Finite Group Databasebeta

← #13166096

Subgroups — #13166096

Order 1728.

460 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 50 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
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593214—31nomaximal
5Dic3×GL(2,𝔽3)—31nomaximal
6#10591571—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6294—41nomaximal
9C3×C3⋊GL(2,𝔽3)C412yescomplemented
10C2×C3⋊GL(2,𝔽3)D311yescharacteristic
11C2×C3×GL(2,𝔽3)—31no—
12C2×C3×GL(2,𝔽3)—61no—
13Q8×C3⋊Dic3D311yescharacteristic, complemented
14C2×C3×#215—31no—
15#63898—31no—
16Dic3×SL(2,𝔽3)—31no—
17Dic3×SL(2,𝔽3)—31no—
18Dic3×SL(2,𝔽3)—61no—
19C3⋊#859D311yescharacteristic, complemented
20C3×#859—31no—
21C3×#859—61no—
22C3×#835—31no—
23#63573—31no—
24#63563—31no—
25#63553—31no—
26C22×C3×C3⋊D3—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30#6294—44no—
31C4×GL(2,𝔽3)—91no—
32Dic3×SD16—91no—
33#4788—91no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C3×D12—31no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—41no—
40C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
41C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
42C2×#624—121no—
43C2×D3×Dic3—61no—
44C2×D3×Dic3—121no—
45C3⋊GL(2,𝔽3)Dic312yescomplemented
46C3×GL(2,𝔽3)—32no—
47C3×GL(2,𝔽3)—62no—
48C4⋊(C3⋊Dic3)—31no—
49C4×C3⋊Dic3—31no—
50C3×#215—32no—
51C3×#215—32no—
52C3×C4⋊Dic3—31no—
53C2×C3×C3⋊C8—31no—
54C2×C3×C3⋊C8—31no—
55(C2×D3)⋊Dic3—61no—
56C22×C33—41no—
57C2×C3×C3⋊D3—42no—
58C2×C3×C3⋊D3—42no—
59C2×C3×C3⋊D3—42no—
60C3×C3⋊Dic3—42no—
61C3×C3⋊Dic3—42no—
62C2×GL(2,𝔽3)—31no—
63C2×GL(2,𝔽3)—61no—
64C2×C3×SD16—91no—
65Q8×Dic3—31no—
66Q8×Dic3—31no—
67Q8×Dic3—61no—
68C2×#215—31no—
69D4×Dic3—91no—
70#893—91no—
71C4×SL(2,𝔽3)—91no—
72#859—31no—
73#859—61no—
74C3×#103—91no—
75#835—31no—
76#832—91no—
77#821—91no—
78#817—91no—
79C8×Dic3—91no—
80C8×Dic3—91no—
81#808—91no—
82C22×C3⋊D3—41no—
83C22×C3×D3—61no—
84C22×C3×D3—121no—
85C22×C3×D3—241no—
86C32×Q8—31no—
87C32×Q8C4×D311yescharacteristic, complemented
88C2×C32×C4—31no—
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3—41no—
91C2×C3⋊Dic3S411yescharacteristic
92C2×C3×Dic3—61no—
93C2×C3×Dic3—121no—
94C2×C3×Dic3—121no—
95C2×C3×Dic3—121no—
96C2×C3×Dic3—241no—
97C2×C3×Dic3—241no—
98C3×D12—32no—
99C3×D12—61no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—21no—
102C3×SL(2,𝔽3)—21no—
103C3×SL(2,𝔽3)—41no—
104C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
105C3×SL(2,𝔽3)C2×Dic311yescharacteristic, complemented
106#624—124no—
107D3×Dic3—124no—
108D3×Dic3—122no—
109C3×C3⋊C8—32no—
110C3×C3⋊C8—61no—
111C4×SD16—271noSylow
112C2×C33—41no—
113C2×C33—41no—
114C2×C33—41no—
115C3×C3⋊D3—44no—
116C2×C3×Q8—21no—
117C2×C3×Q8D3211yescharacteristic
118C2×C3×Q8D3211yescharacteristic
119C2×C3×D4—91no—
120C22×Dic3—181no—
121C2×D12—31no—
122C2×C4×D3—181no—
123C2×C4×D3—361no—
124C2×SL(2,𝔽3)—21no—
125C2×SL(2,𝔽3)—21no—
126C2×SL(2,𝔽3)—41no—
127C2×SL(2,𝔽3)D3211yescharacteristic
128GL(2,𝔽3)—32no—
129GL(2,𝔽3)—62no—
130C3×SD16—92no—
131C3×SD16—92no—
132C2×C3×C8—91no—
133C2×C3×C8—91no—
134C3×C4⋊C4—91no—
135#217—181no—
136#215—32no—
137#215—32no—
138(C2×D3)⋊C4—181no—
139C4⋊Dic3—31no—
140C4⋊Dic3—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—181no—
143C4×Dic3—91no—
144C4×Dic3—91no—
145C4×Dic3—181no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—31no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—81no—
151C22×C32—81no—
152C22×C32—81no—
153C22×C32—161no—
154C22×C32C2×S411yescharacteristic
155C2×C3⋊D3—42no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3—42no—
158C2×C3×D3—62no—
159C2×C3×D3—122no—
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C2×C3×D3—242no—
165C2×C3×D3—242no—
166C2×C3×D3—242no—
167C32×C4—31no—
168C32×C4—31no—
169C3⋊Dic3—42no—
170C3⋊Dic3—61no—
171C3⋊Dic3GL(2,𝔽3)12yescomplemented
172C3×Dic3—122no—
173C3×Dic3—122no—
174C3×Dic3—122no—
175C3×Dic3—121no—
176C3×Dic3—242no—
177C3×Dic3—242no—
178C2×SD16—91no—
179C4×Q8—91no—
180C4×D4—271no—
181#106—271no—
182#103—91no—
183#102—271no—
184C4×C8—271no—
185C33—41noSylow
186C23×C3—181no—
187C22×D3—61no—
188C22×D3—121no—
189C22×D3—241no—
190C3×Q8—21no—
191C3×Q8—31no—
192C3×Q8—31no—
193C3×Q8—61no—
194C3×Q8D3×Dic311yescharacteristic, complemented
195C3×Q8#62411yescharacteristic, complemented
196C3×D4—92no—
197C3×D4—181no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—61no—
201C2×C3×C4—181no—
202C2×C3×C4—361no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—91no—
208C2×Dic3—91no—
209C2×Dic3—121no—
210C2×Dic3—181no—
211C2×Dic3—241no—
212C2×Dic3—362no—
213D12—32no—
214D12—61no—
215C4×D3—362no—
216C4×D3—364no—
217SL(2,𝔽3)—21no—
218SL(2,𝔽3)—21no—
219SL(2,𝔽3)—41no—
220SL(2,𝔽3)D3×Dic311yescharacteristic, complemented
221C3×C8—92no—
222C3×C8—181no—
223C3⋊C8—32no—
224C3⋊C8—61no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—81no—
232C2×C32—81no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32—81no—
239C2×C32—81no—
240C2×C32—161no—
241C2×C32—161no—
242C2×C32—161no—
243C2×C32C4×S411yescharacteristic
244C2×C32C2×GL(2,𝔽3)11yescharacteristic
245C2×C32#98511yescharacteristic
246C3⋊D3—44no—
247C3×D3—124no—
248C3×D3—122no—
249C3×D3—244no—
250C2×Q8#111311yescharacteristic
251C2×D4—91no—
252C22×C4—541no—
253SD16—92no—
254SD16—92no—
255C2×C8—91no—
256C2×C8—91no—
257C4⋊C4—91no—
258C4⋊C4—271no—
259C22⋊C4—541no—
260C42—271no—
261C22×C3—21no—
262C22×C3—41no—
263C22×C3—81no—
264C22×C3—81no—
265C22×C3—161no—
266C22×C3—182no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3D3×S411yescharacteristic
270C22×C3D3×S411yescharacteristic
271C2×D3—62no—
272C2×D3—122no—
273C2×D3—122no—
274C2×D3—122no—
275C2×D3—121no—
276C2×D3—121no—
277C2×D3—242no—
278C2×D3—242no—
279C2×D3—242no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—362no—
287C3×C4—361no—
288Dic3—32no—
289Dic3—32no—
290Dic3—62no—
291Dic3—122no—
292Dic3—121no—
293Dic3—181no—
294Dic3—181no—
295Dic3—242no—
296Dic3—361no—
297C32—41no—
298C32—41no—
299C32—81no—
300C32—81no—
301C32—81no—
302C32—161no—
303C32C4×GL(2,𝔽3)11yescharacteristic, complemented
304C23—181no—
305Q8—31no—
306Q8#629411yescharacteristic, complemented
307D4—92no—
308D4—181no—
309C2×C4—31no—
310C2×C4—91no—
311C2×C4—181no—
312C2×C4—271no—
313C2×C4—1082no—
314C8—92no—
315C8—181no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—81no—
323C2×C3—81no—
324C2×C3—81no—
325C2×C3—81no—
326C2×C3—81no—
327C2×C3—81no—
328C2×C3—161no—
329C2×C3—161no—
330C2×C3—161no—
331C2×C3—362no—
332C2×C3#6419711yescharacteristic
333C2×C3#6419811yescharacteristic
334C2×C3D3×GL(2,𝔽3)11yescharacteristic
335C2×C3D3×GL(2,𝔽3)11yescharacteristic
336C2×C3Dic3×S411yescharacteristic
337C2×C3(C3⋊S4)⋊C411yescharacteristic
338D3—124no—
339D3—122no—
340D3—244no—
341C22—182no—
342C22—361no—
343C22—361no—
344C22#8910111yescharacteristic
345C4—31no—
346C4—31no—
347C4—92no—
348C4—361no—
349C4—541no—
350C3—21no—
351C3—41no—
352C3—81no—
353C3—81no—
354C3—161no—
355C3Dic3×GL(2,𝔽3)11yescharacteristic, complemented
356C3#1059321411yescharacteristic, complemented
357C2—362no—
358C2#1172420711yescharacteristic
359C2#1172421111yescharacteristic
360C2#1172421311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×C3⋊Dic3C2semidirect
3. C3×C3⋊#859C2semidirect
9. C3×C3⋊GL(2,𝔽3)C4semidirect
13. Q8×C3⋊Dic3D3semidirect
19. C3⋊#859D3semidirect
29. C32×SL(2,𝔽3)C2×C4semidirect
45. C3⋊GL(2,𝔽3)Dic3semidirect
87. C32×Q8C4×D3semidirect
104. C3×SL(2,𝔽3)C4×D3semidirect
105. C3×SL(2,𝔽3)C2×Dic3semidirect
171. C3⋊Dic3GL(2,𝔽3)semidirect
194. C3×Q8D3×Dic3semidirect
195. C3×Q8#624semidirect
220. SL(2,𝔽3)D3×Dic3semidirect
303. C32C4×GL(2,𝔽3)semidirect
306. Q8#6294semidirect
355. C3Dic3×GL(2,𝔽3)semidirect
356. C3#10593214semidirect

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
SL(2,𝔽3)×C3⋊Dic32
C3×C3⋊#8592
#105932143
Dic3×GL(2,𝔽3)3
#105915713
C2×#62944