Finite Group Databasebeta

← #13166092

Subgroups — #13166092

Order 1728.

404 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 44 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×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10593208—31nomaximal
5#10593205—31nomaximal
6#10591612—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)—21no—
9C3×C3⋊#226—21no—
10#88814—41nomaximal
11C2×C3⋊GL(2,𝔽3)D311yescharacteristic
12C2×C3⋊#226D311yescharacteristic, complemented
13C2×C3×GL(2,𝔽3)—31no—
14C2×C3×GL(2,𝔽3)—61no—
15C2×C3×#226—31no—
16C2×C3×#226—61no—
17Q8×C3⋊Dic3D311yescharacteristic, complemented
18C2×C3×#216—31no—
19C2×C3×#215—31no—
20#63863—31no—
21Dic3×SL(2,𝔽3)—31no—
22Dic3×SL(2,𝔽3)—31no—
23Dic3×SL(2,𝔽3)—61no—
24#63572—31no—
25#63566—31no—
26#63563—31no—
27C22×C3×C3⋊D3—41no—
28C2×C3×C3⋊Dic3—41no—
29C2×C3×C3⋊Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic
31#5395—91no—
32#5185—91no—
33#5166—91no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C3×D12—31no—
36C2×C3×Dic6—31no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—41no—
41C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
42C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
43C3⋊GL(2,𝔽3)—21no—
44C3⋊#226—21no—
45C3×GL(2,𝔽3)—62no—
46C3×GL(2,𝔽3)—61no—
47C3×#226—62no—
48C3×#226—61no—
49C4⋊(C3⋊Dic3)—31no—
50C4×C3⋊Dic3—31no—
51C3×#216—61no—
52C3×#216—61no—
53C3×#215—61no—
54C3×#215—61no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57#3722—121no—
58(C2×D3)⋊Dic3—61no—
59(C2×D3)⋊Dic3—61no—
60(C2×D3)⋊Dic3—121no—
61C22×C33—41no—
62C2×C3×C3⋊D3—42no—
63C2×C3×C3⋊D3—81no—
64C2×C3×C3⋊D3—81no—
65C3×C3⋊Dic3—81no—
66C3×C3⋊Dic3—81no—
67C2×GL(2,𝔽3)—31no—
68C2×GL(2,𝔽3)—61no—
69C2×#226—31no—
70C2×#226—61no—
71C2×C3×Q16—91no—
72C2×C3×SD16—91no—
73#947—91no—
74Q8×Dic3—31no—
75Q8×Dic3—31no—
76Q8×Dic3—61no—
77C2×#216—31no—
78C2×#215—31no—
79#936—91no—
80C4×SL(2,𝔽3)—91no—
81#835—91no—
82#832—91no—
83#821—91no—
84#816—91no—
85Dic3⋊C8—91no—
86Dic3⋊C8—91no—
87C22×C3⋊D3—41no—
88C22×C3×D3—61no—
89C22×C3×D3—121no—
90C22×C3×D3—241no—
91C32×Q8—31no—
92C32×Q8D1211yescharacteristic
93C2×C32×C4—31no—
94C2×C3⋊Dic3—31no—
95C2×C3⋊Dic3—41no—
96C2×C3⋊Dic3S411yescharacteristic
97C2×C3×Dic3—61no—
98C2×C3×Dic3—121no—
99C2×C3×Dic3—121no—
100C2×C3×Dic3—121no—
101C2×C3×Dic3—241no—
102C2×C3×Dic3—241no—
103C3×D12—61no—
104C3×D12—61no—
105C3×Dic6—61no—
106C3×Dic6—61no—
107C3×SL(2,𝔽3)—21no—
108C3×SL(2,𝔽3)—21no—
109C3×SL(2,𝔽3)—21no—
110C3×SL(2,𝔽3)—41no—
111C3×SL(2,𝔽3)#6711yescharacteristic
112C3×SL(2,𝔽3)#6711yescharacteristic
113C3×C3⋊C8—61no—
114C3×C3⋊C8—61no—
115#464—271noSylow
116C2×C33—41no—
117C2×C33—41no—
118C2×C33—41no—
119C3×C3⋊D3—82no—
120C2×C3×Q8—21no—
121C2×C3×Q8—91no—
122C2×C3×Q8D3211yescharacteristic
123C2×C3×Q8D3211yescharacteristic
124C2×C3×D4—91no—
125C2×D12—31no—
126C2×Dic6—31no—
127C2×SL(2,𝔽3)—21no—
128C2×SL(2,𝔽3)—21no—
129C2×SL(2,𝔽3)—41no—
130C2×SL(2,𝔽3)D3211yescharacteristic
131GL(2,𝔽3)—62no—
132GL(2,𝔽3)—61no—
133#226—62no—
134#226—61no—
135C3×Q16—181no—
136C3×Q16—181no—
137C3×SD16—181no—
138C3×SD16—181no—
139C2×C3×C8—91no—
140C2×C3×C8—91no—
141#217—181no—
142#217—181no—
143#216—61no—
144#216—61no—
145#215—61no—
146#215—61no—
147(C2×D3)⋊C4—181no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—361no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152C4⋊Dic3—181no—
153C4×Dic3—91no—
154C4×Dic3—91no—
155C4×Dic3—181no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32—81no—
161C22×C32—81no—
162C22×C32—81no—
163C22×C32—161no—
164C22×C32C2×S411yescharacteristic
165C2×C3⋊D3—42no—
166C2×C3⋊D3—81no—
167C2×C3⋊D3—81no—
168C2×C3×D3—122no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C2×C3×D3—241no—
173C2×C3×D3—241no—
174C2×C3×D3—242no—
175C2×C3×D3—242no—
176C2×C3×D3—242no—
177C32×C4—31no—
178C32×C4—31no—
179C3⋊Dic3—21no—
180C3⋊Dic3—61no—
181C3⋊Dic3—81no—
182C3×Dic3—121no—
183C3×Dic3—241no—
184C3×Dic3—241no—
185C3×Dic3—242no—
186C3×Dic3—241no—
187C3×Dic3—242no—
188C2×Q16—91no—
189C2×SD16—91no—
190#124—271no—
191C4×Q8—91no—
192C4⋊C8—271no—
193#103—271no—
194#102—271no—
195C33—41noSylow
196C23×C3—181no—
197C22×D3—61no—
198C22×D3—121no—
199C22×D3—241no—
200C3×Q8—21no—
201C3×Q8—31no—
202C3×Q8—31no—
203C3×Q8—61no—
204C3×Q8—181no—
205C3×Q8—181no—
206C3×Q8#62611yescharacteristic
207C3×Q8#62611yescharacteristic
208C3×D4—181no—
209C3×D4—181no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—61no—
213C2×C3×C4—181no—
214C2×C3×C4—361no—
215C2×Dic3—31no—
216C2×Dic3—31no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—91no—
220C2×Dic3—91no—
221C2×Dic3—121no—
222C2×Dic3—181no—
223C2×Dic3—241no—
224D12—61no—
225D12—61no—
226Dic6—61no—
227Dic6—61no—
228SL(2,𝔽3)—21no—
229SL(2,𝔽3)—21no—
230SL(2,𝔽3)—41no—
231SL(2,𝔽3)#62511yescharacteristic
232C3×C8—181no—
233C3×C8—181no—
234C3⋊C8—61no—
235C3⋊C8—61no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—81no—
243C2×C32—81no—
244C2×C32—81no—
245C2×C32—81no—
246C2×C32—81no—
247C2×C32—81no—
248C2×C32—81no—
249C2×C32—81no—
250C2×C32—81no—
251C2×C32—161no—
252C2×C32—161no—
253C2×C32—161no—
254C2×C32C4⋊S411yescharacteristic
255C2×C32#98311yescharacteristic
256C2×C32#98511yescharacteristic
257C3⋊D3—82no—
258C3×D3—242no—
259C3×D3—244no—
260C3×D3—241no—
261C2×Q8—91no—
262C2×Q8#111311yescharacteristic
263C2×D4—91no—
264Q16—181no—
265Q16—181no—
266SD16—181no—
267SD16—181no—
268C2×C8—91no—
269C2×C8—91no—
270C4⋊C4—271no—
271C22⋊C4—541no—
272C22⋊C4—541no—
273C42—271no—
274C22×C3—21no—
275C22×C3—41no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—161no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3—361no—
282C22×C3D3×S411yescharacteristic
283C22×C3D3×S411yescharacteristic
284C2×D3—122no—
285C2×D3—121no—
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—241no—
289C2×D3—241no—
290C2×D3—242no—
291C2×D3—242no—
292C2×D3—242no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—361no—
300C3×C4—721no—
301Dic3—62no—
302Dic3—61no—
303Dic3—61no—
304Dic3—121no—
305Dic3—181no—
306Dic3—181no—
307Dic3—241no—
308Dic3—242no—
309Dic3—361no—
310C32—41no—
311C32—41no—
312C32—81no—
313C32—81no—
314C32—81no—
315C32—161no—
316C32#539511yescharacteristic, complemented
317C23—181no—
318Q8—31no—
319Q8—181no—
320Q8—181no—
321Q8#629511yescharacteristic
322D4—181no—
323D4—181no—
324C2×C4—31no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—271no—
328C8—181no—
329C8—181no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3—161no—
343C2×C3—161no—
344C2×C3—161no—
345C2×C3—721no—
346C2×C3#6419711yescharacteristic
347C2×C3#6419811yescharacteristic
348C2×C3#6420111yescharacteristic
349C2×C3#6420211yescharacteristic
350C2×C3#6420711yescharacteristic
351C2×C3#6420711yescharacteristic
352D3—242no—
353D3—244no—
354D3—241no—
355C22—361no—
356C22—361no—
357C22—361no—
358C22#8910111yescharacteristic
359C4—31no—
360C4—31no—
361C4—181no—
362C4—361no—
363C4—541no—
364C3—21no—
365C3—41no—
366C3—81no—
367C3—81no—
368C3—161no—
369C3#1059320511yescharacteristic, complemented
370C3#1059320811yescharacteristic, complemented
371C2—721no—
372C2#1172420711yescharacteristic
373C2#1172421011yescharacteristic
374C2#1172421411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#226C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
12. C2×C3⋊#226D3semidirect
17. Q8×C3⋊Dic3D3semidirect
316. C32#5395semidirect
369. C3#10593205semidirect
370. C3#10593208semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊GL(2,𝔽3)2
C2×C3×C3⋊#2262
SL(2,𝔽3)×C3⋊Dic32
#105932083
#105932053
#105916123
#888144