Finite Group Databasebeta

← #10596586⋊C3 #13164430

Subgroups — #10596586⋊C3 #13164430

Order 1728.

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

normal subgroups only

This group has 589 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64267⋊C3C211yesmaximal, characteristic, complemented
3#64266⋊C3C212yesmaximal, complemented
4#64265⋊C3C211yesmaximal, characteristic, complemented
5#64264⋊C3C212yesmaximal, complemented
6#10596586C311yesmaximal, characteristic, complemented
7C3×#5822—31nomaximal
8#3828⋊C3—41nomaximal
9#41⋊H(3,𝔽3)C2211yescharacteristic
10#41⋊H(3,𝔽3)C2211yescharacteristic, complemented
11C2×Q8⋊H(3,𝔽3)C2211yescharacteristic, complemented
12(C3⋊GL(2,𝔽3))⋊C3C2212yescomplemented
13(C3⋊#226)⋊C3C2212yescomplemented
14C2×C32×#41D311yescharacteristic, complemented
15C2×C3×#235—31no—
16C2×C3×#41⋊C3—21no—
17C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
18#64267C2×C311yescharacteristic, complemented
19#64266C2×C312yescomplemented
20#64265C2×C311yescharacteristic, complemented
21#64264C2×C312yescomplemented
22C3×#951—31no—
23C3×#951—31no—
24C3×#950—32no—
25C3×#950—32no—
26C3×#949—31no—
27C3×#949—31no—
28C3×#942—32no—
29C3×#932—32no—
30C2×C3×#208—31no—
31C2×C4×H(3,𝔽3)—41no—
32#638⋊C3—42no—
33C4⋊((C3⋊D3)⋊C3)—41no—
34C4×(C3⋊D3)⋊C3—42no—
35#634⋊C3—41no—
36Q8⋊H(3,𝔽3)C2311yescharacteristic
37#5923—91no—
38C3×#575—91no—
39#5822—31no—
40C32×#41—31no—
41C32×#41—31no—
42C32×#41C2×D311yescharacteristic
43C32×#41C2×D311yescharacteristic
44C2×C32×Q8C2×D311yescharacteristic
45C2×C32×D4—31no—
46C22×C32×C4—31no—
47#3828—41no—
48C2×C3×#67—61no—
49C3×#235—32no—
50C3×#235—32no—
51C3×#235—61no—
52C3×#235—61no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—61no—
55C2×C3×Dic6—31no—
56C3×#41⋊C3—21no—
57C3×#41⋊C3—21no—
58C3×#41⋊C3C22×C311yescharacteristic
59C3×#41⋊C3C22×C311yescharacteristic, complemented
60C2×C3×SL(2,𝔽3)—21no—
61C2×C3×SL(2,𝔽3)C22×C311yescharacteristic, complemented
62C3⋊GL(2,𝔽3)C22×C312yescomplemented
63C3⋊#226C22×C312yescomplemented
64C3×#216—32no—
65C3×#216—32no—
66C3×#215—32no—
67C3×#215—32no—
68C3×#214—32no—
69C3×#214—32no—
70C3×#213—32no—
71C3×#213—32no—
72C3×#208—32no—
73C3×#208—31no—
74C3×#208—31no—
75C2×C3×C3⋊C8—32no—
76C22×H(3,𝔽3)—41no—
77C2×(C3⋊D3)⋊C3—42no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)—41no—
80(C3⋊Dic3)⋊C3—42no—
81C2×C3×#41—31no—
82C2×C3×#41—91no—
83C2×C3×#41C3×D311yescharacteristic
84C2×#235—31no—
85C2×#41⋊C3—31no—
86C2×#41⋊C3—61no—
87#986—91no—
88#985—92no—
89#984—91no—
90#983—92no—
91C3×#137—92no—
92C3×#137—91no—
93C3×#137—91no—
94C3×AGL(1,ℤ/8)—92no—
95C3×AGL(1,ℤ/8)—91no—
96C3×AGL(1,ℤ/8)—91no—
97C3×#135—92no—
98C3×#135—92no—
99C2×C3×M16—91no—
100#951—31no—
101#951—31no—
102#950—32no—
103#950—32no—
104#949—31no—
105#949—31no—
106#942—32no—
107#932—32no—
108C2×#208—31no—
109C23×C32—31no—
110C22×C3×D3—61no—
111C32×Q8—31no—
112C32×Q8C22×D311yescharacteristic
113C32×D4—31no—
114C32×D4—31no—
115C32×D4—61no—
116C2×C32×C4—31no—
117C2×C32×C4—31no—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4—41no—
122C2×C32×C4—81no—
123C2×C32×C4S411yescharacteristic
124#638—42no—
125C4⋊(C3⋊D3)—41no—
126C4×C3⋊D3—42no—
127#634—41no—
128C3×#67—62no—
129C3×#67—62no—
130C2×C3×Dic3—61no—
131C3×D12—32no—
132C3×D12—61no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×Dic6—32no—
136C3×Dic6—61no—
137C3×SL(2,𝔽3)—21no—
138C3×SL(2,𝔽3)C23×C311yescharacteristic
139C3×C3⋊C8—32no—
140C3×C3⋊C8—32no—
141#575—91noSylow
142C2×H(3,𝔽3)—41no—
143C2×H(3,𝔽3)—81no—
144(C3⋊D3)⋊C3—82no—
145C3×#41—31no—
146C3×#41—31no—
147C3×#41—31no—
148C3×#41—31no—
149C3×#41—31no—
150C3×#41—31no—
151C3×#41—61no—
152C3×#41—61no—
153C3×#41—92no—
154C3×#41—92no—
155C3×#41—181no—
156C3×#41—181no—
157C3×#41C2×C3×D311yescharacteristic
158C3×#41C2×C3×D311yescharacteristic
159C2×C3×Q8—31no—
160C2×C3×Q8—91no—
161C2×C3×Q8C2×C3×D311yescharacteristic
162C2×C3×D4—31no—
163C2×C3×D4—31no—
164C2×C3×D4—61no—
165C2×C3×D4—91no—
166C2×C3×D4—181no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—61no—
170C22×C3×C4—181no—
171C2×#67—61no—
172#235—32no—
173#235—32no—
174#235—61no—
175#235—61no—
176#235—361no—
177C2×D12—31no—
178C2×C4×D3—61no—
179C2×Dic6—31no—
180#41⋊C3—31no—
181#41⋊C3—31no—
182#41⋊C3—61no—
183#41⋊C3—61no—
184C2×SL(2,𝔽3)—31no—
185C2×SL(2,𝔽3)—61no—
186GL(2,𝔽3)—92no—
187#226—92no—
188C3×Q16—92no—
189C3×Q16—92no—
190C3×SD16—92no—
191C3×SD16—92no—
192C3×SD16—92no—
193C3×SD16—92no—
194C3×D8—92no—
195C3×D8—92no—
196C3×M16—92no—
197C3×M16—91no—
198C3×M16—91no—
199C2×C3×C8—92no—
200#216—32no—
201#216—32no—
202#215—32no—
203#215—32no—
204#214—32no—
205#214—32no—
206#213—32no—
207#213—32no—
208#208—32no—
209#208—31no—
210#208—31no—
211C2×C3⋊C8—32no—
212C22×C32—31no—
213C22×C32—31no—
214C22×C32—41no—
215C22×C32—81no—
216C22×C32—121no—
217C22×C32C2×S411yescharacteristic
218C2×C3⋊D3—42no—
219C2×C3×D3—62no—
220C2×C3×D3—122no—
221C32×C4—31no—
222C32×C4—31no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—81no—
226C32×C4—81no—
227C32×C4C2×S411yescharacteristic
228C32×C4C2×S411yescharacteristic
229C3⋊Dic3—42no—
230C3×Dic3—62no—
231C2×#41—91no—
232C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
233#137—91no—
234#137—91no—
235#137—92no—
236AGL(1,ℤ/8)—92no—
237AGL(1,ℤ/8)—91no—
238AGL(1,ℤ/8)—91no—
239#135—92no—
240#135—92no—
241C2×M16—91no—
242H(3,𝔽3)—41noSylow
243C23×C3—31no—
244C23×C3—31no—
245C23×C3—61no—
246C23×C3—181no—
247C22×D3—61no—
248C3×Q8—31no—
249C3×Q8—31no—
250C3×Q8—31no—
251C3×Q8—61no—
252C3×Q8—92no—
253C3×Q8—181no—
254C3×Q8C22×C3×D311yescharacteristic
255C3×D4—31no—
256C3×D4—31no—
257C3×D4—31no—
258C3×D4—31no—
259C3×D4—61no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—61no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—92no—
266C3×D4—181no—
267C3×D4—182no—
268C3×D4—182no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—121no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288C2×C3×C4—181no—
289C2×C3×C4—241no—
290C2×C3×C4C3×S411yescharacteristic
291#67—62no—
292#67—62no—
293#67—362no—
294C2×Dic3—61no—
295D12—32no—
296D12—61no—
297D12—361no—
298C4×D3—62no—
299C4×D3—62no—
300C4×D3—362no—
301Dic6—32no—
302Dic6—61no—
303Dic6—361no—
304SL(2,𝔽3)—31no—
305SL(2,𝔽3)—61no—
306C3×C8—92no—
307C3×C8—92no—
308C3⋊C8—32no—
309C3⋊C8—32no—
310C2×C32—21no—
311C2×C32—41no—
312C2×C32—61no—
313C2×C32—61no—
314C2×C32—81no—
315C2×C32—82no—
316C2×C32—81no—
317C2×C32C22×S411yescharacteristic
318C3⋊D3—82no—
319C3×D3—122no—
320#41—31no—
321#41—31no—
322#41—92no—
323#41—92no—
324#41—181no—
325#41—181no—
326#41C2×(C3⋊D3)⋊C311yescharacteristic
327#41C2×(C3⋊D3)⋊C311yescharacteristic
328C2×Q8—91no—
329C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
330C2×D4—31no—
331C2×D4—91no—
332C2×D4—181no—
333C22×C4—31no—
334C22×C4—181no—
335Q16—92no—
336Q16—92no—
337SD16—92no—
338SD16—92no—
339SD16—92no—
340SD16—92no—
341D8—92no—
342D8—92no—
343M16—92no—
344M16—91no—
345M16—91no—
346C2×C8—92no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—122no—
357C22×C3—121no—
358C22×C3—182no—
359C22×C3—241no—
360C22×C3—362no—
361C22×C3C2×C3×S411yescharacteristic
362C2×D3—62no—
363C2×D3—122no—
364C2×D3—362no—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—182no—
376C3×C4—241no—
377C3×C4—241no—
378C3×C4C2×C3×S411yescharacteristic
379C3×C4C2×C3×S411yescharacteristic
380Dic3—62no—
381Dic3—362no—
382C32—41no—
383C32—81no—
384C32#592311yescharacteristic, complemented
385C23—31no—
386C23—181no—
387Q8—31no—
388Q8—92no—
389Q8—181no—
390Q8C22×(C3⋊D3)⋊C311yescharacteristic
391D4—31no—
392D4—31no—
393D4—61no—
394D4—92no—
395D4—182no—
396D4—182no—
397D4—181no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—182no—
404C2×C4—181no—
405C2×C4—182no—
406C2×C4(C3⋊S4)⋊C311yescharacteristic
407C8—92no—
408C8—92no—
409C2×C3—21no—
410C2×C3—31no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—121no—
417C2×C3—121no—
418C2×C3—121no—
419C2×C3—241no—
420C2×C3—242no—
421C2×C3—241no—
422C2×C3—362no—
423C2×C3C22×C3×S411yescharacteristic
424D3—122no—
425D3—722no—
426C22—31no—
427C22—31no—
428C22—121no—
429C22—182no—
430C22—362no—
431C22C2×(C3⋊S4)⋊C311yescharacteristic
432C4—31no—
433C4—31no—
434C4—182no—
435C4C2×(C3⋊S4)⋊C311yescharacteristic
436C4C2×(C3⋊S4)⋊C311yescharacteristic
437C3—31no—
438C3—121no—
439C3—241no—
440C3C3×#592311yescharacteristic
441C2—21no—
442C2—61no—
443C2—61no—
444C2—362no—
445C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC4C2×(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
Frattini, socleC2×C3C22×C3×S4
FittingC2×C32×#41D3
hypercenterC2×C4(C3⋊S4)⋊C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. #64267⋊C3C2semidirect
3. #64266⋊C3C2semidirect
4. #64265⋊C3C2semidirect
5. #64264⋊C3C2semidirect
6. #10596586C3semidirect
10. #41⋊H(3,𝔽3)C22semidirect
11. C2×Q8⋊H(3,𝔽3)C22semidirect
12. (C3⋊GL(2,𝔽3))⋊C3C22semidirect
13. (C3⋊#226)⋊C3C22semidirect
14. C2×C32×#41D3semidirect
17. C2×C3×#41⋊C3C2×C3semidirect
18. #64267C2×C3semidirect
19. #64266C2×C3semidirect
20. #64265C2×C3semidirect
21. #64264C2×C3semidirect
59. C3×#41⋊C3C22×C3semidirect
61. C2×C3×SL(2,𝔽3)C22×C3semidirect
62. C3⋊GL(2,𝔽3)C22×C3semidirect
63. C3⋊#226C22×C3semidirect
232. C2×#41(C3⋊D3)⋊C3semidirect
384. C32#5923semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#64267⋊C32
#64266⋊C32
#64265⋊C32
#64264⋊C32
#105965863
C3×#58223
#3828⋊C34