Finite Group Databasebeta

← D3×C22⋊PSU(3,𝔽2) #13180646

Subgroups — D3×C22⋊PSU(3,𝔽2) #13180646

Order 1728.

810 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C22×D3×#157C211yesmaximal, characteristic
2C2×D3×PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
3#11724369C211yesmaximal, characteristic, complemented
4#11724368C212yesmaximal, complemented
5C3×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
6D3×C22⋊#157C212yesmaximal
7#11724151C212yesmaximal, complemented
8#11724150C211yesmaximal, characteristic, complemented
9#11724149C211yesmaximal, characteristic, complemented
10D3×#3777C212yesmaximal, complemented
11D3×#3777C211yesmaximal, characteristic, complemented
12C2×C22⋊PSU(3,𝔽2)—31nomaximal
13C22×D3×C3⋊D3C2211yescharacteristic
14C22×C3⋊#157C2211yescharacteristic
15C22×C3×#157C2211yescharacteristic
16C2×C3⋊PSU(3,𝔽2)C2212yescomplemented
17C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic, complemented
18C2×C3×PSU(3,𝔽2)C2211yescharacteristic, complemented
19C2×D3×#157—24no—
20C2×D3×#157C2211yescharacteristic
21C2×D3×#157C2212yes—
22C2×D3×#157C2212yes—
23C2×D3×#157C2211yescharacteristic
24C2×D3×#157C2211yescharacteristic
25C2×D3×#157C2211yescharacteristic
26D3×PSU(3,𝔽2)—24no—
27#88994C2212yes—
28C3×C22⋊#157C2212yes—
29#88946C2212yescomplemented
30#88946C2211yescharacteristic, complemented
31#88945C2212yescomplemented
32#88945C2212yescomplemented
33#88945C2212yescomplemented
34C3×#3777C2211yescharacteristic, complemented
35C3×#3777C2212yescomplemented
36(C2×D3)⋊#157C2212yes—
37(C2×D3)⋊#157C2212yes—
38C23×#157—31no—
39C22×PSU(3,𝔽2)—31no—
40C2×C22⋊#157—32no—
41C22⋊PSU(3,𝔽2)—32no—
42C22⋊PSU(3,𝔽2)—31no—
43C22⋊PSU(3,𝔽2)—32no—
44C22⋊PSU(3,𝔽2)—31no—
45C22⋊PSU(3,𝔽2)—31no—
46C22⋊PSU(3,𝔽2)D311yescharacteristic, direct product factor
47C2×#3777—31no—
48C2×#3777—32no—
49C22×C3⋊(C3⋊D3)Q811yescharacteristic, complemented
50C22×C3×C3⋊D3C2311yescharacteristic
51C22×C32×D3Q811yescharacteristic, complemented
52C2×D3×C3⋊D3—22no—
53C2×D3×C3⋊D3—22no—
54C2×D3×C3⋊D3Q811yescharacteristic, complemented
55C2×D3×C3⋊D3Q811yescharacteristic, complemented
56C2×D3×C3⋊D3C2311yescharacteristic
57C2×D3×C3⋊D3C2311yescharacteristic
58C2×C3⋊#157—22no—
59C2×C3⋊#157C2311yescharacteristic
60C2×C3⋊#157C2312yes—
61C2×C3⋊#157C2312yes—
62C2×C3⋊#157C2311yescharacteristic
63C2×C3×#157—22no—
64C2×C3×#157C2311yescharacteristic
65C2×C3×#157C2312yes—
66C2×C3×#157C2312yes—
67C2×C3×#157C2311yescharacteristic
68C3⋊PSU(3,𝔽2)—24no—
69C3⋊PSU(3,𝔽2)—22no—
70C3×PSU(3,𝔽2)—22no—
71D3×#157—24no—
72D3×#157—22no—
73D3×#157—24no—
74D3×#157—22no—
75D3×#157—22no—
76D3×#157D414yes—
77D3×C22⋊Q8—91nomaximal
78C23×C3⋊D3—31no—
79C22×D32—121no—
80C22×#157—31no—
81C22×#157—31no—
82C22×#157—32no—
83C22×#157—32no—
84C22×#157—31no—
85C22×#157—31no—
86C22×#157—31no—
87C22×#157—64no—
88C22×#157C2×D311yescharacteristic
89C2×PSU(3,𝔽2)—32no—
90C2×PSU(3,𝔽2)—31no—
91C2×PSU(3,𝔽2)—64no—
92C2×PSU(3,𝔽2)C2×D311yescharacteristic, complemented
93C22⋊#157—32no—
94C22⋊#157—32no—
95C22⋊#157—32no—
96C22⋊#157C2×D312yes—
97#3777—32no—
98#3777—32no—
99#3777—32no—
100#3777—32no—
101#3777—31no—
102#3777C2×D312yescomplemented
103#3777C2×D311yescharacteristic, complemented
104C22×C33C2×Q811yescharacteristic, complemented
105C2×C3⋊(C3⋊D3)—22no—
106C2×C3⋊(C3⋊D3)C2×Q811yescharacteristic
107C2×C3⋊(C3⋊D3)C2×Q811yescharacteristic, complemented
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3C2×Q811yescharacteristic, complemented
111C2×C3×C3⋊D3C2411yescharacteristic
112C2×C32×D3—22no—
113C2×C32×D3C2×Q811yescharacteristic
114C2×C32×D3C2×Q811yescharacteristic, complemented
115D3×C3⋊D3—22no—
116D3×C3⋊D3—21no—
117D3×C3⋊D3—21no—
118D3×C3⋊D3—22no—
119D3×C3⋊D3C2×D412yes—
120D3×C3⋊D3#4112yes—
121C3⋊#157—21no—
122C3⋊#157—22no—
123C3⋊#157—22no—
124C3⋊#157C2×D412yes—
125C3×#157—21no—
126C3×#157—22no—
127C3×#157—22no—
128C3×#157C2×D412yes—
129C2×D3×Q8—91no—
130C22×C4×D3—91no—
131C3×C22⋊Q8—91no—
132#946—91no—
133#924—91no—
134#897—91no—
135#896—92no—
136D3×C4⋊C4—91no—
137D3×C4⋊C4—92no—
138D3×C22⋊C4—92no—
139#878—92no—
140C23×C32—31no—
141C22×C3⋊D3—31no—
142C22×C3⋊D3—31no—
143C22×C3⋊D3—31no—
144C22×C3⋊D3—31no—
145C22×C3⋊D3—31no—
146C22×C3⋊D3—62no—
147C22×C3⋊D3—62no—
148C22×C3⋊D3—121no—
149C22×C3⋊D3—241no—
150C22×C3⋊D3C22×D311yescharacteristic
151C22×C3×D3—41no—
152C22×C3×D3—121no—
153C2×D32—121no—
154C2×D32—121no—
155C2×D32—121no—
156C2×D32—121no—
157C2×D32—124no—
158C2×D32—124no—
159C2×#157—22no—
160C2×#157—31no—
161C2×#157—34no—
162C2×#157—32no—
163C2×#157—31no—
164C2×#157—32no—
165C2×#157—62no—
166C2×#157—64no—
167C2×#157—62no—
168C2×#157—64no—
169C2×#157—62no—
170C2×#157—62no—
171C2×#157C22×D312yes—
172C2×#157C22×D311yescharacteristic
173C2×#157C22×D312yes—
174C2×#157C22×D311yescharacteristic
175PSU(3,𝔽2)—22no—
176PSU(3,𝔽2)—64no—
177PSU(3,𝔽2)—62no—
178C2×C22⋊Q8—271noSylow
179C2×C33—21no—
180C2×C33C22×Q811yescharacteristic
181C3⋊(C3⋊D3)—21no—
182C3⋊(C3⋊D3)C22⋊Q812yescomplemented
183C3×C3⋊D3—21no—
184C3×C3⋊D3C22×D411yescharacteristic
185C3×C3⋊D3C2×#4111yescharacteristic
186C32×D3—21no—
187C32×D3C22⋊Q812yescomplemented
188C23×D3—91no—
189C23×D3—361no—
190C2×C3×Q8—91no—
191C22×C3×C4—91no—
192C22×Dic3—91no—
193D3×Q8—184no—
194C2×C4×D3—91no—
195C2×C4×D3—92no—
196C2×C4×D3—92no—
197C2×C4×D3—91no—
198C2×C4×D3—91no—
199C2×C4×D3—91no—
200C2×C4×D3—184no—
201C2×Dic6—92no—
202C2×Dic6—91no—
203C3×C4⋊C4—92no—
204C3×C4⋊C4—91no—
205C3×C22⋊C4—92no—
206#217—92no—
207(C2×D3)⋊C4—92no—
208(C2×D3)⋊C4—92no—
209C4⋊Dic3—92no—
210C4⋊Dic3—91no—
211Dic3⋊C4—92no—
212Dic3⋊C4—92no—
213Dic3⋊C4—92no—
214C22×C32—31no—
215C22×C32—31no—
216C22×C32—41no—
217C22×C32—62no—
218C22×C32—81no—
219C22×C32D3×Q811yescharacteristic, complemented
220C2×C3⋊D3—21no—
221C2×C3⋊D3—21no—
222C2×C3⋊D3—31no—
223C2×C3⋊D3—32no—
224C2×C3⋊D3—32no—
225C2×C3⋊D3—31no—
226C2×C3⋊D3—62no—
227C2×C3⋊D3—62no—
228C2×C3⋊D3—62no—
229C2×C3⋊D3—61no—
230C2×C3⋊D3—61no—
231C2×C3⋊D3—121no—
232C2×C3⋊D3—121no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—244no—
235C2×C3⋊D3—241no—
236C2×C3⋊D3—241no—
237C2×C3⋊D3D3×Q811yescharacteristic, complemented
238C2×C3⋊D3C23×D311yescharacteristic
239C2×C3×D3—44no—
240C2×C3×D3—41no—
241C2×C3×D3—41no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246D32—122no—
247D32—122no—
248D32—124no—
249D32—122no—
250D32—122no—
251D32—124no—
252#157—22no—
253#157—21no—
254#157—22no—
255#157—32no—
256#157—62no—
257#157—61no—
258#157—62no—
259#157D3×D412yes—
260C22×Q8—271no—
261C23×C4—271no—
262C22⋊Q8—91no—
263C22⋊Q8—272no—
264C22⋊Q8—272no—
265C22⋊Q8—271no—
266C22⋊Q8—271no—
267C22⋊Q8—271no—
268C2×C4⋊C4—271no—
269C2×C4⋊C4—272no—
270C2×C22⋊C4—272no—
271C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
272C23×C3—91no—
273C23×C3—121no—
274C22×D3—91no—
275C22×D3—91no—
276C22×D3—91no—
277C22×D3—91no—
278C22×D3—91no—
279C22×D3—121no—
280C22×D3—182no—
281C22×D3—182no—
282C22×D3—361no—
283C22×D3—361no—
284C22×D3—361no—
285C22×D3—361no—
286C22×D3—361no—
287C22×D3—364no—
288C22×D3—364no—
289C22×D3—721no—
290C22×D3PSU(3,𝔽2)11yescharacteristic, complemented
291C3×Q8—182no—
292C2×C3×C4—91no—
293C2×C3×C4—91no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—182no—
297C2×Dic3—91no—
298C2×Dic3—92no—
299C2×Dic3—91no—
300C2×Dic3—92no—
301C2×Dic3—182no—
302C4×D3—94no—
303C4×D3—182no—
304C4×D3—182no—
305C4×D3—184no—
306C4×D3—182no—
307C4×D3—184no—
308Dic6—184no—
309Dic6—182no—
310C2×C32—21no—
311C2×C32—32no—
312C2×C32—42no—
313C2×C32—41no—
314C2×C32—61no—
315C2×C32—82no—
316C2×C32—81no—
317C2×C32C2×D3×Q811yescharacteristic
318C3⋊D3—21no—
319C3⋊D3—32no—
320C3⋊D3—61no—
321C3⋊D3—122no—
322C3⋊D3—122no—
323C3⋊D3—242no—
324C3⋊D3—242no—
325C3⋊D3C2×D3×D411yescharacteristic
326C3⋊D3D3×#4111yescharacteristic
327C3×D3—42no—
328C3×D3—42no—
329C3×D3—121no—
330C3×D3—121no—
331C3×D3—122no—
332C24—271no—
333C2×Q8—91no—
334C2×Q8—271no—
335C2×Q8—272no—
336C2×Q8—544no—
337C22×C4—91no—
338C22×C4—271no—
339C22×C4—272no—
340C22×C4—272no—
341C22×C4—271no—
342C22×C4—271no—
343C22×C4—271no—
344C22×C4—271no—
345C22×C4—544no—
346C4⋊C4—91no—
347C4⋊C4—92no—
348C4⋊C4—272no—
349C4⋊C4—272no—
350C4⋊C4—272no—
351C4⋊C4—271no—
352C4⋊C4—272no—
353C22⋊C4—92no—
354C22⋊C4—272no—
355C22⋊C4—272no—
356C22⋊C4—272no—
357C22×C3—41no—
358C22×C3—81no—
359C22×C3—91no—
360C22×C3—91no—
361C22×C3—121no—
362C22×C3—121no—
363C22×C3—124no—
364C22×C3—181no—
365C22×C3—181no—
366C22×C3C2×PSU(3,𝔽2)11yescharacteristic, complemented
367C2×D3—22no—
368C2×D3—91no—
369C2×D3—92no—
370C2×D3—92no—
371C2×D3—91no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—122no—
375C2×D3—122no—
376C2×D3—182no—
377C2×D3—181no—
378C2×D3—181no—
379C2×D3—182no—
380C2×D3—182no—
381C2×D3—362no—
382C2×D3—362no—
383C2×D3—361no—
384C2×D3—361no—
385C2×D3—364no—
386C2×D3—362no—
387C2×D3—362no—
388C2×D3—364no—
389C2×D3—364no—
390C2×D3—721no—
391C2×D3—721no—
392C2×D3—724no—
393C2×D3C2×PSU(3,𝔽2)11yescharacteristic
394C2×D3C2×PSU(3,𝔽2)11yescharacteristic, complemented
395C3×C4—92no—
396C3×C4—182no—
397C3×C4—181no—
398C3×C4—182no—
399Dic3—92no—
400Dic3—182no—
401Dic3—181no—
402Dic3—182no—
403C32—41no—
404C32—81no—
405C32D3×C22⋊Q811yescharacteristic, complemented
406C23—31no—
407C23—91no—
408C23—271no—
409C23—271no—
410C23—271no—
411C23—271no—
412C23—271no—
413C23—542no—
414C23—542no—
415Q8—182no—
416Q8—542no—
417Q8—544no—
418C2×C4—91no—
419C2×C4—92no—
420C2×C4—92no—
421C2×C4—91no—
422C2×C4—182no—
423C2×C4—274no—
424C2×C4—271no—
425C2×C4—272no—
426C2×C4—272no—
427C2×C4—271no—
428C2×C4—542no—
429C2×C4—542no—
430C2×C4—542no—
431C2×C4—544no—
432C2×C4—544no—
433C2×C4—542no—
434C2×C3—21no—
435C2×C3—42no—
436C2×C3—41no—
437C2×C3—82no—
438C2×C3—81no—
439C2×C3—91no—
440C2×C3—91no—
441C2×C3—122no—
442C2×C3—122no—
443C2×C3—181no—
444C2×C3C22×PSU(3,𝔽2)11yescharacteristic
445D3—21no—
446D3—92no—
447D3—121no—
448D3—121no—
449D3—122no—
450D3—181no—
451D3—362no—
452D3—362no—
453D3—722no—
454D3—722no—
455D3C22⋊PSU(3,𝔽2)12yesdirect product factor
456C22—31no—
457C22—31no—
458C22—62no—
459C22—91no—
460C22—91no—
461C22—181no—
462C22—181no—
463C22—272no—
464C22—272no—
465C22—271no—
466C22—271no—
467C22—542no—
468C22—542no—
469C22—541no—
470C22—541no—
471C22—542no—
472C22D3×PSU(3,𝔽2)11yescharacteristic, complemented
473C4—92no—
474C4—181no—
475C4—182no—
476C4—182no—
477C4—272no—
478C4—541no—
479C4—542no—
480C4—542no—
481C3—41no—
482C3—81no—
483C3C2×C22⋊PSU(3,𝔽2)11yescharacteristic, complemented
484C2—21no—
485C2—32no—
486C2—61no—
487C2—91no—
488C2—91no—
489C2—181no—
490C2—272no—
491C2—541no—
492C2C2×D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×PSU(3,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FittingC22×C33C2×Q8
socleC2×C33C22×Q8
prime radicalC22×D3×C3⋊D3C22
hypercenterC22D3×PSU(3,𝔽2)
hypocenterC33C2×C22⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×D3×PSU(3,𝔽2)C2semidirect
3. #11724369C2semidirect
4. #11724368C2semidirect
5. C3×C22⋊PSU(3,𝔽2)C2semidirect
7. #11724151C2semidirect
8. #11724150C2semidirect
9. #11724149C2semidirect
10. D3×#3777C2semidirect
11. D3×#3777C2semidirect
16. C2×C3⋊PSU(3,𝔽2)C22semidirect
17. C2×C3⋊PSU(3,𝔽2)C22semidirect
18. C2×C3×PSU(3,𝔽2)C22semidirect
29. #88946C22semidirect
30. #88946C22semidirect
31. #88945C22semidirect
32. #88945C22semidirect
33. #88945C22semidirect
34. C3×#3777C22semidirect
35. C3×#3777C22semidirect
46. C22⋊PSU(3,𝔽2)D3direct (N × H)
49. C22×C3⋊(C3⋊D3)Q8semidirect
51. C22×C32×D3Q8semidirect
54. C2×D3×C3⋊D3Q8semidirect
55. C2×D3×C3⋊D3Q8semidirect
92. C2×PSU(3,𝔽2)C2×D3semidirect
102. #3777C2×D3semidirect
103. #3777C2×D3semidirect
104. C22×C33C2×Q8semidirect
107. C2×C3⋊(C3⋊D3)C2×Q8semidirect
110. C2×C3×C3⋊D3C2×Q8semidirect
114. C2×C32×D3C2×Q8semidirect
182. C3⋊(C3⋊D3)C22⋊Q8semidirect
187. C32×D3C22⋊Q8semidirect
219. C22×C32D3×Q8semidirect
237. C2×C3⋊D3D3×Q8semidirect
271. C33C2×C22⋊Q8semidirect
290. C22×D3PSU(3,𝔽2)semidirect
366. C22×C3C2×PSU(3,𝔽2)semidirect
394. C2×D3C2×PSU(3,𝔽2)semidirect
405. C32D3×C22⋊Q8semidirect
455. D3C22⋊PSU(3,𝔽2)direct (N × H)
472. C22D3×PSU(3,𝔽2)semidirect
483. C3C2×C22⋊PSU(3,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×D3×PSU(3,𝔽2)2
#117243692
#117243682
C3×C22⋊PSU(3,𝔽2)2
D3×C22⋊#1572
#117241512
#117241502
#117241492
D3×#37772
C2×C22⋊PSU(3,𝔽2)3
D3×C22⋊Q89