Finite Group Databasebeta

← #13180633

Subgroups — #13180633

Order 1728.

514 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 118 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
2#11724368C212yesmaximal
3C3×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic
4#11724261C212yesmaximal
5C2×Dic3×#157C211yesmaximal, characteristic
6#11724154C211yesmaximal, characteristic, complemented
7#11724153C211yesmaximal, characteristic, complemented
8#11724152C212yesmaximal, complemented
9#11724148C212yesmaximal, complemented
10#11724148C211yesmaximal, characteristic, complemented
11Dic3×PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
12#10596542—31nomaximal
13C22×C3⋊#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C2×C3⋊PSU(3,𝔽2)C2212yes—
16C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
17C2×C3×PSU(3,𝔽2)C2211yescharacteristic
18C2×Dic3×C3⋊D3C2211yescharacteristic
19#88994C2212yes—
20C3×C22⋊#157C2212yes—
21#88946C2212yes—
22#88946C2211yescharacteristic
23#88945C2212yes—
24#88945C2212yes—
25#88945C2212yes—
26C3×#3777C2212yes—
27C3×#3777C2211yescharacteristic
28Dic3⋊#157C2212yes—
29Dic3⋊#157C2212yes—
30Dic3×#157C2211yescharacteristic
31Dic3×#157C2212yes—
32Dic3×#157C2211yescharacteristic
33Dic3×#157C2212yes—
34Dic3×#157C2211yescharacteristic
35Dic3×#157C2211yescharacteristic
36#64286—32no—
37C2×C4×#157—31no—
38C22⋊PSU(3,𝔽2)—32no—
39C22⋊PSU(3,𝔽2)—31no—
40C22⋊PSU(3,𝔽2)D311yescharacteristic
41C4⋊PSU(3,𝔽2)—31no—
42C4⋊PSU(3,𝔽2)—31no—
43C4×PSU(3,𝔽2)—31no—
44C4×PSU(3,𝔽2)—32no—
45C4×PSU(3,𝔽2)—31no—
46#64243—32no—
47C22×C3×C3⋊D3C2311yescharacteristic
48C2×C3⋊#157—22no—
49C2×C3⋊#157C2312yes—
50C2×C3⋊#157C2312yes—
51C2×C3⋊#157C2311yescharacteristic
52C2×C3⋊#157C2311yescharacteristic
53C2×C3×#157—22no—
54C2×C3×#157C2312yes—
55C2×C3×#157C2312yes—
56C2×C3×#157C2311yescharacteristic
57C2×C3×#157C2311yescharacteristic
58C3⋊PSU(3,𝔽2)—24no—
59C3⋊PSU(3,𝔽2)—22no—
60C3×PSU(3,𝔽2)—22no—
61C2×C3⋊(C3⋊Dic3)Q811yescharacteristic, complemented
62C2×C32×Dic3Q811yescharacteristic, complemented
63Dic3×C3⋊D3Q811yescharacteristic, complemented
64Dic3×C3⋊D3Q811yescharacteristic, complemented
65Dic3×C3⋊D3C2311yescharacteristic
66Dic3×C3⋊D3C2311yescharacteristic
67#5622—91nomaximal
68C22×#157—31no—
69C22×#157C2×D311yescharacteristic
70C2×PSU(3,𝔽2)—32no—
71C2×PSU(3,𝔽2)—31no—
72C2×PSU(3,𝔽2)C2×D311yescharacteristic
73C2×C4×C3⋊D3—31no—
74C2×D3×Dic3—121no—
75C22⋊#157—32no—
76C22⋊#157C2×D312yes—
77C4⋊#157—32no—
78C4⋊#157—32no—
79C4×#157—31no—
80C4×#157—31no—
81C4×#157—31no—
82C4×#157—31no—
83C4×#157—32no—
84C4×#157—32no—
85#3777—32no—
86#3777—32no—
87#3777—32no—
88#3777—32no—
89#3777—31no—
90#3777C2×D312yes—
91#3777C2×D311yescharacteristic
92C22×C33C2×Q811yescharacteristic
93C2×C3×C3⋊D3—21no—
94C2×C3×C3⋊D3—21no—
95C2×C3×C3⋊D3C2×Q811yescharacteristic
96C2×C3×C3⋊D3C2411yescharacteristic
97C3⋊#157—22no—
98C3⋊#157—22no—
99C3⋊#157—21no—
100C3⋊#157#4112yes—
101C3×#157—22no—
102C3×#157—22no—
103C3×#157—21no—
104C3×#157#4112yes—
105C3⋊(C3⋊Dic3)C2×Q811yescharacteristic
106C3⋊(C3⋊Dic3)C2×Q811yescharacteristic
107C32×Dic3C2×Q811yescharacteristic
108C32×Dic3C2×Q811yescharacteristic
109C3×C22⋊Q8—91no—
110Q8×Dic3—91no—
111#944—91no—
112#924—91no—
113C2×C4×Dic3—91no—
114#889—92no—
115#888—91no—
116#887—91no—
117#887—92no—
118#878—92no—
119#877—92no—
120C22×C3⋊D3C22×D311yescharacteristic
121C22×C3×D3—121no—
122C2×#157—22no—
123C2×#157—31no—
124C2×#157—32no—
125C2×#157—32no—
126C2×#157—31no—
127C2×#157—62no—
128C2×#157C22×D312yes—
129C2×#157C22×D312yes—
130C2×#157C22×D311yescharacteristic
131C2×#157C22×D311yescharacteristic
132PSU(3,𝔽2)—22no—
133PSU(3,𝔽2)—64no—
134PSU(3,𝔽2)—62no—
135C2×C32×C4—31no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—121no—
138C2×C3⋊Dic3—241no—
139C4×C3⋊D3—31no—
140C4×C3⋊D3—31no—
141C4×C3⋊D3—31no—
142C4×C3⋊D3—31no—
143C2×C3×Dic3—41no—
144D3×Dic3—121no—
145D3×Dic3—121no—
146D3×Dic3—121no—
147D3×Dic3—121no—
148#533—271noSylow
149C2×C33—21no—
150C2×C33C22×Q811yescharacteristic
151C3×C3⋊D3—21no—
152C3×C3⋊D3C2×#4111yescharacteristic
153C3×C3⋊D3C2×#4111yescharacteristic
154C2×C3×Q8—91no—
155C22×C3×C4—91no—
156C22×Dic3—91no—
157C22×Dic3—91no—
158C2×C4×D3—361no—
159C2×Dic6—92no—
160C2×Dic6—91no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—92no—
163C3×C22⋊C4—92no—
164#217—92no—
165C4⋊Dic3—92no—
166C4⋊Dic3—91no—
167Dic3⋊C4—92no—
168Dic3⋊C4—92no—
169Dic3⋊C4—92no—
170Dic3⋊C4—92no—
171Dic3⋊C4—92no—
172C4×Dic3—92no—
173C4×Dic3—92no—
174C4×Dic3—91no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C4×Dic3—91no—
178C22×C32—41no—
179C22×C32—81no—
180C22×C32D3×Q811yescharacteristic
181C2×C3⋊D3—21no—
182C2×C3⋊D3—21no—
183C2×C3⋊D3D3×Q811yescharacteristic
184C2×C3⋊D3C23×D311yescharacteristic
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C2×C3×D3—122no—
188C2×C3×D3—122no—
189#157—22no—
190#157—21no—
191#157—22no—
192#157—32no—
193#157—62no—
194#157—62no—
195#157—61no—
196#157#23712yes—
197C32×C4—31no—
198C32×C4—31no—
199C3⋊Dic3—31no—
200C3⋊Dic3—31no—
201C3⋊Dic3—121no—
202C3⋊Dic3—121no—
203C3⋊Dic3—241no—
204C3⋊Dic3—241no—
205C3×Dic3—41no—
206C3×Dic3—41no—
207C4⋊Q8—271no—
208C4⋊Q8—271no—
209#125—272no—
210C22⋊Q8—91no—
211C22⋊Q8—272no—
212C22⋊Q8—271no—
213C4×Q8—272no—
214C4×Q8—271no—
215C4×Q8—271no—
216#117—272no—
217C2×C42—271no—
218C33#53311yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—121no—
221C3×Q8—182no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—92no—
225C2×C3×C4—92no—
226C2×C3×C4—121no—
227C2×C3×C4—182no—
228C2×Dic3—91no—
229C2×Dic3—91no—
230C2×Dic3—91no—
231C2×Dic3—91no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—92no—
236C2×Dic3—92no—
237C2×Dic3—182no—
238C2×Dic3—361no—
239C2×Dic3—721no—
240C2×Dic3PSU(3,𝔽2)11yescharacteristic, complemented
241C4×D3—361no—
242C4×D3—361no—
243C4×D3—361no—
244C4×D3—361no—
245Dic6—184no—
246Dic6—182no—
247C2×C32—21no—
248C2×C32—42no—
249C2×C32—41no—
250C2×C32—82no—
251C2×C32—81no—
252C2×C32C2×D3×Q811yescharacteristic
253C3⋊D3—21no—
254C3⋊D3C2×#23711yescharacteristic
255C3⋊D3D3×#4111yescharacteristic
256C3×D3—121no—
257C3×D3—121no—
258C3×D3—122no—
259C2×Q8—91no—
260C2×Q8—271no—
261C2×Q8—272no—
262C22×C4—91no—
263C22×C4—271no—
264C22×C4—271no—
265C4⋊C4—92no—
266C4⋊C4—91no—
267C4⋊C4—272no—
268C4⋊C4—272no—
269C4⋊C4—272no—
270C4⋊C4—272no—
271C4⋊C4—272no—
272C4⋊C4—272no—
273C4⋊C4—271no—
274C22⋊C4—92no—
275C22⋊C4—272no—
276C42—271no—
277C42—271no—
278C42—271no—
279C42—271no—
280C42—272no—
281C42—272no—
282C22×C3—41no—
283C22×C3—81no—
284C22×C3—91no—
285C22×C3—91no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3C2×PSU(3,𝔽2)11yescharacteristic
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—122no—
292C2×D3—122no—
293C3×C4—92no—
294C3×C4—121no—
295C3×C4—121no—
296C3×C4—182no—
297C3×C4—181no—
298C3×C4—182no—
299Dic3—92no—
300Dic3—91no—
301Dic3—91no—
302Dic3—182no—
303Dic3—182no—
304Dic3—181no—
305Dic3—361no—
306Dic3—361no—
307Dic3—721no—
308Dic3—721no—
309Dic3C2×PSU(3,𝔽2)11yescharacteristic
310Dic3C2×PSU(3,𝔽2)11yescharacteristic
311C32—41no—
312C32—81no—
313C32#562211yescharacteristic, complemented
314C23—91no—
315Q8—182no—
316Q8—542no—
317Q8—544no—
318C2×C4—31no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4—92no—
322C2×C4—92no—
323C2×C4—182no—
324C2×C4—271no—
325C2×C4—271no—
326C2×C4—271no—
327C2×C4—271no—
328C2×C4—271no—
329C2×C4—271no—
330C2×C4—271no—
331C2×C4—272no—
332C2×C4—272no—
333C2×C4—542no—
334C2×C3—21no—
335C2×C3—42no—
336C2×C3—41no—
337C2×C3—82no—
338C2×C3—81no—
339C2×C3—91no—
340C2×C3—91no—
341C2×C3—181no—
342C2×C3C22×PSU(3,𝔽2)11yescharacteristic
343D3—121no—
344D3—121no—
345D3—122no—
346C22—91no—
347C22—91no—
348C22—181no—
349C22—181no—
350C22D3×PSU(3,𝔽2)11yescharacteristic
351C4—31no—
352C4—31no—
353C4—92no—
354C4—181no—
355C4—182no—
356C4—182no—
357C4—272no—
358C4—271no—
359C4—271no—
360C4—541no—
361C4—542no—
362C4—542no—
363C3—41no—
364C3—81no—
365C3#1059654211yescharacteristic, complemented
366C2—21no—
367C2—91no—
368C2—91no—
369C2—181no—
370C2C2×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×C3×C3⋊D3C23
hypercenterC22D3×PSU(3,𝔽2)
hypocenterC33#533
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
6. #11724154C2semidirect
7. #11724153C2semidirect
8. #11724152C2semidirect
9. #11724148C2semidirect
10. #11724148C2semidirect
11. Dic3×PSU(3,𝔽2)C2semidirect
61. C2×C3⋊(C3⋊Dic3)Q8semidirect
62. C2×C32×Dic3Q8semidirect
63. Dic3×C3⋊D3Q8semidirect
64. Dic3×C3⋊D3Q8semidirect
218. C33#533semidirect
240. C2×Dic3PSU(3,𝔽2)semidirect
313. C32#5622semidirect
365. C3#10596542semidirect

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
#117242612
C2×Dic3×#1572
#117241542
#117241532
#117241522
#117241482
Dic3×PSU(3,𝔽2)2
#105965423
#56229