Finite Group Databasebeta

← C89×#60331 #36438179

Subgroups — C89×#60331 #36438179

Order 22784.

1792 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3482C211yesmaximal, characteristic
2C89×#2994C216yesmaximal, complemented
3C89×#2991C211yesmaximal, characteristic, complemented
4C89×#2990C213yesmaximal, complemented
5C89×#2986C213yesmaximal, complemented
6C89×#2984C212yesmaximal, complemented
7C89×#2966C216yesmaximal, complemented
8C8×#41×C89C213yesmaximal, complemented
9C8×#41×C89C211yesmaximal, characteristic, complemented
10C89×#2960C213yesmaximal, complemented
11C2×C8×Q8×C89C212yesmaximal, complemented
12#557×C89C411yescharacteristic
13#556×C89C413yes—
14#552×C89C416yes—
15D4×Q8×C89C411yescharacteristic
16#548×C89C412yes—
17#547×C89C413yes—
18#533×C89C2213yes—
19#529×C89C2216yes—
20C4×#41×C89C2213yes—
21C4×#41×C89C2211yescharacteristic
22C2×C4×Q8×C89C2212yes—
23#446×C89C2216yes—
24#446×C89C2216yes—
25#446×C89C2216yes—
26C8×Q8×C89C2211yescharacteristic
27C8×Q8×C89C2216yes—
28C8×Q8×C89C2212yes—
29C8×Q8×C89C2216yes—
30C8×Q8×C89C2213yes—
31C8×Q8×C89C2211yescharacteristic
32C8×Q8×C89C2213yes—
33#436×C89C2216yes—
34#435×C89C22112yes—
35C8×D4×C89C2211yescharacteristic
36C8×D4×C89C2213yes—
37C8×D4×C89C2216yes—
38C8×D4×C89C2213yes—
39C8×D4×C89C2213yes—
40C8×D4×C89C2216yes—
41#433×C89C22112yes—
42#432×C89C2216yes—
43#431×C89C2216yes—
44#431×C89C2213yes—
45#431×C89C2213yes—
46#424×C89C2216yes—
47#423×C89C2213yes—
48C2×C4⋊C8×C89C2216yes—
49#405×C89C2216yes—
50C2×C4×C8×C89C2216yes—
51C2×C4×C8×C89C2213yes—
52C2×#41×C89C2×C413yes—
53C2×#41×C89C2×C411yescharacteristic
54C22×Q8×C89C2×C412yes—
55C2×M16×C89C2316yes—
56C22×C8×C89C2312yes—
57C22×C8×C89C2313yes—
58C22×C8×C89C2316yes—
59C22×C8×C89C2313yes—
60C4⋊Q8×C89C2×C413yes—
61C4⋊Q8×C89C2×C413yes—
62#126×C89C2×C4112yes—
63#125×C89C2×C416yes—
64#125×C89C2×C416yes—
65#124×C89C2×C416yes—
66#123×C89C2×C4112yes—
67C22⋊Q8×C89C2×C416yes—
68C22⋊Q8×C89C2×C416yes—
69C22⋊Q8×C89C2×C416yes—
70#121×C89C2×C416yes—
71C4×Q8×C89C2×C412yes—
72C4×Q8×C89C2×C416yes—
73C4×Q8×C89C2×C411yescharacteristic
74C4×Q8×C89C2×C413yes—
75C4×Q8×C89C2311yescharacteristic
76C4×Q8×C89C2316yes—
77C4×Q8×C89C2313yes—
78C4×D4×C89C2×C413yes—
79C4×D4×C89C2×C413yes—
80C4×D4×C89C2×C416yes—
81C4×D4×C89C2311yescharacteristic
82C4×D4×C89C2316yes—
83C4×D4×C89C2313yes—
84#117×C89C2×C413yes—
85#117×C89C2×C416yes—
86#117×C89C2×C413yes—
87#117×C89C2316yes—
88C2×C4⋊C4×C89C2×C416yes—
89C2×C42×C89C2316yes—
90C2×C42×C89C2313yes—
91C4⋊C8×C89C2313yes—
92C4⋊C8×C89C2316yes—
93C4⋊C8×C89C23112yes—
94C4⋊C8×C89C2313yes—
95C4⋊C8×C89C2316yes—
96C4⋊C8×C89C2311yescharacteristic
97C4⋊C8×C89C2313yes—
98C4⋊C8×C89C2316yes—
99C22⋊C8×C89C2312yes—
100C22⋊C8×C89C2316yes—
101C22⋊C8×C89C2316yes—
102C22⋊C8×C89C2316yes—
103#97×C89C23112yes—
104C4×C8×C89C2313yes—
105C4×C8×C89C2313yes—
106C4×C8×C89C2316yes—
107C4×C8×C89C2316yes—
108C4×C8×C89C2313yes—
109C4×C8×C89C2311yescharacteristic
110C4×C8×C89C2316yes—
111#41×C89—24no—
112#41×C89—212no—
113C2×Q8×C89—28no—
114C2×Q8×C89C22×C416yes—
115C2×Q8×C89C22×C413yes—
116C2×Q8×C89C22×C411yescharacteristic
117C2×D4×C89C22×C416yes—
118C2×D4×C89C22×C413yes—
119C2×D4×C89C22×C411yescharacteristic
120C22×C4×C89C22×C416yes—
121C22×C4×C89C22×C416yes—
122C22×C4×C89C22×C413yes—
123C22×C4×C89C22×C413yes—
124C22×C4×C89C2412yes—
125C22×C4×C89C2413yes—
126M16×C89—46no—
127C2×C8×C89—24no—
128C2×C8×C89—26no—
129C2×C8×C89—43no—
130C2×C8×C89—46no—
131C2×C8×C89C2412yes—
132C2×C8×C89C2411yescharacteristic
133C2×C8×C89C2413yes—
134C2×C8×C89C2413yes—
135C2×C8×C89C2416yes—
136C2×C8×C89C2411yescharacteristic
137C4⋊C4×C89C22×C413yes—
138C4⋊C4×C89C22×C416yes—
139C4⋊C4×C89C22×C4112yes—
140C4⋊C4×C89C22×C411yescharacteristic
141C4⋊C4×C89C22×C416yes—
142C4⋊C4×C89C22×C413yes—
143C4⋊C4×C89C22×C413yes—
144C4⋊C4×C89C22×C416yes—
145C4⋊C4×C89C22×C416yes—
146C4⋊C4×C89C22×C416yes—
147C22⋊C4×C89C22×C412yes—
148C22⋊C4×C89C22×C416yes—
149C22⋊C4×C89C22×C416yes—
150C22⋊C4×C89C22×C416yes—
151C22⋊C4×C89C22×C4112yes—
152C42×C89C22×C413yes—
153C42×C89C22×C416yes—
154C42×C89C22×C413yes—
155C42×C89C22×C416yes—
156C42×C89C2411yescharacteristic
157C42×C89C2413yes—
158C42×C89C2416yes—
159C23×C89C23×C413yes—
160C23×C89C23×C412yes—
161Q8×C89—212no—
162Q8×C89—26no—
163Q8×C89#13114yes—
164D4×C89—212no—
165D4×C89—26no—
166D4×C89#13114yes—
167C2×C4×C89—212no—
168C2×C4×C89—212no—
169C2×C4×C89—26no—
170C2×C4×C89—26no—
171C2×C4×C89—22no—
172C2×C4×C89—23no—
173C2×C4×C89—22no—
174C2×C4×C89—23no—
175C2×C4×C89C23×C412yes—
176C2×C4×C89C23×C411yescharacteristic
177C2×C4×C89C23×C413yes—
178C2×C4×C89C23×C413yes—
179C2×C4×C89C23×C416yes—
180C2×C4×C89C23×C416yes—
181C2×C4×C89C23×C413yes—
182C2×C4×C89C23×C411yescharacteristic
183C2×C4×C89C2511yescharacteristic
184C8×C89—22no—
185C8×C89—23no—
186C8×C89—21no—
187C8×C89—23no—
188C8×C89—26no—
189C8×C892−1+412yes—
190C22×C89—22no—
191C22×C89—22no—
192C22×C89—23no—
193C22×C89—23no—
194C22×C89—23no—
195C22×C89C2×#13114yes—
196C22×C89C24×C411yescharacteristic
197C4×C89—22no—
198C4×C89—23no—
199C4×C89—23no—
200C4×C89—21no—
201C4×C89—26no—
202C4×C89—26no—
203C4×C89C2×#13116yes—
204C4×C89C2×#13112yes—
205C4×C89C2×2−1+411yescharacteristic
206C4×C89#58511yescharacteristic
207#60331C8911yesmaximal, characteristic, direct product factor, Sylow
208C2×C89—24no—
209C2×C89—43no—
210C2×C89C4×2−1+411yescharacteristic
211C2×C89C22×#13111yescharacteristic
212C2×C89C22×#13111yescharacteristic
213#3482C2×C8911yescharacteristic
214#2994C2×C8916yescomplemented
215#2991C2×C8911yescharacteristic, complemented
216#2990C2×C8913yescomplemented
217#2986C2×C8913yescomplemented
218#2984C2×C8912yescomplemented
219#2966C2×C8916yescomplemented
220C8×#41C2×C8913yescomplemented
221C8×#41C2×C8911yescharacteristic, complemented
222#2960C2×C8913yescomplemented
223C2×C8×Q8C2×C8912yescomplemented
224C89#6033111yescharacteristic, direct product factor, Sylow
225#557C4×C8911yescharacteristic
226#556C4×C8913yes—
227#552C4×C8916yes—
228D4×Q8C4×C8911yescharacteristic
229#548C4×C8912yes—
230#547C4×C8913yes—
231#533C22×C8913yes—
232#529C22×C8916yes—
233C4×#41C22×C8913yes—
234C4×#41C22×C8911yescharacteristic
235C2×C4×Q8C22×C8912yes—
236#446C22×C8916yes—
237#446C22×C8916yes—
238#446C22×C8916yes—
239C8×Q8C22×C8911yescharacteristic
240C8×Q8C22×C8916yes—
241C8×Q8C22×C8912yes—
242C8×Q8C22×C8916yes—
243C8×Q8C22×C8913yes—
244C8×Q8C22×C8911yescharacteristic
245C8×Q8C22×C8913yes—
246#436C22×C8916yes—
247#435C22×C89112yes—
248C8×D4C22×C8911yescharacteristic
249C8×D4C22×C8913yes—
250C8×D4C22×C8916yes—
251C8×D4C22×C8913yes—
252C8×D4C22×C8913yes—
253C8×D4C22×C8916yes—
254#433C22×C89112yes—
255#432C22×C8916yes—
256#431C22×C8916yes—
257#431C22×C8913yes—
258#431C22×C8913yes—
259#424C22×C8916yes—
260#423C22×C8913yes—
261C2×C4⋊C8C22×C8916yes—
262#405C22×C8916yes—
263C2×C4×C8C22×C8916yes—
264C2×C4×C8C22×C8913yes—
265C2×#41C2×C4×C8913yes—
266C2×#41C2×C4×C8911yescharacteristic
267C22×Q8C2×C4×C8912yes—
268C2×M16C23×C8916yes—
269C22×C8C23×C8912yes—
270C22×C8C23×C8913yes—
271C22×C8C23×C8916yes—
272C22×C8C23×C8913yes—
273C4⋊Q8C2×C4×C8913yes—
274C4⋊Q8C2×C4×C8913yes—
275#126C2×C4×C89112yes—
276#125C2×C4×C8916yes—
277#125C2×C4×C8916yes—
278#124C2×C4×C8916yes—
279#123C2×C4×C89112yes—
280C22⋊Q8C2×C4×C8916yes—
281C22⋊Q8C2×C4×C8916yes—
282C22⋊Q8C2×C4×C8916yes—
283#121C2×C4×C8916yes—
284C4×Q8C2×C4×C8912yes—
285C4×Q8C2×C4×C8916yes—
286C4×Q8C2×C4×C8911yescharacteristic
287C4×Q8C2×C4×C8913yes—
288C4×Q8C23×C8911yescharacteristic
289C4×Q8C23×C8916yes—
290C4×Q8C23×C8913yes—
291C4×D4C2×C4×C8913yes—
292C4×D4C2×C4×C8913yes—
293C4×D4C2×C4×C8916yes—
294C4×D4C23×C8911yescharacteristic
295C4×D4C23×C8916yes—
296C4×D4C23×C8913yes—
297#117C2×C4×C8913yes—
298#117C2×C4×C8916yes—
299#117C2×C4×C8913yes—
300#117C23×C8916yes—
301C2×C4⋊C4C2×C4×C8916yes—
302C2×C42C23×C8916yes—
303C2×C42C23×C8913yes—
304C4⋊C8C23×C8913yes—
305C4⋊C8C23×C8916yes—
306C4⋊C8C23×C89112yes—
307C4⋊C8C23×C8913yes—
308C4⋊C8C23×C8916yes—
309C4⋊C8C23×C8911yescharacteristic
310C4⋊C8C23×C8913yes—
311C4⋊C8C23×C8916yes—
312C22⋊C8C23×C8912yes—
313C22⋊C8C23×C8916yes—
314C22⋊C8C23×C8916yes—
315C22⋊C8C23×C8916yes—
316#97C23×C89112yes—
317C4×C8C23×C8913yes—
318C4×C8C23×C8911yescharacteristic
319C4×C8C23×C8916yes—
320C4×C8C23×C8913yes—
321C4×C8C23×C8913yes—
322C4×C8C23×C8916yes—
323C4×C8C23×C8916yes—
324#41—24no—
325#41—212no—
326C2×Q8—28no—
327C2×Q8C22×C4×C8916yes—
328C2×Q8C22×C4×C8913yes—
329C2×Q8C22×C4×C8911yescharacteristic
330C2×D4C22×C4×C8916yes—
331C2×D4C22×C4×C8913yes—
332C2×D4C22×C4×C8911yescharacteristic
333C22×C4C22×C4×C8916yes—
334C22×C4C22×C4×C8916yes—
335C22×C4C22×C4×C8913yes—
336C22×C4C22×C4×C8913yes—
337C22×C4C24×C8912yes—
338C22×C4C24×C8913yes—
339M16—46no—
340C2×C8—24no—
341C2×C8—26no—
342C2×C8—43no—
343C2×C8—46no—
344C2×C8C24×C8912yes—
345C2×C8C24×C8911yescharacteristic
346C2×C8C24×C8913yes—
347C2×C8C24×C8913yes—
348C2×C8C24×C8916yes—
349C2×C8C24×C8911yescharacteristic
350C4⋊C4C22×C4×C8913yes—
351C4⋊C4C22×C4×C8916yes—
352C4⋊C4C22×C4×C89112yes—
353C4⋊C4C22×C4×C8911yescharacteristic
354C4⋊C4C22×C4×C8916yes—
355C4⋊C4C22×C4×C8913yes—
356C4⋊C4C22×C4×C8913yes—
357C4⋊C4C22×C4×C8916yes—
358C4⋊C4C22×C4×C8916yes—
359C4⋊C4C22×C4×C8916yes—
360C22⋊C4C22×C4×C8912yes—
361C22⋊C4C22×C4×C8916yes—
362C22⋊C4C22×C4×C8916yes—
363C22⋊C4C22×C4×C8916yes—
364C22⋊C4C22×C4×C89112yes—
365C42C22×C4×C8913yes—
366C42C22×C4×C8916yes—
367C42C22×C4×C8913yes—
368C42C22×C4×C8916yes—
369C42C24×C8911yescharacteristic
370C42C24×C8913yes—
371C42C24×C8916yes—
372C23C23×C4×C8913yes—
373C23C23×C4×C8912yes—
374Q8—212no—
375Q8—26no—
376Q8#131×C8914yes—
377D4—212no—
378D4—26no—
379D4#131×C8914yes—
380C2×C4—212no—
381C2×C4—212no—
382C2×C4—26no—
383C2×C4—26no—
384C2×C4—22no—
385C2×C4—23no—
386C2×C4—22no—
387C2×C4—23no—
388C2×C4C23×C4×C8912yes—
389C2×C4C23×C4×C8911yescharacteristic
390C2×C4C23×C4×C8913yes—
391C2×C4C23×C4×C8913yes—
392C2×C4C23×C4×C8916yes—
393C2×C4C23×C4×C8916yes—
394C2×C4C23×C4×C8913yes—
395C2×C4C23×C4×C8911yescharacteristic
396C2×C4C25×C8911yescharacteristic
397C8—22no—
398C8—23no—
399C8—21no—
400C8—23no—
401C8—26no—
402C82−1+4×C8912yes—
403C22—22no—
404C22—22no—
405C22—23no—
406C22—23no—
407C22—23no—
408C22C2×#131×C8914yes—
409C22C24×C4×C8911yescharacteristic
410C4—22no—
411C4—23no—
412C4—23no—
413C4—21no—
414C4—26no—
415C4—26no—
416C4C2×#131×C8916yes—
417C4C2×#131×C8912yes—
418C4C2×2−1+4×C8911yescharacteristic
419C4#585×C8911yescharacteristic
420C2—24no—
421C2—43no—
422C2C4×2−1+4×C8911yescharacteristic
423C2C22×#131×C8911yescharacteristic
424C2C22×#131×C8911yescharacteristic
subgroupG/H
centerC2×C8×C89C24
commutatorC22C24×C4×C89
Frattini, cosocleC2×C4C25×C89
socleC22×C89C24×C4
prime radicalC89×#3482C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2994C2semidirect
3. C89×#2991C2semidirect
4. C89×#2990C2semidirect
5. C89×#2986C2semidirect
6. C89×#2984C2semidirect
7. C89×#2966C2semidirect
8. C8×#41×C89C2semidirect
9. C8×#41×C89C2semidirect
10. C89×#2960C2semidirect
11. C2×C8×Q8×C89C2semidirect
207. #60331C89direct (N × H)
214. #2994C2×C89semidirect
215. #2991C2×C89semidirect
216. #2990C2×C89semidirect
217. #2986C2×C89semidirect
218. #2984C2×C89semidirect
219. #2966C2×C89semidirect
220. C8×#41C2×C89semidirect
221. C8×#41C2×C89semidirect
222. #2960C2×C89semidirect
223. C2×C8×Q8C2×C89semidirect
224. C89#60331direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#34822
C89×#29942
C89×#29912
C89×#29902
C89×#29862
C89×#29842
C89×#29662
C8×#41×C892
C89×#29602
C2×C8×Q8×C892
#6033189