Finite Group Databasebeta

← #10593098⋊C3 #13164426

Subgroups — #10593098⋊C3 #13164426

Order 1728.

540 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 65 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×#88691C211yesmaximal, characteristic, complemented
2#63692⋊C3C211yesmaximal, characteristic, complemented
3#63691⋊C3C212yesmaximal, complemented
4#63690⋊C3C211yesmaximal, characteristic, complemented
5#63689⋊C3C212yesmaximal, complemented
6C3×#5822—31nomaximal
7#10593098C311yesmaximal, characteristic, complemented
8#3697⋊C3—41nomaximal
9#88691C2211yescharacteristic
10#88691C2211yescharacteristic, complemented
11C2×#6204C2211yescharacteristic, complemented
12#3689⋊C3C2212yescomplemented
13#3688⋊C3C2212yescomplemented
14C2×C32×#41D311yescharacteristic
15C2×C3×#235—31no—
16C3×#951—31no—
17C3×#951—31no—
18C3×#950—32no—
19C3×#950—32no—
20C3×#949—31no—
21C3×#949—31no—
22C3×#942—32no—
23C3×#932—32no—
24C2×C3×#208—31no—
25C2×#41⋊C9—21no—
26C2×#41⋊C9C2×C311yescharacteristic, complemented
27#63692C2×C311yescharacteristic, complemented
28#63691C2×C312yescomplemented
29#63690C2×C311yescharacteristic, complemented
30#63689C2×C312yescomplemented
31C2×C4×3−1+2—41no—
32#611⋊C3—42no—
33C4⋊AGL(1,ℤ/9)—41no—
34C4×AGL(1,ℤ/9)—42no—
353−1+2⋊Q8—41no—
36#6204C2311yescharacteristic
37C3×#575—91no—
38#5822—31no—
39C32×#41—31no—
40C32×#41—31no—
41C32×#41C2×D311yescharacteristic
42C32×#41C2×D311yescharacteristic
43C2×C32×Q8C2×D311yescharacteristic
44C2×C32×D4—31no—
45C22×C32×C4—31no—
46C2×C3×#67—61no—
47C3×#235—32no—
48C3×#235—32no—
49C3×#235—61no—
50C3×#235—61no—
51C2×C3×D12—31no—
52C2×C3×C4×D3—61no—
53C2×C3×Dic6—31no—
54C3×#216—32no—
55C3×#216—32no—
56C3×#215—32no—
57C3×#215—32no—
58C3×#214—32no—
59C3×#214—32no—
60C3×#213—32no—
61C3×#213—32no—
62C3×#208—32no—
63C3×#208—31no—
64C3×#208—31no—
65C2×C3×C3⋊C8—32no—
66#3697—41no—
67#41⋊C9—21no—
68#41⋊C9—21no—
69#41⋊C9C22×C311yescharacteristic
70#41⋊C9C22×C311yescharacteristic, complemented
71C2×Q8⋊C9—21no—
72C2×Q8⋊C9C22×C311yescharacteristic, complemented
73#3689C22×C312yescomplemented
74#3688C22×C312yescomplemented
75C22×3−1+2—41no—
76C2×AGL(1,ℤ/9)—42no—
77C4×3−1+2—41no—
78C4×3−1+2—41no—
793−1+2⋊C4—42no—
80C2×C3×#41—31no—
81C2×C3×#41—91no—
82C2×C3×#41C3×D311yescharacteristic
83C2×#235—31no—
84C3×#137—92no—
85C3×#137—91no—
86C3×#137—91no—
87C3×AGL(1,ℤ/8)—92no—
88C3×AGL(1,ℤ/8)—91no—
89C3×AGL(1,ℤ/8)—91no—
90C3×#135—92no—
91C3×#135—92no—
92C2×C3×M16—91no—
93#951—31no—
94#951—31no—
95#950—32no—
96#950—32no—
97#949—31no—
98#949—31no—
99#942—32no—
100#932—32no—
101C2×#208—31no—
102C23×C32—31no—
103C22×C3×D3—61no—
104C32×Q8—31no—
105C32×Q8C22×D311yescharacteristic
106C32×D4—31no—
107C32×D4—31no—
108C32×D4—61no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4S411yescharacteristic
115C3×#67—62no—
116C3×#67—62no—
117C2×C3×Dic3—61no—
118C3×D12—32no—
119C3×D12—61no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×Dic6—32no—
123C3×Dic6—61no—
124C3×C3⋊C8—32no—
125C3×C3⋊C8—32no—
126C2×C4×C9—41no—
127C2×C4×C9—81no—
128#611—42no—
129D36—41no—
130C4×D9—42no—
131Dic18—41no—
132Q8⋊C9—21no—
133Q8⋊C9C23×C311yescharacteristic
134#575—91noSylow
135C2×3−1+2—41no—
136C2×3−1+2—81no—
137AGL(1,ℤ/9)—82no—
138C3×#41—31no—
139C3×#41—31no—
140C3×#41—31no—
141C3×#41—31no—
142C3×#41—31no—
143C3×#41—31no—
144C3×#41—61no—
145C3×#41—61no—
146C3×#41—92no—
147C3×#41—92no—
148C3×#41—181no—
149C3×#41—181no—
150C3×#41C2×C3×D311yescharacteristic
151C3×#41C2×C3×D311yescharacteristic
152C2×C3×Q8—31no—
153C2×C3×Q8—91no—
154C2×C3×Q8C2×C3×D311yescharacteristic
155C2×C3×D4—31no—
156C2×C3×D4—31no—
157C2×C3×D4—61no—
158C2×C3×D4—91no—
159C2×C3×D4—181no—
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—61no—
163C22×C3×C4—181no—
164C2×#67—61no—
165#235—32no—
166#235—32no—
167#235—61no—
168#235—61no—
169C2×D12—31no—
170C2×C4×D3—61no—
171C2×Dic6—31no—
172C3×Q16—92no—
173C3×Q16—92no—
174C3×SD16—92no—
175C3×SD16—92no—
176C3×SD16—92no—
177C3×SD16—92no—
178C3×D8—92no—
179C3×D8—92no—
180C3×M16—92no—
181C3×M16—91no—
182C3×M16—91no—
183C2×C3×C8—92no—
184#216—32no—
185#216—32no—
186#215—32no—
187#215—32no—
188#214—32no—
189#214—32no—
190#213—32no—
191#213—32no—
192#208—32no—
193#208—31no—
194#208—31no—
195C2×C3⋊C8—32no—
196C22×C32—31no—
197C22×C32—31no—
198C22×C32—121no—
199C22×C32C2×S411yescharacteristic
200C2×C3×D3—62no—
201C2×C3×D3—122no—
202C32×C4—31no—
203C32×C4—31no—
204C32×C4C2×S411yescharacteristic
205C32×C4C2×S411yescharacteristic
206C3×Dic3—62no—
207C22×C9—41no—
208C22×C9—81no—
209C2×D9—42no—
210C4×C9—41no—
211C4×C9—41no—
212C4×C9—81no—
213C4×C9—81no—
214Dic9—42no—
215C2×#41—91no—
216C2×#41AGL(1,ℤ/9)11yescharacteristic, complemented
217#137—91no—
218#137—91no—
219#137—92no—
220AGL(1,ℤ/8)—92no—
221AGL(1,ℤ/8)—91no—
222AGL(1,ℤ/8)—91no—
223#135—92no—
224#135—92no—
225C2×M16—91no—
2263−1+2—41noSylow
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—61no—
230C23×C3—181no—
231C22×D3—61no—
232C3×Q8—31no—
233C3×Q8—31no—
234C3×Q8—31no—
235C3×Q8—61no—
236C3×Q8—92no—
237C3×Q8—181no—
238C3×Q8C22×C3×D311yescharacteristic
239C3×D4—31no—
240C3×D4—31no—
241C3×D4—31no—
242C3×D4—31no—
243C3×D4—61no—
244C3×D4—61no—
245C3×D4—61no—
246C3×D4—61no—
247C3×D4—61no—
248C3×D4—61no—
249C3×D4—92no—
250C3×D4—181no—
251C3×D4—182no—
252C3×D4—182no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—182no—
270C2×C3×C4—182no—
271C2×C3×C4—181no—
272C2×C3×C4C3×S411yescharacteristic
273#67—62no—
274#67—62no—
275C2×Dic3—61no—
276D12—32no—
277D12—61no—
278C4×D3—62no—
279C4×D3—62no—
280Dic6—32no—
281Dic6—61no—
282C3×C8—92no—
283C3×C8—92no—
284C3⋊C8—32no—
285C3⋊C8—32no—
286C2×C32—21no—
287C2×C32—61no—
288C2×C32—61no—
289C2×C32C22×S411yescharacteristic
290C3×D3—122no—
291C2×C9—41no—
292C2×C9—81no—
293C2×C9—82no—
294C2×C9—81no—
295D9—82no—
296#41—31no—
297#41—31no—
298#41—92no—
299#41—92no—
300#41—181no—
301#41—181no—
302#41C2×AGL(1,ℤ/9)11yescharacteristic
303#41C2×AGL(1,ℤ/9)11yescharacteristic
304C2×Q8—91no—
305C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
306C2×D4—31no—
307C2×D4—91no—
308C2×D4—181no—
309C22×C4—31no—
310C22×C4—181no—
311Q16—92no—
312Q16—92no—
313SD16—92no—
314SD16—92no—
315SD16—92no—
316SD16—92no—
317D8—92no—
318D8—92no—
319M16—92no—
320M16—91no—
321M16—91no—
322C2×C8—92no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—122no—
333C22×C3—182no—
334C22×C3—362no—
335C22×C3C2×C3×S411yescharacteristic
336C2×D3—62no—
337C2×D3—122no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—182no—
347C3×C4C2×C3×S411yescharacteristic
348C3×C4C2×C3×S411yescharacteristic
349Dic3—62no—
350C32#592311yescharacteristic
351C9—41no—
352C9—81no—
353C23—31no—
354C23—181no—
355Q8—31no—
356Q8—92no—
357Q8—181no—
358Q8C22×AGL(1,ℤ/9)11yescharacteristic
359D4—31no—
360D4—31no—
361D4—61no—
362D4—92no—
363D4—182no—
364D4—182no—
365D4—181no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—182no—
372C2×C4—181no—
373C2×C4—182no—
374C2×C4((C22⋊C9)⋊C2)⋊C311yescharacteristic
375C8—92no—
376C8—92no—
377C2×C3—21no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—121no—
385C2×C3—121no—
386C2×C3—362no—
387C2×C3C22×C3×S411yescharacteristic
388D3—122no—
389C22—31no—
390C22—31no—
391C22—121no—
392C22—182no—
393C22—362no—
394C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
395C4—31no—
396C4—31no—
397C4—182no—
398C4C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
399C4C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
400C3—31no—
401C3C3×#592311yescharacteristic
402C2—21no—
403C2—61no—
404C2—61no—
405C2—362no—
406C2C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
centerC4C2×((C22⋊C9)⋊C2)⋊C3
commutator, cosocle, hypocenterQ8⋊C9C23×C3
Frattini, socleC2×C3C22×C3×S4
FittingC2×C32×#41D3
hypercenterC2×C4((C22⋊C9)⋊C2)⋊C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88691C2semidirect
2. #63692⋊C3C2semidirect
3. #63691⋊C3C2semidirect
4. #63690⋊C3C2semidirect
5. #63689⋊C3C2semidirect
7. #10593098C3semidirect
10. #88691C22semidirect
11. C2×#6204C22semidirect
12. #3689⋊C3C22semidirect
13. #3688⋊C3C22semidirect
26. C2×#41⋊C9C2×C3semidirect
27. #63692C2×C3semidirect
28. #63691C2×C3semidirect
29. #63690C2×C3semidirect
30. #63689C2×C3semidirect
70. #41⋊C9C22×C3semidirect
72. C2×Q8⋊C9C22×C3semidirect
73. #3689C22×C3semidirect
74. #3688C22×C3semidirect
216. C2×#41AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×#886912
#63692⋊C32
#63691⋊C32
#63690⋊C32
#63689⋊C32
C3×#58223
#105930983
#3697⋊C34