Finite Group Databasebeta

← C89×#33916 #36411764

Subgroups — C89×#33916 #36411764

Order 22784.

464 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3342C211yesmaximal, characteristic
2C89×#3336C211yesmaximal, characteristic
3C89×#2255C211yesmaximal, characteristic, complemented
4C89×#2237C211yesmaximal, characteristic, complemented
5C89×#2237C211yesmaximal, characteristic, complemented
6C89×#2237C211yesmaximal, characteristic, complemented
7C89×#2236C211yesmaximal, characteristic, complemented
8C89×#2229C211yesmaximal, characteristic, complemented
9C89×#2227C211yesmaximal, characteristic, complemented
10C89×#2219C211yesmaximal, characteristic, complemented
11C89×#2216C211yesmaximal, characteristic, complemented
12C89×#2176C211yesmaximal, characteristic, complemented
13C89×#2168C211yesmaximal, characteristic
14C89×#2159C211yesmaximal, characteristic, complemented
15C89×#2158C211yesmaximal, characteristic, complemented
16#556×C89—21no—
17#552×C89—21no—
18C2×Q32×C89C2211yescharacteristic
19C2×M32×C89C2211yescharacteristic
20C22×C16×C89C2211yescharacteristic
21#500×C89C2211yescharacteristic
22#499×C89C2211yescharacteristic
23#485×C89—21no—
24#485×C89—21no—
25#485×C89—21no—
26#483×C89—21no—
27#479×C89—21no—
28#475×C89—21no—
29#467×C89C2211yescharacteristic
30#467×C89C2211yescharacteristic
31#467×C89C2211yescharacteristic
32#467×C89C2211yescharacteristic
33C4×Q16×C89C2211yescharacteristic
34C4×Q16×C89C2211yescharacteristic
35C8×D4×C89C2211yescharacteristic
36#427×C89C2211yescharacteristic
37#427×C89C2211yescharacteristic
38#427×C89C2211yescharacteristic
39C2×#107×C89C2211yescharacteristic
40#367×C89C2211yescharacteristic
41#367×C89C2211yescharacteristic
42#367×C89C2211yescharacteristic
43#367×C89C2211yescharacteristic
44#366×C89C2211yescharacteristic
45#366×C89C2211yescharacteristic
46#366×C89C2211yescharacteristic
47#366×C89C2211yescharacteristic
48#366×C89C2211yescharacteristic
49C4⋊C16×C89C2211yescharacteristic
50#358×C89C2211yescharacteristic
51#358×C89C2211yescharacteristic
52#358×C89C2211yescharacteristic
53#358×C89C2211yescharacteristic
54#358×C89C2211yescharacteristic
55#358×C89C2211yescharacteristic
56C22⋊C16×C89C2211yescharacteristic
57C22⋊C16×C89C2211yescharacteristic
58#346×C89C2211yescharacteristic
59C2×Q16×C89C2311yescharacteristic
60C2×Q16×C89C2311yescharacteristic
61C22×C8×C89C2311yescharacteristic
62C22×C8×C89C2311yescharacteristic
63C4⋊Q8×C89—21no—
64#126×C89—41no—
65#126×C89—41no—
66#126×C89—41no—
67#125×C89—21no—
68#125×C89—41no—
69#124×C89—41no—
70#123×C89—41no—
71C22⋊Q8×C89—21no—
72C22⋊Q8×C89—21no—
73C22⋊Q8×C89—21no—
74C22⋊Q8×C89—21no—
75C4×Q8×C89—21no—
76C4×Q8×C89—21no—
77C4×Q8×C89—41no—
78C4×D4×C89D411yescharacteristic
79#117×C89—21no—
80#117×C89—21no—
81#117×C89—21no—
82#117×C89—41no—
83C2×C4⋊C4×C89—21no—
84Q32×C89—22no—
85M32×C89—22no—
86C2×C16×C89—22no—
87C2×C16×C89C2311yescharacteristic
88C2×C16×C89C2311yescharacteristic
89C2×C16×C89C2311yescharacteristic
90C2×C16×C89C2311yescharacteristic
91#107×C89—21no—
92#107×C89—21no—
93#107×C89C2311yescharacteristic
94#107×C89C2311yescharacteristic
95#107×C89C2311yescharacteristic
96#107×C89C2311yescharacteristic
97#107×C89C2311yescharacteristic
98#107×C89C2311yescharacteristic
99#106×C89—21no—
100#106×C89—21no—
101#106×C89—21no—
102#106×C89—21no—
103C4⋊C8×C89D411yescharacteristic
104#103×C89—21no—
105#103×C89—21no—
106#103×C89—21no—
107#103×C89—21no—
108#103×C89—21no—
109#103×C89—21no—
110C22⋊C8×C89D411yescharacteristic
111C22⋊C8×C89D411yescharacteristic
112C4×C8×C89C2311yescharacteristic
113C2×Q8×C89—21no—
114C2×Q8×C89—21no—
115C2×Q8×C89—41no—
116C2×D4×C89D811yescharacteristic
117C22×C4×C89—41no—
118C22×C4×C89C2×D411yescharacteristic
119C22×C4×C89C2×D411yescharacteristic
120Q16×C89—21no—
121Q16×C89—21no—
122Q16×C89—21no—
123Q16×C89—21no—
124C2×C8×C89—21no—
125C2×C8×C89—21no—
126C2×C8×C89—21no—
127C2×C8×C89—21no—
128C2×C8×C89C2×D411yescharacteristic
129C2×C8×C89C2×D411yescharacteristic
130C2×C8×C89C2×D411yescharacteristic
131C2×C8×C89C2411yescharacteristic
132C4⋊C4×C89—21no—
133C4⋊C4×C89—21no—
134C4⋊C4×C89—21no—
135C4⋊C4×C89—21no—
136C4⋊C4×C89—21no—
137C4⋊C4×C89—21no—
138C4⋊C4×C89—41no—
139C4⋊C4×C89—41no—
140C4⋊C4×C89—41no—
141C4⋊C4×C89—41no—
142C4⋊C4×C89—41no—
143C4⋊C4×C89—41no—
144C4⋊C4×C89—41no—
145C4⋊C4×C89—41no—
146C4⋊C4×C89—41no—
147C4⋊C4×C89—41no—
148C4⋊C4×C89D811yescharacteristic
149C22⋊C4×C89—41no—
150C22⋊C4×C89—41no—
151C22⋊C4×C89—41no—
152C22⋊C4×C89—41no—
153C22⋊C4×C89—41no—
154C22⋊C4×C89—41no—
155C22⋊C4×C89—41no—
156C22⋊C4×C89D811yescharacteristic
157C22⋊C4×C89D811yescharacteristic
158C42×C89—41no—
159C42×C89—41no—
160C42×C89—41no—
161C42×C89—41no—
162C42×C89—41no—
163C42×C89C2×D411yescharacteristic
164C16×C89—21no—
165C16×C89—21no—
166C16×C89—21no—
167C16×C89#4112yes—
168C23×C89C2×D811yescharacteristic
169C23×C89C2×D811yescharacteristic
170Q8×C89—41no—
171Q8×C89—41no—
172Q8×C89—41no—
173Q8×C89—41no—
174Q8×C89—82no—
175D4×C89—41no—
176C2×C4×C89—21no—
177C2×C4×C89—21no—
178C2×C4×C89—21no—
179C2×C4×C89—21no—
180C2×C4×C89—41no—
181C2×C4×C89—41no—
182C2×C4×C89—41no—
183C2×C4×C89—41no—
184C2×C4×C89—41no—
185C2×C4×C89—41no—
186C2×C4×C89—41no—
187C2×C4×C89—41no—
188C2×C4×C89—82no—
189C2×C4×C89C2×D811yescharacteristic
190C2×C4×C89C2×D811yescharacteristic
191C2×C4×C89C2×D811yescharacteristic
192C2×C4×C89C2×D811yescharacteristic
193C2×C4×C89C22×D411yescharacteristic
194C8×C89—21no—
195C8×C89—21no—
196C8×C89—21no—
197C8×C89C2×#4111yescharacteristic
198C8×C892−1+411yescharacteristic
199C22×C89—21no—
200C22×C89—21no—
201C22×C89—21no—
202C22×C89—21no—
203C22×C89—21no—
204C22×C89—21no—
205C22×C89C22×D811yescharacteristic
206C4×C89—21no—
207C4×C89—21no—
208C4×C89—21no—
209C4×C89—21no—
210C4×C89—81no—
211C4×C89—81no—
212C4×C89—81no—
213C4×C89—81no—
214C4×C89—81no—
215C4×C89—81no—
216C4×C89—81no—
217C4×C89—81no—
218C4×C89#54711yescharacteristic
219C4×C89#57811yescharacteristic
220#33916C8911yesmaximal, characteristic, direct product factor, Sylow
221C2×C89—41no—
222C2×C89—41no—
223C2×C89#333411yescharacteristic
224C2×C89#341411yescharacteristic
225C2×C89#341711yescharacteristic
226#3342C2×C8911yescharacteristic
227#3336C2×C8911yescharacteristic
228#2255C2×C8911yescharacteristic, complemented
229#2237C2×C8911yescharacteristic, complemented
230#2237C2×C8911yescharacteristic, complemented
231#2237C2×C8911yescharacteristic, complemented
232#2236C2×C8911yescharacteristic, complemented
233#2229C2×C8911yescharacteristic, complemented
234#2227C2×C8911yescharacteristic, complemented
235#2219C2×C8911yescharacteristic, complemented
236#2216C2×C8911yescharacteristic, complemented
237#2176C2×C8911yescharacteristic, complemented
238#2168C2×C8911yescharacteristic
239#2159C2×C8911yescharacteristic, complemented
240#2158C2×C8911yescharacteristic, complemented
241C89#3391611yescharacteristic, direct product factor, Sylow
242#556—21no—
243#552—21no—
244C2×Q32C22×C8911yescharacteristic
245C2×M32C22×C8911yescharacteristic
246C22×C16C22×C8911yescharacteristic
247#500C22×C8911yescharacteristic
248#499C22×C8911yescharacteristic
249#485—21no—
250#485—21no—
251#485—21no—
252#483—21no—
253#479—21no—
254#475—21no—
255#467C22×C8911yescharacteristic
256#467C22×C8911yescharacteristic
257#467C22×C8911yescharacteristic
258#467C22×C8911yescharacteristic
259C4×Q16C22×C8911yescharacteristic
260C4×Q16C22×C8911yescharacteristic
261C8×D4C22×C8911yescharacteristic
262#427C22×C8911yescharacteristic
263#427C22×C8911yescharacteristic
264#427C22×C8911yescharacteristic
265C2×#107C22×C8911yescharacteristic
266#367C22×C8911yescharacteristic
267#367C22×C8911yescharacteristic
268#367C22×C8911yescharacteristic
269#367C22×C8911yescharacteristic
270#366C22×C8911yescharacteristic
271#366C22×C8911yescharacteristic
272#366C22×C8911yescharacteristic
273#366C22×C8911yescharacteristic
274#366C22×C8911yescharacteristic
275C4⋊C16C22×C8911yescharacteristic
276#358C22×C8911yescharacteristic
277#358C22×C8911yescharacteristic
278#358C22×C8911yescharacteristic
279#358C22×C8911yescharacteristic
280#358C22×C8911yescharacteristic
281#358C22×C8911yescharacteristic
282C22⋊C16C22×C8911yescharacteristic
283C22⋊C16C22×C8911yescharacteristic
284#346C22×C8911yescharacteristic
285C2×Q16C23×C8911yescharacteristic
286C2×Q16C23×C8911yescharacteristic
287C22×C8C23×C8911yescharacteristic
288C22×C8C23×C8911yescharacteristic
289C4⋊Q8—21no—
290#126—41no—
291#126—41no—
292#126—41no—
293#125—21no—
294#125—41no—
295#124—41no—
296#123—41no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—41no—
304C4×D4D4×C8911yescharacteristic
305#117—21no—
306#117—21no—
307#117—21no—
308#117—41no—
309C2×C4⋊C4—21no—
310Q32—22no—
311M32—22no—
312C2×C16—22no—
313C2×C16C23×C8911yescharacteristic
314C2×C16C23×C8911yescharacteristic
315C2×C16C23×C8911yescharacteristic
316C2×C16C23×C8911yescharacteristic
317#107—21no—
318#107—21no—
319#107C23×C8911yescharacteristic
320#107C23×C8911yescharacteristic
321#107C23×C8911yescharacteristic
322#107C23×C8911yescharacteristic
323#107C23×C8911yescharacteristic
324#107C23×C8911yescharacteristic
325#106—21no—
326#106—21no—
327#106—21no—
328#106—21no—
329C4⋊C8D4×C8911yescharacteristic
330#103—21no—
331#103—21no—
332#103—21no—
333#103—21no—
334#103—21no—
335#103—21no—
336C22⋊C8D4×C8911yescharacteristic
337C22⋊C8D4×C8911yescharacteristic
338C4×C8C23×C8911yescharacteristic
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—41no—
342C2×D4D8×C8911yescharacteristic
343C22×C4—41no—
344C22×C4C2×D4×C8911yescharacteristic
345C22×C4C2×D4×C8911yescharacteristic
346Q16—21no—
347Q16—21no—
348Q16—21no—
349Q16—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8C2×D4×C8911yescharacteristic
355C2×C8C2×D4×C8911yescharacteristic
356C2×C8C2×D4×C8911yescharacteristic
357C2×C8C24×C8911yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4D8×C8911yescharacteristic
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4D8×C8911yescharacteristic
383C22⋊C4D8×C8911yescharacteristic
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42C2×D4×C8911yescharacteristic
390C16—21no—
391C16—21no—
392C16—21no—
393C16#41×C8912yes—
394C23C2×D8×C8911yescharacteristic
395C23C2×D8×C8911yescharacteristic
396Q8—41no—
397Q8—41no—
398Q8—41no—
399Q8—41no—
400Q8—82no—
401D4—41no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4C2×D8×C8911yescharacteristic
416C2×C4C2×D8×C8911yescharacteristic
417C2×C4C2×D8×C8911yescharacteristic
418C2×C4C2×D8×C8911yescharacteristic
419C2×C4C22×D4×C8911yescharacteristic
420C8—21no—
421C8—21no—
422C8—21no—
423C8C2×#41×C8911yescharacteristic
424C82−1+4×C8911yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22C22×D8×C8911yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4#547×C8911yescharacteristic
445C4#578×C8911yescharacteristic
446C2—41no—
447C2—41no—
448C2C89×#333411yescharacteristic
449C2C89×#341411yescharacteristic
450C2C89×#341711yescharacteristic
subgroupG/H
center, socleC22×C89C22×D8
commutator, Frattini, cosocleC2×C8C24×C89
prime radicalC2×D4×C89D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C4×D4×C89 ◁ C8×D4×C89 ◁ G

Complements

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

NH ≅ G/Ntype
3. C89×#2255C2semidirect
4. C89×#2237C2semidirect
5. C89×#2237C2semidirect
6. C89×#2237C2semidirect
7. C89×#2236C2semidirect
8. C89×#2229C2semidirect
9. C89×#2227C2semidirect
10. C89×#2219C2semidirect
11. C89×#2216C2semidirect
12. C89×#2176C2semidirect
14. C89×#2159C2semidirect
15. C89×#2158C2semidirect
220. #33916C89direct (N × H)
228. #2255C2×C89semidirect
229. #2237C2×C89semidirect
230. #2237C2×C89semidirect
231. #2237C2×C89semidirect
232. #2236C2×C89semidirect
233. #2229C2×C89semidirect
234. #2227C2×C89semidirect
235. #2219C2×C89semidirect
236. #2216C2×C89semidirect
237. #2176C2×C89semidirect
239. #2159C2×C89semidirect
240. #2158C2×C89semidirect
241. C89#33916direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#33422
C89×#33362
C89×#22552
C89×#22372
C89×#22362
C89×#22292
C89×#22272
C89×#22192
C89×#22162
C89×#21762
C89×#21682
C89×#21592
C89×#21582
#3391689