Finite Group Databasebeta

← #13147283

Subgroups — #13147283

Order 1728.

497 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 56 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×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88623C211yesmaximal, characteristic, complemented
3#11720765C211yesmaximal, characteristic, complemented
4Q8×#638C311yesmaximal, characteristic, complemented
5SL(2,𝔽3)×#67—31nomaximal
6C2×#638⋊C3—41nomaximal
7C2×Q8⋊H(3,𝔽3)—21no—
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9#88623—21no—
10C22×C32×Q8C2×C311yescharacteristic, complemented
11C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—21no—
13C22×C3×SL(2,𝔽3)D311yescharacteristic, complemented
14C2×D3×SL(2,𝔽3)—31no—
15#64155—31no—
16Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
17#64153—31no—
18C4×#638—31no—
19#64133—31no—
20Dic3×SL(2,𝔽3)—31no—
21C23×H(3,𝔽3)—41no—
22C22×(C3⋊D3)⋊C3—41no—
23#638⋊C3—44no—
24C2×(C3⋊Dic3)⋊C3—41no—
25Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
26Q8×#67—31no—
27Q8×#67—91no—
28D4×SL(2,𝔽3)—91no—
29C2×C32×Q8—21no—
30C2×C32×Q8—31no—
31C2×C32×Q8—61no—
32C2×C32×Q8C22×C311yescharacteristic
33C22×C32×C4—31no—
34C2×#638A411yescharacteristic
35Q8×C3⋊D3—21no—
36Q8×C3⋊D3—61no—
37C2×C4×C3⋊D3—31no—
38C2×#634—31no—
39C2×C3×#67—121no—
40C2×C3×SL(2,𝔽3)—22no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
44D3×SL(2,𝔽3)—61no—
45#3757—31no—
46#3752—31no—
47C4⋊(C3⋊Dic3)—31no—
48#3750—32no—
49#3750—31no—
50C4×C3⋊Dic3—31no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)—42no—
53C22×H(3,𝔽3)—81no—
54C22×H(3,𝔽3)—81no—
55C2×(C3⋊D3)⋊C3—42no—
56C2×(C3⋊D3)⋊C3—81no—
57C2×(C3⋊D3)⋊C3—81no—
58(C3⋊Dic3)⋊C3—42no—
59C22×C3×Q8—31no—
60C22×C3×Q8C3×D311yescharacteristic
61C2×D3×Q8—31no—
62C2×D3×Q8—91no—
63C22×SL(2,𝔽3)—31no—
64C22×SL(2,𝔽3)—61no—
65C22×SL(2,𝔽3)—91no—
66#946—93no—
67#946—91no—
68Q8×Dic3—31no—
69Q8×Dic3—91no—
70#944—93no—
71#944—91no—
72C4×#67—91no—
73C4×#67—93no—
74#924—93no—
75#924—91no—
76C4×SL(2,𝔽3)—91no—
77C23×C32—41no—
78C23×C32—81no—
79C23×C32C2×A411yescharacteristic
80C22×C3⋊D3C2×A411yescharacteristic
81C22×C3×D3—121no—
82C32×Q8—31no—
83C32×Q8—61no—
84C32×Q8—61no—
85C32×Q8C3×D411yescharacteristic, complemented
86C2×C32×C4—31no—
87C2×C32×C4—31no—
88C2×C32×C4—61no—
89C2×C32×C4—61no—
90#638SL(2,𝔽3)14yescomplemented
91C2×C3⋊Dic3—31no—
92C2×C3⋊Dic3—31no—
93C2×C3⋊Dic3C2×A411yescharacteristic
94C4×C3⋊D3—61no—
95C4×C3⋊D3—61no—
96#634—61no—
97#634—61no—
98C3×#67—124no—
99C2×C3×Dic3—121no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)#6711yescharacteristic, complemented
102D4×Q8—91noSylow
103C2×H(3,𝔽3)—41no—
104C2×H(3,𝔽3)—41no—
105C2×H(3,𝔽3)—41no—
106C2×H(3,𝔽3)—82no—
107(C3⋊D3)⋊C3—82no—
108C2×C3×Q8—21no—
109C2×C3×Q8—33no—
110C2×C3×Q8—31no—
111C2×C3×Q8—31no—
112C2×C3×Q8—61no—
113C2×C3×Q8—63no—
114C2×C3×Q8—61no—
115C2×C3×Q8C2×C3×D311yescharacteristic
116C2×C3×D4—361no—
117C22×C3×C4—33no—
118C22×C3×C4—31no—
119C2×#67—31no—
120C2×#67—91no—
121D3×Q8—61no—
122D3×Q8—181no—
123D3×Q8—183no—
124D3×Q8—181no—
125C2×C4×D3—91no—
126C2×C4×D3—93no—
127C2×Dic6—93no—
128C2×Dic6—91no—
129C2×SL(2,𝔽3)—31no—
130C2×SL(2,𝔽3)—61no—
131C2×SL(2,𝔽3)—62no—
132C2×SL(2,𝔽3)—61no—
133C2×SL(2,𝔽3)—181no—
134#217—93no—
135#217—91no—
136(C2×D3)⋊C4—91no—
137(C2×D3)⋊C4—93no—
138C4⋊Dic3—93no—
139C4⋊Dic3—91no—
140Dic3⋊C4—91no—
141Dic3⋊C4—96no—
142Dic3⋊C4—92no—
143Dic3⋊C4—93no—
144C4×Dic3—91no—
145C4×Dic3—93no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32—42no—
150C22×C32—81no—
151C22×C32—81no—
152C22×C32—82no—
153C22×C32—82no—
154C22×C32—81no—
155C22×C32—82no—
156C22×C32C2×SL(2,𝔽3)12yescomplemented
157C22×C32C22×A411yescharacteristic
158C2×C3⋊D3—21no—
159C2×C3⋊D3—21no—
160C2×C3⋊D3C2×SL(2,𝔽3)12yescomplemented
161C2×C3×D3—122no—
162C2×C3×D3—241no—
163C2×C3×D3—241no—
164C32×C4—31no—
165C32×C4—31no—
166C32×C4—61no—
167C3⋊Dic3—61no—
168C3⋊Dic3—61no—
169C3⋊Dic3C2×SL(2,𝔽3)12yescomplemented
170C3×Dic3—122no—
171C22×Q8—91no—
172C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
173C4⋊Q8—271no—
174C22⋊Q8—271no—
175C22⋊Q8—271no—
176C4×Q8—91no—
177C4×D4—271no—
178H(3,𝔽3)—41noSylow
179C23×C3—31no—
180C23×C3—121no—
181C23×C3—241no—
182C23×C3—361no—
183C23×C3D3×A411yescharacteristic
184C22×D3—31no—
185C22×D3—91no—
186C3×Q8—31no—
187C3×Q8—33no—
188C3×Q8—31no—
189C3×Q8—61no—
190C3×Q8—61no—
191C3×Q8—63no—
192C3×Q8—63no—
193C3×Q8C3×#6711yescharacteristic
194C3×D4—364no—
195C2×C3×C4—33no—
196C2×C3×C4—31no—
197C2×C3×C4—33no—
198C2×C3×C4—31no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—63no—
202C2×C3×C4—63no—
203C2×C3×C4—361no—
204#67—34no—
205#67—94no—
206C2×Dic3—31no—
207C2×Dic3—91no—
208C2×Dic3—93no—
209C2×Dic3—93no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C4×D3—181no—
213C4×D3—181no—
214C4×D3—183no—
215C4×D3—183no—
216Dic6—183no—
217Dic6—183no—
218Dic6—181no—
219Dic6—181no—
220SL(2,𝔽3)—31no—
221SL(2,𝔽3)—61no—
222C2×C32—22no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—82no—
227C2×C32—84no—
228C2×C32—81no—
229C2×C32—81no—
230C2×C32—81no—
231C2×C32D4×A411yescharacteristic
232C2×C32C22×SL(2,𝔽3)11yescharacteristic
233C2×C322−1+4⋊C311yescharacteristic
234C3⋊D3—22no—
235C3×D3—242no—
236C2×Q8—21no—
237C2×Q8—31no—
238C2×Q8—61no—
239C2×Q8—181no—
240C2×Q8—271no—
241C2×Q8—541no—
242C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
243C2×D4—91no—
244C22×C4—31no—
245C22×C4—271no—
246C4⋊C4—271no—
247C4⋊C4—271no—
248C4⋊C4—272no—
249C22⋊C4—271no—
250C22⋊C4—271no—
251C42—271no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—32no—
255C22×C3—31no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3—122no—
259C22×C3—121no—
260C22×C3—241no—
261C22×C3—241no—
262C22×C3—242no—
263C22×C3—242no—
264C22×C3—242no—
265C22×C3—241no—
266C22×C3—362no—
267C22×C3—721no—
268C22×C3—721no—
269C22×C3D3×SL(2,𝔽3)12yes—
270C22×C3C2×D3×A411yescharacteristic
271C2×D3—32no—
272C2×D3—61no—
273C2×D3—61no—
274C2×D3—92no—
275C2×D3—181no—
276C2×D3—181no—
277C3×C4—33no—
278C3×C4—33no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—63no—
282C3×C4—61no—
283C3×C4—362no—
284Dic3—32no—
285Dic3—92no—
286Dic3—183no—
287Dic3—183no—
288Dic3—181no—
289Dic3—181no—
290C32—41no—
291C32—81no—
292C32D4×SL(2,𝔽3)11yescharacteristic, complemented
293C23—91no—
294C23(C3⋊D3)⋊A411yescharacteristic
295Q8—31no—
296Q8—61no—
297Q8—61no—
298Q8—541no—
299Q8—541no—
300Q8#638⋊C311yescharacteristic, complemented
301D4—94no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—271no—
308C2×C4—271no—
309C2×C4—541no—
310C2×C4—541no—
311C2×C3—22no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—62no—
316C2×C3—121no—
317C2×C3—121no—
318C2×C3—121no—
319C2×C3—242no—
320C2×C3—244no—
321C2×C3—241no—
322C2×C3—241no—
323C2×C3—241no—
324C2×C3—722no—
325C2×C3C2×D3×SL(2,𝔽3)11yescharacteristic
326C2×C3#1007⋊C311yescharacteristic
327C2×C3A4×#6711yescharacteristic
328D3—62no—
329D3—182no—
330C22—21no—
331C22—21no—
332C22—92no—
333C22—181no—
334C22—181no—
335C22#8862312yescomplemented
336C22C2×(C3⋊D3)⋊A411yescharacteristic
337C4—31no—
338C4—31no—
339C4—61no—
340C4—92no—
341C4—541no—
342C4—541no—
343C3—31no—
344C3—121no—
345C3—241no—
346C3SL(2,𝔽3)×#6711yescharacteristic
347C2—22no—
348C2—182no—
349C2C2×#8862311yescharacteristic
350C2#1172243211yescharacteristic
351C2#638⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocleC2×C32×Q8C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC22×C32×Q8C2×C3
hypercenterC23(C3⋊D3)⋊A4
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. C2×#88623C2semidirect
3. #11720765C2semidirect
4. Q8×#638C3semidirect
10. C22×C32×Q8C2×C3semidirect
11. C2×Q8×C3⋊D3C2×C3semidirect
13. C22×C3×SL(2,𝔽3)D3semidirect
16. Q8×C3⋊Dic3C2×C3semidirect
25. Q8⋊H(3,𝔽3)D4semidirect
85. C32×Q8C3×D4semidirect
90. #638SL(2,𝔽3)semidirect
101. C3×SL(2,𝔽3)#67semidirect
156. C22×C32C2×SL(2,𝔽3)semidirect
160. C2×C3⋊D3C2×SL(2,𝔽3)semidirect
169. C3⋊Dic3C2×SL(2,𝔽3)semidirect
172. C22×Q8(C3⋊D3)⋊C3semidirect
292. C32D4×SL(2,𝔽3)semidirect
300. Q8#638⋊C3semidirect
335. C22#88623semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×#886232
#117207652
Q8×#6383
SL(2,𝔽3)×#673
C2×#638⋊C34