Finite Group Databasebeta

← #14554318

Subgroups — #14554318

Order 2187.

2018 conjugacy classes of subgroups; only the 311 normal subgroups are shown.

normal subgroups only

This group has 2018 conjugacy classes of subgroups — too many to draw. Only its 311 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
1C3×3−1+4C311yesmaximal, characteristic, complemented
2C3×#6912C311yesmaximal, characteristic, complemented
3C3×#6912C311yesmaximal, characteristic, complemented
4C3×#6912C311yesmaximal, characteristic, complemented
5C3×#6912C311yesmaximal, characteristic, complemented
6C3×#6910C311yesmaximal, characteristic, complemented
7C3×#6910C311yesmaximal, characteristic, complemented
8C3×#6910C311yesmaximal, characteristic, complemented
9C3×#6909C311yesmaximal, characteristic, complemented
10C3×#6909C311yesmaximal, characteristic, complemented
11C3×#6909C311yesmaximal, characteristic, complemented
12#10624881C311yesmaximal, characteristic, complemented
13#10624881C311yesmaximal, characteristic, complemented
14#10624750C3127yesmaximal, complemented
153−1+4—33no—
163−1+4—33no—
173−1+4—33no—
183−1+4—33no—
193−1+4—33no—
203−1+4—33no—
213−1+4—33no—
223−1+4—33no—
233−1+4C3219yescomplemented
24C3×#737—33no—
25C3×#737—33no—
26C3×#737—33no—
27C32×3−1+2—33no—
28C32×3−1+2C3211yescharacteristic
29C32×3−1+2C3211yescharacteristic
30C32×3−1+2C3211yescharacteristic
31C32×3−1+2C3211yescharacteristic
32C32×3−1+2C3211yescharacteristic
33C32×3−1+2C3211yescharacteristic
34C32×3−1+2C3211yescharacteristic
35C32×3−1+2C3211yescharacteristic
36C32×3−1+2C3211yescharacteristic
37C32×3−1+2C3211yescharacteristic
38C32×H(3,𝔽3)C3211yescharacteristic
39C32×H(3,𝔽3)C3211yescharacteristic
40C32×H(3,𝔽3)C3211yescharacteristic
41#6912—33no—
42#6912—33no—
43#6912—33no—
44#6912—33no—
45#6912—33no—
46#6912—33no—
47#6912—33no—
48#6912—33no—
49#6912—33no—
50#6912C3219yescomplemented
51#6912C3219yescomplemented
52#6912C3219yescomplemented
53#6910—33no—
54#6910—33no—
55#6910—33no—
56#6910—33no—
57#6910—33no—
58#6910—33no—
59#6910—33no—
60#6910C3219yescomplemented
61#6910C3219yescomplemented
62#6909—33no—
63#6909—33no—
64#6909—33no—
65#6909—33no—
66#6909—33no—
67#6909—33no—
68#6909C3219yescomplemented
69#6909C3219yescomplemented
70#6909C3219yescomplemented
71C3×#733—33no—
72C3×#733—33no—
73C3×#733—33no—
74C3×#733C3219yescomplemented
75C3×#731—33no—
76C3×#731—33no—
77C3×#731C3219yescomplemented
78C3×AΣL(1,𝔽27)—33no—
79C3×AΣL(1,𝔽27)—33no—
80C3×AΣL(1,𝔽27)—33no—
81#6893—33no—
82#6893—33no—
83#6893—33no—
84#6893—33no—
85#6893C3219yescomplemented
86#6893C3219yescomplemented
87#6891—33no—
88#6891—33no—
89C34—31no—
90C34—31no—
91C34H(3,𝔽3)13yes—
92C34C3311yescharacteristic
93#737—327no—
94#737—327no—
95#737—327no—
96#737—327no—
97#737—327no—
98#737—327no—
99#737—327no—
100#737—327no—
101#737—327no—
102C3×3−1+2—318no—
103C3×3−1+2—327no—
104C3×3−1+2—327no—
105C3×3−1+2—31no—
106C3×3−1+2—31no—
107C3×3−1+2—31no—
108C3×3−1+2—327no—
109C3×3−1+2—31no—
110C3×3−1+2—31no—
111C3×3−1+2—31no—
112C3×3−1+2—31no—
113C3×3−1+2—31no—
114C3×3−1+2—31no—
115C3×3−1+2—31no—
116C3×3−1+2—31no—
117C3×3−1+2—31no—
118C3×3−1+2—31no—
119C3×3−1+2—31no—
120C3×3−1+2—31no—
121C3×3−1+2—31no—
122C3×3−1+2—31no—
123C3×3−1+2—31no—
124C3×3−1+2—31no—
125C3×3−1+2—31no—
126C3×3−1+2—31no—
127C3×3−1+2—31no—
128C3×3−1+2—31no—
129C3×3−1+2—31no—
130C3×3−1+2—31no—
131C3×3−1+2—31no—
132C3×3−1+2—31no—
133C3×3−1+2—31no—
134C3×3−1+2—31no—
135C3×3−1+2—31no—
136C3×3−1+2—31no—
137C3×3−1+2—31no—
138C3×3−1+2—31no—
139C3×3−1+2—31no—
140C3×3−1+2—31no—
141C3×3−1+2—31no—
142C3×3−1+2—31no—
143C3×3−1+2—31no—
144C3×3−1+2—31no—
145C3×3−1+2—31no—
146C3×3−1+2—31no—
147C3×3−1+2—31no—
148C3×3−1+2—31no—
149C3×3−1+2—31no—
150C3×3−1+2—31no—
151C3×3−1+2—31no—
152C3×3−1+2—31no—
153C3×3−1+2—31no—
154C3×3−1+2—31no—
155C3×3−1+2—31no—
156C3×3−1+2—31no—
157C3×3−1+2—31no—
158C3×3−1+2—31no—
159C3×3−1+2—31no—
160C3×3−1+2—31no—
161C3×3−1+2—31no—
162C3×3−1+2—31no—
163C3×3−1+2—31no—
164C3×3−1+2—31no—
165C3×3−1+2—31no—
166C3×3−1+2—31no—
167C3×3−1+2—31no—
168C3×3−1+2—31no—
169C3×3−1+2—31no—
170C3×3−1+2—31no—
171C3×3−1+2—31no—
172C3×3−1+2—31no—
173C3×3−1+2—31no—
174C3×3−1+2—31no—
175C3×3−1+2—31no—
176C3×3−1+2—31no—
177C3×3−1+2—31no—
178C3×3−1+2—31no—
179C3×3−1+2—31no—
180C3×3−1+2—31no—
181C3×3−1+2—31no—
182C3×3−1+2—31no—
183C3×3−1+2—31no—
184C3×3−1+2—31no—
185C3×3−1+2—31no—
186C3×3−1+2—31no—
187C3×3−1+2—31no—
188C3×3−1+2—31no—
189C3×3−1+2—31no—
190C3×3−1+2—31no—
191C3×3−1+2—31no—
192C3×3−1+2—31no—
193C3×3−1+2—31no—
194C3×3−1+2—31no—
195C3×3−1+2—31no—
196C3×3−1+2—31no—
197C3×3−1+2—31no—
198C3×3−1+2—31no—
199C3×3−1+2—31no—
200C3×3−1+2—31no—
201C3×3−1+2—31no—
202C3×3−1+2—31no—
203C3×3−1+2—31no—
204C3×3−1+2—31no—
205C3×3−1+2—31no—
206C3×3−1+2—31no—
207C3×3−1+2—31no—
208C3×3−1+2—31no—
209C3×3−1+2H(3,𝔽3)13yes—
210C3×3−1+2H(3,𝔽3)13yes—
211C3×3−1+2H(3,𝔽3)13yes—
212C3×3−1+2H(3,𝔽3)13yes—
213C3×3−1+2H(3,𝔽3)13yes—
214C3×3−1+2H(3,𝔽3)13yes—
215C3×3−1+2H(3,𝔽3)13yes—
216C3×3−1+2H(3,𝔽3)13yes—
217C3×3−1+2C3313yes—
218C3×3−1+2C3313yes—
219C3×3−1+2C3313yes—
220C3×3−1+2C3313yes—
221C3×3−1+2C3313yes—
222C3×3−1+2C3313yes—
223C3×3−1+2C3313yes—
224C3×3−1+2C3313yes—
225C3×3−1+2C3313yes—
226C3×H(3,𝔽3)—31no—
227C3×H(3,𝔽3)—31no—
228C3×H(3,𝔽3)—31no—
229C3×H(3,𝔽3)—31no—
230C3×H(3,𝔽3)—31no—
231C3×H(3,𝔽3)—31no—
232C3×H(3,𝔽3)—31no—
233C3×H(3,𝔽3)—31no—
234C3×H(3,𝔽3)—31no—
235C3×H(3,𝔽3)—31no—
236C3×H(3,𝔽3)—31no—
237C3×H(3,𝔽3)—31no—
238C3×H(3,𝔽3)—31no—
239C3×H(3,𝔽3)—31no—
240C3×H(3,𝔽3)—31no—
241C3×H(3,𝔽3)—31no—
242C3×H(3,𝔽3)—31no—
243C3×H(3,𝔽3)—31no—
244C3×H(3,𝔽3)—31no—
245C3×H(3,𝔽3)—31no—
246C3×H(3,𝔽3)—31no—
247C3×H(3,𝔽3)—31no—
248C3×H(3,𝔽3)—31no—
249C3×H(3,𝔽3)—31no—
250C3×H(3,𝔽3)—31no—
251C3×H(3,𝔽3)—31no—
252C3×H(3,𝔽3)—31no—
253C3×H(3,𝔽3)—31no—
254C3×H(3,𝔽3)—31no—
255C3×H(3,𝔽3)—31no—
256C3×H(3,𝔽3)—31no—
257C3×H(3,𝔽3)H(3,𝔽3)13yes—
258C3×H(3,𝔽3)H(3,𝔽3)13yes—
259C3×H(3,𝔽3)C3313yes—
260C3×H(3,𝔽3)C3313yes—
261C3×H(3,𝔽3)C3313yes—
262C32×C9—39no—
263C32×C9—31no—
264C32×C9—31no—
265C32×C9—31no—
266C32×C9—31no—
267C32×C9—31no—
268C32×C9—31no—
269C32×C9—31no—
270C32×C9—31no—
271C32×C9H(3,𝔽3)13yes—
272C32×C9C3313yes—
273#733—327no—
274#733—327no—
275#733—327no—
276#733—327no—
277#731—327no—
278#731—327no—
279#731—327no—
280AΣL(1,𝔽27)—327no—
281AΣL(1,𝔽27)—327no—
282AΣL(1,𝔽27)—327no—
283C32⋊C9—327no—
284C32⋊C9—327no—
285C33—33no—
286C33—33no—
287C33—33no—
288C33—33no—
289C33—33no—
290C33—91no—
291C33—91no—
292C33—91no—
293C33—91no—
294C33—91no—
295C33—91no—
296C33—271no—
297C33—273no—
298C33—273no—
299C33—273no—
300C33C3×H(3,𝔽3)19yes—
301C33C3×H(3,𝔽3)11yescharacteristic
302C33C3×H(3,𝔽3)11yescharacteristic
303C33C3×H(3,𝔽3)11yescharacteristic
304C33C3×H(3,𝔽3)11yescharacteristic
305C33C3×H(3,𝔽3)11yescharacteristic
306C33C3×H(3,𝔽3)11yescharacteristic
307C33C3×H(3,𝔽3)11yescharacteristic
308C33C3×H(3,𝔽3)11yescharacteristic
309C33C3×H(3,𝔽3)11yescharacteristic
310C33C3×H(3,𝔽3)11yescharacteristic
311C33C3×H(3,𝔽3)11yescharacteristic
312C33C3×H(3,𝔽3)11yescharacteristic
313C33C3411yescharacteristic
3143−1+2—3162no—
3153−1+2—33no—
3163−1+2—33no—
3173−1+2—33no—
3183−1+2—33no—
3193−1+2—33no—
3203−1+2—33no—
3213−1+2—33no—
3223−1+2—33no—
3233−1+2—33no—
3243−1+2—33no—
3253−1+2—33no—
3263−1+2—33no—
3273−1+2—33no—
3283−1+2—33no—
3293−1+2—33no—
3303−1+2—33no—
3313−1+2—33no—
3323−1+2—33no—
3333−1+2—33no—
3343−1+2—33no—
3353−1+2—33no—
3363−1+2—33no—
3373−1+2—33no—
3383−1+2—33no—
3393−1+2—33no—
3403−1+2—33no—
3413−1+2—33no—
3423−1+2—33no—
3433−1+2—33no—
3443−1+2—33no—
3453−1+2—33no—
3463−1+2—33no—
3473−1+2—33no—
3483−1+2—33no—
3493−1+2—33no—
3503−1+2—33no—
3513−1+2—33no—
3523−1+2—33no—
3533−1+2—33no—
3543−1+2—33no—
3553−1+2—33no—
3563−1+2—33no—
3573−1+2—33no—
3583−1+2—91no—
3593−1+2—91no—
3603−1+2—91no—
3613−1+2—91no—
3623−1+2—91no—
3633−1+2—91no—
3643−1+2—91no—
3653−1+2—91no—
3663−1+2—91no—
3673−1+2—91no—
3683−1+2—91no—
3693−1+2—91no—
3703−1+2—91no—
3713−1+2—91no—
3723−1+2—91no—
3733−1+2—91no—
3743−1+2—91no—
3753−1+2—91no—
3763−1+2—91no—
3773−1+2—91no—
3783−1+2—91no—
3793−1+2—91no—
3803−1+2—91no—
3813−1+2—91no—
3823−1+2—91no—
3833−1+2—91no—
3843−1+2—91no—
3853−1+2—91no—
3863−1+2—91no—
3873−1+2—91no—
3883−1+2—91no—
3893−1+2—91no—
3903−1+2—91no—
3913−1+2—91no—
3923−1+2—91no—
3933−1+2—91no—
3943−1+2—91no—
3953−1+2—91no—
3963−1+2—91no—
3973−1+2—91no—
3983−1+2—91no—
3993−1+2—91no—
4003−1+2—91no—
4013−1+2—91no—
4023−1+2—91no—
4033−1+2C3×H(3,𝔽3)19yes—
4043−1+2C3×H(3,𝔽3)19yes—
405H(3,𝔽3)—33no—
406H(3,𝔽3)—33no—
407H(3,𝔽3)—33no—
408H(3,𝔽3)—33no—
409H(3,𝔽3)—33no—
410H(3,𝔽3)—33no—
411H(3,𝔽3)—33no—
412H(3,𝔽3)—33no—
413H(3,𝔽3)—33no—
414H(3,𝔽3)—33no—
415H(3,𝔽3)—33no—
416H(3,𝔽3)—33no—
417H(3,𝔽3)—33no—
418H(3,𝔽3)—33no—
419H(3,𝔽3)—33no—
420H(3,𝔽3)—91no—
421H(3,𝔽3)—91no—
422H(3,𝔽3)—91no—
423H(3,𝔽3)—91no—
424H(3,𝔽3)—91no—
425H(3,𝔽3)—91no—
426H(3,𝔽3)—91no—
427H(3,𝔽3)—91no—
428H(3,𝔽3)—91no—
429H(3,𝔽3)—91no—
430H(3,𝔽3)—91no—
431H(3,𝔽3)—91no—
432C3×C9—381no—
433C3×C9—33no—
434C3×C9—33no—
435C3×C9—33no—
436C3×C9—33no—
437C3×C9—33no—
438C3×C9—33no—
439C3×C9—33no—
440C3×C9—33no—
441C3×C9—33no—
442C3×C9—33no—
443C3×C9—33no—
444C3×C9—33no—
445C3×C9—33no—
446C3×C9—33no—
447C3×C9—33no—
448C3×C9—33no—
449C3×C9—33no—
450C3×C9—33no—
451C3×C9—33no—
452C3×C9—33no—
453C3×C9—33no—
454C3×C9—91no—
455C3×C9—91no—
456C3×C9—91no—
457C3×C9—91no—
458C3×C9—91no—
459C3×C9—91no—
460C3×C9—91no—
461C3×C9—91no—
462C3×C9—91no—
463C3×C9—91no—
464C3×C9—91no—
465C3×C9—91no—
466C3×C9—91no—
467C3×C9—91no—
468C3×C9—91no—
469C3×C9—91no—
470C3×C9—91no—
471C3×C9—91no—
472C3×C9C3×H(3,𝔽3)19yes—
473C32—39no—
474C32—91no—
475C32—91no—
476C32—91no—
477C32—91no—
478C32—91no—
479C32—91no—
480C32—91no—
481C32—91no—
482C32—91no—
483C32—91no—
484C32—91no—
485C32—91no—
486C32—91no—
487C32—99no—
488C32—99no—
489C32—99no—
490C32—93no—
491C32—93no—
492C32—273no—
493C32—273no—
494C32—273no—
495C32—273no—
496C32—273no—
497C32—273no—
498C32—273no—
499C32—273no—
500C32—273no—
501C32C32⋊H(3,𝔽3)11yescharacteristic
502C32C32⋊H(3,𝔽3)11yescharacteristic
503C32C32⋊H(3,𝔽3)11yescharacteristic
504C32C32⋊H(3,𝔽3)11yescharacteristic
505C32C32⋊H(3,𝔽3)11yescharacteristic
506C32C32⋊H(3,𝔽3)11yescharacteristic
507C32C32⋊H(3,𝔽3)11yescharacteristic
508C32C32⋊H(3,𝔽3)11yescharacteristic
509C32C32⋊H(3,𝔽3)11yescharacteristic
510C32C32×H(3,𝔽3)11yescharacteristic
511C32C32×H(3,𝔽3)11yescharacteristic
512C32C32×H(3,𝔽3)11yescharacteristic
513C32C32×H(3,𝔽3)11yescharacteristic
514C9—39no—
515C9—39no—
516C9—39no—
517C9—93no—
518C9—93no—
519C9—93no—
520C9—93no—
521C9—93no—
522C9—93no—
523C9—93no—
524C3—31no—
525C3—31no—
526C3—31no—
527C3—31no—
528C3—31no—
529C3—31no—
530C3—31no—
531C3—31no—
532C3—31no—
533C3—31no—
534C3—31no—
535C3—31no—
536C3—31no—
537C3—99no—
538C3—273no—
539C3—273no—
540C3C3×C32⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
center, socleC3C3×C32⋊H(3,𝔽3)
commutator, Frattini, cosocleC33C34
LCSG ▷ C33 ▷ C3 ▷ C1
UCSC1 ◁ C3 ◁ C34 ◁ G
LECSG ▷ C33 ▷ C3 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C3×3−1+4C3semidirect
2. C3×#6912C3semidirect
3. C3×#6912C3semidirect
4. C3×#6912C3semidirect
5. C3×#6912C3semidirect
6. C3×#6910C3semidirect
7. C3×#6910C3semidirect
8. C3×#6910C3semidirect
9. C3×#6909C3semidirect
10. C3×#6909C3semidirect
11. C3×#6909C3semidirect
12. #10624881C3semidirect
13. #10624881C3semidirect
14. #10624750C3semidirect
23. 3−1+4C32semidirect
50. #6912C32semidirect
51. #6912C32semidirect
52. #6912C32semidirect
60. #6910C32semidirect
61. #6910C32semidirect
68. #6909C32semidirect
69. #6909C32semidirect
70. #6909C32semidirect
74. C3×#733C32semidirect
77. C3×#731C32semidirect
85. #6893C32semidirect
86. #6893C32semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+43
C3×#69123
C3×#69103
C3×#69093
#106248813
#106247503