Finite Group Databasebeta

← #10588830⋊C3 #13143317

Subgroups — #10588830⋊C3 #13143317

Order 1728.

458 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×3−1+2C211yesmaximal, characteristic, complemented
2#63502⋊C3C211yesmaximal, characteristic, complemented
3D4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
4C2×#3673⋊C3C211yesmaximal, characteristic, complemented
5#63392⋊C3C211yesmaximal, characteristic, complemented
6#63380⋊C3C211yesmaximal, characteristic, complemented
7(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic, complemented
8C3×#5150—31nomaximal
9#10588830C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C2×D4×3−1+2C2211yescharacteristic
12C2×#611⋊C3—21no—
13C2×#611⋊C3—21no—
14C22×3−1+2⋊C4—21no—
15C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
16D8×3−1+2—22no—
17C2×C8×3−1+2C2211yescharacteristic
18#3676⋊C3—21no—
19#3673⋊C3—22no—
20C4⋊(3−1+2⋊C4)C2211yescharacteristic
21C4×3−1+2⋊C4C2211yescharacteristic
22C2×3−1+2⋊C8C2211yescharacteristic
23C2×C32×D8D311yescharacteristic
24C3×#940—31no—
25C3×D4×Dic3—31no—
26C2×C3×#213—31no—
27C3×#832—31no—
28C3×#821—31no—
29C3×Dic3⋊C8—31no—
30C2×C9×D8—21no—
31C2×C9×D8C2×C311yescharacteristic, complemented
32#63502C2×C311yescharacteristic, complemented
33D4×Dic9C2×C311yescharacteristic, complemented
34C2×#3673C2×C311yescharacteristic, complemented
35#63392C2×C311yescharacteristic, complemented
36#63380C2×C311yescharacteristic, complemented
37Dic9⋊C8C2×C311yescharacteristic, complemented
38C23×3−1+2—21no—
39C23×3−1+2—21no—
40C22×AGL(1,ℤ/9)—21no—
41D4×3−1+2—21no—
42D4×3−1+2—21no—
43D4×3−1+2—21no—
44D4×3−1+2D412yescomplemented
45C2×C4×3−1+2C2311yescharacteristic
46#611⋊C3—42no—
47#611⋊C3—42no—
48C2×3−1+2⋊C4—21no—
49C2×3−1+2⋊C4—42no—
50C2×3−1+2⋊C4D411yescharacteristic
51C2×3−1+2⋊C4D411yescharacteristic
52C4⋊AGL(1,ℤ/9)—21no—
53C4⋊AGL(1,ℤ/9)—21no—
54C8×3−1+2—21no—
553−1+2⋊C8—21no—
56C3×#459—91no—
57#5150—31no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×C3×#67—61no—
61C2×C3×#67—61no—
62C22×C3×Dic3—61no—
63C2×C3×D12—31no—
64C32×D8—22no—
65C2×C32×C8C2×D311yescharacteristic
66C3×#217—61no—
67C3×#213—62no—
68C3×C4⋊Dic3—31no—
69C3×C4×Dic3—31no—
70C2×C3×C3⋊C8—31no—
71C2×D4×C9—21no—
72C2×D4×C9—21no—
73C2×D4×C9C22×C311yescharacteristic
74C2×D4×C9C22×C311yescharacteristic
75C2×#611—21no—
76C2×#611—21no—
77C22×Dic9—21no—
78C2×D36C22×C311yescharacteristic
79C9×D8—24no—
80C9×D8—22no—
81C2×C8×C9—21no—
82C2×C8×C9C22×C311yescharacteristic
83#3676—21no—
84#3673—22no—
85C4⋊Dic9C22×C311yescharacteristic
86C4×Dic9C22×C311yescharacteristic
87C2×C9⋊C8C22×C311yescharacteristic
88C22×3−1+2—22no—
89C22×3−1+2—41no—
90C22×3−1+2—41no—
91C22×3−1+2—41no—
92C22×3−1+2—41no—
93C22×3−1+2—41no—
94C22×3−1+2C2×D411yescharacteristic
95C2×AGL(1,ℤ/9)—41no—
96C2×AGL(1,ℤ/9)—41no—
97C2×AGL(1,ℤ/9)—41no—
98C4×3−1+2C2×D411yescharacteristic
99C4×3−1+2#4111yescharacteristic
1003−1+2⋊C4—21no—
1013−1+2⋊C4—41no—
1023−1+2⋊C4D812yescomplemented
103C2×C3×D8—31no—
104C2×C3×D8—91no—
105C2×C3×D8C3×D311yescharacteristic
106C3×#127—91no—
107C3×C4×D4—91no—
108#940—31no—
109D4×Dic3—31no—
110C2×#213—31no—
111C3×C4⋊C8—91no—
112C3×#102—91no—
113C3×#102—91no—
114#832—31no—
115#821—31no—
116Dic3⋊C8—31no—
117C23×C32—21no—
118C23×C32—21no—
119C22×C3×D3—61no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4#6712yes—
124C2×C32×C4C22×D311yescharacteristic
125C3×#67—122no—
126C3×#67—122no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—122no—
131C3×D12—61no—
132C3×D12—61no—
133C23×C9—21no—
134C23×C9—21no—
135C23×C9—41no—
136C23×C9—41no—
137C22×D9—21no—
138C32×C8—21no—
139C3×C3⋊C8—61no—
140D4×C9—22no—
141D4×C9—21no—
142D4×C9—21no—
143D4×C9—21no—
144D4×C9—22no—
145D4×C9—41no—
146D4×C9—41no—
147D4×C9C3×D412yescomplemented
148C2×C4×C9—21no—
149C2×C4×C9C23×C311yescharacteristic
150#611—42no—
151#611—42no—
152C2×Dic9—21no—
153C2×Dic9—42no—
154C2×Dic9C3×D411yescharacteristic
155C2×Dic9C3×D411yescharacteristic
156D36—21no—
157D36—21no—
158C8×C9—22no—
159C8×C9—21no—
160C9⋊C8—21no—
161#459—91noSylow
162C2×3−1+2—42no—
163C2×3−1+2—81no—
164C2×3−1+2#12111yescharacteristic
165C2×3−1+2C2×D811yescharacteristic
166C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
167AGL(1,ℤ/9)—81no—
168C2×C3×D4—31no—
169C2×C3×D4—31no—
170C2×C3×D4—91no—
171C2×C3×D4—181no—
172C2×C3×D4—181no—
173C2×C3×D4C2×C3×D311yescharacteristic
174C2×C3×D4C2×C3×D311yescharacteristic
175C22×C3×C4—181no—
176C2×#67—61no—
177C2×#67—61no—
178C22×Dic3—61no—
179C2×D12—31no—
180C3×D8—22no—
181C3×D8—62no—
182C3×D8—182no—
183C2×C3×C8—31no—
184C2×C3×C8—91no—
185C2×C3×C8C2×C3×D311yescharacteristic
186C3×C4⋊C4—91no—
187C3×C22⋊C4—181no—
188C3×C42—91no—
189#217—61no—
190#213—62no—
191C4⋊Dic3—31no—
192C4×Dic3—31no—
193C2×C3⋊C8—31no—
194C22×C32—22no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32D3×D411yescharacteristic
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C32×C4#23711yescharacteristic
205C32×C4C2×#6711yescharacteristic
206C3×Dic3—32no—
207C3×Dic3—61no—
208C3×Dic3—121no—
209C22×C9—22no—
210C22×C9—21no—
211C22×C9—42no—
212C22×C9—42no—
213C22×C9—41no—
214C22×C9—41no—
215C22×C9—41no—
216C22×C9—41no—
217C22×C9—41no—
218C22×C9—81no—
219C22×C9—81no—
220C22×C9—81no—
221C22×C9—81no—
222C22×C9C2×C3×D411yescharacteristic
223C2×D9—41no—
224C2×D9—41no—
225C2×D9—41no—
226C4×C9—21no—
227C4×C9—21no—
228C4×C9C2×C3×D411yescharacteristic
229C4×C9C3×#4111yescharacteristic
230Dic9—21no—
231Dic9—41no—
232Dic9C3×D812yescomplemented
233C2×D8—91no—
234C2×D8AGL(1,ℤ/9)11yescharacteristic, complemented
235#127—91no—
236C4×D4—91no—
237C4⋊C8—91no—
238#102—91no—
239#102—91no—
2403−1+2#45911yescharacteristic, complemented, Sylow
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—181no—
246C22×D3—61no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—32no—
251C3×D4—61no—
252C3×D4—61no—
253C3×D4—61no—
254C3×D4—181no—
255C3×D4—181no—
256C3×D4—362no—
257C3×D4—362no—
258C3×D4C3×#6712yes—
259C2×C3×C4—31no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—181no—
263C2×C3×C4—362no—
264C2×C3×C4C22×C3×D311yescharacteristic
265#67—122no—
266#67—122no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—122no—
271D12—61no—
272D12—61no—
273C3×C8—21no—
274C3×C8—61no—
275C3×C8—181no—
276C3⋊C8—61no—
277C2×C32—42no—
278C2×C32—81no—
279C2×C32D3×D811yescharacteristic
280C2×C32#91111yescharacteristic
281C2×C32#93911yescharacteristic
282C3×D3—241no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9—42no—
287C2×C9—82no—
288C2×C9—82no—
289C2×C9—81no—
290C2×C9C3×#12111yescharacteristic
291C2×C9C2×C3×D811yescharacteristic
292C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
293D9—81no—
294C2×D4—91no—
295C2×D4—181no—
296C2×D4—181no—
297C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
298C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
299C22×C4—181no—
300D8—22no—
301D8—182no—
302C2×C8—91no—
303C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
304C4⋊C4—91no—
305C22⋊C4—181no—
306C42—91no—
307C22×C3—22no—
308C22×C3—31no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—62no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3C3×D3×D411yescharacteristic
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C3×C4—31no—
328C3×C4—31no—
329C3×C4—92no—
330C3×C4—181no—
331C3×C4—361no—
332C3×C4C3×#23711yescharacteristic
333C3×C4C2×C3×#6711yescharacteristic
334Dic3—32no—
335Dic3—61no—
336Dic3—121no—
337C32#515011yescharacteristic
338C9—21no—
339C9C3×#45911yescharacteristic, complemented
340C23—21no—
341C23—21no—
342C23—181no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—181no—
347D4—181no—
348D4—362no—
349D4—362no—
350D4#611⋊C312yescomplemented
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—181no—
354C2×C4—362no—
355C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
356C8—21no—
357C8—181no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—42no—
362C2×C3—81no—
363C2×C3—122no—
364C2×C3—241no—
365C2×C3—721no—
366C2×C3C3×D3×D811yescharacteristic
367C2×C3C3×#91111yescharacteristic
368C2×C3C3×#93911yescharacteristic
369D3—241no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—361no—
377C22—361no—
378C22—361no—
379C22D4×AGL(1,ℤ/9)11yescharacteristic
380C4—92no—
381C4—181no—
382C4—361no—
383C4#3699⋊C311yescharacteristic
384C4C2×#611⋊C311yescharacteristic
385C3—31no—
386C3C3×#515011yescharacteristic
387C2—42no—
388C2—81no—
389C2—721no—
390C2D8×AGL(1,ℤ/9)11yescharacteristic
391C2#63473⋊C311yescharacteristic
392C2#63501⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×D8×3−1+2C2
socleC22×C3C3×D3×D4
hypercenterC2×D8AGL(1,ℤ/9)
hypocenterC9C3×#459
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×D8×3−1+2C2semidirect
2. #63502⋊C3C2semidirect
3. D4×3−1+2⋊C4C2semidirect
4. C2×#3673⋊C3C2semidirect
5. #63392⋊C3C2semidirect
6. #63380⋊C3C2semidirect
7. (3−1+2⋊C4)⋊C8C2semidirect
9. #10588830C3semidirect
31. C2×C9×D8C2×C3semidirect
32. #63502C2×C3semidirect
33. D4×Dic9C2×C3semidirect
34. C2×#3673C2×C3semidirect
35. #63392C2×C3semidirect
36. #63380C2×C3semidirect
37. Dic9⋊C8C2×C3semidirect
44. D4×3−1+2D4semidirect
102. 3−1+2⋊C4D8semidirect
147. D4×C9C3×D4semidirect
232. Dic9C3×D8semidirect
234. C2×D8AGL(1,ℤ/9)semidirect
240. 3−1+2#459semidirect
339. C9C3×#459semidirect
350. D4#611⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D8×3−1+22
#63502⋊C32
D4×3−1+2⋊C42
C2×#3673⋊C32
#63392⋊C32
#63380⋊C32
(3−1+2⋊C4)⋊C82
C3×#51503
#105888303