Finite Group Databasebeta

← C89×#62000 #36439848

Subgroups — C89×#62000 #36439848

Order 22784.

1818 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3503C211yesmaximal, characteristic, complemented
2C89×#3482C211yesmaximal, characteristic
3C89×#3395C212yesmaximal, complemented
4C89×#3359C212yesmaximal, complemented
5C89×#3328C216yesmaximal, complemented
6C89×#3276C216yesmaximal, complemented
7C89×#3255C213yesmaximal, complemented
8C89×#3209C213yesmaximal, complemented
9C89×#3107C213yesmaximal, complemented
10C89×#3039C213yesmaximal, complemented
11C89×#2991C211yesmaximal, characteristic, complemented
12C2×#137×C89C2216yes—
13Q82×C89—21no—
14#557×C89C2212yes—
15#557×C89C2211yescharacteristic
16#556×C89C2213yes—
17#554×C89—23no—
18#552×C89—23no—
19D4×Q8×C89C2216yes—
20D4×Q8×C89C2211yescharacteristic
21#548×C89—21no—
22#547×C89C2213yes—
23#544×C89—23no—
24#533×C89C2213yes—
25#533×C89C2213yes—
26C2×C4⋊Q8×C89C2213yes—
27#529×C89—23no—
28#520×C89—21no—
29C4×#41×C89C2213yes—
30C4×#41×C89C2211yescharacteristic
31C2×C4×Q8×C89—21no—
32#494×C89C2216yes—
33#487×C89C2216yes—
34#485×C89C2216yes—
35#475×C89C2216yes—
36#475×C89C2213yes—
37#474×C89C2211yescharacteristic
38#470×C89C22112yes—
39#462×C89C2216yes—
40#462×C89C2213yes—
41#462×C89C2216yes—
42#462×C89C2216yes—
43#461×C89C2216yes—
44#461×C89C2213yes—
45#452×C89C2216yes—
46#451×C89C22112yes—
47C8×Q8×C89C2212yes—
48C4×Q16×C89C2216yes—
49C4×SD16×C89C2216yes—
50#436×C89C2216yes—
51#431×C89C2213yes—
52#423×C89C2213yes—
53#419×C89C2216yes—
54C2×#41×C89D411yescharacteristic
55C2×#41×C89C2313yes—
56C22×Q8×C89—21no—
57C22×Q8×C89C2316yes—
58#137×C89—412no—
59C2×Q16×C89C2316yes—
60C2×Q16×C89C23112yes—
61C2×SD16×C89C2316yes—
62C2×M16×C89C2316yes—
63C4⋊Q8×C89—26no—
64C4⋊Q8×C89—23no—
65C4⋊Q8×C89—23no—
66C4⋊Q8×C89C2313yes—
67C4⋊Q8×C89C2316yes—
68C4⋊Q8×C89C2313yes—
69C4⋊Q8×C89C2311yescharacteristic
70C4⋊Q8×C89C2313yes—
71#126×C89—26no—
72#125×C89—23no—
73#125×C89—26no—
74#125×C89—23no—
75#125×C89—23no—
76#124×C89—23no—
77#123×C89—26no—
78C22⋊Q8×C89—23no—
79C22⋊Q8×C89—23no—
80C22⋊Q8×C89—23no—
81C22⋊Q8×C89—212no—
82C22⋊Q8×C89C2316yes—
83C22⋊Q8×C89C2316yes—
84#121×C89—23no—
85C4×Q8×C89—21no—
86C4×Q8×C89—22no—
87C4×Q8×C89—23no—
88C4×Q8×C89—23no—
89C4×Q8×C89—23no—
90C4×Q8×C89D411yescharacteristic
91C4×Q8×C89C2316yes—
92C4×Q8×C89C2313yes—
93C4×Q8×C89C2313yes—
94C4×Q8×C89C2311yescharacteristic
95C4×D4×C89—26no—
96C4×D4×C89—23no—
97C4×D4×C89—23no—
98C4×D4×C89D413yes—
99C4×D4×C89C2313yes—
100C4×D4×C89C2311yescharacteristic
101C4×D4×C89C2313yes—
102#117×C89—23no—
103#117×C89—23no—
104#117×C89—23no—
105#117×C89D413yes—
106#117×C89C2313yes—
107C2×C4⋊C4×C89—23no—
108C2×C4⋊C4×C89—23no—
109C2×C4⋊C4×C89—23no—
110C2×C42×C89—23no—
111C2×C42×C89C2313yes—
112#107×C89C2312yes—
113#106×C89C2316yes—
114C4⋊C8×C89C2313yes—
115C4⋊C8×C89C2316yes—
116C4⋊C8×C89C2311yescharacteristic
117#103×C89C2316yes—
118#103×C89C23112yes—
119#103×C89C2316yes—
120#103×C89C2316yes—
121#102×C89C2312yes—
122C22⋊C8×C89C2316yes—
123C4×C8×C89C2316yes—
124#41×C89—24no—
125#41×C89—26no—
126#41×C89—43no—
127C2×Q8×C89—21no—
128C2×Q8×C89—23no—
129C2×Q8×C89—26no—
130C2×Q8×C89—46no—
131C2×Q8×C89—46no—
132C2×Q8×C89—44no—
133C2×Q8×C89C2×D411yescharacteristic
134C2×Q8×C89C2416yes—
135C2×Q8×C89C2413yes—
136C2×D4×C89—23no—
137C2×D4×C89C2×D413yes—
138C2×D4×C89C2411yescharacteristic
139C22×C4×C89—26no—
140C22×C4×C89—21no—
141C22×C4×C89—23no—
142C22×C4×C89—23no—
143C22×C4×C89C2×D413yes—
144C22×C4×C89C2×D413yes—
145C22×C4×C89C2413yes—
146Q16×C89—46no—
147Q16×C89—412no—
148SD16×C89—212no—
149M16×C89—46no—
150C2×C8×C89C2412yes—
151C2×C8×C89C2416yes—
152C4⋊C4×C89—23no—
153C4⋊C4×C89—23no—
154C4⋊C4×C89—26no—
155C4⋊C4×C89—23no—
156C4⋊C4×C89—26no—
157C4⋊C4×C89—26no—
158C4⋊C4×C89—21no—
159C4⋊C4×C89—23no—
160C4⋊C4×C89—23no—
161C4⋊C4×C89—23no—
162C4⋊C4×C89—23no—
163C4⋊C4×C89—26no—
164C4⋊C4×C89—26no—
165C4⋊C4×C89C2×D413yes—
166C4⋊C4×C89C2×D413yes—
167C4⋊C4×C89C2412yes—
168C4⋊C4×C89C2413yes—
169C4⋊C4×C89C2411yescharacteristic
170C22⋊C4×C89—26no—
171C22⋊C4×C89—26no—
172C22⋊C4×C89—23no—
173C22⋊C4×C89—21no—
174C22⋊C4×C89—23no—
175C22⋊C4×C89C2×D416yes—
176C42×C89—23no—
177C42×C89—23no—
178C42×C89—23no—
179C42×C89—26no—
180C42×C89C2×D413yes—
181C42×C89C2×D413yes—
182C42×C89C2413yes—
183C42×C89C2411yescharacteristic
184C23×C89—21no—
185C23×C89C22×D413yes—
186Q8×C89—26no—
187Q8×C89—23no—
188Q8×C89—23no—
189Q8×C89—26no—
190Q8×C89—46no—
191Q8×C89—42no—
192Q8×C89—86no—
193Q8×C89#13714yescomplemented
194D4×C89—26no—
195D4×C89—21no—
196D4×C89—46no—
197D4×C89H(5,𝔽2)12yes—
198C2×C4×C89—26no—
199C2×C4×C89—26no—
200C2×C4×C89—23no—
201C2×C4×C89—23no—
202C2×C4×C89—21no—
203C2×C4×C89—22no—
204C2×C4×C89—23no—
205C2×C4×C89—23no—
206C2×C4×C89—26no—
207C2×C4×C89—46no—
208C2×C4×C89—46no—
209C2×C4×C89—42no—
210C2×C4×C89—86no—
211C2×C4×C89C22×D413yes—
212C2×C4×C89C22×D411yescharacteristic
213C2×C4×C89C22×D413yes—
214C2×C4×C89C22×D413yes—
215C2×C4×C89C22×D411yescharacteristic
216C2×C4×C89C2511yescharacteristic
217C8×C89—26no—
218C8×C89—22no—
219C22×C89—23no—
220C22×C89—22no—
221C22×C89—23no—
222C22×C89—23no—
223C22×C89—41no—
224C22×C89—41no—
225C22×C89C23×D411yescharacteristic
226C4×C89—23no—
227C4×C89—21no—
228C4×C89—23no—
229C4×C89—43no—
230C4×C89—46no—
231C4×C89—43no—
232C4×C89—41no—
233C4×C89—42no—
234C4×C89C2×#13716yes—
235C4×C89#57812yes—
236C4×C89C2×H(5,𝔽2)11yescharacteristic
237C4×C89#58511yescharacteristic
238#62000C8911yesmaximal, characteristic, direct product factor, Sylow
239C2×C89—43no—
240C2×C89—42no—
241C2×C89#348811yescharacteristic
242C2×C89C22×#13711yescharacteristic
243C2×C89C2×#57811yescharacteristic
244#3503C2×C8911yescharacteristic, complemented
245#3482C2×C8911yescharacteristic
246#3395C2×C8912yescomplemented
247#3359C2×C8912yescomplemented
248#3328C2×C8916yescomplemented
249#3276C2×C8916yescomplemented
250#3255C2×C8913yescomplemented
251#3209C2×C8913yescomplemented
252#3107C2×C8913yescomplemented
253#3039C2×C8913yescomplemented
254#2991C2×C8911yescharacteristic, complemented
255C89#6200011yescharacteristic, direct product factor, Sylow
256C2×#137C22×C8916yes—
257Q82—21no—
258#557C22×C8912yes—
259#557C22×C8911yescharacteristic
260#556C22×C8913yes—
261#554—23no—
262#552—23no—
263D4×Q8C22×C8916yes—
264D4×Q8C22×C8911yescharacteristic
265#548—21no—
266#547C22×C8913yes—
267#544—23no—
268#533C22×C8913yes—
269#533C22×C8913yes—
270C2×C4⋊Q8C22×C8913yes—
271#529—23no—
272#520—21no—
273C4×#41C22×C8913yes—
274C4×#41C22×C8911yescharacteristic
275C2×C4×Q8—21no—
276#494C22×C8916yes—
277#487C22×C8916yes—
278#485C22×C8916yes—
279#475C22×C8916yes—
280#475C22×C8913yes—
281#474C22×C8911yescharacteristic
282#470C22×C89112yes—
283#462C22×C8916yes—
284#462C22×C8913yes—
285#462C22×C8916yes—
286#462C22×C8916yes—
287#461C22×C8916yes—
288#461C22×C8913yes—
289#452C22×C8916yes—
290#451C22×C89112yes—
291C8×Q8C22×C8912yes—
292C4×Q16C22×C8916yes—
293C4×SD16C22×C8916yes—
294#436C22×C8916yes—
295#431C22×C8913yes—
296#423C22×C8913yes—
297#419C22×C8916yes—
298C2×#41D4×C8911yescharacteristic
299C2×#41C23×C8913yes—
300C22×Q8—21no—
301C22×Q8C23×C8916yes—
302#137—412no—
303C2×Q16C23×C89112yes—
304C2×Q16C23×C8916yes—
305C2×SD16C23×C8916yes—
306C2×M16C23×C8916yes—
307C4⋊Q8—26no—
308C4⋊Q8—23no—
309C4⋊Q8—23no—
310C4⋊Q8C23×C8913yes—
311C4⋊Q8C23×C8916yes—
312C4⋊Q8C23×C8913yes—
313C4⋊Q8C23×C8911yescharacteristic
314C4⋊Q8C23×C8913yes—
315#126—26no—
316#125—23no—
317#125—26no—
318#125—23no—
319#125—23no—
320#124—23no—
321#123—26no—
322C22⋊Q8—23no—
323C22⋊Q8—23no—
324C22⋊Q8—23no—
325C22⋊Q8—212no—
326C22⋊Q8C23×C8916yes—
327C22⋊Q8C23×C8916yes—
328#121—23no—
329C4×Q8—21no—
330C4×Q8—22no—
331C4×Q8—23no—
332C4×Q8—23no—
333C4×Q8—23no—
334C4×Q8D4×C8911yescharacteristic
335C4×Q8C23×C8913yes—
336C4×Q8C23×C8913yes—
337C4×Q8C23×C8916yes—
338C4×Q8C23×C8911yescharacteristic
339C4×D4—26no—
340C4×D4—23no—
341C4×D4—23no—
342C4×D4D4×C8913yes—
343C4×D4C23×C8911yescharacteristic
344C4×D4C23×C8913yes—
345C4×D4C23×C8913yes—
346#117—23no—
347#117—23no—
348#117—23no—
349#117D4×C8913yes—
350#117C23×C8913yes—
351C2×C4⋊C4—23no—
352C2×C4⋊C4—23no—
353C2×C4⋊C4—23no—
354C2×C42—23no—
355C2×C42C23×C8913yes—
356#107C23×C8912yes—
357#106C23×C8916yes—
358C4⋊C8C23×C8913yes—
359C4⋊C8C23×C8916yes—
360C4⋊C8C23×C8911yescharacteristic
361#103C23×C8916yes—
362#103C23×C89112yes—
363#103C23×C8916yes—
364#103C23×C8916yes—
365#102C23×C8912yes—
366C22⋊C8C23×C8916yes—
367C4×C8C23×C8916yes—
368#41—24no—
369#41—26no—
370#41—43no—
371C2×Q8—21no—
372C2×Q8—23no—
373C2×Q8—26no—
374C2×Q8—46no—
375C2×Q8—46no—
376C2×Q8—44no—
377C2×Q8C2×D4×C8911yescharacteristic
378C2×Q8C24×C8916yes—
379C2×Q8C24×C8913yes—
380C2×D4—23no—
381C2×D4C2×D4×C8913yes—
382C2×D4C24×C8911yescharacteristic
383C22×C4—26no—
384C22×C4—21no—
385C22×C4—23no—
386C22×C4—23no—
387C22×C4C2×D4×C8913yes—
388C22×C4C2×D4×C8913yes—
389C22×C4C24×C8913yes—
390Q16—412no—
391Q16—46no—
392SD16—212no—
393M16—46no—
394C2×C8C24×C8912yes—
395C2×C8C24×C8916yes—
396C4⋊C4—23no—
397C4⋊C4—23no—
398C4⋊C4—26no—
399C4⋊C4—23no—
400C4⋊C4—26no—
401C4⋊C4—26no—
402C4⋊C4—21no—
403C4⋊C4—23no—
404C4⋊C4—23no—
405C4⋊C4—23no—
406C4⋊C4—23no—
407C4⋊C4—26no—
408C4⋊C4—26no—
409C4⋊C4C2×D4×C8913yes—
410C4⋊C4C2×D4×C8913yes—
411C4⋊C4C24×C8912yes—
412C4⋊C4C24×C8913yes—
413C4⋊C4C24×C8911yescharacteristic
414C22⋊C4—26no—
415C22⋊C4—26no—
416C22⋊C4—23no—
417C22⋊C4—21no—
418C22⋊C4—23no—
419C22⋊C4C2×D4×C8916yes—
420C42—23no—
421C42—23no—
422C42—23no—
423C42—26no—
424C42C2×D4×C8913yes—
425C42C2×D4×C8913yes—
426C42C24×C8913yes—
427C42C24×C8911yescharacteristic
428C23—21no—
429C23C22×D4×C8913yes—
430Q8—26no—
431Q8—23no—
432Q8—23no—
433Q8—26no—
434Q8—46no—
435Q8—42no—
436Q8—86no—
437Q8#137×C8914yescomplemented
438D4—26no—
439D4—21no—
440D4—46no—
441D4H(5,𝔽2)×C8912yes—
442C2×C4—26no—
443C2×C4—26no—
444C2×C4—23no—
445C2×C4—23no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—23no—
449C2×C4—23no—
450C2×C4—26no—
451C2×C4—46no—
452C2×C4—46no—
453C2×C4—42no—
454C2×C4—86no—
455C2×C4C22×D4×C8913yes—
456C2×C4C22×D4×C8911yescharacteristic
457C2×C4C22×D4×C8913yes—
458C2×C4C22×D4×C8913yes—
459C2×C4C22×D4×C8911yescharacteristic
460C2×C4C25×C8911yescharacteristic
461C8—26no—
462C8—22no—
463C22—23no—
464C22—23no—
465C22—23no—
466C22—22no—
467C22—41no—
468C22—41no—
469C22C23×D4×C8911yescharacteristic
470C4—21no—
471C4—23no—
472C4—23no—
473C4—43no—
474C4—41no—
475C4—42no—
476C4—43no—
477C4—46no—
478C4C2×#137×C8916yes—
479C4#578×C8912yes—
480C4C2×H(5,𝔽2)×C8911yescharacteristic
481C4#585×C8911yescharacteristic
482C2—43no—
483C2—42no—
484C2C89×#348811yescharacteristic
485C2C22×#137×C8911yescharacteristic
486C2C2×#578×C8911yescharacteristic
subgroupG/H
center, socleC22×C89C23×D4
commutator, Frattini, cosocleC2×C4C25×C89
prime radicalC89×#3482C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3503C2semidirect
3. C89×#3395C2semidirect
4. C89×#3359C2semidirect
5. C89×#3328C2semidirect
6. C89×#3276C2semidirect
7. C89×#3255C2semidirect
8. C89×#3209C2semidirect
9. C89×#3107C2semidirect
10. C89×#3039C2semidirect
11. C89×#2991C2semidirect
193. Q8×C89#137semidirect
238. #62000C89direct (N × H)
244. #3503C2×C89semidirect
246. #3395C2×C89semidirect
247. #3359C2×C89semidirect
248. #3328C2×C89semidirect
249. #3276C2×C89semidirect
250. #3255C2×C89semidirect
251. #3209C2×C89semidirect
252. #3107C2×C89semidirect
253. #3039C2×C89semidirect
254. #2991C2×C89semidirect
255. C89#62000direct (N × H)
437. Q8#137×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#35032
C89×#34822
C89×#33952
C89×#33592
C89×#33282
C89×#32762
C89×#32552
C89×#32092
C89×#31072
C89×#30392
C89×#29912
#6200089