Finite Group Databasebeta

← #36462620

Subgroups — #36462620

Order 22784.

814 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
1C22×#15667944C211yesmaximal, characteristic
2C2×#19151361C212yesmaximal
3C2×#19151361C211yesmaximal, characteristic
4#26018482C212yesmaximal
5C2×#336×C89C211yesmaximal, characteristic
6C22×C4⋊Dic89C2212yes—
7C22×C4⋊Dic89C2212yes—
8C2×#15667944—24no—
9C2×#15667944—22no—
10C2×#15667919—22no—
11C2×#15667919—22no—
12#19151361—24no—
13#19151361C414yescomplemented
14C23×C8×C89C2211yescharacteristic
15C23×C8×C89C2211yescharacteristic
16C22×C42×C89C2211yescharacteristic
17#336×C89—22no—
18C23×Dic89—22no—
19C23×Dic89—22no—
20C2×C4⋊Dic89—22no—
21C2×C4⋊Dic89—22no—
22C2×C4⋊Dic89—22no—
23C2×C4⋊Dic89—22no—
24C2×C4⋊Dic89—22no—
25C2×C4⋊Dic89—22no—
26C2×C4⋊Dic89—22no—
27C2×C4⋊Dic89—22no—
28C2×C4⋊Dic89—22no—
29C2×C4⋊Dic89—22no—
30C2×C4⋊Dic89C2×C414yes—
31C2×C4⋊Dic89D414yes—
32#15667944—28no—
33#15667919—44no—
34#15667919—44no—
35C23×C4×C89D412yes—
36C23×C4×C89C2311yescharacteristic
37C22×C8×C89—22no—
38C22×C8×C89—22no—
39C22×C8×C89—22no—
40C22×C8×C89—21no—
41C22×C8×C89—22no—
42C22×C8×C89—21no—
43C22×C8×C89C2×C412yes—
44C22×C8×C89D412yes—
45C22×C8×C89Q812yes—
46C22×C8×C89Q812yes—
47C2×C42×C89—21no—
48C2×C42×C89—22no—
49C2×C42×C89—22no—
50C2×C42×C89—21no—
51C22×Dic89—24no—
52C22×Dic89—24no—
53C22×Dic89—42no—
54C22×Dic89—42no—
55C22×Dic89—42no—
56C22×Dic89—42no—
57C22×Dic89—42no—
58C22×Dic89—42no—
59C22×Dic89—42no—
60C22×Dic89—42no—
61C22×Dic89—42no—
62C22×Dic89—42no—
63C22×Dic89—42no—
64C22×Dic89—42no—
65C4⋊Dic89—24no—
66C4⋊Dic89—24no—
67C4⋊Dic89—24no—
68C4⋊Dic89—24no—
69C4⋊Dic89—42no—
70C4⋊Dic89—42no—
71C4⋊Dic89—42no—
72C4⋊Dic89—42no—
73C24×C89C2×D411yescharacteristic
74C22×C4×C89—22no—
75C22×C4×C89—22no—
76C22×C4×C89—22no—
77C22×C4×C89—22no—
78C22×C4×C89—22no—
79C22×C4×C89—22no—
80C22×C4×C89SD1614yes—
81C22×C4×C89C22×C411yescharacteristic
82C22×C4×C89C2×D411yescharacteristic
83C22×C4×C89C2×D412yes—
84C22×C4×C89C2×Q811yescharacteristic
85C22×C4×C89C2×Q811yescharacteristic
86C22×C4×C89#4112yes—
87C22×C4×C89#4112yes—
88C22×C4×C89#4112yes—
89C22×C4×C89#4111yescharacteristic
90C22×C4×C89#4111yescharacteristic
91C2×C8×C89—24no—
92C2×C8×C89—22no—
93C2×C8×C89—24no—
94C2×C8×C89—22no—
95C2×C8×C89—24no—
96C2×C8×C89—22no—
97C2×C8×C89—41no—
98C2×C8×C89—42no—
99C2×C8×C89—41no—
100C2×C8×C89C4⋊C414yes—
101C42×C89—41no—
102C42×C89—41no—
103C42×C89—42no—
104C23×C89C4×D411yescharacteristic
105C23×C89#12111yescharacteristic
106C23×C89#12112yes—
107C23×C89#12312yes—
108C23×C89C4⋊Q811yescharacteristic
109C23×C89C2×SD1612yes—
110C23×C89#13512yes—
111C23×C89AGL(1,ℤ/8)12yes—
112C23×C89#13712yes—
113C2×C4×C89—24no—
114C2×C4×C89—24no—
115C2×C4×C89—24no—
116C2×C4×C89—42no—
117C2×C4×C89—42no—
118C2×C4×C89—42no—
119C2×C4×C89—42no—
120C2×C4×C89C2×C4⋊C411yescharacteristic
121C2×C4×C89#11712yes—
122C2×C4×C89C4×D412yes—
123C2×C4×C89C4×Q811yescharacteristic
124C2×C4×C89#12011yescharacteristic
125C2×C4×C89C22⋊Q812yes—
126C2×C4×C89C22⋊Q812yes—
127C2×C4×C89C22⋊Q811yescharacteristic
128C2×C4×C89#12311yescharacteristic
129C2×C4×C89#12312yes—
130C2×C4×C89#12412yes—
131C2×C4×C89#12411yescharacteristic
132C2×C4×C89#12411yescharacteristic
133C2×C4×C89#12512yes—
134C2×C4×C89#12512yes—
135C2×C4×C89#12511yescharacteristic
136C2×C4×C89#12612yes—
137C2×C4×C89#12612yes—
138C2×Dic89—44no—
139C2×Dic89—44no—
140C2×Dic89—44no—
141C2×Dic89—44no—
142C2×Dic89—44no—
143C2×Dic89—44no—
144C2×Dic89—82no—
145C2×Dic89—82no—
146C2×Dic89—82no—
147C2×Dic89—82no—
148C2×Dic89—82no—
149C2×Dic89—82no—
150C2×Dic89—82no—
151C2×Dic89—82no—
152C8×C89—24no—
153C8×C89—42no—
154C22×C89#38812yes—
155C22×C89#38911yescharacteristic
156C22×C89#39411yescharacteristic
157C22×C89#39711yescharacteristic
158C22×C89#39812yes—
159C22×C89C4×SD1612yes—
160C22×C89#44111yescharacteristic
161C22×C89#44211yescharacteristic
162C22×C89#46012yes—
163C22×C89#46112yes—
164C22×C89#46312yes—
165C22×C89#46412yes—
166C22×C89#46512yes—
167C22×C89#46812yes—
168C22×C89#48112yes—
169C22×C89#48212yes—
170C22×C89#48312yes—
171C22×C89#48512yes—
172C22×C89#49811yescharacteristic
173C22×C89#49911yescharacteristic
174C22×C89#50112yes—
175C4×C89—44no—
176C4×C89#38211yescharacteristic
177C4×C89#38511yescharacteristic
178C4×C89#38712yes—
179C4×C89#39311yescharacteristic
180C4×C89#40011yescharacteristic
181C4×C89#40012yes—
182Dic89—84no—
183Dic89—84no—
184#11257—891nomaximal, Sylow
185C2×C89#144211yescharacteristic
186C2×C89#192612yes—
187C2×C89#192712yes—
188C2×C89#194111yescharacteristic
189C2×C89#194311yescharacteristic
190C2×C89#205512yes—
191C2×C89#207712yes—
192C2×C89#208412yes—
193C2×C89#208512yes—
194C22×#106—891no—
195C2×#340—891no—
196C2×#340—892no—
197C2×#336D8911yescharacteristic
198C4⋊#95—892no—
199C89#1125711yescharacteristic, complemented, Sylow
200C23×C8C2×D8911yescharacteristic
201C23×C8C2×D8911yescharacteristic
202C22×C4⋊C4—892no—
203C22×C4⋊C4—892no—
204C22×C42C2×D8911yescharacteristic
205C2×#106—1784no—
206C2×#106—1782no—
207C2×#95—1782no—
208C2×#95—1782no—
209#340—894no—
210#340—1784no—
211#336—22no—
212C23×C4—1782no—
213C23×C4—1782no—
214C23×C4D35612yes—
215C23×C4C22×D8911yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8Dic17812yes—
223C22×C8Dic17812yes—
224C22×C8C4×D8912yes—
225C22×C8C22⋊D8912yes—
226C2×C4⋊C4—894no—
227C2×C4⋊C4—894no—
228C2×C4⋊C4—1782no—
229C2×C4⋊C4—1782no—
230C2×C4⋊C4—1782no—
231C2×C4⋊C4—1782no—
232C2×C4⋊C4—1782no—
233C2×C4⋊C4—1782no—
234C2×C4⋊C4—1782no—
235C2×C4⋊C4—1782no—
236C2×C4⋊C4—1782no—
237C2×C4⋊C4—1782no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242#106—1788no—
243#95—3564no—
244#95—3564no—
245C24C2×D35611yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—3562no—
253C22×C4—3562no—
254C22×C4—3562no—
255C22×C4—3562no—
256C22×C4—3562no—
257C22×C4—3562no—
258C22×C4—3562no—
259C22×C4—3562no—
260C22×C4—3562no—
261C22×C4—3562no—
262C22×C4—3562no—
263C22×C4—3562no—
264C22×C4—1784no—
265C22×C4—1784no—
266C22×C4#1308356314yes—
267C22×C4C2×C4×D8911yescharacteristic
268C22×C4D4×D8912yes—
269C22×C4C2×C22⋊D8911yescharacteristic
270C22×C4C2×Dic17811yescharacteristic
271C22×C4C2×Dic17811yescharacteristic
272C22×C4#1308357312yes—
273C22×C4#1308357312yes—
274C22×C4#1308357411yescharacteristic
275C22×C4#1308357411yescharacteristic
276C22×C4#1308357512yes—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8#1308355314yes—
287C4⋊C4—3562no—
288C4⋊C4—3562no—
289C4⋊C4—3562no—
290C4⋊C4—3562no—
291C4⋊C4—1784no—
292C4⋊C4—1784no—
293C4⋊C4—1784no—
294C4⋊C4—1784no—
295C42—41no—
296C42—41no—
297C42—42no—
298C23C4×D35611yescharacteristic
299C23#1566798611yescharacteristic
300C23#1566798712yes—
301C23#1566799612yes—
302C23#1566801211yescharacteristic
303C23C2×#1308356312yes—
304C23#1566803412yes—
305C23#1566804012yes—
306C23#1566804512yes—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—3564no—
315C2×C4—3564no—
316C2×C4—3564no—
317C2×C4—3564no—
318C2×C4—3564no—
319C2×C4—3564no—
320C2×C4—7122no—
321C2×C4—7122no—
322C2×C4—7122no—
323C2×C4—7122no—
324C2×C4—7122no—
325C2×C4—7122no—
326C2×C4—7122no—
327C2×C4—7122no—
328C2×C4C2×#1308355311yescharacteristic
329C2×C4#1566797012yes—
330C2×C4#1566797212yes—
331C2×C4C4×Dic17811yescharacteristic
332C2×C4#1566798011yescharacteristic
333C2×C4#1566798912yes—
334C2×C4#1566798912yes—
335C2×C4#1566799111yescharacteristic
336C2×C4#1566799311yescharacteristic
337C2×C4#1566799712yes—
338C2×C4#1566799812yes—
339C2×C4#1566799911yescharacteristic
340C2×C4#1566800111yescharacteristic
341C2×C4#1566800312yes—
342C2×C4#1566800312yes—
343C2×C4#1566800411yescharacteristic
344C2×C4#1566800512yes—
345C2×C4#1566800712yes—
346C8—24no—
347C8—42no—
348C22#1915149012yes—
349C22#1915149611yescharacteristic
350C22#1915151511yescharacteristic
351C22#1915152311yescharacteristic
352C22#1915153012yes—
353C22C4×#1308356312yes—
354C22#1915170611yescharacteristic
355C22#1915171111yescharacteristic
356C22#1915179912yes—
357C22#1915180412yes—
358C22#1915181412yes—
359C22#1915182112yes—
360C22#1915182712yes—
361C22#1915184412yes—
362C22#1915191712yes—
363C22#1915192312yes—
364C22#1915192912yes—
365C22#1915194012yes—
366C22#1915199611yescharacteristic
367C22#1915200011yescharacteristic
368C22#1915200712yes—
369C4—44no—
370C4—7124no—
371C4—7124no—
372C4#1915146311yescharacteristic
373C4#1915147011yescharacteristic
374C4#1915148512yes—
375C4#1915151011yescharacteristic
376C4#1915153711yescharacteristic
377C4#1915154012yes—
378C2#2601850811yescharacteristic
379C2#2602088012yes—
380C2#2602088712yes—
381C2#2602098011yescharacteristic
382C2#2602099011yescharacteristic
383C2#2602157012yes—
384C2#2602168512yes—
385C2#2602173012yes—
386C2#2602173512yes—
subgroupG/H
centerC24C2×D356
commutatorC22×C4×C89C22×C4
FrattiniC23×C4C22×D89
FittingC2×#336×C89C2
socle, prime radicalC24×C89C2×D4
cosocleC23×C4×C89C23
hypercenterC2×#336D89
hypocenterC89#11257
LCSG ▷ C22×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
13. #19151361C4semidirect
199. C89#11257semidirect

Maximal subgroups

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

subgroupindex
C22×#156679442
C2×#191513612
#260184822
C2×#336×C892
#1125789