Finite Group Databasebeta

← #36448740

Subgroups — #36448740

Order 22784.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26028820C211yesmaximal, characteristic
2#26019024C211yesmaximal, characteristic, complemented
3#26019007C211yesmaximal, characteristic, complemented
4#26018634C211yesmaximal, characteristic
5#26018155C211yesmaximal, characteristic, complemented
6#26018093C211yesmaximal, characteristic, complemented
7C89×#1806C211yesmaximal, characteristic
8C2×#15668019—21no—
9#19151651C2211yescharacteristic
10#19151625—22no—
11#19151625—21no—
12#19151622—21no—
13#19151622C2211yescharacteristic
14C2×C4⋊#10623494—21no—
15C2×C4⋊#10623494C2211yescharacteristic
16C2×C4×#10623494C2211yescharacteristic
17#19151348—22no—
18#19151348—21no—
19#19151348—21no—
20Q8⋊#10623494C411yescharacteristic, complemented
21Q8⋊#10623494C411yescharacteristic, complemented
22D4⋊#10623494C411yescharacteristic, complemented
23D4⋊#10623494C411yescharacteristic, complemented
24C4×#41×C89C2211yescharacteristic
25#423×C89C2211yescharacteristic
26C2×#104×C89—21no—
27C4×C4⋊C4×C89C2211yescharacteristic
28#339×C89—21no—
29#15668019—42no—
30#15668019—42no—
31C2×#13083558—22no—
32C2×#13083558—21no—
33C22×#10623494—22no—
34C22×#10623494—21no—
35C22×#10623494—21no—
36C4⋊#10623494—22no—
37C4⋊#10623494—22no—
38C4⋊#10623494—21no—
39C4⋊#10623494—21no—
40C4⋊#10623494C2×C411yescharacteristic
41C4⋊#10623494C2×C411yescharacteristic
42C4⋊#10623494D411yescharacteristic
43C4⋊#10623494D411yescharacteristic
44C4⋊#10623494D412yes—
45C22⋊#10623494—21no—
46C4×#10623494—21no—
47C4×#10623494—21no—
48C4×#10623494C2×C411yescharacteristic
49C4×#10623494C2×C411yescharacteristic
50C2×#41×C89D411yescharacteristic
51C2×M16×C89—21no—
52C4×Q8×C89C2×C411yescharacteristic
53C4×Q8×C89Q811yescharacteristic
54C4×D4×C89—21no—
55C4×D4×C89—21no—
56C4×D4×C89C2×C411yescharacteristic
57C4×D4×C89Q811yescharacteristic
58#117×C89—21no—
59#117×C89D411yescharacteristic
60C2×C4⋊C4×C89—21no—
61C2×C42×C89—21no—
62C2×C42×C89—21no—
63C2×C42×C89C2311yescharacteristic
64C4⋊C8×C89—21no—
65C4⋊C8×C89—21no—
66#104×C89—42no—
67#95×C89—21no—
68#13083558—44no—
69#13083558—41no—
70#13083558—41no—
71C2×#10623494—21no—
72C2×#10623494—21no—
73C2×#10623494—21no—
74C2×#10623494—21no—
75C2×#10623494—21no—
76C2×#10623494—22no—
77C2×#10623494—21no—
78C2×#10623494—42no—
79C2×#10623494—44no—
80C2×#10623494—41no—
81C2×#10623494—41no—
82#41×C89—21no—
83#41×C89—21no—
84#41×C89SD1612yes—
85#41×C89SD1612yes—
86C2×Q8×C89C4⋊C411yescharacteristic
87C2×D4×C89—21no—
88C2×D4×C89C4⋊C411yescharacteristic
89C22×C4×C89—21no—
90C22×C4×C89—21no—
91C22×C4×C89—21no—
92C22×C4×C89—41no—
93C22×C4×C89C2×D411yescharacteristic
94C22×C4×C89#4111yescharacteristic
95C22×C4×C89#4111yescharacteristic
96M16×C89—41no—
97M16×C89—41no—
98C2×C8×C89—41no—
99C4⋊C4×C89—21no—
100C4⋊C4×C89—21no—
101C4⋊C4×C89—22no—
102C4⋊C4×C89—41no—
103C4⋊C4×C89C4⋊C411yescharacteristic
104C4⋊C4×C89C4⋊C411yescharacteristic
105C22⋊C4×C89—21no—
106C22⋊C4×C89—21no—
107C22⋊C4×C89—21no—
108C42×C89—21no—
109C42×C89—21no—
110C42×C89—21no—
111C42×C89—41no—
112C42×C89—41no—
113C42×C89C22×C411yescharacteristic
114C42×C89C2×D411yescharacteristic
115C42×C89C2×D411yescharacteristic
116C42×C89C2×Q811yescharacteristic
117C23×C89—21no—
118C23×C89#12311yescharacteristic
119Q8×C89—21no—
120Q8×C89#10612yes—
121D4×C89—22no—
122D4×C89—22no—
123D4×C89—21no—
124D4×C89#10612yes—
125C2×C4×C89—21no—
126C2×C4×C89—21no—
127C2×C4×C89—21no—
128C2×C4×C89—22no—
129C2×C4×C89—22no—
130C2×C4×C89—21no—
131C2×C4×C89—21no—
132C2×C4×C89—41no—
133C2×C4×C89—41no—
134C2×C4×C89—41no—
135C2×C4×C89—42no—
136C2×C4×C89—81no—
137C2×C4×C89—81no—
138C2×C4×C89—81no—
139C2×C4×C89C2×C4⋊C411yescharacteristic
140C2×C4×C89C4×D411yescharacteristic
141C2×C4×C89C4×D411yescharacteristic
142C2×C4×C89#12011yescharacteristic
143C2×C4×C89C22⋊Q811yescharacteristic
144C2×C4×C89C22⋊Q811yescharacteristic
145C2×C4×C89C2×SD1611yescharacteristic
146C2×C4×C89C2×SD1611yescharacteristic
147C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C89#13711yescharacteristic
149C8×C89—81no—
150#10623494—22no—
151#10623494—41no—
152#10623494—41no—
153#10623494—41no—
154#10623494—41no—
155#10623494—42no—
156#10623494—41no—
157C22×C89—21no—
158C22×C89—21no—
159C22×C89—22no—
160C22×C89—41no—
161C22×C89—41no—
162C22×C89#38511yescharacteristic
163C22×C89#48111yescharacteristic
164C22×C89#48311yescharacteristic
165C4×C89—22no—
166C4×C89—21no—
167C4×C89—21no—
168C4×C89—21no—
169C4×C89—41no—
170C4×C89—41no—
171C4×C89—82no—
172C4×C89C2×#10611yescharacteristic
173C4×C89#42811yescharacteristic
174C4×C89#44311yescharacteristic
175C4×C89#44411yescharacteristic
176C4×C89#44811yescharacteristic
177C4×C89#45011yescharacteristic
178#8904—891nomaximal, Sylow
179C2×C89—21no—
180C2×C89—21no—
181C2×C89—42no—
182C2×C89#185311yescharacteristic
183C2×C89#186111yescharacteristic
184C2×C89#186611yescharacteristic
185#2906—891no—
186#1806D8911yescharacteristic
187#1570—891no—
188#1566—891no—
189#1476—891no—
190#1303—891no—
191#1279—891no—
192C89#890411yescharacteristic, complemented, Sylow
193C2×#131—1781no—
194C4×#41C2×D8911yescharacteristic
195#431—891no—
196#424—1782no—
197#424—1781no—
198#423—891no—
199#423—1781no—
200#423C2×D8911yescharacteristic
201C2×C4⋊C8—891no—
202C2×C4⋊C8—1781no—
203C2×#104—21no—
204C2×C4×C8—891no—
205C4×C4⋊C4C2×D8911yescharacteristic
206#339—21no—
207#334—1782no—
208#334—1781no—
209#334—1781no—
210Q8⋊C8—891no—
211Q8⋊C8—891no—
212D4⋊C8—891no—
213D4⋊C8—891no—
214C2×#41C22⋊D8911yescharacteristic
215#131—3562no—
216#131—3562no—
217C2×M16—21no—
218C2×M16—1782no—
219C2×M16—1781no—
220C22×C8—1782no—
221C22×C8—1781no—
222C22×C8—1781no—
223C4×Q8Dic17811yescharacteristic
224C4×Q8C4×D8911yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic17811yescharacteristic
228C4×D4C4×D8911yescharacteristic
229#117—21no—
230#117C22⋊D8911yescharacteristic
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×D8911yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—891no—
238C4⋊C8—891no—
239C4⋊C8—891no—
240C4⋊C8—891no—
241C4⋊C8—892no—
242C4⋊C8—1782no—
243C4⋊C8—1782no—
244C4⋊C8—1781no—
245C4⋊C8—1781no—
246#104—42no—
247C22⋊C8—1781no—
248C4×C8—891no—
249C4×C8—891no—
250C4×C8—1781no—
251C4×C8—1781no—
252#95—21no—
253#41—21no—
254#41—21no—
255#41#1308356112yes—
256#41#1308356212yes—
257C2×Q8#1308355311yescharacteristic
258C2×D4—21no—
259C2×D4#1308355311yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4C2×C22⋊D8911yescharacteristic
265C22×C4#1308357311yescharacteristic
266C22×C4#1308357311yescharacteristic
267M16—41no—
268M16—41no—
269M16—3564no—
270M16—3561no—
271M16—3561no—
272C2×C8—41no—
273C2×C8—3564no—
274C2×C8—3561no—
275C2×C8—3561no—
276C2×C8—3562no—
277C2×C8—1781no—
278C2×C8—1781no—
279C2×C8—1781no—
280C2×C8—1781no—
281C2×C8—1781no—
282C2×C8—1782no—
283C2×C8—1781no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4#1308355311yescharacteristic
289C4⋊C4#1308355311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42C2×C4×D8911yescharacteristic
299C42D4×D8911yescharacteristic
300C42D4×D8911yescharacteristic
301C42C2×Dic17811yescharacteristic
302C23—21no—
303C23#1566799311yescharacteristic
304Q8—21no—
305Q8#1566794312yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1566794312yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4C2×#1308355311yescharacteristic
325C2×C4#1566797211yescharacteristic
326C2×C4#1566797211yescharacteristic
327C2×C4#1566798011yescharacteristic
328C2×C4#1566798911yescharacteristic
329C2×C4#1566798911yescharacteristic
330C2×C4C2×#1308356111yescharacteristic
331C2×C4C2×#1308356211yescharacteristic
332C2×C4#1566803711yescharacteristic
333C2×C4#1566804211yescharacteristic
334C8—81no—
335C8—3561no—
336C8—3561no—
337C8—3561no—
338C8—3561no—
339C8—3562no—
340C8—3561no—
341C8—1782no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1915147011yescharacteristic
348C22#1915191411yescharacteristic
349C22#1915192611yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4C2×#1566794311yescharacteristic
358C4#1915164111yescharacteristic
359C4#1915171211yescharacteristic
360C4#1915171711yescharacteristic
361C4#1915173611yescharacteristic
362C4#1915174811yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2602044411yescharacteristic
367C2#2602049011yescharacteristic
368C2#2602052511yescharacteristic
subgroupG/H
centerC2×C4#15667980
commutatorC42×C89C22×C4
FrattiniC2×C42C22×D89
FittingC89×#1806C2
socleC22×C89#385
prime radicalC2×#41×C89D4
cosocleC2×C42×C89C23
hypercenter#1806D89
hypocenterC89#8904
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #26019024C2semidirect
3. #26019007C2semidirect
5. #26018155C2semidirect
6. #26018093C2semidirect
20. Q8⋊#10623494C4semidirect
21. Q8⋊#10623494C4semidirect
22. D4⋊#10623494C4semidirect
23. D4⋊#10623494C4semidirect
192. C89#8904semidirect

Maximal subgroups

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

subgroupindex
#260288202
#260190242
#260190072
#260186342
#260181552
#260180932
C89×#18062
#890489