Finite Group Databasebeta

← #13159679

Subgroups — #13159679

Order 1728.

864 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 192 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×#3744C211yesmaximal, characteristic
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic, complemented
4#11722188C211yesmaximal, characteristic, complemented
5C32×#815C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic
8#10592409—31nomaximal
9#10590766—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×#3744C414yescomplemented
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20#88803—21no—
21Dic3×C3⋊Dic3—21no—
22C2×#3744D311yescharacteristic
23C4×C4⋊(C3⋊D3)—31no—
24C4×#634—31no—
25C2×C3×#204—34no—
26Dic3×D12—34no—
27Dic3×Dic6—34no—
28C32×#97—31no—
29#63648—31no—
30#63645—31no—
31#63645—34no—
32#63645—64no—
33#63643—31no—
34C3×#815D314yescomplemented
35#63566—34no—
36#63563—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)C2×C412yes—
44C3×#634—21no—
45C3×#634C2×C412yescomplemented
46C2×C32×Dic3D411yescharacteristic
47C2×C32×Dic3D411yescharacteristic
48Dic3×C3⋊D3—42no—
49C33×C8C2×C412yescomplemented
50C32×C3⋊C8—21no—
51#5164—91no—
52#4710—94no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3—61no—
55C2×#634C2×D311yescharacteristic
56C2×C3×D12—34no—
57C2×C3×Dic6—34no—
58C2×D3×Dic3—64no—
59C2×C32×C8—24no—
60C2×C32×C8—31no—
61C2×C32×C8C2×D314yes—
62C2×C32×C8C2×D311yescharacteristic
63C32×C42—31no—
64#3752—61no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—64no—
70C4⋊(C3⋊Dic3)—64no—
71#3750—61no—
72C4×C3⋊Dic3—61no—
73#3744Dic314yescomplemented
74C3×C4×Dic3C2×D314yes—
75C2×C3×C3⋊C8C2×D314yes—
76C3×#204—316no—
77Dic3⋊Dic3—64no—
78(C2×D3)⋊Dic3—64no—
79Dic32—64no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—41no—
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—22no—
89C32×Dic3—21no—
90C32×Dic3—21no—
91C2×C3×SD16—91no—
92Q8×Dic3—91no—
93D4×Dic3—91no—
94C2×#204—34no—
95C4×D12—94no—
96C4×Dic6—94no—
97C3×#97—34no—
98#835—91no—
99#832—91no—
100#821—94no—
101#818—94no—
102#818—91no—
103#818—184no—
104#816—94no—
105#815C3⋊D311yescharacteristic, complemented
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C2×C32×C4—24no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119C2×C3⋊Dic3—124no—
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)C2×Dic312yes—
122C4×C3⋊D3—122no—
123#634—21no—
124#634C2×Dic312yescomplemented
125C2×C3×Dic3—64no—
126C2×C3×Dic3D1214yes—
127C2×C3×Dic3D1214yes—
128C3×D12—38no—
129C3×D12—64no—
130C3×Dic6—38no—
131C3×Dic6—64no—
132D3×Dic3—128no—
133C32×C8—28no—
134C32×C8—61no—
135C32×C8C4×D318yescomplemented
136C32×C8C2×Dic312yescomplemented
137C3×C3⋊C8—24no—
138#440—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C2×C33#13711yescharacteristic
142C3×C3⋊D3—42no—
143C2×C3×Q8—91no—
144C2×C3×D4—91no—
145C22×Dic3—181no—
146C2×D12—34no—
147C2×C4×D3—184no—
148C2×Dic6—34no—
149C3×SD16—94no—
150C2×C3×C8—24no—
151C2×C3×C8—34no—
152C2×C3×C8D3214yes—
153C2×C3×C8C2×C3⋊D311yescharacteristic
154C3×C42—34no—
155#217—181no—
156(C2×D3)⋊C4—184no—
157C4⋊Dic3—94no—
158C4⋊Dic3—94no—
159C4⋊Dic3—91no—
160C4⋊Dic3—91no—
161C4⋊Dic3—184no—
162C4⋊Dic3—184no—
163C4⋊Dic3—181no—
164Dic3⋊C4—184no—
165C4×Dic3—184no—
166C4×Dic3—181no—
167C4×Dic3C2×C3⋊D311yescharacteristic
168C2×C3⋊C8C2×C3⋊D311yescharacteristic
169#204—316no—
170C22×C32—24no—
171C22×C32C2×D1214yes—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—22no—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—41no—
176C2×C3×D3—68no—
177C2×C3×D3—124no—
178C2×C3×D3—124no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—61no—
182C32×C4—61no—
183C32×C4C2×C4×D314yes—
184C32×C4#23514yes—
185C32×C4#23711yescharacteristic
186C32×C4C22×Dic311yescharacteristic
187C3⋊Dic3—22no—
188C3⋊Dic3—121no—
189C3⋊Dic3—121no—
190C3⋊Dic3—124no—
191C3⋊Dic3—124no—
192C3⋊Dic3—244no—
193C3⋊Dic3—244no—
194C3×Dic3—24no—
195C3×Dic3—24no—
196C3×Dic3—68no—
197C2×SD16—91no—
198C4×Q8—271no—
199C4×D4—271no—
200#107—271no—
201#103—271no—
202#102—271no—
203#97—31no—
204C33#44011yescharacteristic, complemented, Sylow
205C23×C3—181no—
206C22×D3—64no—
207C3×Q8—92no—
208C3×Q8—181no—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—24no—
212C2×C3×C4—34no—
213C2×C3×C4—34no—
214C2×C3×C4—181no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—64no—
218C2×Dic3—184no—
219C2×Dic3—184no—
220C2×Dic3—181no—
221C2×Dic3—181no—
222C2×Dic3—364no—
223C2×Dic3—364no—
224C2×Dic3—362no—
225C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
226C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
227D12—38no—
228D12—64no—
229C4×D3—368no—
230Dic6—38no—
231Dic6—64no—
232C3×C8—28no—
233C3×C8—64no—
234C3×C8D3×Dic318yescomplemented
235C3×C8C4×C3⋊D312yescomplemented
236C3⋊C8—21no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32C4×D1214yes—
241C2×C32#90814yes—
242C2×C32#90914yes—
243C2×C32#91411yescharacteristic
244C2×C32#91511yescharacteristic
245C2×C32D4×Dic311yescharacteristic
246C3⋊D3—42no—
247C3×D3—128no—
248C2×Q8—91no—
249C2×D4—91no—
250C22×C4—541no—
251SD16—94no—
252C2×C8—31no—
253C2×C8D3×C3⋊D311yescharacteristic
254C4⋊C4—271no—
255C4⋊C4—271no—
256C4⋊C4—541no—
257C22⋊C4—541no—
258C42—31no—
259C42—541no—
260C22×C3—24no—
261C22×C3—182no—
262C22×C3—361no—
263C22×C3—361no—
264C22×C3D3×D1214yes—
265C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
266C2×D3—68no—
267C2×D3—124no—
268C2×D3—124no—
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—64no—
272C3×C4—64no—
273C3×C4—182no—
274C3×C4#379514yes—
275C3×C4C2×D3×Dic314yes—
276C3×C4C2×C4×C3⋊D311yescharacteristic
277C3×C4#382811yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—68no—
281Dic3—364no—
282Dic3—364no—
283Dic3—361no—
284Dic3—361no—
285Dic3—724no—
286Dic3—724no—
287C32—24no—
288C32#471014yescomplemented
289C32#516411yescharacteristic, complemented
290C23—181no—
291Q8—92no—
292Q8—181no—
293D4—92no—
294D4—181no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—181no—
298C2×C4—541no—
299C2×C4—541no—
300C2×C4—1082no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—61no—
303C8Dic3×C3⋊D312yescomplemented
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—362no—
308C2×C3#6379414yes—
309C2×C3#6380014yes—
310C2×C3Dic3×D1214yes—
311C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
312C2×C3#6411711yescharacteristic
313C2×C3#6411811yescharacteristic
314D3—128no—
315C22—182no—
316C22—361no—
317C22—361no—
318C22D3×C4⋊(C3⋊D3)11yescharacteristic
319C4—61no—
320C4—61no—
321C4—182no—
322C4—1081no—
323C4—1081no—
324C4#8902111yescharacteristic
325C4C2×Dic3×C3⋊D311yescharacteristic
326C3—24no—
327C3#1059076614yescomplemented
328C3#1059240911yescharacteristic, complemented
329C2—362no—
330C2#1172331511yescharacteristic
331C2#1172332611yescharacteristic
332C2Dic3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. Dic3×#634C2semidirect
4. #11722188C2semidirect
5. C32×#815C2semidirect
6. #11722040C2semidirect
16. C3×#3744C4semidirect
34. C3×#815D3semidirect
45. C3×#634C2×C4semidirect
49. C33×C8C2×C4semidirect
73. #3744Dic3semidirect
105. #815C3⋊D3semidirect
124. #634C2×Dic3semidirect
135. C32×C8C4×D3semidirect
136. C32×C8C2×Dic3semidirect
204. C33#440semidirect
234. C3×C8D3×Dic3semidirect
235. C3×C8C4×C3⋊D3semidirect
288. C32#4710semidirect
289. C32#5164semidirect
303. C8Dic3×C3⋊D3semidirect
327. C3#10590766semidirect
328. C3#10592409semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
Dic3×C4⋊(C3⋊D3)2
Dic3×#6342
#117221882
C32×#8152
#117220402
#117220342
#105924093
#105907663