Finite Group Databasebeta

← #36447541

Subgroups — #36447541

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151675C211yesmaximal, characteristic
2#26020379C211yesmaximal, characteristic, complemented
3#26020253C211yesmaximal, characteristic
4#26020249C211yesmaximal, characteristic
5C2×#19151348C211yesmaximal, characteristic, complemented
6#26018136C211yesmaximal, characteristic, complemented
7C89×#1296C211yesmaximal, characteristic, complemented
8C22×#13083557—21no—
9C2×C4×D356C2211yescharacteristic
10#19151675—22no—
11#19151675C414yescomplemented
12C2×C4⋊#10623494C2211yescharacteristic
13C2×#15667934—21no—
14C2×#15667932—21no—
15C2×#15667924—21no—
16C2×C4×#10623494C2211yescharacteristic
17#19151468C2211yescharacteristic
18#19151361—21no—
19#19151360—21no—
20#19151360—21no—
21#19151360—21no—
22#19151348—22no—
23C2×C4⋊C8×C89C2211yescharacteristic
24C2×C4⋊C8×C89C2211yescharacteristic
25#384×C89C2211yescharacteristic
26C22×D356D411yescharacteristic
27C22×C4×D89—21no—
28C2×#13083557—24no—
29C2×#13083557—42no—
30C2×#13083557—42no—
31C22×#10623494—21no—
32C22×#10623494D411yescharacteristic
33C22×#10623494D411yescharacteristic
34C4×D356—21no—
35C4×D356—21no—
36C4×D356C2×C412yes—
37C4×D356Q812yes—
38C2×C4⋊Dic89—21no—
39C2×C4⋊Dic89—21no—
40C2×C4⋊Dic89—21no—
41C2×C4⋊Dic89D411yescharacteristic
42C2×#13083551—21no—
43C4⋊#10623494—21no—
44C4⋊#10623494—21no—
45#15667934—42no—
46#15667932—42no—
47#15667924—24no—
48C4×#10623494C2×C412yes—
49C4×#10623494D412yes—
50C22×C8×C89—21no—
51C22×C8×C89—21no—
52C2×C4⋊C4×C89—21no—
53C2×C4⋊C4×C89—21no—
54C2×C4⋊C4×C89—21no—
55C2×C42×C89C2311yescharacteristic
56C4⋊C8×C89—21no—
57C4⋊C8×C89—21no—
58C4⋊C8×C89—21no—
59C4⋊C8×C89C2×C412yes—
60C23×D89—21no—
61C2×D356—21no—
62C2×D356—21no—
63C2×D356—21no—
64C2×D356—21no—
65C2×D356—21no—
66C2×D356C4⋊C412yes—
67C22×Dic89—21no—
68C22×Dic89—41no—
69C2×C4×D89—22no—
70C2×C4×D89—22no—
71C2×C4×D89—41no—
72C2×C4×D89—41no—
73C2×C4×D89—41no—
74C2×C4×D89—41no—
75#13083557—48no—
76C2×#10623494—21no—
77C2×#10623494—21no—
78C2×#10623494—21no—
79C2×#10623494—21no—
80C2×#10623494—41no—
81C2×#10623494—41no—
82C2×#10623494—41no—
83C2×#10623494C22⋊C412yes—
84C2×#10623494C22⋊C412yes—
85C4⋊Dic89—21no—
86C4⋊Dic89—41no—
87C4⋊Dic89—41no—
88C4⋊Dic89—41no—
89C4⋊Dic89—41no—
90C4⋊Dic89—81no—
91C4⋊Dic89C4⋊C412yes—
92#13083551—22no—
93#13083551—22no—
94C22×C4×C89—41no—
95C22×C4×C89C2×D411yescharacteristic
96C22×C4×C89C2×D411yescharacteristic
97C22×C4×C89#4111yescharacteristic
98C2×C8×C89—22no—
99C2×C8×C89—41no—
100C2×C8×C89—41no—
101C2×C8×C89—41no—
102C2×C8×C89—41no—
103C2×C8×C89—41no—
104C4⋊C4×C89—41no—
105C4⋊C4×C89—41no—
106C4⋊C4×C89—41no—
107C4⋊C4×C89—41no—
108C4⋊C4×C89—81no—
109C42×C89C22×C411yescharacteristic
110C42×C89C2×D411yescharacteristic
111C42×C89C2×Q811yescharacteristic
112C42×C89#4111yescharacteristic
113C23×C89#12111yescharacteristic
114C22×D89—22no—
115C22×D89—41no—
116C22×D89—41no—
117C22×D89—41no—
118C22×D89—41no—
119C22×D89—41no—
120C22×D89—41no—
121C2×C4×C89—21no—
122C2×C4×C89—21no—
123C2×C4×C89—81no—
124C2×C4×C89—81no—
125C2×C4×C89—81no—
126C2×C4×C89C2×C22⋊C411yescharacteristic
127C2×C4×C89C2×C4⋊C411yescharacteristic
128C2×C4×C89#11711yescharacteristic
129C2×C4×C89#12111yescharacteristic
130C2×C4×C89C22⋊Q811yescharacteristic
131C2×C4×C89C22⋊Q811yescharacteristic
132C2×C4×C89#13511yescharacteristic
133C2×C4×C89#13511yescharacteristic
134C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C89#13711yescharacteristic
139C2×C4×C89#13711yescharacteristic
140C2×Dic89—22no—
141C2×Dic89—41no—
142C2×Dic89—41no—
143C2×Dic89—81no—
144C2×Dic89—81no—
145C2×Dic89—81no—
146D356—22no—
147D356—22no—
148D356—41no—
149D356—41no—
150C4×D89—42no—
151C4×D89—42no—
152C4×D89—42no—
153C4×D89—42no—
154C8×C89—42no—
155C8×C89—81no—
156#10623494—22no—
157#10623494—22no—
158#10623494—81no—
159C22×C89#38011yescharacteristic
160C22×C89#46311yescharacteristic
161C22×C89#46311yescharacteristic
162C22×C89#46911yescharacteristic
163C22×C89#47011yescharacteristic
164C22×C89#47111yescharacteristic
165C22×C89#47111yescharacteristic
166C2×D89—42no—
167C2×D89—42no—
168C2×D89—42no—
169C2×D89—81no—
170C2×D89—81no—
171C2×D89—81no—
172C2×D89—81no—
173C4×C89—21no—
174C4×C89—21no—
175C4×C89—161no—
176C4×C89#41611yescharacteristic
177C4×C89#41811yescharacteristic
178C4×C89#42811yescharacteristic
179C4×C89#42811yescharacteristic
180C4×C89#46811yescharacteristic
181C4×C89#46811yescharacteristic
182C4×C89#47811yescharacteristic
183C4×C89#47811yescharacteristic
184Dic89—42no—
185Dic89—161no—
186#8656—891nomaximal, Sylow
187C2×C89#170111yescharacteristic
188C2×C89#170111yescharacteristic
189C2×C89#170311yescharacteristic
190C2×C89#170311yescharacteristic
191C2×C89#180411yescharacteristic
192C2×C89#180911yescharacteristic
193C2×C89#181511yescharacteristic
194D89—82no—
195C2×#436—891no—
196#1834—891no—
197#1802—891no—
198#1802—891no—
199C2×#334—891no—
200#1296—891no—
201#1296D8911yescharacteristic, complemented
202C89#865611yescharacteristic, complemented, Sylow
203C22×M16—1781no—
204C2×C4×D4—891no—
205#436—894no—
206#436—1782no—
207C2×C4⋊C8—891no—
208C2×C4⋊C8C2×D8911yescharacteristic
209C2×C4⋊C8C2×D8911yescharacteristic
210C2×#102—1781no—
211C2×#102—1781no—
212C2×C22⋊C8—1781no—
213C2×C4×C8—891no—
214#384—891no—
215#384C2×D8911yescharacteristic
216#340—1781no—
217#340—1781no—
218#340—1781no—
219#340—1781no—
220#334—1782no—
221C22×D4—891no—
222C23×C4—1781no—
223C2×M16—3562no—
224C2×M16—3562no—
225C2×M16—1784no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—891no—
229C22×C8—891no—
230C22×C8—1781no—
231C4×D4—892no—
232C4×D4—892no—
233C4×D4—1781no—
234C4×D4—1781no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—891no—
239C2×C4⋊C4—1781no—
240C2×C4⋊C4—1781no—
241C2×C4⋊C4—1781no—
242C2×C22⋊C4—1781no—
243C2×C42C22×D8911yescharacteristic
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—1781no—
248C4⋊C8—1781no—
249C4⋊C8C4×D8912yes—
250#102—3562no—
251#102—3562no—
252C22⋊C8—1784no—
253C4×C8—892no—
254C4×C8—892no—
255C24—1781no—
256C2×D4—892no—
257C2×D4—1781no—
258C2×D4—1781no—
259C2×D4—1781no—
260C2×D4—1781no—
261C2×D4—1781no—
262C22×C4—41no—
263C22×C4—3561no—
264C22×C4—3561no—
265C22×C4—3561no—
266C22×C4—3561no—
267C22×C4—3561no—
268C22×C4—1781no—
269C22×C4—1782no—
270C22×C4—1782no—
271C22×C4D4×D8911yescharacteristic
272C22×C4D4×D8911yescharacteristic
273C22×C4#1308357411yescharacteristic
274M16—3568no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—892no—
282C2×C8—892no—
283C2×C8—3561no—
284C2×C8—3561no—
285C2×C8—3561no—
286C2×C8—1781no—
287C2×C8—1781no—
288C2×C8—1781no—
289C2×C8—1781no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C4⋊C4—892no—
296C4⋊C4—3561no—
297C4⋊C4—3561no—
298C4⋊C4—3561no—
299C4⋊C4—3561no—
300C4⋊C4—1781no—
301C4⋊C4—7121no—
302C22⋊C4—1782no—
303C22⋊C4—1782no—
304C42C2×C4×D8911yescharacteristic
305C42D4×D8911yescharacteristic
306C42Q8×D8911yescharacteristic
307C42#1308357411yescharacteristic
308C23—3561no—
309C23—3561no—
310C23—3561no—
311C23—3561no—
312C23—3561no—
313C23—3561no—
314C23—1782no—
315C23#1566798511yescharacteristic
316D4—3561no—
317D4—3561no—
318D4—1782no—
319D4—1782no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—3561no—
326C2×C4—3561no—
327C2×C4—3562no—
328C2×C4—3562no—
329C2×C4—3562no—
330C2×C4—3562no—
331C2×C4—1782no—
332C2×C4—7121no—
333C2×C4—7121no—
334C2×C4—7121no—
335C2×C4C22⋊C4×D8911yescharacteristic
336C2×C4C4⋊C4×D8911yescharacteristic
337C2×C4#1566796911yescharacteristic
338C2×C4#1566798511yescharacteristic
339C2×C4#1566799011yescharacteristic
340C2×C4#1566799011yescharacteristic
341C2×C4#1566803311yescharacteristic
342C2×C4#1566803511yescharacteristic
343C2×C4#1566803611yescharacteristic
344C2×C4#1566803611yescharacteristic
345C2×C4#1566803911yescharacteristic
346C2×C4#1566803911yescharacteristic
347C2×C4#1566804111yescharacteristic
348C2×C4#1566804411yescharacteristic
349C8—42no—
350C8—81no—
351C8—1782no—
352C8—1782no—
353C8—7121no—
354C22—3562no—
355C22—3562no—
356C22—3562no—
357C22—7121no—
358C22—7121no—
359C22—7121no—
360C22—7121no—
361C22#1915145711yescharacteristic
362C22#1915181211yescharacteristic
363C22#1915181611yescharacteristic
364C22#1915185011yescharacteristic
365C22#1915185511yescharacteristic
366C22#1915186011yescharacteristic
367C22#1915186411yescharacteristic
368C4—21no—
369C4—21no—
370C4—161no—
371C4—3562no—
372C4—14241no—
373C4#1915159311yescharacteristic
374C4#1915160111yescharacteristic
375C4#1915163711yescharacteristic
376C4#1915163911yescharacteristic
377C4#1915184311yescharacteristic
378C4#1915184711yescharacteristic
379C4#1915189711yescharacteristic
380C4#1915190111yescharacteristic
381C2—7122no—
382C2#2601981211yescharacteristic
383C2#2601981611yescharacteristic
384C2#2601982611yescharacteristic
385C2#2601983011yescharacteristic
386C2#2602026111yescharacteristic
387C2#2602028411yescharacteristic
388C2#2602031011yescharacteristic
subgroupG/H
centerC23#15667985
commutatorC42×C89C22×C4
FrattiniC2×C42C22×D89
FittingC89×#1296C2
socleC23×C89#121
prime radicalC22×D356D4
cosocleC2×C42×C89C23
hypercenter#1296D89
hypocenterC89#8656
LCSG ▷ C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #26020379C2semidirect
5. C2×#19151348C2semidirect
6. #26018136C2semidirect
7. C89×#1296C2semidirect
11. #19151675C4semidirect
201. #1296D89semidirect
202. C89#8656semidirect

Maximal subgroups

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

subgroupindex
C2×#191516752
#260203792
#260202532
#260202492
C2×#191513482
#260181362
C89×#12962
#865689