Finite Group Databasebeta

← #36564658

Subgroups — #36564658

Order 22784.

1258 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151562C211yesmaximal, characteristic
2C22×C22⋊#10623494C211yesmaximal, characteristic
3#26020437C214yesmaximal, complemented
4#26020432C214yesmaximal, complemented
5#26020232C214yesmaximal, complemented
6C2×C4×C22⋊C4×C89C211yesmaximal, characteristic
7C22×#13083558C2212yes—
8C23×#10623494C2212yes—
9C2×C4⋊#10623494C2218yescomplemented
10#19151562—24no—
11C2×C22⋊#10623494—24no—
12C2×C22⋊#10623494C2212yes—
13C2×C22⋊#10623494C2212yes—
14C2×C22⋊#10623494C2212yes—
15C2×C22⋊#10623494C2212yes—
16#19151353C2218yescomplemented
17C24×C4×C89C2211yescharacteristic
18C22×C22⋊C4×C89C412yes—
19C22×C42×C89C2212yes—
20C4×C22⋊C4×C89C411yescharacteristic
21C4×C22⋊C4×C89C412yes—
22C4×C22⋊C4×C89C411yescharacteristic
23C4×C22⋊C4×C89C2212yes—
24C4×C22⋊C4×C89C2212yes—
25C2×#95×C89C412yes—
26C2×#13083558—24no—
27C2×#13083558—44no—
28C22×#10623494—24no—
29C22×#10623494—44no—
30C22×#10623494C2314yes—
31C22×#10623494C2314yes—
32C4⋊#10623494—48no—
33C22⋊#10623494—24no—
34C22⋊#10623494—24no—
35C22⋊#10623494—24no—
36C22⋊#10623494—24no—
37C25×C89C2×C411yescharacteristic
38C23×C4×C89—44no—
39C23×C4×C89C2×C414yes—
40C23×C4×C89C2×C411yescharacteristic
41C23×C4×C89C2×C411yescharacteristic
42C23×C4×C89C2×C411yescharacteristic
43C23×C4×C89D414yes—
44C23×C4×C89Q814yes—
45C23×C4×C89C2311yescharacteristic
46C23×C4×C89C2311yescharacteristic
47C23×C4×C89C2311yescharacteristic
48C2×C22⋊C4×C89—28no—
49C2×C22⋊C4×C89C2×C412yes—
50C2×C22⋊C4×C89C2×C414yes—
51C2×C22⋊C4×C89C2×C412yes—
52C2×C42×C89—44no—
53C2×C42×C89C2×C412yes—
54C2×C42×C89C2×C412yes—
55C2×C42×C89C2314yes—
56#95×C89C2×C412yes—
57#95×C89C2×C414yes—
58#95×C89C2×C412yes—
59#13083558—48no—
60C2×#10623494—24no—
61C2×#10623494—24no—
62C2×#10623494—28no—
63C2×#10623494—28no—
64C2×#10623494—48no—
65C24×C89—24no—
66C24×C89—24no—
67C24×C89—42no—
68C24×C89C4⋊C414yes—
69C24×C89C22×C411yescharacteristic
70C24×C89C22×C411yescharacteristic
71C24×C89C22×C411yescharacteristic
72C22×C4×C89—24no—
73C22×C4×C89—24no—
74C22×C4×C89—24no—
75C22×C4×C89—28no—
76C22×C4×C89—28no—
77C22×C4×C89—48no—
78C22×C4×C89—48no—
79C22×C4×C89—44no—
80C22×C4×C89—42no—
81C22×C4×C89—42no—
82C22×C4×C89—42no—
83C22×C4×C89C4⋊C414yes—
84C22×C4×C89C4⋊C414yes—
85C22×C4×C89C4⋊C414yes—
86C22×C4×C89C22×C414yes—
87C22×C4×C89C22×C411yescharacteristic
88C22×C4×C89C22×C411yescharacteristic
89C22×C4×C89C22×C411yescharacteristic
90C22×C4×C89C22×C414yes—
91C22×C4×C89C2×D412yes—
92C22×C4×C89C2×D412yes—
93C22×C4×C89C2×D412yes—
94C22×C4×C89C2×Q812yes—
95C22×C4×C89C2×Q812yes—
96C22×C4×C89C2×Q812yes—
97C22×C4×C89C2411yescharacteristic
98C22⋊C4×C89—24no—
99C22⋊C4×C89—28no—
100C22⋊C4×C89M1618yescomplemented
101C42×C89—28no—
102C42×C89—42no—
103C42×C89—42no—
104C23×C89—22no—
105C23×C89—22no—
106C23×C89—22no—
107C23×C89—22no—
108C23×C89—22no—
109C23×C89—22no—
110C23×C89—22no—
111C23×C89—28no—
112C23×C89—28no—
113C23×C89—44no—
114C23×C89—416no—
115C23×C89C2×C4⋊C412yes—
116C23×C89C2×C4⋊C412yes—
117C23×C89C2×C4⋊C412yes—
118C23×C89C2×M1614yes—
119C23×C89#13114yes—
120C23×C89C23×C411yescharacteristic
121C2×C4×C89—24no—
122C2×C4×C89—22no—
123C2×C4×C89—22no—
124C2×C4×C89—22no—
125C2×C4×C89—24no—
126C2×C4×C89—24no—
127C2×C4×C89—24no—
128C2×C4×C89—24no—
129C2×C4×C89—24no—
130C2×C4×C89—24no—
131C2×C4×C89—24no—
132C2×C4×C89—24no—
133C2×C4×C89—24no—
134C2×C4×C89—24no—
135C2×C4×C89—44no—
136C2×C4×C89—44no—
137C2×C4×C89—44no—
138C2×C4×C89—416no—
139C2×C4×C89C2×C4⋊C412yes—
140C2×C4×C89C2×C4⋊C412yes—
141C2×C4×C89C2×C4⋊C412yes—
142C2×C4×C89C2×M1618yes—
143C2×C4×C89C22×D411yescharacteristic
144C2×C4×C89C22×Q811yescharacteristic
145C2×C4×C89H(5,𝔽2)14yes—
146#10623494—44no—
147#10623494—44no—
148C22×C89—22no—
149C22×C89—22no—
150C22×C89—22no—
151C22×C89—22no—
152C22×C89—28no—
153C22×C89—28no—
154C22×C89—22no—
155C22×C89—22no—
156C22×C89—22no—
157C22×C89—416no—
158C22×C89—44no—
159C22×C89#42314yes—
160C22×C89#42414yes—
161C22×C89C22×C4⋊C411yescharacteristic
162C22×C89#51812yes—
163C22×C89C22×M1612yes—
164C22×C89C2×#13112yes—
165C4×C89—22no—
166C4×C89—22no—
167C4×C89—22no—
168C4×C89—28no—
169C4×C89—44no—
170C4×C89#53412yes—
171C4×C89#54312yes—
172#21932—891nomaximal, Sylow
173C2×C89—24no—
174C2×C89—24no—
175C2×C89—42no—
176C2×C89#231511yescharacteristic
177C2×C89C2×#42311yescharacteristic
178C2×C89C2×#42411yescharacteristic
179C2×C89#297414yes—
180C2×#407—891no—
181C22×C22⋊C8—891no—
182C2×C4×C22⋊C4D8911yescharacteristic
183#1851—894no—
184#1850—894no—
185#1797—894no—
186C89#2193211yescharacteristic, complemented, Sylow
187C24×C4C2×D8911yescharacteristic
188C22×M16—892no—
189C23×C8—892no—
190C22×C22⋊C4Dic8912yes—
191C22×C42C2×D8912yes—
192C2×C4⋊C8—898no—
193#407—1784no—
194C2×C22⋊C8—892no—
195C2×C22⋊C8—892no—
196C2×C22⋊C8—892no—
197C2×C22⋊C8—892no—
198C2×C22⋊C8—1784no—
199C4×C22⋊C4Dic8911yescharacteristic
200C4×C22⋊C4Dic8912yes—
201C4×C22⋊C4Dic8911yescharacteristic
202C4×C22⋊C4C2×D8912yes—
203C4×C22⋊C4C2×D8912yes—
204C2×#95Dic8912yes—
205#336—898no—
206C25C2×Dic8911yescharacteristic
207C23×C4—44no—
208C23×C4Dic17814yes—
209C23×C4D35614yes—
210C23×C4C2×Dic8914yes—
211C23×C4C2×Dic8911yescharacteristic
212C23×C4C2×Dic8911yescharacteristic
213C23×C4C2×Dic8911yescharacteristic
214C23×C4C22×D8911yescharacteristic
215C23×C4C22×D8911yescharacteristic
216C23×C4C22×D8911yescharacteristic
217C2×M16—3564no—
218C2×M16—1784no—
219C22×C8—894no—
220C22×C8—894no—
221C22×C8—3564no—
222C22×C8—1784no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4C2×Dic8912yes—
225C2×C22⋊C4C2×Dic8914yes—
226C2×C22⋊C4C2×Dic8912yes—
227C2×C42—44no—
228C2×C42C2×Dic8912yes—
229C2×C42C2×Dic8912yes—
230C2×C42C22×D8914yes—
231C4⋊C8—3568no—
232C22⋊C8—1784no—
233C22⋊C8—1784no—
234C22⋊C8—1784no—
235C22⋊C8—1784no—
236#95C2×Dic8912yes—
237#95C2×Dic8914yes—
238#95C2×Dic8912yes—
239C24—24no—
240C24—24no—
241C24—42no—
242C24C4⋊Dic8914yes—
243C24C22×Dic8911yescharacteristic
244C24C22×Dic8911yescharacteristic
245C24C22×Dic8911yescharacteristic
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C4⋊Dic8914yes—
258C22×C4C4⋊Dic8914yes—
259C22×C4C4⋊Dic8914yes—
260C22×C4C22×Dic8914yes—
261C22×C4C22×Dic8914yes—
262C22×C4C22×Dic8911yescharacteristic
263C22×C4C22×Dic8911yescharacteristic
264C22×C4C22×Dic8911yescharacteristic
265C22×C4C2×D35612yes—
266C22×C4C2×D35612yes—
267C22×C4C2×D35612yes—
268C22×C4C2×Dic17812yes—
269C22×C4C2×Dic17812yes—
270C22×C4C2×Dic17812yes—
271C22×C4C23×D8911yescharacteristic
272M16—3568no—
273C2×C8—3568no—
274C2×C8—1784no—
275C2×C8—1784no—
276C2×C8—1788no—
277C2×C8—1788no—
278C22⋊C4—24no—
279C22⋊C4—28no—
280C22⋊C4#1308355818yescomplemented
281C42—28no—
282C42—42no—
283C42—42no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—28no—
289C23—28no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—44no—
294C23—416no—
295C23C2×C4⋊Dic8912yes—
296C23C2×C4⋊Dic8912yes—
297C23C2×C4⋊Dic8912yes—
298C23C2×#1308355814yes—
299C23#1566801914yes—
300C23C23×Dic8911yescharacteristic
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—416no—
319C2×C4C2×C4⋊Dic8912yes—
320C2×C4C2×C4⋊Dic8912yes—
321C2×C4C2×C4⋊Dic8912yes—
322C2×C4C2×#1308355818yes—
323C2×C4C22×D35611yescharacteristic
324C2×C4C22×Dic17811yescharacteristic
325C2×C4#1566805714yes—
326C8—3564no—
327C8—3564no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—28no—
333C22—28no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—416no—
338C22—44no—
339C22#1915162214yes—
340C22#1915162514yes—
341C22C22×C4⋊Dic8911yescharacteristic
342C22#1915207312yes—
343C22C22×#1308355812yes—
344C22C2×#1566801912yes—
345C4—22no—
346C4—22no—
347C4—22no—
348C4—28no—
349C4—44no—
350C4#1915216012yes—
351C4#1915222412yes—
352C2—24no—
353C2—24no—
354C2—42no—
355C2#2602286611yescharacteristic
356C2C2×#1915162211yescharacteristic
357C2C2×#1915162511yescharacteristic
358C2#2602928714yes—
subgroupG/H
center, FrattiniC22×C4C23×D89
commutator, socleC23×C89C23×C4
FittingC2×C4×C22⋊C4×C89C2
prime radicalC25×C89C2×C4
cosocleC22×C4×C89C24
hypercenterC2×C4×C22⋊C4D89
hypocenterC89#21932
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C22×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #26020437C2semidirect
4. #26020432C2semidirect
5. #26020232C2semidirect
9. C2×C4⋊#10623494C22semidirect
16. #19151353C22semidirect
100. C22⋊C4×C89M16semidirect
186. C89#21932semidirect
280. C22⋊C4#13083558semidirect

Maximal subgroups

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

subgroupindex
C2×#191515622
C22×C22⋊#106234942
#260204372
#260204322
#260202322
C2×C4×C22⋊C4×C892
#2193289