Finite Group Databasebeta

← #11882308

Subgroups — #11882308

Order 1152.

597 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
1C9×#529C211yesmaximal, characteristic, complemented
2#10589579C211yesmaximal, characteristic, complemented
3#10588747C212yesmaximal, complemented
4#10588738C212yesmaximal, complemented
5#10588731C212yesmaximal, complemented
6#10588721C212yesmaximal, complemented
7#10588710C211yesmaximal, characteristic, complemented
8#10588703C211yesmaximal, characteristic, complemented
9#10588698C211yesmaximal, characteristic, complemented
10#10588691C211yesmaximal, characteristic, complemented
11C2×C4⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
12#83148—31nomaximal
13C2×#3697—21no—
14C9×#126—21no—
15C9×#125C2211yescharacteristic
16C9×#124C2211yescharacteristic
17C9×#123—21no—
18C9×C22⋊Q8C2211yescharacteristic
19C9×#121C2211yescharacteristic
20C4×Q8×C9C2211yescharacteristic
21C4×D4×C9—21no—
22C4×D4×C9C2211yescharacteristic
23C9×#117—21no—
24C2×C42×C9C2211yescharacteristic
25C2×#3675C2212yes—
26C2×#3673C2212yes—
27#63492—21no—
28#63492C2212yes—
29C22×C9⋊C8C2212yes—
30#63437—21no—
31#63436C2211yescharacteristic
32C4×D36—21no—
33C4×D36C2212yes—
34#63431C2211yescharacteristic
35#63395C2212yes—
36#63392C2212yes—
37#63369C2212yes—
38#63369C2212yes—
39#63367C2212yes—
40#63366C2212yes—
41C4⋊(C9⋊C8)C2211yescharacteristic
42C4⋊(C9⋊C8)C2211yescharacteristic
43C4⋊(C9⋊C8)C2212yes—
44C3×#529D311yescharacteristic
45#5477—31no—
46#5066—32no—
47#5057—32no—
48#5048—32no—
49#5038—32no—
50#5027—31no—
51#5021—31no—
52#5015—31no—
53#5008—31no—
54C2×C4⋊(C3⋊C8)—31no—
55C2×Q8×C9C2311yescharacteristic
56C2×D4×C9—21no—
57C2×D4×C9C2311yescharacteristic
58C22×C4×C9—21no—
59C22×C4×C9D411yescharacteristic
60C22×C4×C9D411yescharacteristic
61C22×C4×C9C2311yescharacteristic
62C2×#611—22no—
63#3697—44no—
64C2×D36—21no—
65C2×D36—21no—
66C2×D36C2312yes—
67C2×C4×D9—22no—
68C2×Dic18—21no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4—21no—
72C9×C4⋊C4—21no—
73C9×C4⋊C4C2311yescharacteristic
74C9×C4⋊C4C2311yescharacteristic
75C9×C22⋊C4—21no—
76C9×C22⋊C4—21no—
77C9×C22⋊C4—21no—
78C9×C22⋊C4—21no—
79C9×C22⋊C4—21no—
80C42×C9—21no—
81C42×C9D411yescharacteristic
82C42×C9D411yescharacteristic
83C42×C9C2311yescharacteristic
84C42×C9C2311yescharacteristic
85#3675—42no—
86#3673—42no—
87(C2×D9)⋊C4—22no—
88(C2×D9)⋊C4—22no—
89C4⋊Dic9—21no—
90C4⋊Dic9C2312yes—
91C2×C9⋊C8—42no—
92C2×C9⋊C8C2312yes—
93C2×C9⋊C8C2312yes—
94#3247—91noSylow
95C2×#235—61no—
96C3×#126—21no—
97C3×#125C2×D311yescharacteristic
98C3×#124C2×D311yescharacteristic
99C3×#123—21no—
100C3×C22⋊Q8C2×D311yescharacteristic
101C3×#121C2×D311yescharacteristic
102C3×C4×Q8C2×D311yescharacteristic
103C3×C4×D4—21no—
104C3×C4×D4C2×D311yescharacteristic
105C3×#117—21no—
106C2×C3×C42C2×D311yescharacteristic
107C2×#215—32no—
108C2×#213—32no—
109#930—32no—
110#930—61no—
111C22×C3⋊C8—32no—
112#875—61no—
113#874—31no—
114C4×D12—32no—
115C4×D12—61no—
116#869—31no—
117#835—32no—
118#832—32no—
119#809—32no—
120#809—32no—
121#808—32no—
122#807—32no—
123C4⋊(C3⋊C8)—31no—
124C4⋊(C3⋊C8)—31no—
125C4⋊(C3⋊C8)—32no—
126C23×C9—21no—
127C23×C9C2×D411yescharacteristic
128C22×D9—22no—
129Q8×C9—21no—
130Q8×C9—21no—
131D4×C9—21no—
132D4×C9—21no—
133D4×C9—42no—
134C2×C4×C9—22no—
135C2×C4×C9—22no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—42no—
142C2×C4×C9C2×D411yescharacteristic
143C2×C4×C9C2×D411yescharacteristic
144C2×C4×C9C2×D411yescharacteristic
145C2×C4×C9C2×D411yescharacteristic
146C2×C4×C9C2×D411yescharacteristic
147C2×C4×C9C2411yescharacteristic
148#611—44no—
149C2×Dic9—22no—
150D36—22no—
151D36—22no—
152D36—42no—
153D36—42no—
154C4×D9—44no—
155Dic18—42no—
156C9⋊C8—22no—
157C9⋊C8—22no—
158#532—91no—
159#529D911yescharacteristic, complemented
160#478—92no—
161#477—91no—
162#476—91no—
163#466—92no—
164#465—92no—
165#463—92no—
166#460—91no—
167#459—91no—
168C2×C4⋊C8—91no—
169C2×C3×Q8C22×D311yescharacteristic
170C2×C3×D4—21no—
171C2×C3×D4C22×D311yescharacteristic
172C22×C3×C4—21no—
173C22×C3×C4#6711yescharacteristic
174C22×C3×C4#6711yescharacteristic
175C22×C3×C4C22×D311yescharacteristic
176C2×#67—62no—
177#235—124no—
178C2×D12—32no—
179C2×D12—61no—
180C2×D12—61no—
181C2×C4×D3—62no—
182C2×Dic6—61no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4C22×D311yescharacteristic
188C3×C4⋊C4C22×D311yescharacteristic
189C3×C22⋊C4—21no—
190C3×C22⋊C4—21no—
191C3×C22⋊C4—21no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—21no—
194C3×C42—21no—
195C3×C42#6711yescharacteristic
196C3×C42#6711yescharacteristic
197C3×C42C22×D311yescharacteristic
198C3×C42C22×D311yescharacteristic
199#215—122no—
200#213—122no—
201(C2×D3)⋊C4—62no—
202(C2×D3)⋊C4—62no—
203C4⋊Dic3—32no—
204C4⋊Dic3—61no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—32no—
207C2×C3⋊C8—122no—
208C22×C9—21no—
209C22×C9—21no—
210C22×C9—41no—
211C22×C9—41no—
212C22×C9—41no—
213C22×C9AGL(1,ℤ/8)12yes—
214C22×C9C22×D411yescharacteristic
215C2×D9—42no—
216C2×D9—42no—
217C2×D9—42no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—21no—
222C4×C9—41no—
223C4×C9—41no—
224C4×C9—41no—
225C4×C9#13512yes—
226C4×C9H(5,𝔽2)11yescharacteristic
227C4×C92−1+411yescharacteristic
228Dic9—42no—
229C2×#41—181no—
230C2×SD16—92no—
231C2×D8—92no—
232C22×C8—92no—
233C4⋊Q8—91no—
234#127—91no—
235#126—21no—
236#125C2×D911yescharacteristic
237#124—181no—
238#124C2×D911yescharacteristic
239#123—21no—
240C22⋊Q8C2×D911yescharacteristic
241#121—92no—
242#121—181no—
243#121C2×D911yescharacteristic
244C4×Q8C2×D911yescharacteristic
245C4×D4—21no—
246C4×D4—92no—
247C4×D4—181no—
248C4×D4C2×D911yescharacteristic
249#117—21no—
250C2×C42C2×D911yescharacteristic
251#107—92no—
252#106—92no—
253C4⋊C8—91no—
254C4⋊C8—91no—
255C4⋊C8—92no—
256#103—92no—
257#102—92no—
258#102—92no—
259#102—92no—
260C23×C3—21no—
261C23×C3C2×#6711yescharacteristic
262C22×D3—62no—
263C3×Q8—21no—
264C3×Q8—21no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—42no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—42no—
276C2×C3×C4C2×#6711yescharacteristic
277C2×C3×C4C2×#6711yescharacteristic
278C2×C3×C4C2×#6711yescharacteristic
279C2×C3×C4C2×#6711yescharacteristic
280C2×C3×C4C2×#6711yescharacteristic
281C2×C3×C4C23×D311yescharacteristic
282#67—124no—
283C2×Dic3—62no—
284D12—62no—
285D12—62no—
286D12—122no—
287D12—122no—
288C4×D3—124no—
289Dic6—122no—
290C3⋊C8—62no—
291C3⋊C8—62no—
292C2×C9—22no—
293C2×C9—81no—
294C2×C9#53711yescharacteristic
295C2×C9C2×#13511yescharacteristic
296C2×C9C2×AGL(1,ℤ/8)11yescharacteristic
297D9—82no—
298#41—364no—
299C2×Q8—181no—
300C2×Q8C22×D911yescharacteristic
301C2×D4—21no—
302C2×D4—92no—
303C2×D4—182no—
304C2×D4—181no—
305C2×D4—181no—
306C2×D4C22×D911yescharacteristic
307C22×C4—21no—
308C22×C4—182no—
309C22×C4#61111yescharacteristic
310C22×C4#61111yescharacteristic
311C22×C4C22×D911yescharacteristic
312SD16—362no—
313D8—362no—
314C2×C8—92no—
315C2×C8—92no—
316C2×C8—362no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—92no—
322C4⋊C4—181no—
323C4⋊C4C22×D911yescharacteristic
324C4⋊C4C22×D911yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—182no—
331C22⋊C4—182no—
332C42—21no—
333C42#61111yescharacteristic
334C42#61111yescharacteristic
335C42C22×D911yescharacteristic
336C42C22×D911yescharacteristic
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3#94912yes—
343C22×C3C22×#6711yescharacteristic
344C2×D3—122no—
345C2×D3—122no—
346C2×D3—122no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—41no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4#95012yes—
355C3×C4#100911yescharacteristic
356C3×C4#101011yescharacteristic
357Dic3—122no—
358C9#324711yescharacteristic, complemented, Sylow
359C23—21no—
360C23—182no—
361C23C2×#61111yescharacteristic
362Q8—21no—
363Q8—21no—
364Q8—362no—
365D4—21no—
366D4—21no—
367D4—42no—
368D4—182no—
369D4—182no—
370D4—362no—
371D4—364no—
372D4—362no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—182no—
382C2×C4—364no—
383C2×C4C2×#61111yescharacteristic
384C2×C4C2×#61111yescharacteristic
385C2×C4C2×#61111yescharacteristic
386C2×C4C2×#61111yescharacteristic
387C2×C4C2×#61111yescharacteristic
388C2×C4C23×D911yescharacteristic
389C8—182no—
390C8—182no—
391C2×C3—22no—
392C2×C3—81no—
393C2×C3C2×#94911yescharacteristic
394C2×C3C2×#95011yescharacteristic
395C2×C3#583311yescharacteristic
396D3—242no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—362no—
403C22—362no—
404C22—362no—
405C22#6351212yes—
406C22C22×#61111yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—362no—
415C4#6351312yes—
416C4#6371511yescharacteristic
417C4#6371611yescharacteristic
418C3#8314811yescharacteristic
419C2—22no—
420C2—81no—
421C2—722no—
422C2C2×#6351211yescharacteristic
423C2C2×#6351311yescharacteristic
424C2#1058993511yescharacteristic
subgroupG/H
centerC22C22×#611
commutator, cosocleC2×C4×C9C24
FrattiniC2×C3×C4C23×D3
FittingC9×#529C2
socleC22×C3C22×#67
hypercenter#529D9
hypocenterC9#3247
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C9×#529C2semidirect
2. #10589579C2semidirect
3. #10588747C2semidirect
4. #10588738C2semidirect
5. #10588731C2semidirect
6. #10588721C2semidirect
7. #10588710C2semidirect
8. #10588703C2semidirect
9. #10588698C2semidirect
10. #10588691C2semidirect
11. C2×C4⋊(C9⋊C8)C2semidirect
159. #529D9semidirect
358. C9#3247semidirect

Maximal subgroups

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

subgroupindex
C9×#5292
#105895792
#105887472
#105887382
#105887312
#105887212
#105887102
#105887032
#105886982
#105886912
C2×C4⋊(C9⋊C8)2
#831483