Finite Group Databasebeta

← C89×#9167 #36387015

Subgroups — C89×#9167 #36387015

Order 22784.

478 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3079C211yesmaximal, characteristic
2C89×#2906C211yesmaximal, characteristic
3C89×#1548C211yesmaximal, characteristic, complemented
4C2×#332×C89C211yesmaximal, characteristic, complemented
5C89×#1477C211yesmaximal, characteristic
6C89×#1301C211yesmaximal, characteristic, complemented
7C89×#1292C211yesmaximal, characteristic, complemented
8C2×#131×C89—21no—
9C2×C4⋊Q8×C89C2211yescharacteristic
10C4×#41×C89C2211yescharacteristic
11#477×C89C412yes—
12#476×C89C412yes—
13#475×C89—21no—
14#474×C89—21no—
15#431×C89C2211yescharacteristic
16#428×C89—21no—
17#424×C89—21no—
18#424×C89—22no—
19#423×C89—21no—
20#423×C89C2211yescharacteristic
21#423×C89C2211yescharacteristic
22C2×C4⋊C8×C89—21no—
23C2×C4⋊C8×C89C2211yescharacteristic
24C2×C4⋊C8×C89C2211yescharacteristic
25#417×C89—21no—
26#333×C89—22no—
27#332×C89—21no—
28#332×C89—21no—
29Q8⋊C8×C89—21no—
30D4⋊C8×C89—21no—
31C2×#41×C89D411yescharacteristic
32C22×Q8×C89—21no—
33#131×C89—42no—
34#131×C89—42no—
35C2×M16×C89—22no—
36C2×M16×C89—21no—
37C2×M16×C89—21no—
38C22×C8×C89—22no—
39C22×C8×C89—21no—
40C22×C8×C89—21no—
41C4⋊Q8×C89—21no—
42C4⋊Q8×C89—21no—
43C4⋊Q8×C89—21no—
44C4⋊Q8×C89C2×C412yes—
45C4×Q8×C89C2×C411yescharacteristic
46C4×Q8×C89D411yescharacteristic
47C4×D4×C89—21no—
48C4×D4×C89—21no—
49C4×D4×C89C2×C411yescharacteristic
50C4×D4×C89D411yescharacteristic
51#117×C89—21no—
52#117×C89D411yescharacteristic
53C2×C4⋊C4×C89—21no—
54C2×C4⋊C4×C89—21no—
55C2×C42×C89—21no—
56C2×C42×C89C2311yescharacteristic
57#107×C89—22no—
58#106×C89—22no—
59C4⋊C8×C89—22no—
60C4⋊C8×C89—22no—
61C4⋊C8×C89—21no—
62C4⋊C8×C89—21no—
63C4⋊C8×C89—21no—
64C4⋊C8×C89—21no—
65C4⋊C8×C89—21no—
66C4⋊C8×C89—21no—
67C4⋊C8×C89C2×C412yes—
68C4⋊C8×C89D411yescharacteristic
69C4⋊C8×C89D411yescharacteristic
70C4⋊C8×C89D412yes—
71#103×C89—22no—
72#102×C89—22no—
73C22⋊C8×C89—21no—
74C4×C8×C89—21no—
75#41×C89—21no—
76#41×C89—21no—
77#41×C89SD1612yes—
78#41×C89Q1612yes—
79C2×Q8×C89—22no—
80C2×Q8×C89—41no—
81C2×Q8×C89—81no—
82C2×Q8×C89—81no—
83C2×Q8×C89C22⋊C411yescharacteristic
84C2×D4×C89—21no—
85C2×D4×C89C22⋊C411yescharacteristic
86C22×C4×C89—21no—
87C22×C4×C89—21no—
88C22×C4×C89—21no—
89C22×C4×C89—41no—
90C22×C4×C89C2×D411yescharacteristic
91C22×C4×C89#4111yescharacteristic
92C22×C4×C89#4111yescharacteristic
93M16×C89—44no—
94M16×C89—41no—
95M16×C89—41no—
96M16×C89—41no—
97M16×C89—41no—
98C2×C8×C89—22no—
99C2×C8×C89—21no—
100C2×C8×C89—22no—
101C2×C8×C89—21no—
102C2×C8×C89—44no—
103C2×C8×C89—41no—
104C2×C8×C89—41no—
105C2×C8×C89—41no—
106C2×C8×C89—41no—
107C2×C8×C89—81no—
108C4⋊C4×C89—21no—
109C4⋊C4×C89—21no—
110C4⋊C4×C89—22no—
111C4⋊C4×C89—22no—
112C4⋊C4×C89—41no—
113C4⋊C4×C89—41no—
114C4⋊C4×C89C22⋊C411yescharacteristic
115C4⋊C4×C89C22⋊C411yescharacteristic
116C22⋊C4×C89—21no—
117C22⋊C4×C89—21no—
118C22⋊C4×C89—21no—
119C42×C89—21no—
120C42×C89—21no—
121C42×C89—21no—
122C42×C89C22×C411yescharacteristic
123C42×C89C2×D411yescharacteristic
124C42×C89C2×D411yescharacteristic
125C42×C89C2×D411yescharacteristic
126C23×C89—21no—
127C23×C89#12311yescharacteristic
128Q8×C89—21no—
129Q8×C89—82no—
130Q8×C89—81no—
131Q8×C89—81no—
132Q8×C89#10312yes—
133D4×C89—22no—
134D4×C89—22no—
135D4×C89—21no—
136D4×C89#10312yes—
137C2×C4×C89—21no—
138C2×C4×C89—22no—
139C2×C4×C89—22no—
140C2×C4×C89—21no—
141C2×C4×C89—21no—
142C2×C4×C89—21no—
143C2×C4×C89—21no—
144C2×C4×C89—41no—
145C2×C4×C89—41no—
146C2×C4×C89—41no—
147C2×C4×C89—42no—
148C2×C4×C89—81no—
149C2×C4×C89—82no—
150C2×C4×C89C2×C22⋊C411yescharacteristic
151C2×C4×C89C4×D411yescharacteristic
152C2×C4×C89C4×D411yescharacteristic
153C2×C4×C89#12011yescharacteristic
154C2×C4×C89#12111yescharacteristic
155C2×C4×C89#12111yescharacteristic
156C2×C4×C89C2×SD1611yescharacteristic
157C2×C4×C89C2×Q1611yescharacteristic
158C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C89#13711yescharacteristic
160C8×C89—42no—
161C8×C89—41no—
162C8×C89—42no—
163C8×C89—41no—
164C8×C89—81no—
165C8×C89—81no—
166C22×C89—21no—
167C22×C89—21no—
168C22×C89—22no—
169C22×C89—41no—
170C22×C89—41no—
171C22×C89#38611yescharacteristic
172C22×C89#48311yescharacteristic
173C22×C89#48411yescharacteristic
174C4×C89—22no—
175C4×C89—21no—
176C4×C89—21no—
177C4×C89—21no—
178C4×C89—41no—
179C4×C89—41no—
180C4×C89—82no—
181C4×C89C2×#10311yescharacteristic
182C4×C89#41711yescharacteristic
183C4×C89#45011yescharacteristic
184C4×C89#45111yescharacteristic
185C4×C89#47111yescharacteristic
186C4×C89#47211yescharacteristic
187#9167C8911yesmaximal, characteristic, direct product factor, Sylow
188C2×C89—21no—
189C2×C89—21no—
190C2×C89—42no—
191C2×C89#187211yescharacteristic
192C2×C89#190211yescharacteristic
193C2×C89#190711yescharacteristic
194#3079C2×C8911yescharacteristic
195#2906C2×C8911yescharacteristic
196#1548C2×C8911yescharacteristic, complemented
197C2×#332C2×C8911yescharacteristic, complemented
198#1477C2×C8911yescharacteristic
199#1301C2×C8911yescharacteristic, complemented
200#1292C2×C8911yescharacteristic, complemented
201C89#916711yescharacteristic, direct product factor, Sylow
202C2×#131—21no—
203C2×C4⋊Q8C22×C8911yescharacteristic
204C4×#41C22×C8911yescharacteristic
205#477C4×C8912yes—
206#476C4×C8912yes—
207#475—21no—
208#474—21no—
209#431C22×C8911yescharacteristic
210#428—21no—
211#424—21no—
212#424—22no—
213#423—21no—
214#423C22×C8911yescharacteristic
215#423C22×C8911yescharacteristic
216C2×C4⋊C8—21no—
217C2×C4⋊C8C22×C8911yescharacteristic
218C2×C4⋊C8C22×C8911yescharacteristic
219#417—21no—
220#333—22no—
221#332—21no—
222#332—21no—
223Q8⋊C8—21no—
224D4⋊C8—21no—
225C2×#41D4×C8911yescharacteristic
226C22×Q8—21no—
227#131—42no—
228#131—42no—
229C2×M16—22no—
230C2×M16—21no—
231C2×M16—21no—
232C22×C8—22no—
233C22×C8—21no—
234C22×C8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C2×C4×C8912yes—
239C4×Q8C2×C4×C8911yescharacteristic
240C4×Q8D4×C8911yescharacteristic
241C4×D4—21no—
242C4×D4—21no—
243C4×D4C2×C4×C8911yescharacteristic
244C4×D4D4×C8911yescharacteristic
245#117—21no—
246#117D4×C8911yescharacteristic
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C42—21no—
250C2×C42C23×C8911yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8C2×C4×C8912yes—
262C4⋊C8D4×C8911yescharacteristic
263C4⋊C8D4×C8911yescharacteristic
264C4⋊C8D4×C8912yes—
265#103—22no—
266#102—22no—
267C22⋊C8—21no—
268C4×C8—21no—
269#41—21no—
270#41—21no—
271#41SD16×C8912yes—
272#41Q16×C8912yes—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×Q8C22⋊C4×C8911yescharacteristic
278C2×D4—21no—
279C2×D4C22⋊C4×C8911yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4C2×D4×C8911yescharacteristic
285C22×C4#41×C8911yescharacteristic
286C22×C4#41×C8911yescharacteristic
287M16—44no—
288M16—41no—
289M16—41no—
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—44no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—81no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4C22⋊C4×C8911yescharacteristic
309C4⋊C4C22⋊C4×C8911yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42C22×C4×C8911yescharacteristic
317C42C2×D4×C8911yescharacteristic
318C42C2×D4×C8911yescharacteristic
319C42C2×D4×C8911yescharacteristic
320C23—21no—
321C23#123×C8911yescharacteristic
322Q8—21no—
323Q8—82no—
324Q8—81no—
325Q8—81no—
326Q8#103×C8912yes—
327D4—22no—
328D4—22no—
329D4—21no—
330D4#103×C8912yes—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4C2×C22⋊C4×C8911yescharacteristic
345C2×C4C4×D4×C8911yescharacteristic
346C2×C4C4×D4×C8911yescharacteristic
347C2×C4#120×C8911yescharacteristic
348C2×C4#121×C8911yescharacteristic
349C2×C4#121×C8911yescharacteristic
350C2×C4C2×SD16×C8911yescharacteristic
351C2×C4C2×Q16×C8911yescharacteristic
352C2×C4AGL(1,ℤ/8)×C8911yescharacteristic
353C2×C4#137×C8911yescharacteristic
354C8—42no—
355C8—41no—
356C8—42no—
357C8—41no—
358C8—81no—
359C8—81no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22#386×C8911yescharacteristic
366C22#483×C8911yescharacteristic
367C22#484×C8911yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—82no—
375C4C2×#103×C8911yescharacteristic
376C4#417×C8911yescharacteristic
377C4#450×C8911yescharacteristic
378C4#451×C8911yescharacteristic
379C4#471×C8911yescharacteristic
380C4#472×C8911yescharacteristic
381C2—21no—
382C2—21no—
383C2—42no—
384C2C89×#187211yescharacteristic
385C2C89×#190211yescharacteristic
386C2C89×#190711yescharacteristic
subgroupG/H
center, socleC22×C89#386
commutatorC42C22×C4×C89
Frattini, cosocleC2×C42C23×C89
prime radicalC2×#41×C89D4
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
3. C89×#1548C2semidirect
4. C2×#332×C89C2semidirect
6. C89×#1301C2semidirect
7. C89×#1292C2semidirect
187. #9167C89direct (N × H)
196. #1548C2×C89semidirect
197. C2×#332C2×C89semidirect
199. #1301C2×C89semidirect
200. #1292C2×C89semidirect
201. C89#9167direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#30792
C89×#29062
C89×#15482
C2×#332×C892
C89×#14772
C89×#13012
C89×#12922
#916789