Finite Group Databasebeta

← C3×C4⋊AGL(2,𝔽3) #72259826

Subgroups — C3×C4⋊AGL(2,𝔽3) #72259826

Order 5184.

591 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 40 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×C4×PSU(3,𝔽2)⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×AGL(2,𝔽3)C212yesmaximal, complemented
3C3×#10596410—31nomaximal
4C4⋊AGL(2,𝔽3)C311yesmaximal, characteristic, direct product factor
5C2×C3×PSU(3,𝔽2)⋊C3C2211yescharacteristic
6C3×AGL(2,𝔽3)—22no—
7C3×#88654—41nomaximal
8C2×C3×AΓL(1,𝔽9)—32no—
9C2×AGL(2,𝔽3)C2×C312yescomplemented
10C3×C4×PSU(3,𝔽2)D311yescharacteristic, complemented
11C3×#64231—31no—
12C3×C4⋊AGL(1,𝔽9)—31no—
13C3×GO+(4,𝔽2)⋊C4—32no—
14C4×PSU(3,𝔽2)⋊C3—21no—
15C4×PSU(3,𝔽2)⋊C3C2×C311yescharacteristic, complemented
16C3×PSU(3,𝔽2)⋊C3D411yescharacteristic, complemented
17C2×C3×((C3⋊D3)⋊C3)⋊C2—42no—
18C3×C4⋊#272—41no—
19C3×C4⋊((C3⋊D3)⋊C3)—41no—
20C3×C4×(C3⋊D3)⋊C3—41no—
21C3×#6200—42no—
22#10596410—31no—
23C3×C4⋊GL(2,𝔽3)—91nomaximal
24C2×C3×PSU(3,𝔽2)C2×D311yescharacteristic
25C2×C3×GO+(4,𝔽2)—32no—
26C2×C3×GO+(4,𝔽2)—61no—
27C2×C3×AGL(1,𝔽9)—32no—
28C3×AΓL(1,𝔽9)—62no—
29C3×AΓL(1,𝔽9)—62no—
30C2×PSU(3,𝔽2)⋊C3—21no—
31C2×PSU(3,𝔽2)⋊C3C22×C311yescharacteristic
32AGL(2,𝔽3)—22no—
33C3×#3802—61no—
34C3×D3×D12—121no—
35C3×C4×#157—31no—
36C3×#3777—31no—
37#88654—41no—
38C2×C3×#272—42no—
39C2×C3×(C3⋊D3)⋊C3—42no—
40C2×C3×(C3⋊D3)⋊C3—41no—
41C3×((C3⋊D3)⋊C3)⋊C2—82no—
42C3×((C3⋊D3)⋊C3)⋊C2—82no—
43C3×C4×H(3,𝔽3)—41no—
44C3×(C3⋊Dic3)⋊C3—41no—
45C2×AΓL(1,𝔽9)—32no—
46C2×C3×GL(2,𝔽3)—92no—
47C4×PSU(3,𝔽2)C3×D311yescharacteristic, complemented
48#64231—31no—
49C4⋊AGL(1,𝔽9)—31no—
50GO+(4,𝔽2)⋊C4—32no—
51C3×C4×SL(2,𝔽3)—91no—
52C2×C3×D32—62no—
53C2×C3×D32—122no—
54C2×C3×#157—31no—
55C2×C3×#157—31no—
56C3×PSU(3,𝔽2)—31no—
57C3×PSU(3,𝔽2)D1211yescharacteristic, complemented
58C3×GO+(4,𝔽2)—62no—
59C3×GO+(4,𝔽2)—62no—
60C3×GO+(4,𝔽2)—122no—
61C3×AGL(1,𝔽9)—62no—
62PSU(3,𝔽2)⋊C3—21no—
63PSU(3,𝔽2)⋊C3C3×D411yescharacteristic, complemented
64C3×C4⋊(C3⋊D3)—41no—
65C3×C4×C3⋊D3S411yescharacteristic
66C32×D12—121no—
67C32×D12—121no—
68C32×D12—241no—
69C32×C4×D3—121no—
70C3×#626—122no—
71C3×#625—121no—
72C2×((C3⋊D3)⋊C3)⋊C2—42no—
73C4⋊#272—41no—
74C4⋊((C3⋊D3)⋊C3)—41no—
75C4⋊((C3⋊D3)⋊C3)—81no—
76C4×(C3⋊D3)⋊C3—41no—
77C4×(C3⋊D3)⋊C3—81no—
78#6200—42no—
79C4⋊GL(2,𝔽3)—91no—
80C3×#460—271no—
81C2×C3×H(3,𝔽3)—41no—
82C3×#272—82no—
83C3×(C3⋊D3)⋊C3—41no—
84C3×(C3⋊D3)⋊C3—41no—
85C3×(C3⋊D3)⋊C3—82no—
86C2×PSU(3,𝔽2)C2×C3×D311yescharacteristic
87C2×GO+(4,𝔽2)—32no—
88C2×GO+(4,𝔽2)—61no—
89C2×AGL(1,𝔽9)—32no—
90AΓL(1,𝔽9)—62no—
91AΓL(1,𝔽9)—62no—
92C3×D3×D4—361no—
93C2×C3×D12—361no—
94C2×C3×SL(2,𝔽3)—91no—
95#3802—61no—
96D3×D12—121no—
97C4×#157—31no—
98C3×GL(2,𝔽3)—182no—
99#3777—31no—
100C2×C3×C3⋊D3—42no—
101C2×C3×C3⋊D3C2×S411yescharacteristic
102C2×C32×D3—121no—
103C2×C32×D3—122no—
104C2×C32×D3—122no—
105C2×C32×D3—242no—
106C3×D32—122no—
107C3×D32—122no—
108C3×D32—242no—
109C3×D32—242no—
110C3×#157—31no—
111C3×#157—31no—
112C3×#157—61no—
113C33×C4—41no—
114C33×C4—81no—
115C33×C4GL(2,𝔽3)11yescharacteristic, complemented
116C3×C3⋊Dic3GL(2,𝔽3)11yescharacteristic, complemented
117C32×Dic3—121no—
118C2×#272—42no—
119C2×(C3⋊D3)⋊C3—41no—
120C2×(C3⋊D3)⋊C3—42no—
121C2×(C3⋊D3)⋊C3—81no—
122C2×(C3⋊D3)⋊C3—82no—
123((C3⋊D3)⋊C3)⋊C2—82no—
124((C3⋊D3)⋊C3)⋊C2—82no—
125C4×H(3,𝔽3)—41no—
126C4×H(3,𝔽3)—81no—
127C4×H(3,𝔽3)—81no—
128C4×H(3,𝔽3)—161no—
129(C3⋊Dic3)⋊C3—41no—
130(C3⋊Dic3)⋊C3—81no—
131C2×GL(2,𝔽3)—92no—
132C2×C3×SD16—272no—
133C3×#127—271no—
134C3×C4×Q8—91no—
135C4×SL(2,𝔽3)—91no—
136C4×SL(2,𝔽3)—181no—
137C3×C4⋊C8—271no—
138C3×#102—272no—
139C3×H(3,𝔽3)—41noSylow
140C22×C3×D3—362no—
141C22×C3×D3—362no—
142C2×D32—62no—
143C2×D32—122no—
144C2×#157—31no—
145C2×#157—31no—
146PSU(3,𝔽2)—31no—
147PSU(3,𝔽2)C3×D1211yescharacteristic, complemented
148GO+(4,𝔽2)—62no—
149GO+(4,𝔽2)—62no—
150GO+(4,𝔽2)—122no—
151AGL(1,𝔽9)—62no—
152C32×D4—361no—
153C2×C32×C4—361no—
154C4⋊(C3⋊D3)—41no—
155C4×C3⋊D3C3×S411yescharacteristic
156C3×#67—362no—
157C3×D12—121no—
158C3×D12—121no—
159C3×D12—121no—
160C3×D12—121no—
161C3×D12—241no—
162C3×D12—241no—
163C3×D12—241no—
164C3×D12—241no—
165C3×D12—242no—
166C3×D12—361no—
167C3×D12—362no—
168C3×C4×D3—121no—
169C3×C4×D3—121no—
170C3×C4×D3—241no—
171C3×SL(2,𝔽3)—91no—
172#626—122no—
173#625—121no—
174#460—271noSylow
175C2×C33—41no—
176C2×C33—81no—
177C2×C33C2×GL(2,𝔽3)11yescharacteristic
178C3×C3⋊D3—82no—
179C3×C3⋊D3C4⋊S411yescharacteristic
180C3×C3⋊D3#98611yescharacteristic
181C32×D3—121no—
182C32×D3—121no—
183C32×D3—242no—
184C32×D3—242no—
185C32×D3—482no—
186C2×H(3,𝔽3)—41no—
187C2×H(3,𝔽3)—81no—
188C2×H(3,𝔽3)—81no—
189C2×H(3,𝔽3)—161no—
190#272—82no—
191(C3⋊D3)⋊C3—41no—
192(C3⋊D3)⋊C3—41no—
193(C3⋊D3)⋊C3—82no—
194(C3⋊D3)⋊C3—81no—
195(C3⋊D3)⋊C3—81no—
196(C3⋊D3)⋊C3—162no—
197C2×C3×Q8—91no—
198C2×C3×D4—272no—
199C2×C3×D4—541no—
200C2×C3×D4—541no—
201D3×D4—361no—
202C2×D12—361no—
203C2×SL(2,𝔽3)—91no—
204C2×SL(2,𝔽3)—181no—
205GL(2,𝔽3)—182no—
206C3×SD16—542no—
207C3×SD16—542no—
208C2×C3×C8—272no—
209C3×C4⋊C4—271no—
210C3×C42—271no—
211C22×C32—361no—
212C22×C32—362no—
213C2×C3⋊D3—42no—
214C2×C3⋊D3C2×C3×S411yescharacteristic
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C2×C3×D3—122no—
218C2×C3×D3—122no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—241no—
222C2×C3×D3—242no—
223C2×C3×D3—242no—
224C2×C3×D3—242no—
225C2×C3×D3—242no—
226C2×C3×D3—244no—
227C2×C3×D3—362no—
228C2×C3×D3—722no—
229C2×C3×D3—722no—
230C2×C3×D3—722no—
231C2×C3×D3—722no—
232D32—122no—
233D32—122no—
234D32—242no—
235D32—242no—
236#157—31no—
237#157—31no—
238#157—61no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—81no—
242C32×C4—81no—
243C32×C4—81no—
244C32×C4—121no—
245C32×C4—161no—
246C32×C4—241no—
247C32×C4—242no—
248C32×C4—241no—
249C32×C4—361no—
250C32×C4C3×GL(2,𝔽3)11yescharacteristic, complemented
251C3⋊Dic3C3×GL(2,𝔽3)11yescharacteristic, complemented
252C3×Dic3—121no—
253C3×Dic3—121no—
254C3×Dic3—241no—
255C2×SD16—272no—
256#127—271no—
257C4×Q8—91no—
258C4⋊C8—271no—
259#102—272no—
260C33—41no—
261C33—81no—
262C33C4⋊GL(2,𝔽3)11yescharacteristic, complemented
263H(3,𝔽3)—41no—
264H(3,𝔽3)—81no—
265H(3,𝔽3)—81no—
266H(3,𝔽3)—161no—
267C23×C3—542no—
268C22×D3—362no—
269C22×D3—362no—
270C3×Q8—91no—
271C3×Q8—271no—
272C3×D4—361no—
273C3×D4—361no—
274C3×D4—542no—
275C3×D4—542no—
276C3×D4—721no—
277C3×D4—1081no—
278C3×D4—1082no—
279C2×C3×C4—91no—
280C2×C3×C4—271no—
281C2×C3×C4—271no—
282C2×C3×C4—361no—
283C2×C3×C4—721no—
284#67—362no—
285D12—121no—
286D12—121no—
287D12—241no—
288D12—361no—
289D12—362no—
290C4×D3—121no—
291SL(2,𝔽3)—91no—
292SL(2,𝔽3)—181no—
293C3×C8—542no—
294C2×C32—41no—
295C2×C32—41no—
296C2×C32—81no—
297C2×C32—81no—
298C2×C32—81no—
299C2×C32—121no—
300C2×C32—161no—
301C2×C32—241no—
302C2×C32—242no—
303C2×C32—241no—
304C2×C32—361no—
305C2×C32—361no—
306C2×C32—722no—
307C2×C32C2×C3×GL(2,𝔽3)11yescharacteristic
308C3⋊D3—82no—
309C3⋊D3C3×C4⋊S411yescharacteristic
310C3⋊D3C3×#98611yescharacteristic
311C3×D3—121no—
312C3×D3—121no—
313C3×D3—121no—
314C3×D3—121no—
315C3×D3—241no—
316C3×D3—241no—
317C3×D3—242no—
318C3×D3—242no—
319C3×D3—242no—
320C3×D3—242no—
321C3×D3—482no—
322C3×D3—482no—
323C3×D3—482no—
324C3×D3—482no—
325C3×D3—484no—
326C3×D3—722no—
327C2×Q8—91no—
328C2×D4—272no—
329C2×D4—541no—
330C2×D4—541no—
331SD16—542no—
332SD16—542no—
333C2×C8—272no—
334C4⋊C4—271no—
335C42—271no—
336C22×C3—91no—
337C22×C3—361no—
338C22×C3—362no—
339C22×C3—362no—
340C22×C3—721no—
341C22×C3—722no—
342C22×C3—1082no—
343C22×C3—1082no—
344C2×D3—121no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—242no—
348C2×D3—362no—
349C2×D3—722no—
350C2×D3—722no—
351C2×D3—722no—
352C2×D3—722no—
353C3×C4—41no—
354C3×C4—81no—
355C3×C4—91no—
356C3×C4—121no—
357C3×C4—241no—
358C3×C4—241no—
359C3×C4—271no—
360C3×C4—271no—
361C3×C4—361no—
362C3×C4—481no—
363C3×C4—541no—
364C3×C4—721no—
365C3×C4AGL(2,𝔽3)11yescharacteristic, complemented
366Dic3—121no—
367C32—41no—
368C32—41no—
369C32—81no—
370C32—81no—
371C32—81no—
372C32—121no—
373C32—161no—
374C32—241no—
375C32—242no—
376C32—241no—
377C32C3×C4⋊GL(2,𝔽3)11yescharacteristic, complemented
378C23—542no—
379Q8—91no—
380Q8—271no—
381D4—361no—
382D4—542no—
383D4—542no—
384D4—1081no—
385D4—1082no—
386C2×C4—91no—
387C2×C4—271no—
388C2×C4—271no—
389C8—542no—
390C2×C3—41no—
391C2×C3—81no—
392C2×C3—91no—
393C2×C3—91no—
394C2×C3—121no—
395C2×C3—241no—
396C2×C3—241no—
397C2×C3—361no—
398C2×C3—361no—
399C2×C3—481no—
400C2×C3—721no—
401C2×C3—721no—
402C2×C3—722no—
403C2×C3—722no—
404C2×C3—1442no—
405C2×C3C2×AGL(2,𝔽3)11yescharacteristic
406D3—121no—
407D3—121no—
408D3—242no—
409D3—242no—
410D3—482no—
411D3—722no—
412C22—91no—
413C22—362no—
414C22—1082no—
415C22—1082no—
416C4—91no—
417C4—271no—
418C4—271no—
419C4—541no—
420C4C3×AGL(2,𝔽3)11yescharacteristic, complemented
421C3—41no—
422C3—81no—
423C3—121no—
424C3—241no—
425C3—241no—
426C3—481no—
427C3C4⋊AGL(2,𝔽3)11yescharacteristic, direct product factor
428C2—91no—
429C2—91no—
430C2—722no—
431C2C2×C3×AGL(2,𝔽3)11yescharacteristic
subgroupG/H
centerC2×C3C2×AGL(2,𝔽3)
commutator, cosocleC2×PSU(3,𝔽2)⋊C3C22×C3
FrattiniC2C2×C3×AGL(2,𝔽3)
FittingC33×C4GL(2,𝔽3)
socleC2×C33C2×GL(2,𝔽3)
hypercenterC3×C4AGL(2,𝔽3)
hypocenterPSU(3,𝔽2)⋊C3C3×D4
LCSG ▷ C2×PSU(3,𝔽2)⋊C3 ▷ PSU(3,𝔽2)⋊C3
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×C4×PSU(3,𝔽2)⋊C3C2semidirect
2. C2×C3×AGL(2,𝔽3)C2semidirect
4. C4⋊AGL(2,𝔽3)C3direct (N × H)
9. C2×AGL(2,𝔽3)C2×C3semidirect
10. C3×C4×PSU(3,𝔽2)D3semidirect
15. C4×PSU(3,𝔽2)⋊C3C2×C3semidirect
16. C3×PSU(3,𝔽2)⋊C3D4semidirect
47. C4×PSU(3,𝔽2)C3×D3semidirect
57. C3×PSU(3,𝔽2)D12semidirect
63. PSU(3,𝔽2)⋊C3C3×D4semidirect
115. C33×C4GL(2,𝔽3)semidirect
116. C3×C3⋊Dic3GL(2,𝔽3)semidirect
147. PSU(3,𝔽2)C3×D12semidirect
250. C32×C4C3×GL(2,𝔽3)semidirect
251. C3⋊Dic3C3×GL(2,𝔽3)semidirect
262. C33C4⋊GL(2,𝔽3)semidirect
365. C3×C4AGL(2,𝔽3)semidirect
377. C32C3×C4⋊GL(2,𝔽3)semidirect
420. C4C3×AGL(2,𝔽3)semidirect
427. C3C4⋊AGL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×PSU(3,𝔽2)⋊C32
C2×C3×AGL(2,𝔽3)2
C3×#105964103
C4⋊AGL(2,𝔽3)3
C3×#886544
C3×C4⋊GL(2,𝔽3)9