Finite Group Databasebeta

← C2×C32×AGL(2,𝔽3) #72228425

Subgroups — C2×C32×AGL(2,𝔽3) #72228425

Order 7776.

1354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 84 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×C32×AΓL(1,𝔽9)—31nomaximal
2C2×C32×PSU(3,𝔽2)⋊C3C211yesmaximal, characteristic, complemented
3C2×C3×AGL(2,𝔽3)C314yesmaximal, direct product factor
4C32×AGL(2,𝔽3)C212yesmaximal, direct product factor
5C32×PSU(3,𝔽2)⋊C3C2211yescharacteristic, complemented
6C2×C32×((C3⋊D3)⋊C3)⋊C2—41nomaximal
7C2×C32×PSU(3,𝔽2)D311yescharacteristic, complemented
8C2×C32×GO+(4,𝔽2)—31no—
9C2×C32×AGL(1,𝔽9)—31no—
10C32×AΓL(1,𝔽9)—32no—
11C32×AΓL(1,𝔽9)—32no—
12C2×C3×PSU(3,𝔽2)⋊C3—24no—
13C2×C3×PSU(3,𝔽2)⋊C3C2×C314yescomplemented
14C3×AGL(2,𝔽3)C2×C318yesdirect product factor
15C2×C32×#272—41no—
16C2×C32×(C3⋊D3)⋊C3—41no—
17C2×C32×(C3⋊D3)⋊C3—41no—
18C32×((C3⋊D3)⋊C3)⋊C2—42no—
19C32×((C3⋊D3)⋊C3)⋊C2—42no—
20C2×C3×AΓL(1,𝔽9)—34no—
21C2×AGL(2,𝔽3)C3211yescharacteristic, direct product factor
22C2×C32×GL(2,𝔽3)—91nomaximal
23C2×C32×D32—61no—
24C2×C32×D32—121no—
25C2×C32×#157—31no—
26C32×PSU(3,𝔽2)—31no—
27C32×PSU(3,𝔽2)C2×D311yescharacteristic, complemented
28C32×GO+(4,𝔽2)—32no—
29C32×GO+(4,𝔽2)—61no—
30C32×AGL(1,𝔽9)—32no—
31C3×PSU(3,𝔽2)⋊C3—24no—
32C3×PSU(3,𝔽2)⋊C3C22×C314yescomplemented
33C2×C3×((C3⋊D3)⋊C3)⋊C2—44no—
34C2×C32×H(3,𝔽3)—41no—
35C32×#272—42no—
36C32×(C3⋊D3)⋊C3—42no—
37C32×(C3⋊D3)⋊C3—41no—
38C32×(C3⋊D3)⋊C3—41no—
39C2×C3×PSU(3,𝔽2)C3×D314yescomplemented
40C2×C3×GO+(4,𝔽2)—34no—
41C2×C3×AGL(1,𝔽9)—34no—
42C3×AΓL(1,𝔽9)—38no—
43C3×AΓL(1,𝔽9)—38no—
44C2×PSU(3,𝔽2)⋊C3—24no—
45C2×PSU(3,𝔽2)⋊C3C2×C3211yescharacteristic, complemented
46AGL(2,𝔽3)C2×C3212yesdirect product factor
47C2×C32×SL(2,𝔽3)—91no—
48C32×GL(2,𝔽3)—92no—
49C2×C32×C3⋊D3—41no—
50C2×C32×C3⋊D3S411yescharacteristic
51C2×C33×D3—121no—
52C2×C33×D3—121no—
53C2×C33×D3—121no—
54C2×C33×D3—241no—
55C32×D32—62no—
56C32×D32—122no—
57C32×D32—122no—
58C32×D32—121no—
59C32×#157—31no—
60C32×#157—31no—
61C2×C3×#272—44no—
62C2×C3×(C3⋊D3)⋊C3—44no—
63C2×C3×(C3⋊D3)⋊C3—44no—
64C2×C3×(C3⋊D3)⋊C3—84no—
65C2×C3×(C3⋊D3)⋊C3—84no—
66C3×((C3⋊D3)⋊C3)⋊C2—48no—
67C3×((C3⋊D3)⋊C3)⋊C2—48no—
68C2×AΓL(1,𝔽9)—31no—
69C2×C3×GL(2,𝔽3)—94no—
70C2×C32×SD16—271no—
71C32×H(3,𝔽3)—41noSylow
72C22×C32×D3—361no—
73C22×C32×D3—361no—
74C2×C3×D32—64no—
75C2×C3×D32—124no—
76C2×C3×#157—34no—
77C3×PSU(3,𝔽2)—34no—
78C3×PSU(3,𝔽2)C2×C3×D314yescomplemented
79C3×GO+(4,𝔽2)—38no—
80C3×GO+(4,𝔽2)—64no—
81C3×AGL(1,𝔽9)—38no—
82PSU(3,𝔽2)⋊C3—24no—
83PSU(3,𝔽2)⋊C3C22×C3211yescharacteristic, complemented
84C32×SL(2,𝔽3)—91no—
85C2×((C3⋊D3)⋊C3)⋊C2—41no—
86C2×C34—41no—
87C2×C34—81no—
88C2×C34GL(2,𝔽3)11yescharacteristic, complemented
89C32×C3⋊D3—42no—
90C32×C3⋊D3GL(2,𝔽3)11yescharacteristic, complemented
91C32×C3⋊D3C2×S411yescharacteristic
92C33×D3—121no—
93C33×D3—121no—
94C33×D3—122no—
95C33×D3—122no—
96C33×D3—242no—
97C2×C3×H(3,𝔽3)—44no—
98C2×C3×H(3,𝔽3)—84no—
99C2×C3×H(3,𝔽3)—84no—
100C2×C3×H(3,𝔽3)—164no—
101C3×#272—48no—
102C3×(C3⋊D3)⋊C3—48no—
103C3×(C3⋊D3)⋊C3—44no—
104C3×(C3⋊D3)⋊C3—44no—
105C3×(C3⋊D3)⋊C3—88no—
106C3×(C3⋊D3)⋊C3—84no—
107C3×(C3⋊D3)⋊C3—84no—
108C2×PSU(3,𝔽2)C32×D311yescharacteristic, complemented
109C2×GO+(4,𝔽2)—31no—
110C2×AGL(1,𝔽9)—31no—
111AΓL(1,𝔽9)—32no—
112AΓL(1,𝔽9)—32no—
113C2×C32×Q8—91no—
114C2×C32×D4—271no—
115C2×C3×SL(2,𝔽3)—94no—
116C2×C3×SL(2,𝔽3)—184no—
117C3×GL(2,𝔽3)—98no—
118C32×SD16—272no—
119C32×SD16—272no—
120C2×C32×C8—271no—
121C22×C33—361no—
122C22×C33—361no—
123C2×C3×C3⋊D3—44no—
124C2×C3×C3⋊D3C3×S414yes—
125C2×C32×D3—121no—
126C2×C32×D3—121no—
127C2×C32×D3—124no—
128C2×C32×D3—124no—
129C2×C32×D3—121no—
130C2×C32×D3—124no—
131C2×C32×D3—244no—
132C2×C32×D3—244no—
133C2×C32×D3—244no—
134C2×C32×D3—244no—
135C2×C32×D3—241no—
136C2×C32×D3—248no—
137C2×C32×D3—362no—
138C2×C32×D3—362no—
139C2×C32×D3—362no—
140C2×C32×D3—362no—
141C2×C32×D3—361no—
142C3×D32—68no—
143C3×D32—128no—
144C3×D32—128no—
145C3×D32—124no—
146C3×#157—34no—
147C3×#157—34no—
148C2×#272—41no—
149C2×(C3⋊D3)⋊C3—41no—
150C2×(C3⋊D3)⋊C3—41no—
151C2×(C3⋊D3)⋊C3—84no—
152C2×(C3⋊D3)⋊C3—84no—
153((C3⋊D3)⋊C3)⋊C2—42no—
154((C3⋊D3)⋊C3)⋊C2—42no—
155C2×GL(2,𝔽3)—91no—
156C2×C3×SD16—274no—
157C34—41no—
158C34—81no—
159C34C2×GL(2,𝔽3)11yescharacteristic, complemented
160C3×H(3,𝔽3)—44no—
161C3×H(3,𝔽3)—84no—
162C3×H(3,𝔽3)—84no—
163C3×H(3,𝔽3)—164no—
164C23×C32—541no—
165C22×C3×D3—364no—
166C22×C3×D3—364no—
167C2×D32—61no—
168C2×D32—121no—
169C2×#157—31no—
170PSU(3,𝔽2)—31no—
171PSU(3,𝔽2)C2×C32×D311yescharacteristic, complemented
172GO+(4,𝔽2)—32no—
173GO+(4,𝔽2)—61no—
174AGL(1,𝔽9)—32no—
175C32×Q8—91no—
176C32×Q8—271no—
177C32×D4—272no—
178C32×D4—541no—
179C2×C32×C4—271no—
180C3×SL(2,𝔽3)—94no—
181C3×SL(2,𝔽3)—184no—
182C32×C8—272no—
183C2×C33—41no—
184C2×C33—44no—
185C2×C33—84no—
186C2×C33—84no—
187C2×C33—84no—
188C2×C33—121no—
189C2×C33—164no—
190C2×C33—244no—
191C2×C33—248no—
192C2×C33—241no—
193C2×C33—361no—
194C2×C33—361no—
195C2×C33—362no—
196C2×C33C3×GL(2,𝔽3)14yescomplemented
197C3×C3⋊D3—48no—
198C3×C3⋊D3C3×GL(2,𝔽3)14yescomplemented
199C3×C3⋊D3C2×C3×S414yes—
200C32×D3—121no—
201C32×D3—121no—
202C32×D3—122no—
203C32×D3—122no—
204C32×D3—124no—
205C32×D3—124no—
206C32×D3—128no—
207C32×D3—128no—
208C32×D3—244no—
209C32×D3—244no—
210C32×D3—248no—
211C32×D3—248no—
212C32×D3—248no—
213C32×D3—242no—
214C32×D3—2416no—
215C32×D3—362no—
216C2×H(3,𝔽3)—41no—
217C2×H(3,𝔽3)—84no—
218C2×H(3,𝔽3)—84no—
219C2×H(3,𝔽3)—164no—
220C2×H(3,𝔽3)—1612no—
221#272—42no—
222(C3⋊D3)⋊C3—42no—
223(C3⋊D3)⋊C3—41no—
224(C3⋊D3)⋊C3—41no—
225(C3⋊D3)⋊C3—88no—
226(C3⋊D3)⋊C3—84no—
227(C3⋊D3)⋊C3—84no—
228C2×C3×Q8—94no—
229C2×C3×D4—274no—
230C2×SL(2,𝔽3)—91no—
231C2×SL(2,𝔽3)—184no—
232GL(2,𝔽3)—92no—
233C3×SD16—278no—
234C3×SD16—278no—
235C2×C3×C8—274no—
236C22×C32—91no—
237C22×C32—364no—
238C22×C32—361no—
239C22×C32—364no—
240C22×C32—542no—
241C22×C32—724no—
242C22×C32—724no—
243C22×C32—1081no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3C32×S411yescharacteristic
246C2×C3×D3—124no—
247C2×C3×D3—121no—
248C2×C3×D3—121no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—121no—
252C2×C3×D3—244no—
253C2×C3×D3—244no—
254C2×C3×D3—244no—
255C2×C3×D3—241no—
256C2×C3×D3—244no—
257C2×C3×D3—248no—
258C2×C3×D3—364no—
259C2×C3×D3—368no—
260C2×C3×D3—368no—
261C2×C3×D3—368no—
262C2×C3×D3—368no—
263D32—62no—
264D32—122no—
265D32—122no—
266D32—121no—
267#157—31no—
268#157—31no—
269C32×C4—271no—
270C32×C4—271no—
271C2×SD16—271noSylow
272C33—41no—
273C33—44no—
274C33—84no—
275C33—84no—
276C33—84no—
277C33—121no—
278C33—164no—
279C33—244no—
280C33—248no—
281C33—241no—
282C33C2×C3×GL(2,𝔽3)14yescomplemented
283H(3,𝔽3)—41no—
284H(3,𝔽3)—84no—
285H(3,𝔽3)—84no—
286H(3,𝔽3)—164no—
287H(3,𝔽3)—1612no—
288C23×C3—544no—
289C22×D3—361no—
290C22×D3—361no—
291C3×Q8—94no—
292C3×Q8—274no—
293C3×D4—278no—
294C3×D4—544no—
295C2×C3×C4—274no—
296SL(2,𝔽3)—91no—
297SL(2,𝔽3)—184no—
298C3×C8—278no—
299C2×C32—44no—
300C2×C32—41no—
301C2×C32—84no—
302C2×C32—81no—
303C2×C32—84no—
304C2×C32—84no—
305C2×C32—91no—
306C2×C32—91no—
307C2×C32—124no—
308C2×C32—164no—
309C2×C32—244no—
310C2×C32—248no—
311C2×C32—244no—
312C2×C32—244no—
313C2×C32—364no—
314C2×C32—364no—
315C2×C32—362no—
316C2×C32—368no—
317C2×C32—481no—
318C2×C32—484no—
319C2×C32—488no—
320C2×C32—484no—
321C2×C32—724no—
322C2×C32—724no—
323C2×C32—728no—
324C2×C32C32×GL(2,𝔽3)11yescharacteristic, complemented
325C2×C32AGL(2,𝔽3)11yescharacteristic, direct product factor
326C3⋊D3—42no—
327C3⋊D3C32×GL(2,𝔽3)11yescharacteristic, complemented
328C3⋊D3C2×C32×S411yescharacteristic
329C3×D3—124no—
330C3×D3—124no—
331C3×D3—121no—
332C3×D3—121no—
333C3×D3—122no—
334C3×D3—128no—
335C3×D3—128no—
336C3×D3—122no—
337C3×D3—244no—
338C3×D3—244no—
339C3×D3—248no—
340C3×D3—248no—
341C3×D3—242no—
342C3×D3—248no—
343C3×D3—2416no—
344C3×D3—368no—
345C2×Q8—91no—
346C2×D4—271no—
347SD16—272no—
348SD16—272no—
349C2×C8—271no—
350C22×C3—94no—
351C22×C3—361no—
352C22×C3—361no—
353C22×C3—364no—
354C22×C3—548no—
355C22×C3—724no—
356C22×C3—724no—
357C22×C3—1084no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—241no—
362C2×D3—361no—
363C2×D3—362no—
364C2×D3—362no—
365C2×D3—362no—
366C2×D3—362no—
367C3×C4—274no—
368C3×C4—274no—
369C32—44no—
370C32—41no—
371C32—84no—
372C32—81no—
373C32—84no—
374C32—84no—
375C32—124no—
376C32—164no—
377C32—244no—
378C32—248no—
379C32—244no—
380C32—244no—
381C32—481no—
382C32—484no—
383C32—488no—
384C32—484no—
385C32C2×C32×GL(2,𝔽3)11yescharacteristic, complemented
386C32C2×AGL(2,𝔽3)11yescharacteristic, direct product factor
387C23—541no—
388Q8—91no—
389Q8—271no—
390D4—272no—
391D4—541no—
392C2×C4—271no—
393C8—272no—
394C2×C3—41no—
395C2×C3—84no—
396C2×C3—94no—
397C2×C3—94no—
398C2×C3—121no—
399C2×C3—241no—
400C2×C3—244no—
401C2×C3—361no—
402C2×C3—361no—
403C2×C3—362no—
404C2×C3—368no—
405C2×C3—484no—
406C2×C3—724no—
407C2×C3—724no—
408C2×C3—728no—
409C2×C3C3×AGL(2,𝔽3)14yesdirect product factor
410D3—121no—
411D3—121no—
412D3—122no—
413D3—122no—
414D3—242no—
415D3—362no—
416C22—91no—
417C22—361no—
418C22—542no—
419C22—1081no—
420C4—271no—
421C4—271no—
422C3—41no—
423C3—84no—
424C3—121no—
425C3—241no—
426C3—244no—
427C3—484no—
428C3C2×C3×AGL(2,𝔽3)14yesdirect product factor
429C2—91no—
430C2—91no—
431C2—362no—
432C2C32×AGL(2,𝔽3)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C32AGL(2,𝔽3)
commutator, cosocle, hypocenterPSU(3,𝔽2)⋊C3C22×C32
FrattiniC1G
Fitting, socleC2×C34GL(2,𝔽3)
LCSG ▷ PSU(3,𝔽2)⋊C3
UCSC1 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
2. C2×C32×PSU(3,𝔽2)⋊C3C2semidirect
3. C2×C3×AGL(2,𝔽3)C3direct (N × H)
4. C32×AGL(2,𝔽3)C2direct (N × H)
5. C32×PSU(3,𝔽2)⋊C3C22semidirect
7. C2×C32×PSU(3,𝔽2)D3semidirect
13. C2×C3×PSU(3,𝔽2)⋊C3C2×C3semidirect
14. C3×AGL(2,𝔽3)C2×C3direct (N × H)
21. C2×AGL(2,𝔽3)C32direct (N × H)
27. C32×PSU(3,𝔽2)C2×D3semidirect
32. C3×PSU(3,𝔽2)⋊C3C22×C3semidirect
39. C2×C3×PSU(3,𝔽2)C3×D3semidirect
45. C2×PSU(3,𝔽2)⋊C3C2×C32semidirect
46. AGL(2,𝔽3)C2×C32direct (N × H)
78. C3×PSU(3,𝔽2)C2×C3×D3semidirect
83. PSU(3,𝔽2)⋊C3C22×C32semidirect
88. C2×C34GL(2,𝔽3)semidirect
90. C32×C3⋊D3GL(2,𝔽3)semidirect
108. C2×PSU(3,𝔽2)C32×D3semidirect
159. C34C2×GL(2,𝔽3)semidirect
171. PSU(3,𝔽2)C2×C32×D3semidirect
196. C2×C33C3×GL(2,𝔽3)semidirect
198. C3×C3⋊D3C3×GL(2,𝔽3)semidirect
282. C33C2×C3×GL(2,𝔽3)semidirect
324. C2×C32C32×GL(2,𝔽3)semidirect
325. C2×C32AGL(2,𝔽3)direct (N × H)
327. C3⋊D3C32×GL(2,𝔽3)semidirect
385. C32C2×C32×GL(2,𝔽3)semidirect
386. C32C2×AGL(2,𝔽3)direct (N × H)
409. C2×C3C3×AGL(2,𝔽3)direct (N × H)
428. C3C2×C3×AGL(2,𝔽3)direct (N × H)
432. C2C32×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
C2×C32×AΓL(1,𝔽9)3
C2×C32×PSU(3,𝔽2)⋊C32
C2×C3×AGL(2,𝔽3)3
C32×AGL(2,𝔽3)2
C2×C32×((C3⋊D3)⋊C3)⋊C24
C2×C32×GL(2,𝔽3)9