4×C89—216no—
108C42×C89C22⋊C4116yescomplemented
109C23×C89—24no—
110C23×C89—24no—
111C23×C89—28no—
112C23×C89—28no—
113C23×C89—28no—
114C23×C89—28no—
115C23×C89—216no—
116C23×C89—216no—
117C23×C89—44no—
118C23×C89—44no—
119C23×C89—44no—
120C23×C89—44no—
121C23×C89—48no—
122C23×C89—48no—
123C23×C89#11718yes—
124C23×C89C23×C411yescharacteristic
125C23×C89C22×D411yescharacteristic
126C23×C89C22×D411yescharacteristic
127C23×C89C2×#4114yes—
128C23×C89C2×#4114yes—
129C23×C89H(5,𝔽2)14yes—
130D4×C89—232no—
131C2×C4×C89—24no—
132C2×C4×C89—24no—
133C2×C4×C89—24no—
134C2×C4×C89—24no—
135C2×C4×C89—24no—
136C2×C4×C89—24no—
137C2×C4×C89—24no—
138C2×C4×C89—24no—
139C2×C4×C89—24no—
140C2×C4×C89—24no—
141C2×C4×C89—28no—
142C2×C4×C89—432no—
143C2×C4×C89—44no—
144C2×C4×C89—44no—
145C2×C4×C89C2×C22⋊C414yes—
146C2×C4×C89C2×C22⋊C414yes—
147C2×C4×C89C2×C22⋊C414yes—
148C2×Dic89—28no—
149C2×Dic89—216no—
150C2×Dic89—44no—
151C2×Dic89—44no—
152C2×Dic89—48no—
153C2×Dic89—44no—
154C2×Dic89—44no—
155C2×Dic89—44no—
156C2×Dic89—48no—
157C2×Dic89—48no—
158C22×C89—216no—
159C22×C89—28no—
160C22×C89—44no—
161C22×C89—416no—
162C22×C89—416no—
163C22×C89—48no—
164C22×C89—44no—
165C22×C89C22×C22⋊C411yescharacteristic
166C22×C89C2×#11712yes—
167C22×C89C4×#4112yes—
168C22×C89#51812yes—
169C22×C89#52512yes—
170C22×C89#52512yes—
171C22×C89#52512yes—
172C22×C89#53212yes—
173C22×C89#53512yes—
174C22×C89#53512yes—
175C22×C89#55118yes—
176C22×C89#55318yes—
177C4×C89—24no—
178C4×C89—24no—
179C4×C89—24no—
180C4×C89—44no—
181Dic89—48no—
182Dic89—84no—
183#13959—891nomaximal, Sylow
184C2×C89—48no—
185C2×C89#228011yescharacteristic
186C2×C89#229711yescharacteristic
187C2×C89#230911yescharacteristic
188C2×C89#235314yes—
189C2×C89#259814yes—
190C2×C89#260314yes—
191C22×C4×D4D8911yescharacteristic
192C2×#383—891no—
193C2×#376—891no—
194#1445—892no—
195#1438—892no—
196#1437—894no—
197C22⋊#95—894no—
198C89#1395911yescharacteristic, complemented, Sylow
199C23×D4C2×D8911yescharacteristic
200C24×C4C2×D8912yes—
201C2×C4×D4—24no—
202C2×C4×D4—24no—
203C2×C4×D4Dic8918yescomplemented
204C22×C4⋊C4C2×D8911yescharacteristic
205C22×C22⋊C4—898no—
206C22×C22⋊C4C2×D8912yes—
207C22×C42—892no—
208C22×C42C2×D8911yescharacteristic
209#383—1784no—
210#376—1784no—
211C2×#95—898no—
212C2×#95—892no—
213C2×#95—892no—
214C2×#95—892no—
215C2×#95—892no—
216C2×#95—892no—
217C25C22×D8912yes—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×Dic8914yes—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—48no—
225C23×C4—894no—
226C23×C4—894no—
227C23×C4C2×Dic8918yes—
228C23×C4C22×D8912yes—
229C23×C4C22×D8911yescharacteristic
230C23×C4C22×D8911yescharacteristic
231C23×C4C22×D8911yescharacteristic
232C4×D4—232no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C2×Dic8914yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—3568no—
239C2×C22⋊C4—17816no—
240C2×C22⋊C4—17816no—
241C2×C22⋊C4C2×Dic8918yes—
242C2×C42—1788no—
243C2×C42—1784no—
244C2×C42C2×Dic8914yes—
245C2×C42C22⋊D8914yes—
246C2×C42C22⋊D8914yes—
247#95—3568no—
248#95—1784no—
249#95—1784no—
250#95—1784no—
251#95—1784no—
252#95—1784no—
253C24—22no—
254C24—22no—
255C24—28no—
256C24—24no—
257C24—28no—
258C24C22×Dic8914yes—
259C24#1308357418yes—
260C24C23×D8911yescharacteristic
261C2×D4—216no—
262C2×D4—216no—
263C2×D4—216no—
264C2×D4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—216no—
272C22×C4—416no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—898no—
276C22×C4—1784no—
277C22×C4—1788no—
278C22×C4—1788no—
279C22×C4—1784no—
280C22×C4—1784no—
281C22×C4—1784no—
282C22×C4—1784no—
283C22×C4—1788no—
284C22×C4—1788no—
285C22×C4C22×Dic8914yes—
286C22×C4C22×Dic8912yes—
287C22×C4C22×Dic8912yes—
288C22×C4C22×Dic8912yes—
289C22×C4C2×C22⋊D8912yes—
290C22×C4C2×C22⋊D8912yes—
291C22×C4C2×C22⋊D8912yes—
292C22×C4C2×C22⋊D8912yes—
293C22×C4C2×C22⋊D8912yes—
294C22×C4C2×C22⋊D8912yes—
295C4⋊C4—28no—
296C22⋊C4—216no—
297C22⋊C4—35632no—
298C42—17816no—
299C42C22⋊Dic89116yescomplemented
300C23—216no—
301C23—216no—
302C23—24no—
303C23—24no—
304C23—28no—
305C23—28no—
306C23—28no—
307C23—28no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—48no—
312C23—48no—
313C23—44no—
314C23#1566797118yes—
315C23C23×Dic8911yescharacteristic
316C23C22×C22⋊D8911yescharacteristic
317C23C22×C22⋊D8911yescharacteristic
318C23#41×D8914yes—
319C23C2×#1308357414yes—
320C23#1566805714yes—
321D4—232no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—432no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—3564no—
337C2×C4—3568no—
338C2×C4—3564no—
339C2×C4—3564no—
340C2×C4—3564no—
341C2×C4—3568no—
342C2×C4—3568no—
343C2×C4—3564no—
344C2×C4—17816no—
345C2×C4—1788no—
346C2×C4C2×C22⋊Dic8914yes—
347C2×C4C2×C22⋊Dic8914yes—
348C2×C4C2×C22⋊Dic8914yes—
349C22—216no—
350C22—28no—
351C22—48no—
352C22—416no—
353C22—44no—
354C22—416no—
355C22—44no—
356C22C22×C22⋊Dic8911yescharacteristic
357C22C2×#1566797112yes—
358C22#41×Dic8912yes—
359C22#1915207312yes—
360C22#1915210512yes—
361C22#1915210912yes—
362C22#1915210912yes—
363C22#1915214412yes—
364C22#1915216512yes—
365C22#1915216512yes—
366C22#1915227718yes—
367C22#1915229718yes—
368C4—24no—
369C4—24no—
370C4—24no—
371C4—44no—
372C4—3568no—
373C4—7124no—
374C2—48no—
375C2#2602263511yescharacteristic
376C2#2602274311yescharacteristic
377C2#2602283411yescharacteristic
378C2#2602319214yes—
379C2#2602581214yes—
380C2#2602587314yes—
subgroupG/H
center, FrattiniC24C23×D89
commutatorC23×C89C23×C4
FittingC22×C4×D4×C89C2
socle, cosocleC24×C89C24
prime radicalC23×D4×C89C22
hypercenterC22×C4×D4D89
hypocenterC89#13959
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19151466C2semidirect
2. C2×#19151444C2semidirect
3. #26018520C2semidirect
5. #26018489C2semidirect
6. #26018475C2semidirect
22. C2×C4×D4×C89C4semidirect
108. C42×C89C22⋊C4semidirect
198. C89#13959semidirect
203. C2×C4×D4Dic89semidirect
299. C42C22⋊Dic89semidirect

Maximal subgroups

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

subgroupindex
C2×#191514662
C2×#191514442
#260185202
#260184932
#260184892
#260184752
C22×C4×D4×C892
#1395989