Finite Group Databasebeta

← #13180637

Subgroups — #13180637

Order 1728.

650 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#11724369C211yesmaximal, characteristic, complemented
2#11724368C212yesmaximal, complemented
3C3×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
4D3×C22⋊#157C212yesmaximal
5C2×(C2×D3)⋊#157C211yesmaximal, characteristic
6#11724151C212yesmaximal, complemented
7#11724151C212yesmaximal, complemented
8#11724150C212yesmaximal, complemented
9#11724149C212yesmaximal, complemented
10#10596550—31nomaximal
11C22×D3×C3⋊D3C2211yescharacteristic
12C22×C3⋊#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C2×C3⋊PSU(3,𝔽2)C2212yescomplemented
15C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic, complemented
16C2×C3×PSU(3,𝔽2)C2211yescharacteristic, complemented
17C2×D3×#157C2212yes—
18C2×D3×#157C2212yes—
19#88994C2212yes—
20C3×C22⋊#157C2212yes—
21#88946C2212yescomplemented
22#88946C2211yescharacteristic, complemented
23#88945C2212yescomplemented
24#88945C2212yescomplemented
25#88945C2212yescomplemented
26C3×#3777C2212yescomplemented
27C3×#3777C2211yescharacteristic, complemented
28(C2×D3)⋊#157C2212yes—
29(C2×D3)⋊#157C2212yes—
30(C2×D3)⋊#157C2212yes—
31(C2×D3)⋊#157C2212yes—
32C2×C22⋊#157—32no—
33C2×C22⋊#157—31no—
34C22⋊PSU(3,𝔽2)—32no—
35C22⋊PSU(3,𝔽2)—32no—
36C22⋊PSU(3,𝔽2)—32no—
37C22⋊PSU(3,𝔽2)—32no—
38C22⋊PSU(3,𝔽2)—31no—
39C22⋊PSU(3,𝔽2)—32no—
40C22⋊PSU(3,𝔽2)D311yescharacteristic, complemented
41C22×C3⋊(C3⋊D3)Q811yescharacteristic, complemented
42C22×C3×C3⋊D3C2311yescharacteristic
43C22×C32×D3Q811yescharacteristic, complemented
44C2×D3×C3⋊D3—41no—
45C2×D3×C3⋊D3—41no—
46C2×D3×C3⋊D3Q812yescomplemented
47C2×D3×C3⋊D3C2312yes—
48C2×C3⋊#157—41no—
49C2×C3⋊#157C2312yes—
50C2×C3⋊#157C2312yes—
51C2×C3⋊#157C2312yes—
52C2×C3×#157—41no—
53C2×C3×#157C2312yes—
54C2×C3×#157C2312yes—
55C2×C3×#157C2311yescharacteristic
56C2×C3×#157C2311yescharacteristic
57C3⋊PSU(3,𝔽2)—42no—
58C3⋊PSU(3,𝔽2)—41no—
59C3×PSU(3,𝔽2)—41no—
60D3×#157—42no—
61D3×#157—42no—
62#5634—91nomaximal
63C23×C3⋊D3—31no—
64C22×D32—121no—
65C22×#157—31no—
66C22×#157—32no—
67C22×#157—32no—
68C22×#157C2×D311yescharacteristic
69C2×PSU(3,𝔽2)—32no—
70C2×PSU(3,𝔽2)—31no—
71C2×PSU(3,𝔽2)C2×D311yescharacteristic, complemented
72C22⋊#157—32no—
73C22⋊#157—32no—
74C22⋊#157—32no—
75C22⋊#157—32no—
76C22⋊#157—32no—
77C22⋊#157C2×D312yes—
78#3777—32no—
79#3777—32no—
80#3777—32no—
81#3777—31no—
82#3777—32no—
83#3777C2×D311yescharacteristic, complemented
84#3777C2×D312yescomplemented
85C22×C33C2×Q811yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)C2×Q812yes—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3C2×Q811yescharacteristic
91C2×C3×C3⋊D3C2411yescharacteristic
92C2×C32×D3—41no—
93C2×C32×D3C2×Q812yes—
94D3×C3⋊D3—22no—
95D3×C3⋊D3—22no—
96D3×C3⋊D3—42no—
97C3⋊#157—22no—
98C3⋊#157—22no—
99C3⋊#157—22no—
100C3×#157—22no—
101C3×#157—22no—
102C3×#157—21no—
103C3×#157—21no—
104C3×C22⋊Q8—91no—
105#946—92no—
106C2×(C2×D3)⋊C4—91no—
107#924—91no—
108#897—92no—
109#896—92no—
110#896—92no—
111D3×C22⋊C4—92no—
112#878—92no—
113C23×C32—31no—
114C22×C3⋊D3—31no—
115C22×C3⋊D3—32no—
116C22×C3⋊D3—32no—
117C22×C3⋊D3—121no—
118C22×C3⋊D3—121no—
119C22×C3⋊D3—121no—
120C22×C3⋊D3—241no—
121C22×C3⋊D3C22×D311yescharacteristic
122C22×C3×D3—41no—
123C22×C3×D3—121no—
124C2×D32—122no—
125C2×D32—122no—
126C2×D32—124no—
127C2×D32—124no—
128C2×#157—32no—
129C2×#157—32no—
130C2×#157—32no—
131C2×#157—41no—
132C2×#157—122no—
133C2×#157—122no—
134C2×#157—121no—
135C2×#157C22×D312yes—
136C2×#157C22×D311yescharacteristic
137C2×#157C22×D312yes—
138C2×#157C22×D311yescharacteristic
139PSU(3,𝔽2)—41no—
140PSU(3,𝔽2)—121no—
141PSU(3,𝔽2)—122no—
142#543—271noSylow
143C2×C33—21no—
144C2×C33C22×Q811yescharacteristic
145C3⋊(C3⋊D3)—22no—
146C3×C3⋊D3—21no—
147C3×C3⋊D3H(5,𝔽2)11yescharacteristic
148C3×C3⋊D3H(5,𝔽2)11yescharacteristic
149C32×D3—22no—
150C23×D3—91no—
151C23×D3—361no—
152C2×C3×Q8—91no—
153C22×C3×C4—91no—
154C22×Dic3—91no—
155C2×C4×D3—92no—
156C2×C4×D3—92no—
157C2×Dic6—92no—
158C2×Dic6—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—92no—
161C3×C22⋊C4—92no—
162#217—92no—
163(C2×D3)⋊C4—92no—
164(C2×D3)⋊C4—92no—
165(C2×D3)⋊C4—92no—
166(C2×D3)⋊C4—92no—
167C4⋊Dic3—92no—
168C4⋊Dic3—91no—
169Dic3⋊C4—92no—
170Dic3⋊C4—92no—
171Dic3⋊C4—92no—
172C22×C32—32no—
173C22×C32—41no—
174C22×C32—81no—
175C22×C32—121no—
176C22×C32D3×Q811yescharacteristic
177C2×C3⋊D3—21no—
178C2×C3⋊D3—21no—
179C2×C3⋊D3—32no—
180C2×C3⋊D3—62no—
181C2×C3⋊D3—62no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—122no—
184C2×C3⋊D3—121no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—244no—
187C2×C3⋊D3—242no—
188C2×C3⋊D3D3×Q811yescharacteristic
189C2×C3⋊D3C23×D311yescharacteristic
190C2×C3×D3—44no—
191C2×C3×D3—42no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196D32—124no—
197D32—124no—
198D32—128no—
199#157—21no—
200#157—22no—
201#157—21no—
202#157—22no—
203#157—62no—
204#157—62no—
205#157—62no—
206C22⋊Q8—91no—
207C22⋊Q8—272no—
208C22⋊Q8—272no—
209C22⋊Q8—272no—
210C22⋊Q8—272no—
211C22⋊Q8—271no—
212C22⋊Q8—272no—
213C2×C22⋊C4—272no—
214C2×C22⋊C4—271no—
215C33#54311yescharacteristic, complemented, Sylow
216C23×C3—91no—
217C23×C3—121no—
218C22×D3—91no—
219C22×D3—92no—
220C22×D3—92no—
221C22×D3—121no—
222C22×D3—362no—
223C22×D3—362no—
224C22×D3—361no—
225C22×D3—361no—
226C22×D3—361no—
227C22×D3—364no—
228C22×D3—364no—
229C22×D3—721no—
230C22×D3PSU(3,𝔽2)11yescharacteristic, complemented
231C3×Q8—361no—
232C2×C3×C4—91no—
233C2×C3×C4—92no—
234C2×C3×C4—92no—
235C2×C3×C4—91no—
236C2×C3×C4—361no—
237C2×Dic3—92no—
238C2×Dic3—92no—
239C2×Dic3—92no—
240C2×Dic3—361no—
241C4×D3—362no—
242C4×D3—362no—
243Dic6—361no—
244Dic6—362no—
245C2×C32—21no—
246C2×C32—42no—
247C2×C32—41no—
248C2×C32—62no—
249C2×C32—82no—
250C2×C32—81no—
251C2×C32C2×D3×Q811yescharacteristic
252C3⋊D3—21no—
253C3⋊D3—62no—
254C3⋊D3—124no—
255C3⋊D3—244no—
256C3⋊D3#100411yescharacteristic
257C3⋊D3#100911yescharacteristic
258C3×D3—44no—
259C3×D3—121no—
260C3×D3—121no—
261C3×D3—122no—
262C24—271no—
263C2×Q8—91no—
264C2×Q8—272no—
265C2×Q8—271no—
266C22×C4—91no—
267C22×C4—272no—
268C22×C4—272no—
269C22×C4—271no—
270C4⋊C4—91no—
271C4⋊C4—92no—
272C4⋊C4—272no—
273C4⋊C4—272no—
274C4⋊C4—272no—
275C4⋊C4—272no—
276C4⋊C4—271no—
277C22⋊C4—92no—
278C22⋊C4—272no—
279C22⋊C4—272no—
280C22⋊C4—272no—
281C22⋊C4—272no—
282C22⋊C4—272no—
283C22×C3—41no—
284C22×C3—81no—
285C22×C3—91no—
286C22×C3—91no—
287C22×C3—122no—
288C22×C3—124no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3C2×PSU(3,𝔽2)11yescharacteristic
292C2×D3—41no—
293C2×D3—92no—
294C2×D3—121no—
295C2×D3—121no—
296C2×D3—122no—
297C2×D3—122no—
298C2×D3—182no—
299C2×D3—182no—
300C2×D3—362no—
301C2×D3—364no—
302C2×D3—364no—
303C2×D3—362no—
304C2×D3—368no—
305C2×D3—361no—
306C2×D3—364no—
307C2×D3—722no—
308C2×D3—724no—
309C2×D3C2×PSU(3,𝔽2)12yes—
310C3×C4—182no—
311C3×C4—181no—
312C3×C4—182no—
313C3×C4—181no—
314Dic3—182no—
315Dic3—182no—
316Dic3—182no—
317C32—41no—
318C32—81no—
319C32#563411yescharacteristic, complemented
320C23—31no—
321C23—91no—
322C23—272no—
323C23—272no—
324C23—271no—
325C23—1081no—
326C23—1081no—
327Q8—361no—
328Q8—1081no—
329Q8—1082no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—91no—
333C2×C4—91no—
334C2×C4—272no—
335C2×C4—272no—
336C2×C4—272no—
337C2×C4—361no—
338C2×C4—1082no—
339C2×C4—1082no—
340C2×C4—1081no—
341C2×C3—21no—
342C2×C3—42no—
343C2×C3—41no—
344C2×C3—82no—
345C2×C3—81no—
346C2×C3—91no—
347C2×C3—91no—
348C2×C3—124no—
349C2×C3—181no—
350C2×C3C22×PSU(3,𝔽2)11yescharacteristic
351D3—22no—
352D3—121no—
353D3—121no—
354D3—122no—
355D3—182no—
356D3—364no—
357D3—724no—
358C22—32no—
359C22—91no—
360C22—91no—
361C22—121no—
362C22—181no—
363C22—181no—
364C22—272no—
365C22—542no—
366C22—542no—
367C22—1081no—
368C22—1082no—
369C22D3×PSU(3,𝔽2)11yescharacteristic
370C4—181no—
371C4—181no—
372C4—182no—
373C4—182no—
374C4—542no—
375C4—542no—
376C4—542no—
377C3—41no—
378C3—81no—
379C3#1059655011yescharacteristic, complemented
380C2—21no—
381C2—62no—
382C2—91no—
383C2—91no—
384C2—181no—
385C2—542no—
386C2C2×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)
hypocenterC33#543
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11724369C2semidirect
2. #11724368C2semidirect
3. C3×C22⋊PSU(3,𝔽2)C2semidirect
6. #11724151C2semidirect
7. #11724151C2semidirect
8. #11724150C2semidirect
9. #11724149C2semidirect
14. C2×C3⋊PSU(3,𝔽2)C22semidirect
15. C2×C3⋊PSU(3,𝔽2)C22semidirect
16. C2×C3×PSU(3,𝔽2)C22semidirect
21. #88946C22semidirect
22. #88946C22semidirect
23. #88945C22semidirect
24. #88945C22semidirect
25. #88945C22semidirect
26. C3×#3777C22semidirect
27. C3×#3777C22semidirect
40. C22⋊PSU(3,𝔽2)D3semidirect
41. C22×C3⋊(C3⋊D3)Q8semidirect
43. C22×C32×D3Q8semidirect
46. C2×D3×C3⋊D3Q8semidirect
71. C2×PSU(3,𝔽2)C2×D3semidirect
83. #3777C2×D3semidirect
84. #3777C2×D3semidirect
215. C33#543semidirect
230. C22×D3PSU(3,𝔽2)semidirect
319. C32#5634semidirect
379. C3#10596550semidirect

Maximal subgroups

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

subgroupindex
#117243692
#117243682
C3×C22⋊PSU(3,𝔽2)2
D3×C22⋊#1572
C2×(C2×D3)⋊#1572
#117241512
#117241502
#117241492
#105965503
#56349