Finite Group Databasebeta

← #13147219

Subgroups — #13147219

Order 1728.

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

normal subgroups only

This group has 437 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×#88623C211yesmaximal, characteristic, complemented
2C2×#88622C211yesmaximal, characteristic, complemented
3C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#10596047C311yesmaximal, characteristic, complemented
5#5576⋊C3—31nomaximal
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#88623—21no—
8#88622—21no—
9#3752⋊C3—41nomaximal
10C2×#3832C2×C311yescharacteristic, complemented
11C2×Q8×C3⋊D3C2×C311yescharacteristic
12C2×D3×SL(2,𝔽3)—31no—
13C2×C3⋊(#41⋊C3)—31no—
14C32×C4×Q8C2×C311yescharacteristic, complemented
15#64106—31no—
16#64104—31no—
17#64084—31no—
18C4×C4⋊(C3⋊D3)—31no—
19C3×C4×SL(2,𝔽3)—21no—
20C3×C4×SL(2,𝔽3)D311yescharacteristic, complemented
21C22×(C3⋊D3)⋊C3—41no—
22C2×C4×H(3,𝔽3)—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
25#5576—31no—
26#5576—91no—
27#548⋊C3—91no—
28C2×C32×Q8C22×C311yescharacteristic
29#3832—21no—
30#3832—61no—
31Q8×C3⋊D3—21no—
32Q8×C3⋊D3—61no—
33C2×C4⋊(C3⋊D3)—31no—
34C2×C4⋊(C3⋊D3)—31no—
35C2×C4×C3⋊D3—31no—
36C2×C4×C3⋊D3—31no—
37C2×#634—31no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
40D3×SL(2,𝔽3)—61no—
41C3⋊(#41⋊C3)—61no—
42C32×C4⋊C4—31no—
43C32×C42—31no—
44#3752—32no—
45#3752—31no—
46#3752A411yescharacteristic
47C4⋊(C3⋊Dic3)—31no—
48C3×(C2×D3)⋊C4—121no—
49C22×H(3,𝔽3)—41no—
50C2×(C3⋊D3)⋊C3—42no—
51C2×(C3⋊D3)⋊C3—81no—
52C2×(C3⋊D3)⋊C3—81no—
53C4×H(3,𝔽3)—81no—
54(C3⋊Dic3)⋊C3—81no—
55C2×#239—31no—
56C2×#239—91no—
57C2×D3×Q8—31no—
58C2×D3×Q8—91no—
59C2×#41⋊C3—91no—
60C22×SL(2,𝔽3)—91no—
61C3×C4×Q8—31no—
62C3×C4×Q8C3×D311yescharacteristic
63#897—93no—
64#897—91no—
65#895—91no—
66#895—93no—
67#875—93no—
68#875—91no—
69C4×D12—91no—
70C4×D12—93no—
71C4×SL(2,𝔽3)—31no—
72C4×SL(2,𝔽3)—61no—
73C22×C3⋊D3—31no—
74C22×C3⋊D3C2×A411yescharacteristic
75C22×C3×D3—121no—
76C32×Q8—31no—
77C32×Q8C3×D411yescharacteristic
78C2×C32×C4—31no—
79C2×C32×C4—31no—
80C2×C32×C4—41no—
81C2×C32×C4—81no—
82C2×C32×C4C2×A411yescharacteristic
83C2×C3⋊Dic3—31no—
84C2×C3⋊Dic3C2×A411yescharacteristic
85C4⋊(C3⋊D3)—61no—
86C4⋊(C3⋊D3)—61no—
87C4⋊(C3⋊D3)—62no—
88C4×C3⋊D3—61no—
89C4×C3⋊D3—61no—
90C4×C3⋊D3—61no—
91C4×C3⋊D3—61no—
92#634—61no—
93#634—61no—
94C2×C3×Dic3—121no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)D1211yescharacteristic, complemented
97#548—91noSylow
98C2×H(3,𝔽3)—41no—
99C2×H(3,𝔽3)—41no—
100C2×H(3,𝔽3)—41no—
101(C3⋊D3)⋊C3—82no—
102C2×C3×Q8—31no—
103C2×C3×Q8C2×C3×D311yescharacteristic
104#239—61no—
105#239—181no—
106#239—183no—
107#239—181no—
108D3×Q8—61no—
109D3×Q8—181no—
110D3×Q8—183no—
111D3×Q8—181no—
112C2×D12—91no—
113C2×D12—91no—
114C2×D12—93no—
115C2×D12—93no—
116C2×C4×D3—91no—
117C2×C4×D3—91no—
118C2×C4×D3—93no—
119C2×C4×D3—93no—
120C2×Dic6—93no—
121C2×Dic6—91no—
122#41⋊C3—181no—
123C2×SL(2,𝔽3)—31no—
124C2×SL(2,𝔽3)—61no—
125C2×SL(2,𝔽3)—181no—
126C3×C4⋊C4—31no—
127C3×C4⋊C4—33no—
128C3×C22⋊C4—361no—
129C3×C42—33no—
130C3×C42—31no—
131(C2×D3)⋊C4—31no—
132(C2×D3)⋊C4—91no—
133(C2×D3)⋊C4—93no—
134(C2×D3)⋊C4—96no—
135(C2×D3)⋊C4—92no—
136(C2×D3)⋊C4—91no—
137C4⋊Dic3—91no—
138C4⋊Dic3—93no—
139C22×C32—41no—
140C22×C32—81no—
141C22×C32C22×A411yescharacteristic
142C2×C3⋊D3—21no—
143C2×C3⋊D3—21no—
144C2×C3⋊D3—61no—
145C2×C3⋊D3—61no—
146C2×C3⋊D3—61no—
147C2×C3⋊D3#41⋊C312yescomplemented
148C2×C3×D3—122no—
149C2×C3×D3—241no—
150C2×C3×D3—241no—
151C32×C4—21no—
152C32×C4—31no—
153C32×C4—31no—
154C32×C4—61no—
155C32×C4—81no—
156C32×C4—82no—
157C3⋊Dic3—21no—
158C3⋊Dic3—61no—
159C3×Dic3—241no—
160C2×#41—91no—
161C22×Q8—91no—
162#124—271no—
163C22⋊Q8—271no—
164#121—271no—
165C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
166C4×D4—271no—
167H(3,𝔽3)—41noSylow
168C23×C3—361no—
169C22×D3—31no—
170C22×D3—91no—
171C22×D3—93no—
172C22×D3—91no—
173C3×Q8—31no—
174C3×Q8—33no—
175C3×Q8—31no—
176C3×Q8C3×D1211yescharacteristic
177C2×C3×C4—33no—
178C2×C3×C4—31no—
179C2×C3×C4—33no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—121no—
183C2×C3×C4—241no—
184C2×C3×C4—361no—
185C2×C3×C4D3×A411yescharacteristic
186C2×Dic3—31no—
187C2×Dic3—91no—
188C2×Dic3—93no—
189C2×Dic3—91no—
190D12—181no—
191D12—181no—
192D12—183no—
193D12—183no—
194D12—182no—
195D12—186no—
196C4×D3—181no—
197C4×D3—181no—
198C4×D3—181no—
199C4×D3—181no—
200C4×D3—183no—
201C4×D3—183no—
202C4×D3—183no—
203C4×D3—183no—
204Dic6—183no—
205Dic6—183no—
206Dic6—181no—
207Dic6—181no—
208SL(2,𝔽3)—31no—
209SL(2,𝔽3)—61no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—81no—
214C2×C32—81no—
215C2×C32—81no—
216C2×C32D4×A411yescharacteristic
217C2×C32C2×#41⋊C311yescharacteristic
218C2×C322−1+4⋊C311yescharacteristic
219C3⋊D3—22no—
220C3⋊D3—121no—
221C3×D3—242no—
222#41—181no—
223#41—541no—
224C2×Q8—181no—
225C2×Q8—271no—
226C2×Q8—541no—
227C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
228C2×D4—271no—
229C2×D4—271no—
230C22×C4—271no—
231C22×C4—271no—
232C4⋊C4—31no—
233C4⋊C4—271no—
234C22⋊C4—91no—
235C22⋊C4—272no—
236C22⋊C4—271no—
237C42—31no—
238C22×C3—31no—
239C22×C3—121no—
240C22×C3—241no—
241C22×C3—362no—
242C22×C3—721no—
243C22×C3—721no—
244C22×C3C2×D3×A411yescharacteristic
245C2×D3—32no—
246C2×D3—61no—
247C2×D3—61no—
248C2×D3—92no—
249C2×D3—181no—
250C2×D3—181no—
251C2×D3—183no—
252C2×D3—183no—
253C2×D3—183no—
254C2×D3—181no—
255C2×D3—181no—
256C2×D3—181no—
257C3×C4—21no—
258C3×C4—33no—
259C3×C4—33no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—61no—
263C3×C4—63no—
264C3×C4—61no—
265C3×C4—241no—
266C3×C4—242no—
267C3×C4—721no—
268Dic3—61no—
269Dic3—181no—
270Dic3—183no—
271Dic3—181no—
272C32—41no—
273C32—81no—
274C32#548⋊C311yescharacteristic, complemented
275C23—91no—
276C23—271no—
277Q8—31no—
278Q8—541no—
279Q8—541no—
280Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
281D4—541no—
282D4—541no—
283D4—542no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—91no—
287C2×C4—271no—
288C2×C4—541no—
289C2×C4—541no—
290C2×C4—541no—
291C2×C4—541no—
292C2×C4(C3⋊D3)⋊A411yescharacteristic
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—121no—
297C2×C3—121no—
298C2×C3—121no—
299C2×C3—241no—
300C2×C3—241no—
301C2×C3—241no—
302C2×C3—722no—
303C2×C3A4×D1211yescharacteristic
304C2×C3#1007⋊C311yescharacteristic
305C2×C3D3×#41⋊C311yescharacteristic
306D3—62no—
307D3—182no—
308D3—363no—
309D3—361no—
310C22—92no—
311C22—181no—
312C22—181no—
313C22—541no—
314C22—541no—
315C22—541no—
316C22C2×(C3⋊D3)⋊A411yescharacteristic
317C4—21no—
318C4—31no—
319C4—31no—
320C4—61no—
321C4—181no—
322C4—541no—
323C3—31no—
324C3—121no—
325C3—241no—
326C3#5576⋊C311yescharacteristic
327C2—182no—
328C2—1081no—
329C2C4⋊((C3⋊D3)⋊A4)11yescharacteristic
330C2#1172243211yescharacteristic
331C2#1172243711yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocleC2×C32×Q8C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC32×C4×Q8C2×C3
hypercenterC2×C4(C3⋊D3)⋊A4
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88623C2semidirect
2. C2×#88622C2semidirect
3. C4×Q8⋊H(3,𝔽3)C2semidirect
4. #10596047C3semidirect
10. C2×#3832C2×C3semidirect
14. C32×C4×Q8C2×C3semidirect
20. C3×C4×SL(2,𝔽3)D3semidirect
24. Q8⋊H(3,𝔽3)D4semidirect
96. C3×SL(2,𝔽3)D12semidirect
147. C2×C3⋊D3#41⋊C3semidirect
165. C4×Q8(C3⋊D3)⋊C3semidirect
274. C32#548⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#886232
C2×#886222
C4×Q8⋊H(3,𝔽3)2
#105960473
#5576⋊C33
#3752⋊C34