Finite Group Databasebeta

← C89×#36567 #36414415

Subgroups — C89×#36567 #36414415

Order 22784.

1574 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2864C212yesmaximal
2C89×#2859C211yesmaximal, characteristic
3C89×#2590C214yesmaximal
4C89×#2585C214yesmaximal
5C89×#2584C212yesmaximal
6C89×#2507C212yesmaximal
7C89×#2506C212yesmaximal
8C89×#2462C214yesmaximal
9C89×#2370C212yesmaximal
10C89×#2370C214yesmaximal
11C4×#125×C89C211yesmaximal, characteristic
12C4×#125×C89C212yesmaximal
13C42×Q8×C89C211yesmaximal, characteristic
14C2×#125×C89C2212yes—
15C2×#125×C89C2214yes—
16C2×#125×C89C2211yescharacteristic
17C2×C4×Q8×C89C2211yescharacteristic
18C2×C4×Q8×C89C2212yes—
19#400×C89C2214yes—
20#398×C89C2214yes—
21#398×C89C2214yes—
22#398×C89C2214yes—
23#398×C89C2214yes—
24#395×C89C2214yes—
25#389×C89C2212yes—
26#389×C89C2212yes—
27#389×C89C2212yes—
28#389×C89C2214yes—
29#389×C89C2214yes—
30#389×C89C2212yes—
31#389×C89C2212yes—
32#389×C89C2214yes—
33#389×C89C2214yes—
34#389×C89C2214yes—
35#389×C89C2214yes—
36#389×C89C2212yes—
37#389×C89C2212yes—
38#387×C89C2214yes—
39#387×C89C2214yes—
40#387×C89C2214yes—
41#387×C89C2214yes—
42#387×C89C2214yes—
43#387×C89C2214yes—
44#387×C89C2214yes—
45#387×C89C2214yes—
46#387×C89C2214yes—
47#387×C89C2214yes—
48#387×C89C2214yes—
49#384×C89C2211yescharacteristic
50#384×C89C2211yescharacteristic
51#382×C89C2212yes—
52#382×C89C2211yescharacteristic
53C4×C4⋊C4×C89C2212yes—
54C4×C4⋊C4×C89C2214yes—
55C4×C4⋊C4×C89C2214yes—
56C4×C4⋊C4×C89C2214yes—
57C4×C4⋊C4×C89C2212yes—
58C4×C4⋊C4×C89C2212yes—
59C4×C4⋊C4×C89C2212yes—
60C4×C4⋊C4×C89C2212yes—
61C4×C4⋊C4×C89C2214yes—
62C4×C4⋊C4×C89C2211yescharacteristic
63C4×C4⋊C4×C89C2212yes—
64C4×C4⋊C4×C89C2212yes—
65C43×C89C2211yescharacteristic
66C43×C89C2212yes—
67C22×Q8×C89C2311yescharacteristic
68#125×C89—216no—
69#125×C89—28no—
70#125×C89Q818yescomplemented
71C4×Q8×C89—28no—
72C4×Q8×C89—24no—
73C2×C4⋊C4×C89C2312yes—
74C2×C4⋊C4×C89C2312yes—
75C2×C4⋊C4×C89C2312yes—
76C2×C4⋊C4×C89C2311yescharacteristic
77C2×C4⋊C4×C89C2312yes—
78C2×C4⋊C4×C89C2312yes—
79C2×C4⋊C4×C89C2312yes—
80C2×C4⋊C4×C89C2314yes—
81C2×C4⋊C4×C89C2314yes—
82C2×C4⋊C4×C89C2314yes—
83C2×C4⋊C4×C89C2314yes—
84C2×C4⋊C4×C89C2314yes—
85C2×C4⋊C4×C89C2314yes—
86C2×C4⋊C4×C89C2314yes—
87C2×C4⋊C4×C89C2314yes—
88C2×C4⋊C4×C89C2314yes—
89C2×C4⋊C4×C89C2314yes—
90C2×C4⋊C4×C89C2314yes—
91C2×C4⋊C4×C89C2314yes—
92C2×C4⋊C4×C89C2312yes—
93C2×C4⋊C4×C89C2312yes—
94C2×C42×C89C2312yes—
95C2×C42×C89C2312yes—
96C2×C42×C89C2311yescharacteristic
97C2×C42×C89C2312yes—
98C2×C42×C89C2312yes—
99C2×C42×C89C2312yes—
100C2×C42×C89C2314yes—
101C2×C42×C89C2312yes—
102C2×C42×C89C2312yes—
103C2×C42×C89C2312yes—
104C2×C42×C89C2314yes—
105C2×C42×C89C2314yes—
106C2×C42×C89C2311yescharacteristic
107C2×C42×C89C2311yescharacteristic
108#95×C89C2311yescharacteristic
109#95×C89C2311yescharacteristic
110#95×C89C2312yes—
111#95×C89C2312yes—
112#95×C89C2314yes—
113#95×C89C2314yes—
114#95×C89C2314yes—
115#95×C89C2314yes—
116#95×C89C2314yes—
117#95×C89C2314yes—
118#95×C89C2314yes—
119#95×C89C2314yes—
120#95×C89C2314yes—
121#95×C89C2314yes—
122#95×C89C2314yes—
123#95×C89C2314yes—
124#95×C89C2314yes—
125C2×Q8×C89—24no—
126C2×Q8×C89—22no—
127C22×C4×C89C2412yes—
128C22×C4×C89C2414yes—
129C22×C4×C89C2412yes—
130C22×C4×C89C2414yes—
131C22×C4×C89C2414yes—
132C22×C4×C89C2412yes—
133C22×C4×C89C2414yes—
134C22×C4×C89C2412yes—
135C22×C4×C89C2412yes—
136C22×C4×C89C2412yes—
137C22×C4×C89C2411yescharacteristic
138C22×C4×C89C2411yescharacteristic
139C22×C4×C89C2411yescharacteristic
140C4⋊C4×C89—24no—
141C4⋊C4×C89—24no—
142C4⋊C4×C89—24no—
143C4⋊C4×C89—22no—
144C4⋊C4×C89—28no—
145C4⋊C4×C89—28no—
146C4⋊C4×C89—28no—
147C4⋊C4×C89—28no—
148C4⋊C4×C89—28no—
149C4⋊C4×C89—28no—
150C4⋊C4×C89—28no—
151C4⋊C4×C89—24no—
152C4⋊C4×C89—24no—
153C4⋊C4×C89—24no—
154C4⋊C4×C89—44no—
155C4⋊C4×C89—44no—
156C4⋊C4×C89—44no—
157C4⋊C4×C89—44no—
158C4⋊C4×C89C2×Q8116yes—
159C4⋊C4×C89C2×Q818yes—
160C4⋊C4×C89#4118yes—
161C42×C89—24no—
162C42×C89—24no—
163C42×C89—24no—
164C42×C89—24no—
165C42×C89—28no—
166C42×C89—22no—
167C42×C89—24no—
168C42×C89—42no—
169C42×C89—42no—
170C42×C89—44no—
171C42×C89—44no—
172C42×C89—42no—
173C42×C89C2×Q814yes—
174C42×C89#4114yes—
175C23×C89C2511yescharacteristic
176Q8×C89—28no—
177C2×C4×C89—22no—
178C2×C4×C89—24no—
179C2×C4×C89—22no—
180C2×C4×C89—22no—
181C2×C4×C89—22no—
182C2×C4×C89—22no—
183C2×C4×C89—22no—
184C2×C4×C89—22no—
185C2×C4×C89—22no—
186C2×C4×C89—22no—
187C2×C4×C89—24no—
188C2×C4×C89—22no—
189C2×C4×C89—24no—
190C2×C4×C89—22no—
191C2×C4×C89—22no—
192C2×C4×C89—24no—
193C2×C4×C89—24no—
194C2×C4×C89—24no—
195C2×C4×C89—24no—
196C2×C4×C89—22no—
197C2×C4×C89—24no—
198C2×C4×C89—22no—
199C2×C4×C89—22no—
200C2×C4×C89—24no—
201C2×C4×C89—22no—
202C2×C4×C89—24no—
203C2×C4×C89—24no—
204C2×C4×C89C22×Q814yes—
205C2×C4×C89C22×Q818yes—
206C2×C4×C89C22×Q812yes—
207C2×C4×C89C2×#4114yes—
208C2×C4×C89C2×#4114yes—
209C2×C4×C89C2×#4114yes—
210C2×C4×C89C2×#4112yes—
211C2×C4×C89C2×#4112yes—
212C2×C4×C89C2×#4112yes—
213C2×C4×C892−1+412yes—
214C22×C89C23×Q811yescharacteristic
215C22×C89C22×#4111yescharacteristic
216C22×C89C22×#4112yes—
217C22×C89C2×2−1+411yescharacteristic
218C22×C89#58512yes—
219C4×C89—24no—
220C4×C89—22no—
221C4×C89—24no—
222C4×C89—28no—
223C4×C89—22no—
224C4×C89—24no—
225C4×C89—24no—
226C4×C89—44no—
227C4×C89—42no—
228C4×C89—44no—
229C4×C89—42no—
230C4×C89—44no—
231C4×C89#55714yes—
232#36567C8911yesmaximal, characteristic, direct product factor, Sylow
233C2×C89#344411yescharacteristic
234C2×C89C2×#55711yescharacteristic
235C2×C89Q8×#4112yes—
236C2×C89#351211yescharacteristic
237C2×C89#351512yes—
238#2864C2×C8912yes—
239#2859C2×C8911yescharacteristic
240#2590C2×C8914yes—
241#2585C2×C8914yes—
242#2584C2×C8912yes—
243#2507C2×C8912yes—
244#2506C2×C8912yes—
245#2462C2×C8914yes—
246#2370C2×C8912yes—
247#2370C2×C8914yes—
248C4×#125C2×C8911yescharacteristic
249C4×#125C2×C8912yes—
250C42×Q8C2×C8911yescharacteristic
251C89#3656711yescharacteristic, direct product factor, Sylow
252C2×#125C22×C8912yes—
253C2×#125C22×C8914yes—
254C2×#125C22×C8911yescharacteristic
255C2×C4×Q8C22×C8911yescharacteristic
256C2×C4×Q8C22×C8912yes—
257#400C22×C8914yes—
258#398C22×C8914yes—
259#398C22×C8914yes—
260#398C22×C8914yes—
261#398C22×C8914yes—
262#395C22×C8914yes—
263#389C22×C8912yes—
264#389C22×C8912yes—
265#389C22×C8912yes—
266#389C22×C8914yes—
267#389C22×C8914yes—
268#389C22×C8912yes—
269#389C22×C8912yes—
270#389C22×C8914yes—
271#389C22×C8914yes—
272#389C22×C8914yes—
273#389C22×C8914yes—
274#389C22×C8912yes—
275#389C22×C8912yes—
276#387C22×C8914yes—
277#387C22×C8914yes—
278#387C22×C8914yes—
279#387C22×C8914yes—
280#387C22×C8914yes—
281#387C22×C8914yes—
282#387C22×C8914yes—
283#387C22×C8914yes—
284#387C22×C8914yes—
285#387C22×C8914yes—
286#387C22×C8914yes—
287#384C22×C8911yescharacteristic
288#384C22×C8911yescharacteristic
289#382C22×C8912yes—
290#382C22×C8911yescharacteristic
291C4×C4⋊C4C22×C8912yes—
292C4×C4⋊C4C22×C8914yes—
293C4×C4⋊C4C22×C8914yes—
294C4×C4⋊C4C22×C8914yes—
295C4×C4⋊C4C22×C8912yes—
296C4×C4⋊C4C22×C8912yes—
297C4×C4⋊C4C22×C8912yes—
298C4×C4⋊C4C22×C8912yes—
299C4×C4⋊C4C22×C8914yes—
300C4×C4⋊C4C22×C8911yescharacteristic
301C4×C4⋊C4C22×C8912yes—
302C4×C4⋊C4C22×C8912yes—
303C43C22×C8911yescharacteristic
304C43C22×C8912yes—
305C22×Q8C23×C8911yescharacteristic
306#125—216no—
307#125—28no—
308#125Q8×C8918yescomplemented
309C4×Q8—28no—
310C4×Q8—24no—
311C2×C4⋊C4C23×C8912yes—
312C2×C4⋊C4C23×C8912yes—
313C2×C4⋊C4C23×C8912yes—
314C2×C4⋊C4C23×C8911yescharacteristic
315C2×C4⋊C4C23×C8912yes—
316C2×C4⋊C4C23×C8912yes—
317C2×C4⋊C4C23×C8914yes—
318C2×C4⋊C4C23×C8914yes—
319C2×C4⋊C4C23×C8914yes—
320C2×C4⋊C4C23×C8914yes—
321C2×C4⋊C4C23×C8914yes—
322C2×C4⋊C4C23×C8914yes—
323C2×C4⋊C4C23×C8914yes—
324C2×C4⋊C4C23×C8912yes—
325C2×C4⋊C4C23×C8914yes—
326C2×C4⋊C4C23×C8914yes—
327C2×C4⋊C4C23×C8914yes—
328C2×C4⋊C4C23×C8914yes—
329C2×C4⋊C4C23×C8914yes—
330C2×C4⋊C4C23×C8912yes—
331C2×C4⋊C4C23×C8912yes—
332C2×C42C23×C8912yes—
333C2×C42C23×C8912yes—
334C2×C42C23×C8912yes—
335C2×C42C23×C8912yes—
336C2×C42C23×C8914yes—
337C2×C42C23×C8912yes—
338C2×C42C23×C8911yescharacteristic
339C2×C42C23×C8912yes—
340C2×C42C23×C8912yes—
341C2×C42C23×C8912yes—
342C2×C42C23×C8914yes—
343C2×C42C23×C8914yes—
344C2×C42C23×C8911yescharacteristic
345C2×C42C23×C8911yescharacteristic
346#95C23×C8911yescharacteristic
347#95C23×C8911yescharacteristic
348#95C23×C8912yes—
349#95C23×C8912yes—
350#95C23×C8914yes—
351#95C23×C8914yes—
352#95C23×C8914yes—
353#95C23×C8914yes—
354#95C23×C8914yes—
355#95C23×C8914yes—
356#95C23×C8914yes—
357#95C23×C8914yes—
358#95C23×C8914yes—
359#95C23×C8914yes—
360#95C23×C8914yes—
361#95C23×C8914yes—
362#95C23×C8914yes—
363C2×Q8—24no—
364C2×Q8—22no—
365C22×C4C24×C8912yes—
366C22×C4C24×C8914yes—
367C22×C4C24×C8912yes—
368C22×C4C24×C8914yes—
369C22×C4C24×C8914yes—
370C22×C4C24×C8912yes—
371C22×C4C24×C8914yes—
372C22×C4C24×C8912yes—
373C22×C4C24×C8912yes—
374C22×C4C24×C8912yes—
375C22×C4C24×C8911yescharacteristic
376C22×C4C24×C8911yescharacteristic
377C22×C4C24×C8911yescharacteristic
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4C2×Q8×C89116yes—
397C4⋊C4C2×Q8×C8918yes—
398C4⋊C4#41×C8918yes—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—28no—
404C42—22no—
405C42—24no—
406C42—42no—
407C42—42no—
408C42—44no—
409C42—44no—
410C42—42no—
411C42C2×Q8×C8914yes—
412C42#41×C8914yes—
413C23C25×C8911yescharacteristic
414Q8—28no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4C22×Q8×C8914yes—
443C2×C4C22×Q8×C8918yes—
444C2×C4C22×Q8×C8912yes—
445C2×C4C2×#41×C8914yes—
446C2×C4C2×#41×C8914yes—
447C2×C4C2×#41×C8914yes—
448C2×C4C2×#41×C8912yes—
449C2×C4C2×#41×C8912yes—
450C2×C4C2×#41×C8912yes—
451C2×C42−1+4×C8912yes—
452C22C23×Q8×C8911yescharacteristic
453C22C22×#41×C8911yescharacteristic
454C22C22×#41×C8912yes—
455C22C2×2−1+4×C8911yescharacteristic
456C22#585×C8912yes—
457C4—24no—
458C4—22no—
459C4—24no—
460C4—28no—
461C4—22no—
462C4—24no—
463C4—24no—
464C4—44no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—44no—
469C4#557×C8914yes—
470C2C89×#344411yescharacteristic
471C2C2×#557×C8911yescharacteristic
472C2Q8×#41×C8912yes—
473C2C89×#351211yescharacteristic
474C2C89×#351512yes—
subgroupG/H
center, socle, prime radicalC23×C89C25
commutator, Frattini, cosocleC23C25×C89
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C89 ◁ G

Complements

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

NH ≅ G/Ntype
70. #125×C89Q8semidirect
232. #36567C89direct (N × H)
251. C89#36567direct (N × H)
308. #125Q8×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#28642
C89×#28592
C89×#25902
C89×#25852
C89×#25842
C89×#25072
C89×#25062
C89×#24622
C89×#23702
C4×#125×C892
C42×Q8×C892
#3656789