Finite Group Databasebeta

← C2×A4×PSU(3,𝔽2)⋊C3 #72242879

Subgroups — C2×A4×PSU(3,𝔽2)⋊C3 #72242879

Order 5184.

546 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 37 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
1A4×PSU(3,𝔽2)⋊C3C211yesmaximal, characteristic, direct product factor
2C2×A4×PSU(3,𝔽2)C311yesmaximal, characteristic, complemented
3C2×C22⋊(PSU(3,𝔽2)⋊C3)C312yesmaximal, complemented
4C23×PSU(3,𝔽2)⋊C3C311yesmaximal, characteristic, complemented
5C2×C3×PSU(3,𝔽2)⋊C3—41nomaximal
6C2×A4×(C3⋊D3)⋊C3—41nomaximal
7C2×A4×#157—31no—
8A4×PSU(3,𝔽2)—31no—
9A4×PSU(3,𝔽2)C2×C311yescharacteristic, complemented
10C22⋊(PSU(3,𝔽2)⋊C3)C2×C312yescomplemented
11C22×PSU(3,𝔽2)⋊C3—31no—
12C22×PSU(3,𝔽2)⋊C3—31no—
13C22×PSU(3,𝔽2)⋊C3C2×C311yescharacteristic, complemented
14C3×PSU(3,𝔽2)⋊C3—41no—
15C2×A4×H(3,𝔽3)—41no—
16A4×(C3⋊D3)⋊C3—41no—
17A4×(C3⋊D3)⋊C3—41no—
18C23×PSU(3,𝔽2)C3211yescharacteristic, complemented
19C2×A4×SL(2,𝔽3)—91nomaximal
20C2×A4×C3⋊D3A411yescharacteristic
21C2×C3×D3×A4—121no—
22C2×C3×PSU(3,𝔽2)—41no—
23A4×#157—31no—
24A4×#157—31no—
25C2×PSU(3,𝔽2)⋊C3—31no—
26C2×PSU(3,𝔽2)⋊C3—31no—
27C2×PSU(3,𝔽2)⋊C3—42no—
28C2×PSU(3,𝔽2)⋊C3—122no—
29C2×PSU(3,𝔽2)⋊C3A411yescharacteristic, direct product factor
30C23×(C3⋊D3)⋊C3—41no—
31C2×(C3⋊D3)⋊A4—42no—
32C2×C3×(C3⋊D3)⋊C3—161no—
33A4×H(3,𝔽3)—41no—
34C23×#157—31no—
35C22×PSU(3,𝔽2)—31no—
36C22×PSU(3,𝔽2)—31no—
37C22×PSU(3,𝔽2)—31no—
38C22×PSU(3,𝔽2)—91no—
39C22×PSU(3,𝔽2)—91no—
40C22×PSU(3,𝔽2)C2×C3211yescharacteristic, complemented
41A4×SL(2,𝔽3)—91no—
42C2×C32×A4—41no—
43C2×C32×A4—81no—
44C2×C32×A4SL(2,𝔽3)11yescharacteristic, complemented
45C2×C3×#157—121no—
46A4×C3⋊D3SL(2,𝔽3)11yescharacteristic, complemented
47A4×C3⋊D3C2×A411yescharacteristic
48C3×D3×A4—121no—
49C3×D3×A4—121no—
50C3×PSU(3,𝔽2)—41no—
51C3×PSU(3,𝔽2)—121no—
52PSU(3,𝔽2)⋊C3—42no—
53PSU(3,𝔽2)⋊C3—122no—
54PSU(3,𝔽2)⋊C3C2×A411yescharacteristic, direct product factor
55C23×H(3,𝔽3)—41no—
56C22×(C3⋊D3)⋊C3—41no—
57C22×(C3⋊D3)⋊C3—41no—
58C22×(C3⋊D3)⋊C3—121no—
59C22×(C3⋊D3)⋊C3—121no—
60C22×(C3⋊D3)⋊C3—121no—
61C22×(C3⋊D3)⋊C3—121no—
62C2×C22⋊H(3,𝔽3)—42no—
63C2×C22⋊H(3,𝔽3)—82no—
64C2×C22⋊H(3,𝔽3)—81no—
65(C3⋊D3)⋊A4—42no—
66(C3⋊D3)⋊A4—42no—
67C2×C22⋊SL(2,𝔽3)—92no—
68C2×Q8×A4—91no—
69C23×SL(2,𝔽3)—91no—
70C2×C3×H(3,𝔽3)—161no—
71C3×(C3⋊D3)⋊C3—161no—
72C3×(C3⋊D3)⋊C3—161no—
73C23×C3⋊D3C3×A411yescharacteristic
74C23×C3×D3—121no—
75C22×C3×A4—361no—
76C22×#157—31no—
77C22×#157—31no—
78C22×#157—91no—
79C22×#157—91no—
80C22×#157—91no—
81C22×#157—91no—
82C2×D3×A4—121no—
83C2×D3×A4—122no—
84C2×PSU(3,𝔽2)—32no—
85C2×PSU(3,𝔽2)—31no—
86C2×PSU(3,𝔽2)—31no—
87C2×PSU(3,𝔽2)—91no—
88C2×PSU(3,𝔽2)—91no—
89C2×PSU(3,𝔽2)—91no—
90C2×PSU(3,𝔽2)—92no—
91C2×PSU(3,𝔽2)—91no—
92C2×PSU(3,𝔽2)—92no—
93C2×PSU(3,𝔽2)—91no—
94C2×PSU(3,𝔽2)—92no—
95C2×PSU(3,𝔽2)C3×A411yescharacteristic, complemented
96C2×C3×SL(2,𝔽3)—361no—
97C2×C3×C3⋊D3—41no—
98C2×C32×D3—481no—
99C32×A4—41no—
100C32×A4—81no—
101C32×A4C2×SL(2,𝔽3)11yescharacteristic, complemented
102C3×#157—121no—
103C3×#157—121no—
104C22×H(3,𝔽3)—41no—
105C22×H(3,𝔽3)—121no—
106C22×H(3,𝔽3)—121no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3—121no—
109C2×(C3⋊D3)⋊C3—121no—
110C2×(C3⋊D3)⋊C3—121no—
111C2×(C3⋊D3)⋊C3—121no—
112C2×(C3⋊D3)⋊C3—121no—
113C2×(C3⋊D3)⋊C3—122no—
114C2×(C3⋊D3)⋊C3—121no—
115C2×(C3⋊D3)⋊C3—121no—
116C2×(C3⋊D3)⋊C3—162no—
117C22⋊H(3,𝔽3)—42no—
118C22⋊H(3,𝔽3)—82no—
119C22⋊H(3,𝔽3)—81no—
120C22⋊SL(2,𝔽3)—92no—
121Q8×A4—91no—
122Q8×A4—271no—
123C22×SL(2,𝔽3)—91no—
124C22×SL(2,𝔽3)—271no—
125C22×SL(2,𝔽3)—271no—
126C2×C4×A4—271no—
127C3×H(3,𝔽3)—161noSylow
128C23×C32—41no—
129C23×C32—81no—
130C23×C32C3×SL(2,𝔽3)11yescharacteristic, complemented
131C22×C3⋊D3—31no—
132C22×C3⋊D3—31no—
133C22×C3⋊D3—31no—
134C22×C3⋊D3—31no—
135C22×C3⋊D3C3×SL(2,𝔽3)11yescharacteristic, complemented
136C22×C3⋊D3C2×C3×A411yescharacteristic
137C22×C3×D3—121no—
138C22×C3×D3—121no—
139C22×C3×D3—361no—
140C22×C3×D3—361no—
141C22×C3×D3—361no—
142C22×C3×D3—361no—
143C2×C3×A4—41no—
144C2×C3×A4—42no—
145C2×C3×A4—81no—
146C2×C3×A4—82no—
147C2×C3×A4—121no—
148C2×C3×A4—241no—
149C2×C3×A4—242no—
150C2×C3×A4—241no—
151C2×C3×A4—361no—
152C2×C3×A4—361no—
153C2×#157—31no—
154C2×#157—91no—
155C2×#157—91no—
156C2×#157—91no—
157C2×#157—91no—
158C2×#157—92no—
159C2×#157—91no—
160C2×#157—91no—
161C2×#157—91no—
162D3×A4—121no—
163D3×A4—121no—
164D3×A4—122no—
165D3×A4—122no—
166PSU(3,𝔽2)—31no—
167PSU(3,𝔽2)—32no—
168PSU(3,𝔽2)—91no—
169PSU(3,𝔽2)—91no—
170PSU(3,𝔽2)—92no—
171PSU(3,𝔽2)—92no—
172PSU(3,𝔽2)C2×C3×A411yescharacteristic, complemented
173C3×SL(2,𝔽3)—361no—
174C23×Q8—91noSylow
175C2×C33—41no—
176C2×C33—161no—
177C2×C33—321no—
178C3×C3⋊D3—41no—
179C3×C3⋊D3—41no—
180C32×D3—481no—
181C32×D3—481no—
182C2×H(3,𝔽3)—41no—
183C2×H(3,𝔽3)—121no—
184C2×H(3,𝔽3)—121no—
185C2×H(3,𝔽3)—162no—
186C2×H(3,𝔽3)—321no—
187C2×H(3,𝔽3)—322no—
188(C3⋊D3)⋊C3—41no—
189(C3⋊D3)⋊C3—41no—
190(C3⋊D3)⋊C3—121no—
191(C3⋊D3)⋊C3—121no—
192(C3⋊D3)⋊C3—162no—
193(C3⋊D3)⋊C3—162no—
194C24×C3—361no—
195C23×D3—121no—
196C22×A4—91no—
197C22×A4—362no—
198C2×C3×Q8—361no—
199C2×SL(2,𝔽3)—91no—
200C2×SL(2,𝔽3)—271no—
201C2×SL(2,𝔽3)—271no—
202C2×SL(2,𝔽3)—362no—
203C2×SL(2,𝔽3)—1082no—
204C4×A4—271no—
205C4×A4—271no—
206C22×C32—31no—
207C22×C32—31no—
208C22×C32—41no—
209C22×C32—81no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32—241no—
213C22×C32—241no—
214C22×C32—1441no—
215C22×C32C2×C3×SL(2,𝔽3)11yescharacteristic, complemented
216C2×C3⋊D3—32no—
217C2×C3⋊D3—31no—
218C2×C3⋊D3—31no—
219C2×C3⋊D3—31no—
220C2×C3⋊D3—31no—
221C2×C3⋊D3—31no—
222C2×C3⋊D3—31no—
223C2×C3⋊D3—31no—
224C2×C3⋊D3A4211yescharacteristic
225C2×C3×D3—121no—
226C2×C3×D3—361no—
227C2×C3×D3—361no—
228C2×C3×D3—361no—
229C2×C3×D3—361no—
230C2×C3×D3—361no—
231C2×C3×D3—362no—
232C2×C3×D3—361no—
233C2×C3×D3—361no—
234C2×C3×D3—481no—
235C2×C3×D3—482no—
236C3×A4—41no—
237C3×A4—42no—
238C3×A4—81no—
239C3×A4—82no—
240C3×A4—121no—
241C3×A4—241no—
242C3×A4—242no—
243C3×A4—241no—
244#157—31no—
245#157—31no—
246#157—91no—
247#157—91no—
248C22×Q8—91no—
249C22×Q8—271no—
250C22×Q8—271no—
251C22×Q8—271no—
252C22×Q8—811no—
253C22×Q8—811no—
254C23×C4—271no—
255C33—41no—
256C33—161no—
257C33—321no—
258H(3,𝔽3)—41no—
259H(3,𝔽3)—162no—
260H(3,𝔽3)—321no—
261H(3,𝔽3)—322no—
262C23×C3—41no—
263C23×C3—121no—
264C23×C3—241no—
265C23×C3—361no—
266C23×C3—361no—
267C23×C3—1081no—
268C23×C3—1081no—
269C23×C3—1081no—
270C23×C3—1081no—
271C22×D3—121no—
272C22×D3—121no—
273C22×D3—361no—
274C22×D3—361no—
275C22×D3—361no—
276C22×D3—361no—
277C2×A4—81no—
278C2×A4—91no—
279C2×A4—91no—
280C2×A4—122no—
281C2×A4—242no—
282C2×A4—362no—
283C2×A4—362no—
284C2×A4PSU(3,𝔽2)⋊C311yescharacteristic, direct product factor
285C3×Q8—361no—
286C3×Q8—1081no—
287C2×C3×C4—1081no—
288SL(2,𝔽3)—91no—
289SL(2,𝔽3)—362no—
290SL(2,𝔽3)—1082no—
291C2×C32—31no—
292C2×C32—31no—
293C2×C32—41no—
294C2×C32—81no—
295C2×C32—121no—
296C2×C32—121no—
297C2×C32—161no—
298C2×C32—162no—
299C2×C32—241no—
300C2×C32—241no—
301C2×C32—321no—
302C2×C32—322no—
303C2×C32—481no—
304C2×C32—961no—
305C2×C32—961no—
306C2×C32—962no—
307C2×C32—1441no—
308C2×C32—1441no—
309C2×C32A4×SL(2,𝔽3)11yescharacteristic, complemented
310C3⋊D3—31no—
311C3⋊D3—31no—
312C3⋊D3A4×SL(2,𝔽3)11yescharacteristic, complemented
313C3⋊D3C2×A4211yescharacteristic
314C3×D3—121no—
315C3×D3—121no—
316C3×D3—361no—
317C3×D3—361no—
318C3×D3—481no—
319C3×D3—481no—
320C3×D3—482no—
321C3×D3—482no—
322C24—91no—
323C2×Q8—91no—
324C2×Q8—271no—
325C2×Q8—271no—
326C2×Q8—272no—
327C2×Q8—811no—
328C2×Q8—812no—
329C2×Q8—812no—
330C2×Q8—811no—
331C2×Q8—811no—
332C2×Q8—811no—
333C2×Q8—811no—
334C2×Q8—812no—
335C22×C4—271no—
336C22×C4—271no—
337C22×C4—811no—
338C22×C4—811no—
339C22×C4—811no—
340C22×C4—811no—
341C22×C3—41no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—241no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3—721no—
351C22×C3—721no—
352C22×C3—1081no—
353C22×C3—1081no—
354C22×C3—1081no—
355C22×C3—1081no—
356C22×C3—1081no—
357C22×C3—1081no—
358C22×C3—1081no—
359C22×C3—1082no—
360C22×C3—1442no—
361C2×D3—121no—
362C2×D3—361no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C2×D3—361no—
367C2×D3—362no—
368C2×D3—361no—
369C2×D3—361no—
370A4—81no—
371A4—122no—
372A4—242no—
373A4C2×PSU(3,𝔽2)⋊C311yescharacteristic, direct product factor
374C3×C4—1081no—
375C3×C4—1081no—
376C32—41no—
377C32—81no—
378C32—161no—
379C32—162no—
380C32—321no—
381C32—322no—
382C32—481no—
383C32—961no—
384C32—961no—
385C32—962no—
386C32C2×A4×SL(2,𝔽3)11yescharacteristic, complemented
387C23—91no—
388C23—91no—
389C23—271no—
390C23—271no—
391C23—271no—
392C23—271no—
393C23C3×PSU(3,𝔽2)⋊C311yescharacteristic, complemented
394Q8—91no—
395Q8—271no—
396Q8—272no—
397Q8—811no—
398Q8—811no—
399Q8—812no—
400Q8—812no—
401C2×C4—271no—
402C2×C4—811no—
403C2×C4—811no—
404C2×C4—811no—
405C2×C4—811no—
406C2×C4—811no—
407C2×C4—812no—
408C2×C4—811no—
409C2×C4—811no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—121no—
413C2×C3—121no—
414C2×C3—121no—
415C2×C3—241no—
416C2×C3—321no—
417C2×C3—361no—
418C2×C3—361no—
419C2×C3—361no—
420C2×C3—361no—
421C2×C3—361no—
422C2×C3—361no—
423C2×C3—482no—
424C2×C3—721no—
425C2×C3—721no—
426C2×C3—962no—
427C2×C3—1081no—
428C2×C3—1081no—
429C2×C3—1442no—
430C2×C3—1442no—
431D3—121no—
432D3—121no—
433D3—361no—
434D3—361no—
435C22—31no—
436C22—31no—
437C22—91no—
438C22—271no—
439C22—271no—
440C22—271no—
441C22—271no—
442C22—271no—
443C22—271no—
444C22—272no—
445C22—271no—
446C22C2×C3×PSU(3,𝔽2)⋊C311yescharacteristic, complemented
447C4—271no—
448C4—271no—
449C4—811no—
450C4—811no—
451C3—41no—
452C3—41no—
453C3—121no—
454C3—241no—
455C3—321no—
456C3—482no—
457C3—962no—
458C2—31no—
459C2—31no—
460C2—91no—
461C2—91no—
462C2—271no—
463C2—271no—
464C2A4×PSU(3,𝔽2)⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2A4×PSU(3,𝔽2)⋊C3
commutator, cosocle, hypocenterC22×PSU(3,𝔽2)C2×C32
FrattiniC1G
Fitting, socleC23×C32C3×SL(2,𝔽3)
LCSG ▷ C22×PSU(3,𝔽2)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. A4×PSU(3,𝔽2)⋊C3C2direct (N × H)
2. C2×A4×PSU(3,𝔽2)C3semidirect
3. C2×C22⋊(PSU(3,𝔽2)⋊C3)C3semidirect
4. C23×PSU(3,𝔽2)⋊C3C3semidirect
9. A4×PSU(3,𝔽2)C2×C3semidirect
10. C22⋊(PSU(3,𝔽2)⋊C3)C2×C3semidirect
13. C22×PSU(3,𝔽2)⋊C3C2×C3semidirect
18. C23×PSU(3,𝔽2)C32semidirect
29. C2×PSU(3,𝔽2)⋊C3A4direct (N × H)
40. C22×PSU(3,𝔽2)C2×C32semidirect
44. C2×C32×A4SL(2,𝔽3)semidirect
46. A4×C3⋊D3SL(2,𝔽3)semidirect
54. PSU(3,𝔽2)⋊C3C2×A4direct (N × H)
95. C2×PSU(3,𝔽2)C3×A4semidirect
101. C32×A4C2×SL(2,𝔽3)semidirect
130. C23×C32C3×SL(2,𝔽3)semidirect
135. C22×C3⋊D3C3×SL(2,𝔽3)semidirect
172. PSU(3,𝔽2)C2×C3×A4semidirect
215. C22×C32C2×C3×SL(2,𝔽3)semidirect
284. C2×A4PSU(3,𝔽2)⋊C3direct (N × H)
309. C2×C32A4×SL(2,𝔽3)semidirect
312. C3⋊D3A4×SL(2,𝔽3)semidirect
373. A4C2×PSU(3,𝔽2)⋊C3direct (N × H)
386. C32C2×A4×SL(2,𝔽3)semidirect
393. C23C3×PSU(3,𝔽2)⋊C3semidirect
446. C22C2×C3×PSU(3,𝔽2)⋊C3semidirect
464. C2A4×PSU(3,𝔽2)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
A4×PSU(3,𝔽2)⋊C32
C2×A4×PSU(3,𝔽2)3
C2×C22⋊(PSU(3,𝔽2)⋊C3)3
C23×PSU(3,𝔽2)⋊C33
C2×C3×PSU(3,𝔽2)⋊C34
C2×A4×(C3⋊D3)⋊C34
C2×A4×SL(2,𝔽3)9