Finite Group Databasebeta

← #13160917

Subgroups — #13160917

Order 1728.

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

normal subgroups only

This group has 992 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×#3753C211yesmaximal, characteristic
2C32×D4×Dic3C211yesmaximal, characteristic, complemented
3Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4#11722202C211yesmaximal, characteristic, complemented
5#11722054C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10592713—31nomaximal
9#10591430—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C32×Dic3—21no—
13C2×Dic3×C3⋊D3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3753C414yescomplemented
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21#88803—21no—
22C32×C4×D4—31no—
23C2×#3753D311yescharacteristic
24C4×C4⋊(C3⋊D3)—31no—
25C3×D4×Dic3D314yescomplemented
26C2×C3×#213—34no—
27Dic3×D12—34no—
28#63659—31no—
29#63659—34no—
30#63659—64no—
31#63636—31no—
32#63635—31no—
33#63630—31no—
34#63573—34no—
35#63562—34no—
36#63554—34no—
37C23×C33—21no—
38C22×C3×C3⋊D3—21no—
39C33×D4—21no—
40C33×D4C2×C412yescomplemented
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yescomplemented
45C2×C32×Dic3—21no—
46C2×C32×Dic3—42no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49Dic3×C3⋊D3—42no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3×C3⋊(C3⋊C8)C2×C412yescomplemented
52#5152—91no—
53#5014—94no—
54C2×C32×D4—24no—
55C2×C32×D4C2×D314yes—
56C2×C32×D4C2×D311yescharacteristic
57C22×C32×C4—61no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C4×C3⋊D3—61no—
60C22×C3×Dic3—24no—
61C2×C3×D12—34no—
62C2×D3×Dic3—64no—
63C32×C4⋊C4—31no—
64C32×C22⋊C4—61no—
65C32×C42—31no—
66#3753Dic314yescomplemented
67#3752—61no—
68C4⋊(C3⋊Dic3)—31no—
69C4⋊(C3⋊Dic3)—34no—
70C4⋊(C3⋊Dic3)—64no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×#217—24no—
76C3×#213—316no—
77C3×C4⋊Dic3C2×D314yes—
78C3×C4×Dic3C2×D314yes—
79C2×C3×C3⋊C8—34no—
80(C2×D3)⋊Dic3—64no—
81C22×C33—22no—
82C22×C33—41no—
83C22×C33—41no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C33×C4C22×C411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C32×Dic3—21no—
92C32×Dic3—21no—
93C32×Dic3—41no—
94C2×C3×D8—91no—
95C3×C4×D4—34no—
96D4×Dic3—91no—
97D4×Dic3C3⋊D311yescharacteristic, complemented
98C2×#213—34no—
99C4×D12—94no—
100#832—94no—
101#832—91no—
102#832—91no—
103#832—184no—
104#817—91no—
105#815—91no—
106#809—94no—
107#808—94no—
108#803—94no—
109C23×C32—21no—
110C23×C32—24no—
111C23×C32—44no—
112C22×C3⋊D3—21no—
113C22×C3×D3—64no—
114C32×D4—21no—
115C32×D4—28no—
116C32×D4—24no—
117C32×D4—44no—
118C32×D4C4×D318yescomplemented
119C32×D4C2×Dic312yescomplemented
120C2×C32×C4—24no—
121C2×C32×C4—31no—
122C2×C32×C4—31no—
123C2×C32×C4—61no—
124C2×C32×C4—122no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—124no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)C2×Dic312yescomplemented
132C4×C3⋊D3—122no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—48no—
135C2×C3×Dic3#6714yes—
136C2×C3×Dic3#6714yes—
137C3×D12—38no—
138C3×D12—64no—
139D3×Dic3—128no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—124no—
143C3⋊(C3⋊C8)C2×Dic312yescomplemented
144C3×C3⋊C8—38no—
145#442—271noSylow
146C2×C33—42no—
147C2×C33C4×D411yescharacteristic
148C2×C33AGL(1,ℤ/8)11yescharacteristic
149C2×C33AGL(1,ℤ/8)11yescharacteristic
150C3×C3⋊D3—42no—
151C2×C3×D4—24no—
152C2×C3×D4—91no—
153C2×C3×D4D3214yes—
154C2×C3×D4C2×C3⋊D311yescharacteristic
155C22×C3×C4—64no—
156C22×Dic3—21no—
157C22×Dic3—181no—
158C2×D12—34no—
159C2×C4×D3—184no—
160C3×D8—94no—
161C2×C3×C8—91no—
162C3×C4⋊C4—34no—
163C3×C22⋊C4—64no—
164C3×C42—34no—
165#217—21no—
166#217—181no—
167#213—316no—
168(C2×D3)⋊C4—184no—
169C4⋊Dic3—94no—
170C4⋊Dic3—91no—
171C4⋊Dic3—184no—
172C4⋊Dic3C2×C3⋊D311yescharacteristic
173C4×Dic3C2×C3⋊D311yescharacteristic
174C2×C3⋊C8—34no—
175C2×C3⋊C8—94no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—184no—
178C22×C32—22no—
179C22×C32—24no—
180C22×C32—28no—
181C22×C32—48no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—44no—
185C22×C32—44no—
186C22×C32—84no—
187C22×C32—84no—
188C22×C32D3×D411yescharacteristic
189C22×C32C2×#6714yes—
190C2×C3⋊D3—22no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3×D3—68no—
194C2×C3×D3—124no—
195C2×C3×D3—124no—
196C32×C4—24no—
197C32×C4—24no—
198C32×C4—61no—
199C32×C4—61no—
200C32×C4—121no—
201C32×C4C2×C4×D314yes—
202C32×C4#23514yes—
203C32×C4#23711yescharacteristic
204C32×C4C22×Dic311yescharacteristic
205C3⋊Dic3—121no—
206C3⋊Dic3—124no—
207C3⋊Dic3—244no—
208C3×Dic3—24no—
209C3×Dic3—24no—
210C3×Dic3—44no—
211C2×D8—91no—
212C4×D4—31no—
213C4×D4—271no—
214#106—271no—
215#102—271no—
216#102—271no—
217#97—271no—
218C33#44211yescharacteristic, complemented, Sylow
219C23×C3—24no—
220C23×C3—21no—
221C23×C3—44no—
222C23×C3—181no—
223C22×D3—64no—
224C3×D4—24no—
225C3×D4—21no—
226C3×D4—28no—
227C3×D4—44no—
228C3×D4—92no—
229C3×D4—181no—
230C3×D4D3×Dic318yescomplemented
231C3×D4C4×C3⋊D312yescomplemented
232C2×C3×C4—24no—
233C2×C3×C4—34no—
234C2×C3×C4—34no—
235C2×C3×C4—64no—
236C2×C3×C4—128no—
237C2×C3×C4C2×D3214yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—42no—
241C2×Dic3—184no—
242C2×Dic3—181no—
243C2×Dic3—364no—
244C2×Dic3—362no—
245C2×Dic3#63811yescharacteristic
246C2×Dic3#63811yescharacteristic
247D12—38no—
248D12—64no—
249C4×D3—368no—
250C3×C8—92no—
251C3⋊C8—38no—
252C3⋊C8—184no—
253C3⋊C8—181no—
254C3⋊C8—364no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—42no—
259C2×C32—48no—
260C2×C32—88no—
261C2×C32#91111yescharacteristic
262C2×C32#91111yescharacteristic
263C2×C32C4×#6714yes—
264C2×C32#93214yes—
265C2×C32D4×Dic311yescharacteristic
266C2×C32#94914yes—
267C3⋊D3—42no—
268C3×D3—128no—
269C2×D4—91no—
270C2×D4D3×C3⋊D311yescharacteristic
271C22×C4—61no—
272C22×C4—541no—
273D8—94no—
274C2×C8—91no—
275C2×C8—271no—
276C4⋊C4—31no—
277C4⋊C4—271no—
278C22⋊C4—61no—
279C22⋊C4—541no—
280C42—31no—
281C22×C3—22no—
282C22×C3—28no—
283C22×C3—24no—
284C22×C3—48no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—44no—
288C22×C3—44no—
289C22×C3—84no—
290C22×C3—84no—
291C22×C3—182no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3D3×#6714yes—
295C22×C3C2×#63811yescharacteristic
296C2×D3—68no—
297C2×D3—124no—
298C2×D3—124no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—64no—
302C3×C4—64no—
303C3×C4—124no—
304C3×C4C2×D3×Dic314yes—
305C3×C4#380414yes—
306C3×C4C2×C4×C3⋊D311yescharacteristic
307C3×C4#382811yescharacteristic
308Dic3—21no—
309Dic3—21no—
310Dic3—41no—
311Dic3—361no—
312Dic3—364no—
313Dic3—724no—
314C32—24no—
315C32#501414yescomplemented
316C32#515211yescharacteristic, complemented
317C23—21no—
318C23—181no—
319D4—21no—
320D4—92no—
321D4—181no—
322D4Dic3×C3⋊D312yescomplemented
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—61no—
326C2×C4—122no—
327C2×C4—541no—
328C2×C4—1082no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—92no—
331C8—541no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—48no—
336C2×C3—42no—
337C2×C3—88no—
338C2×C3—362no—
339C2×C3#6392514yes—
340C2×C3#6393214yes—
341C2×C3Dic3×#6714yes—
342C2×C3C4×#63811yescharacteristic
343C2×C3#6414111yescharacteristic
344C2×C3#6415811yescharacteristic
345D3—128no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—182no—
350C22—361no—
351C22—361no—
352C22D3×#63811yescharacteristic
353C4—61no—
354C4—61no—
355C4—121no—
356C4—1081no—
357C4C2×Dic3×C3⋊D311yescharacteristic
358C4#8903111yescharacteristic
359C3—24no—
360C3#1059143014yescomplemented
361C3#1059271311yescharacteristic, complemented
362C2—42no—
363C2—362no—
364C2#1172354411yescharacteristic
365C2#1172355511yescharacteristic
366C2Dic3×#63811yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C32×D4×Dic3C2semidirect
3. Dic3×C4⋊(C3⋊D3)C2semidirect
4. #11722202C2semidirect
5. #11722054C2semidirect
6. #11722033C2semidirect
7. #11722019C2semidirect
16. C3×#3753C4semidirect
25. C3×D4×Dic3D3semidirect
40. C33×D4C2×C4semidirect
44. C3×C4⋊(C3⋊D3)C2×C4semidirect
51. C3×C3⋊(C3⋊C8)C2×C4semidirect
66. #3753Dic3semidirect
97. D4×Dic3C3⋊D3semidirect
118. C32×D4C4×D3semidirect
119. C32×D4C2×Dic3semidirect
131. C4⋊(C3⋊D3)C2×Dic3semidirect
143. C3⋊(C3⋊C8)C2×Dic3semidirect
218. C33#442semidirect
230. C3×D4D3×Dic3semidirect
231. C3×D4C4×C3⋊D3semidirect
315. C32#5014semidirect
316. C32#5152semidirect
322. D4Dic3×C3⋊D3semidirect
360. C3#10591430semidirect
361. C3#10592713semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×D4×Dic32
Dic3×C4⋊(C3⋊D3)2
#117222022
#117220542
#117220332
#117220192
#105927133
#105914303