Finite Group Databasebeta

← #13147295

Subgroups — #13147295

Order 1728.

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

normal subgroups only

This group has 441 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88623C211yesmaximal, characteristic, complemented
3#11720765C211yesmaximal, characteristic, complemented
4#10596302C311yesmaximal, characteristic, complemented
5#5831⋊C3—31nomaximal
6#41⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8#88623—21no—
9#3752⋊C3—41nomaximal
10C2×C32×#41C2×C311yescharacteristic, complemented
11C2×Q8×C3⋊D3C2×C311yescharacteristic
12C2×C3×#41⋊C3—21no—
13C2×C3×#41⋊C3D311yescharacteristic, complemented
14C2×D3×SL(2,𝔽3)—31no—
15Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
16#64147—31no—
17#64145—31no—
18C4×#638—31no—
19#64133—31no—
20Dic3×SL(2,𝔽3)—31no—
21C22×(C3⋊D3)⋊C3—41no—
22C2×C4×H(3,𝔽3)—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
25#5831—31no—
26#5831—91no—
27#548⋊C3—91no—
28C32×#41—21no—
29C32×#41—61no—
30C2×C32×Q8C22×C311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×#638—31no—
34Q8×C3⋊D3—21no—
35Q8×C3⋊D3—61no—
36C2×C4×C3⋊D3—31no—
37C2×#634—31no—
38C3×#41⋊C3—22no—
39C3×#41⋊C3—21no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
42D3×SL(2,𝔽3)—61no—
43#3757—32no—
44#3757—31no—
45#3752A411yescharacteristic
46C4⋊(C3⋊Dic3)—31no—
47#3750—31no—
48C4×C3⋊Dic3—31no—
49C3×(C2×D3)⋊C4—121no—
50C22×H(3,𝔽3)—41no—
51C2×(C3⋊D3)⋊C3—42no—
52C2×(C3⋊D3)⋊C3—81no—
53C2×(C3⋊D3)⋊C3—81no—
54C4×H(3,𝔽3)—81no—
55(C3⋊Dic3)⋊C3—81no—
56C2×C3×#41—31no—
57C2×C3×#41C3×D311yescharacteristic
58C2×D3×Q8—31no—
59C2×D3×Q8—91no—
60C2×#41⋊C3—31no—
61C2×#41⋊C3—61no—
62C22×SL(2,𝔽3)—91no—
63Q8×Dic3—31no—
64Q8×Dic3—91no—
65#938—93no—
66#938—91no—
67#936—93no—
68#936—91no—
69C4×#67—91no—
70C4×#67—93no—
71#924—93no—
72#924—91no—
73C4×SL(2,𝔽3)—91no—
74C23×C32—31no—
75C22×C3⋊D3C2×A411yescharacteristic
76C22×C3×D3—121no—
77C32×Q8—31no—
78C32×Q8C3×D411yescharacteristic
79C32×D4—61no—
80C32×D4—61no—
81C2×C32×C4—31no—
82C2×C32×C4—41no—
83C2×C32×C4—61no—
84C2×C32×C4—61no—
85C2×C32×C4—81no—
86C2×C32×C4C2×A411yescharacteristic
87#638—62no—
88C2×C3⋊Dic3—31no—
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3C2×A411yescharacteristic
91C4×C3⋊D3—61no—
92C4×C3⋊D3—61no—
93#634—61no—
94#634—61no—
95C2×C3×Dic3—121no—
96C3×SL(2,𝔽3)—21no—
97C3×SL(2,𝔽3)#6711yescharacteristic, complemented
98#548—91noSylow
99C2×H(3,𝔽3)—41no—
100C2×H(3,𝔽3)—41no—
101C2×H(3,𝔽3)—41no—
102(C3⋊D3)⋊C3—82no—
103C3×#41—21no—
104C3×#41—61no—
105C3×#41—63no—
106C3×#41—61no—
107C2×C3×Q8—31no—
108C2×C3×Q8C2×C3×D311yescharacteristic
109C2×C3×D4—31no—
110C2×C3×D4—33no—
111C22×C3×C4—33no—
112C22×C3×C4—31no—
113C2×#67—91no—
114C2×#67—93no—
115D3×Q8—61no—
116D3×Q8—181no—
117D3×Q8—183no—
118D3×Q8—181no—
119C2×C4×D3—91no—
120C2×C4×D3—93no—
121C2×Dic6—93no—
122C2×Dic6—91no—
123#41⋊C3—61no—
124#41⋊C3—62no—
125C2×SL(2,𝔽3)—31no—
126C2×SL(2,𝔽3)—61no—
127C2×SL(2,𝔽3)—181no—
128C3×C22⋊C4—361no—
129#217—91no—
130#217—96no—
131#217—92no—
132#217—93no—
133(C2×D3)⋊C4—31no—
134(C2×D3)⋊C4—91no—
135C4⋊Dic3—93no—
136C4⋊Dic3—91no—
137Dic3⋊C4—93no—
138Dic3⋊C4—91no—
139C4×Dic3—91no—
140C4×Dic3—93no—
141C22×C32—41no—
142C22×C32—61no—
143C22×C32—61no—
144C22×C32—61no—
145C22×C32—81no—
146C22×C32C22×A411yescharacteristic
147C2×C3⋊D3—21no—
148C2×C3⋊D3—21no—
149C2×C3⋊D3#41⋊C312yescomplemented
150C2×C3×D3—122no—
151C2×C3×D3—241no—
152C2×C3×D3—241no—
153C32×C4—21no—
154C32×C4—31no—
155C32×C4—31no—
156C32×C4—81no—
157C32×C4—82no—
158C3⋊Dic3—21no—
159C3⋊Dic3—61no—
160C3⋊Dic3—61no—
161C3×Dic3—241no—
162C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
163C22×Q8—91no—
164#124—271no—
165C22⋊Q8—271no—
166#121—271no—
167C4×Q8—91no—
168C4×D4—271no—
169H(3,𝔽3)—41noSylow
170C23×C3—33no—
171C23×C3—31no—
172C23×C3—361no—
173C22×D3—31no—
174C22×D3—91no—
175C3×Q8—31no—
176C3×Q8—33no—
177C3×Q8—31no—
178C3×Q8C3×#6711yescharacteristic
179C3×D4—61no—
180C3×D4—61no—
181C3×D4—63no—
182C3×D4—63no—
183C2×C3×C4—33no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—63no—
189C2×C3×C4—63no—
190C2×C3×C4—121no—
191C2×C3×C4—241no—
192C2×C3×C4—361no—
193C2×C3×C4D3×A411yescharacteristic
194#67—182no—
195#67—186no—
196C2×Dic3—31no—
197C2×Dic3—91no—
198C2×Dic3—93no—
199C2×Dic3—93no—
200C2×Dic3—91no—
201C2×Dic3—91no—
202C4×D3—181no—
203C4×D3—181no—
204C4×D3—183no—
205C4×D3—183no—
206Dic6—183no—
207Dic6—183no—
208Dic6—181no—
209Dic6—181no—
210SL(2,𝔽3)—31no—
211SL(2,𝔽3)—61no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—81no—
216C2×C32—81no—
217C2×C32—81no—
218C2×C32—121no—
219C2×C32D4×A411yescharacteristic
220C2×C32C2×#41⋊C311yescharacteristic
221C2×C322−1+4⋊C311yescharacteristic
222C3⋊D3—22no—
223C3×D3—242no—
224#41—21no—
225#41—61no—
226C2×Q8—181no—
227C2×Q8—271no—
228C2×Q8—541no—
229C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
230C2×D4—31no—
231C2×D4—271no—
232C22×C4—31no—
233C22×C4—271no—
234C4⋊C4—271no—
235C4⋊C4—271no—
236C22⋊C4—91no—
237C22⋊C4—271no—
238C22⋊C4—272no—
239C42—271no—
240C22×C3—31no—
241C22×C3—63no—
242C22×C3—63no—
243C22×C3—63no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—121no—
248C22×C3—241no—
249C22×C3—362no—
250C22×C3—721no—
251C22×C3—721no—
252C22×C3C2×D3×A411yescharacteristic
253C2×D3—32no—
254C2×D3—61no—
255C2×D3—61no—
256C2×D3—92no—
257C2×D3—181no—
258C2×D3—181no—
259C3×C4—21no—
260C3×C4—33no—
261C3×C4—33no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—61no—
265C3×C4—241no—
266C3×C4—242no—
267C3×C4—721no—
268Dic3—61no—
269Dic3—181no—
270Dic3—183no—
271Dic3—183no—
272Dic3—181no—
273Dic3—181no—
274C32—41no—
275C32—81no—
276C32#548⋊C311yescharacteristic, complemented
277C23—31no—
278C23—91no—
279Q8—31no—
280Q8—541no—
281Q8—541no—
282Q8#638⋊C311yescharacteristic
283D4—61no—
284D4—61no—
285D4—542no—
286C2×C4—31no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—91no—
290C2×C4—271no—
291C2×C4—271no—
292C2×C4—541no—
293C2×C4—541no—
294C2×C4(C3⋊D3)⋊A411yescharacteristic
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—123no—
299C2×C3—121no—
300C2×C3—121no—
301C2×C3—121no—
302C2×C3—121no—
303C2×C3—241no—
304C2×C3—241no—
305C2×C3—241no—
306C2×C3—722no—
307C2×C3D3×#41⋊C311yescharacteristic
308C2×C3#1010⋊C311yescharacteristic
309C2×C3A4×#6711yescharacteristic
310D3—62no—
311D3—182no—
312C22—61no—
313C22—61no—
314C22—61no—
315C22—92no—
316C22—181no—
317C22—181no—
318C22C2×(C3⋊D3)⋊A411yescharacteristic
319C4—21no—
320C4—31no—
321C4—31no—
322C4—181no—
323C4—541no—
324C4—541no—
325C3—31no—
326C3—121no—
327C3—241no—
328C3#5831⋊C311yescharacteristic
329C2—121no—
330C2—182no—
331C2#1172243711yescharacteristic
332C2#1172243811yescharacteristic
333C2#638⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocleC2×C32×Q8C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC2×C32×#41C2×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×#41⋊H(3,𝔽3)C2semidirect
2. C2×#88623C2semidirect
3. #11720765C2semidirect
4. #10596302C3semidirect
10. C2×C32×#41C2×C3semidirect
13. C2×C3×#41⋊C3D3semidirect
15. Q8×C3⋊Dic3C2×C3semidirect
24. Q8⋊H(3,𝔽3)D4semidirect
97. C3×SL(2,𝔽3)#67semidirect
149. C2×C3⋊D3#41⋊C3semidirect
162. C2×#41(C3⋊D3)⋊C3semidirect
276. C32#548⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
C2×#886232
#117207652
#105963023
#5831⋊C33
#3752⋊C34