Finite Group Databasebeta

← C89×#19416 #36397264

Subgroups — C89×#19416 #36397264

Order 22784.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3226C211yesmaximal, characteristic, complemented
2C89×#3039C211yesmaximal, characteristic, complemented
3C89×#3035C211yesmaximal, characteristic
4C89×#2988C211yesmaximal, characteristic, complemented
5C89×#1682C212yesmaximal, complemented
6C89×#1681C212yesmaximal, complemented
7C89×#1674C212yesmaximal, complemented
8C89×#1673C212yesmaximal, complemented
9C89×#1571C211yesmaximal, characteristic, complemented
10C89×#1547C211yesmaximal, characteristic, complemented
11C2×#332×C89C211yesmaximal, characteristic, complemented
12C2×#137×C89—21no—
13C2×#135×C89—21no—
14#532×C89C2211yescharacteristic
15C2×C4⋊Q8×C89C2211yescharacteristic
16C4×#41×C89C2211yescharacteristic
17#500×C89C2212yes—
18#486×C89C2212yes—
19#480×C89—22no—
20#474×C89—21no—
21#474×C89—21no—
22#464×C89—21no—
23#463×C89—21no—
24#462×C89—21no—
25#462×C89C2211yescharacteristic
26#462×C89C2212yes—
27#461×C89—21no—
28#461×C89C2211yescharacteristic
29#461×C89C2212yes—
30#460×C89C2211yescharacteristic
31#459×C89C2211yescharacteristic
32C8×Q8×C89C2212yes—
33C8×D4×C89—22no—
34C8×D4×C89C2212yes—
35#431×C89—21no—
36#431×C89C2211yescharacteristic
37#423×C89C2211yescharacteristic
38C2×C4⋊C8×C89—21no—
39C2×C4⋊C8×C89C2211yescharacteristic
40C2×C4⋊C8×C89C2211yescharacteristic
41#419×C89—21no—
42#416×C89—21no—
43#335×C89C2212yes—
44#335×C89C2212yes—
45#332×C89C2212yes—
46#332×C89C2212yes—
47#332×C89C2212yes—
48#331×C89C2212yes—
49C2×#41×C89—21no—
50C2×#41×C89D411yescharacteristic
51C22×Q8×C89—21no—
52#137×C89—44no—
53#135×C89—44no—
54C2×Q16×C89—22no—
55C2×Q16×C89—21no—
56C2×SD16×C89—21no—
57C2×SD16×C89—22no—
58C2×SD16×C89—21no—
59C2×D8×C89—21no—
60C2×M16×C89—21no—
61C22×C8×C89—21no—
62C22×C8×C89—21no—
63C22×C8×C89—22no—
64C4⋊Q8×C89—21no—
65C4⋊Q8×C89—21no—
66C4⋊Q8×C89—21no—
67C4⋊Q8×C89C2312yes—
68C4⋊Q8×C89C2311yescharacteristic
69#127×C89C2311yescharacteristic
70#124×C89—21no—
71#121×C89—21no—
72#121×C89—21no—
73C4×Q8×C89D411yescharacteristic
74C4×Q8×C89C2311yescharacteristic
75C4×D4×C89—21no—
76C4×D4×C89—21no—
77C4×D4×C89—21no—
78C4×D4×C89—21no—
79C4×D4×C89D411yescharacteristic
80C4×D4×C89C2311yescharacteristic
81#117×C89—21no—
82#117×C89D411yescharacteristic
83C2×C4⋊C4×C89—21no—
84C2×C4⋊C4×C89—21no—
85C2×C42×C89—21no—
86C2×C42×C89C2311yescharacteristic
87#107×C89—22no—
88#107×C89—22no—
89C4⋊C8×C89—22no—
90C4⋊C8×C89—21no—
91C4⋊C8×C89—21no—
92C4⋊C8×C89—21no—
93C4⋊C8×C89D411yescharacteristic
94C4⋊C8×C89D411yescharacteristic
95C4⋊C8×C89C2312yes—
96C4⋊C8×C89C2311yescharacteristic
97C4⋊C8×C89C2311yescharacteristic
98C4⋊C8×C89C2312yes—
99#103×C89—21no—
100#103×C89—21no—
101#103×C89—22no—
102#103×C89—22no—
103#102×C89—22no—
104#102×C89—22no—
105#102×C89—21no—
106#102×C89—21no—
107C22⋊C8×C89—22no—
108C22⋊C8×C89D412yes—
109C4×C8×C89—22no—
110C4×C8×C89C2312yes—
111#41×C89—22no—
112#41×C89—42no—
113#41×C89—42no—
114#41×C89D814yes—
115C2×Q8×C89—21no—
116C2×Q8×C89—22no—
117C2×Q8×C89—41no—
118C2×Q8×C89—42no—
119C2×Q8×C89—42no—
120C2×Q8×C89C2×D411yescharacteristic
121C2×D4×C89—21no—
122C2×D4×C89—21no—
123C2×D4×C89—21no—
124C2×D4×C89—21no—
125C2×D4×C89—41no—
126C2×D4×C89C2×D411yescharacteristic
127C22×C4×C89—21no—
128C22×C4×C89—21no—
129C22×C4×C89—21no—
130C22×C4×C89—41no—
131C22×C4×C89—41no—
132C22×C4×C89C2×D411yescharacteristic
133C22×C4×C89C2×D411yescharacteristic
134C22×C4×C89#4111yescharacteristic
135Q16×C89—42no—
136Q16×C89—44no—
137SD16×C89—42no—
138SD16×C89—42no—
139SD16×C89—44no—
140D8×C89—42no—
141M16×C89—42no—
142C2×C8×C89—22no—
143C2×C8×C89—21no—
144C2×C8×C89—21no—
145C2×C8×C89—22no—
146C2×C8×C89—42no—
147C2×C8×C89—82no—
148C2×C8×C89—81no—
149C2×C8×C89C2×D412yes—
150C2×C8×C89C2×D412yes—
151C4⋊C4×C89—21no—
152C4⋊C4×C89—21no—
153C4⋊C4×C89—22no—
154C4⋊C4×C89—21no—
155C4⋊C4×C89—22no—
156C4⋊C4×C89—21no—
157C4⋊C4×C89—41no—
158C4⋊C4×C89—41no—
159C4⋊C4×C89C2×D411yescharacteristic
160C4⋊C4×C89C2×D411yescharacteristic
161C22⋊C4×C89—21no—
162C22⋊C4×C89—21no—
163C22⋊C4×C89—21no—
164C22⋊C4×C89—41no—
165C22⋊C4×C89—41no—
166C42×C89—21no—
167C42×C89—21no—
168C42×C89—21no—
169C42×C89C2×D411yescharacteristic
170C42×C89C2×D411yescharacteristic
171C42×C89#4111yescharacteristic
172C42×C89C2411yescharacteristic
173C23×C89—21no—
174C23×C89—41no—
175C23×C89#12111yescharacteristic
176Q8×C89—21no—
177Q8×C89—42no—
178Q8×C89—41no—
179Q8×C89—42no—
180Q8×C89—41no—
181Q8×C89—82no—
182Q8×C89C2×D812yes—
183D4×C89—24no—
184D4×C89—21no—
185D4×C89—41no—
186D4×C89—41no—
187D4×C89—41no—
188D4×C89—41no—
189D4×C89—81no—
190D4×C89—82no—
191D4×C89C2×D812yes—
192C2×C4×C89—21no—
193C2×C4×C89—21no—
194C2×C4×C89—21no—
195C2×C4×C89—24no—
196C2×C4×C89—21no—
197C2×C4×C89—21no—
198C2×C4×C89—41no—
199C2×C4×C89—41no—
200C2×C4×C89—41no—
201C2×C4×C89—41no—
202C2×C4×C89—42no—
203C2×C4×C89—81no—
204C2×C4×C89—82no—
205C2×C4×C89—82no—
206C2×C4×C89#12111yescharacteristic
207C2×C4×C89#12111yescharacteristic
208C2×C4×C89#12111yescharacteristic
209C2×C4×C89C2×D812yes—
210C2×C4×C89#13512yes—
211C2×C4×C89C22×D411yescharacteristic
212C2×C4×C89C22×D411yescharacteristic
213C2×C4×C89C2×#4111yescharacteristic
214C8×C89—22no—
215C8×C89—22no—
216C8×C89—41no—
217C8×C89—41no—
218C8×C89—42no—
219C8×C89—42no—
220C22×C89—21no—
221C22×C89—21no—
222C22×C89—22no—
223C22×C89—41no—
224C22×C89—41no—
225C22×C89—81no—
226C22×C89—81no—
227C22×C89—81no—
228C22×C89#46612yes—
229C22×C89C2×#12111yescharacteristic
230C4×C89—21no—
231C4×C89—22no—
232C4×C89—21no—
233C4×C89—21no—
234C4×C89—41no—
235C4×C89—41no—
236C4×C89—81no—
237C4×C89—82no—
238C4×C89#46612yes—
239C4×C89C22×D811yescharacteristic
240C4×C89C2×#13511yescharacteristic
241C4×C89#57711yescharacteristic
242C4×C89#57811yescharacteristic
243#19416C8911yesmaximal, characteristic, direct product factor, Sylow
244C2×C89—22no—
245C2×C89—42no—
246C2×C89—161no—
247C2×C89#304611yescharacteristic
248C2×C89C2×#46611yescharacteristic
249C2×C89#306911yescharacteristic
250#3226C2×C8911yescharacteristic, complemented
251#3039C2×C8911yescharacteristic, complemented
252#3035C2×C8911yescharacteristic
253#2988C2×C8911yescharacteristic, complemented
254#1682C2×C8912yescomplemented
255#1681C2×C8912yescomplemented
256#1674C2×C8912yescomplemented
257#1673C2×C8912yescomplemented
258#1571C2×C8911yescharacteristic, complemented
259#1547C2×C8911yescharacteristic, complemented
260C2×#332C2×C8911yescharacteristic, complemented
261C89#1941611yescharacteristic, direct product factor, Sylow
262C2×#137—21no—
263C2×#135—21no—
264#532C22×C8911yescharacteristic
265C2×C4⋊Q8C22×C8911yescharacteristic
266C4×#41C22×C8911yescharacteristic
267#500C22×C8912yes—
268#486C22×C8912yes—
269#480—22no—
270#474—21no—
271#474—21no—
272#464—21no—
273#463—21no—
274#462—21no—
275#462C22×C8911yescharacteristic
276#462C22×C8912yes—
277#461—21no—
278#461C22×C8911yescharacteristic
279#461C22×C8912yes—
280#460C22×C8911yescharacteristic
281#459C22×C8911yescharacteristic
282C8×Q8C22×C8912yes—
283C8×D4—22no—
284C8×D4C22×C8912yes—
285#431—21no—
286#431C22×C8911yescharacteristic
287#423C22×C8911yescharacteristic
288C2×C4⋊C8—21no—
289C2×C4⋊C8C22×C8911yescharacteristic
290C2×C4⋊C8C22×C8911yescharacteristic
291#419—21no—
292#416—21no—
293#335C22×C8912yes—
294#335C22×C8912yes—
295#332C22×C8912yes—
296#332C22×C8912yes—
297#332C22×C8912yes—
298#331C22×C8912yes—
299C2×#41—21no—
300C2×#41D4×C8911yescharacteristic
301C22×Q8—21no—
302#137—44no—
303#135—44no—
304C2×Q16—21no—
305C2×Q16—22no—
306C2×SD16—21no—
307C2×SD16—21no—
308C2×SD16—22no—
309C2×D8—21no—
310C2×M16—21no—
311C22×C8—21no—
312C22×C8—21no—
313C22×C8—22no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8C23×C8912yes—
318C4⋊Q8C23×C8911yescharacteristic
319#127C23×C8911yescharacteristic
320#124—21no—
321#121—21no—
322#121—21no—
323C4×Q8D4×C8911yescharacteristic
324C4×Q8C23×C8911yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4D4×C8911yescharacteristic
330C4×D4C23×C8911yescharacteristic
331#117—21no—
332#117D4×C8911yescharacteristic
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C42—21no—
336C2×C42C23×C8911yescharacteristic
337#107—22no—
338#107—22no—
339C4⋊C8—22no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8D4×C8911yescharacteristic
344C4⋊C8D4×C8911yescharacteristic
345C4⋊C8C23×C8912yes—
346C4⋊C8C23×C8912yes—
347C4⋊C8C23×C8911yescharacteristic
348C4⋊C8C23×C8911yescharacteristic
349#103—21no—
350#103—21no—
351#103—22no—
352#103—22no—
353#102—22no—
354#102—22no—
355#102—21no—
356#102—21no—
357C22⋊C8—22no—
358C22⋊C8D4×C8912yes—
359C4×C8—22no—
360C4×C8C23×C8912yes—
361#41—22no—
362#41—42no—
363#41—42no—
364#41D8×C8914yes—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—41no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8C2×D4×C8911yescharacteristic
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4C2×D4×C8911yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4C2×D4×C8911yescharacteristic
383C22×C4C2×D4×C8911yescharacteristic
384C22×C4#41×C8911yescharacteristic
385Q16—42no—
386Q16—44no—
387SD16—42no—
388SD16—42no—
389SD16—44no—
390D8—42no—
391M16—42no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—42no—
397C2×C8—82no—
398C2×C8—81no—
399C2×C8C2×D4×C8912yes—
400C2×C8C2×D4×C8912yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4C2×D4×C8911yescharacteristic
410C4⋊C4C2×D4×C8911yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42C2×D4×C8911yescharacteristic
420C42C2×D4×C8911yescharacteristic
421C42#41×C8911yescharacteristic
422C42C24×C8911yescharacteristic
423C23—21no—
424C23—41no—
425C23#121×C8911yescharacteristic
426Q8—21no—
427Q8—42no—
428Q8—41no—
429Q8—42no—
430Q8—41no—
431Q8—82no—
432Q8C2×D8×C8912yes—
433D4—24no—
434D4—21no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—81no—
440D4—82no—
441D4C2×D8×C8912yes—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#121×C8911yescharacteristic
457C2×C4#121×C8911yescharacteristic
458C2×C4#121×C8911yescharacteristic
459C2×C4C2×D8×C8912yes—
460C2×C4#135×C8912yes—
461C2×C4C22×D4×C8911yescharacteristic
462C2×C4C22×D4×C8911yescharacteristic
463C2×C4C2×#41×C8911yescharacteristic
464C8—22no—
465C8—22no—
466C8—41no—
467C8—41no—
468C8—42no—
469C8—42no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#466×C8912yes—
479C22C2×#121×C8911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—82no—
488C4#466×C8912yes—
489C4C22×D8×C8911yescharacteristic
490C4C2×#135×C8911yescharacteristic
491C4#577×C8911yescharacteristic
492C4#578×C8911yescharacteristic
493C2—22no—
494C2—42no—
495C2—161no—
496C2C89×#304611yescharacteristic
497C2C2×#466×C8911yescharacteristic
498C2C89×#306911yescharacteristic
subgroupG/H
center, socleC22×C89C2×#121
commutator, Frattini, cosocleC42C24×C89
prime radicalC89×#3035C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3226C2semidirect
2. C89×#3039C2semidirect
4. C89×#2988C2semidirect
5. C89×#1682C2semidirect
6. C89×#1681C2semidirect
7. C89×#1674C2semidirect
8. C89×#1673C2semidirect
9. C89×#1571C2semidirect
10. C89×#1547C2semidirect
11. C2×#332×C89C2semidirect
243. #19416C89direct (N × H)
250. #3226C2×C89semidirect
251. #3039C2×C89semidirect
253. #2988C2×C89semidirect
254. #1682C2×C89semidirect
255. #1681C2×C89semidirect
256. #1674C2×C89semidirect
257. #1673C2×C89semidirect
258. #1571C2×C89semidirect
259. #1547C2×C89semidirect
260. C2×#332C2×C89semidirect
261. C89#19416direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#32262
C89×#30392
C89×#30352
C89×#29882
C89×#16822
C89×#16812
C89×#16742
C89×#16732
C89×#15712
C89×#15472
C2×#332×C892
#1941689