Finite Group Databasebeta

← #13156305

Subgroups — #13156305

Order 1728.

624 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 110 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
1D12×C3⋊Dic3C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3#11722078C211yesmaximal, characteristic, complemented
4#11722054C212yesmaximal, complemented
5C3×#63562C212yesmaximal
6#10591495—31nomaximal
7#10590465—32nomaximal
8C2×C32×D12C2211yescharacteristic
9C2×D3×C3⋊Dic3—21no—
10C2×C3×#627—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×C4⋊Dic3C2212yes—
15(C2×D3)⋊(C3⋊Dic3)—21no—
16C3×#3724—21no—
17D4×C3⋊Dic3—31no—
18C3×#888—32no—
19#63919D311yescharacteristic, complemented
20Dic3×D12—32no—
21Dic3×D12—61no—
22#63635—32no—
23C3×#832—32no—
24C3×#809—32no—
25#63573—32no—
26#63573—32no—
27#63573—62no—
28#63572—31no—
29Dic3⋊(C3⋊C8)—32no—
30#63562D312yes—
31C22×C32×D3—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—22no—
37C32×D12—21no—
38C32×D12Q812yescomplemented
39D3×C3⋊Dic3—42no—
40C3×#627—42no—
41C3×C3⋊(C3⋊C8)—22no—
42#5084—91no—
43#4761—92no—
44C2×C32×D4—31no—
45C22×C3⋊Dic3—61no—
46C2×C3×D12—21no—
47C2×C3×D12C2×D312yes—
48C2×C3×Dic6—62no—
49C2×#627—21no—
50C2×D3×Dic3—62no—
51C2×D3×Dic3—62no—
52C32×C4⋊C4—32no—
53#3757—61no—
54C4⋊(C3⋊Dic3)—31no—
55C4⋊(C3⋊Dic3)—32no—
56C4⋊(C3⋊Dic3)—61no—
57C4⋊(C3⋊Dic3)—62no—
58C4⋊(C3⋊Dic3)—121no—
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D312yes—
61C3×C4⋊Dic3—22no—
62C3×C4⋊Dic3C2×D312yes—
63C3×C4⋊Dic3C2×D312yes—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—62no—
70#3724—21no—
71(C2×D3)⋊Dic3—62no—
72(C2×D3)⋊Dic3—62no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3SD1612yes—
82C32×Dic3—42no—
83C3×C4⋊Q8—91no—
84D4×Dic3—92no—
85D4×Dic3—181no—
86#888—32no—
87C4×D12—91no—
88C3×#102—92no—
89#832—32no—
90#817—92no—
91Dic3⋊C8—92no—
92#809—32no—
93#808—92no—
94#808—92no—
95#808—182no—
96C4⋊(C3⋊C8)—91no—
97C23×C32—61no—
98C22×C3×D3—22no—
99C22×C3×D3—22no—
100C32×D4—32no—
101C32×D4—61no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4—62no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—62no—
110C2×C3⋊Dic3—62no—
111C2×C3⋊Dic3—122no—
112C2×C3⋊Dic3—122no—
113C2×C3⋊Dic3—122no—
114C2×C3⋊Dic3#6711yescharacteristic
115C2×C3⋊Dic3#6711yescharacteristic
116C2×C3×Dic3—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—42no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×D12—22no—
124C3×D12—22no—
125C3×D12—22no—
126C3×D12Dic614yescomplemented
127C3×Dic6—124no—
128#627—42no—
129D3×Dic3—124no—
130D3×Dic3—124no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—64no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—62no—
135#476—271noSylow
136C2×C33C22⋊Q811yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C32×D3—42no—
140C2×C3×Q8—181no—
141C2×C3×D4—32no—
142C2×C3×D4—61no—
143C22×Dic3—182no—
144C22×Dic3—182no—
145C2×D12D3211yescharacteristic
146C2×C4×D3—181no—
147C2×Dic6—62no—
148C2×C3×C8—92no—
149C3×C4⋊C4—32no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—62no—
152C3×C4⋊C4—181no—
153C3×C42—91no—
154#217—182no—
155#217—182no—
156(C2×D3)⋊C4—181no—
157C4⋊Dic3—91no—
158C4⋊Dic3—92no—
159C4⋊Dic3—181no—
160C4⋊Dic3—182no—
161C4⋊Dic3—361no—
162C4⋊Dic3D3212yes—
163Dic3⋊C4—62no—
164C4×Dic3—32no—
165C4×Dic3—61no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—62no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32—62no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32D3×D412yes—
176C22×C32C2×#6711yescharacteristic
177C2×C3×D3—24no—
178C2×C3×D3—24no—
179C2×C3×D3—42no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4—122no—
190C32×C4C2×Dic612yes—
191C32×C4#23712yes—
192C32×C4D3×Q811yescharacteristic
193C32×C4#23911yescharacteristic
194C3⋊Dic3—21no—
195C3⋊Dic3—121no—
196C3⋊Dic3—122no—
197C3⋊Dic3—122no—
198C3⋊Dic3—242no—
199C3⋊Dic3—242no—
200C3⋊Dic3#21512yes—
201C3×Dic3—34no—
202C3×Dic3—42no—
203C3×Dic3—42no—
204C3×Dic3—62no—
205C3×Dic3—62no—
206C3×Dic3—82no—
207C3×Dic3—121no—
208C4⋊Q8—91no—
209C4×D4—271no—
210#106—272no—
211C4⋊C8—271no—
212#102—92no—
213C33#47611yescharacteristic, complemented, Sylow
214C23×C3—62no—
215C23×C3—62no—
216C22×D3—21no—
217C3×Q8—362no—
218C3×D4—34no—
219C3×D4—62no—
220C3×D4—62no—
221C3×D4—62no—
222C2×C3×C4—22no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—122no—
230C2×C3×C4C2×D3212yes—
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—22no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—182no—
238C2×Dic3—182no—
239C2×Dic3—181no—
240C2×Dic3—362no—
241C2×Dic3—362no—
242C2×Dic3—364no—
243C2×Dic3—364no—
244D12—21no—
245D12#62712yescomplemented
246C4×D3—362no—
247Dic6—124no—
248C3×C8—182no—
249C3⋊C8—64no—
250C3⋊C8—62no—
251C3⋊C8—62no—
252C2×C32—22no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—122no—
262C2×C32#87812yes—
263C2×C32#91112yes—
264C2×C32D3×SD1612yes—
265C2×C32#93211yescharacteristic
266C2×C32C2×#21511yescharacteristic
267C2×C32#94611yescharacteristic
268C3×D3—44no—
269C3×D3—44no—
270C2×Q8—181no—
271C2×D4—31no—
272C22×C4—541no—
273C2×C8—92no—
274C4⋊C4—32no—
275C4⋊C4—181no—
276C4⋊C4—271no—
277C22⋊C4—541no—
278C42—91no—
279C22×C3—21no—
280C22×C3—22no—
281C22×C3—41no—
282C22×C3—64no—
283C22×C3—64no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—122no—
287C22×C3—122no—
288C22×C3D3×#6712yes—
289C22×C3#381111yescharacteristic
290C2×D3—22no—
291C2×D3—41no—
292C2×D3—41no—
293C3×C4—21no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—92no—
300C3×C4—122no—
301C3×C4—122no—
302C3×C4—181no—
303C3×C4—242no—
304C3×C4D3×Dic612yes—
305C3×C4#379612yes—
306C3×C4#380511yescharacteristic
307C3×C4C2×#62711yescharacteristic
308Dic3—34no—
309Dic3—42no—
310Dic3—62no—
311Dic3—62no—
312Dic3—121no—
313Dic3—362no—
314Dic3—361no—
315Dic3—362no—
316Dic3—722no—
317Dic3—722no—
318C32—22no—
319C32—21no—
320C32—41no—
321C32#476112yescomplemented
322C32#508411yescharacteristic, complemented
323C23—61no—
324Q8—362no—
325D4—32no—
326D4—61no—
327C2×C4—62no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—541no—
331C2×C4—1082no—
332C2×C4D3311yescharacteristic
333C8—182no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—124no—
344C2×C3—124no—
345C2×C3#6388712yes—
346C2×C3#6392711yescharacteristic
347C2×C3#6393011yescharacteristic
348C2×C3#6393212yes—
349C2×C3D3×#21512yes—
350C2×C3#6398211yescharacteristic
351D3—42no—
352C22—62no—
353C22—121no—
354C22—121no—
355C22#8895211yescharacteristic
356C4—92no—
357C4—122no—
358C4—181no—
359C4—1081no—
360C4D3×#62711yescharacteristic
361C4#8896411yescharacteristic
362C3—21no—
363C3—22no—
364C3—41no—
365C3#1059046512yescomplemented
366C3#1059149511yescharacteristic, complemented
367C2—122no—
368C2#1172283811yescharacteristic
369C2#1172287211yescharacteristic
370C2#1172297611yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#476
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
3. #11722078C2semidirect
4. #11722054C2semidirect
19. #63919D3semidirect
38. C32×D12Q8semidirect
126. C3×D12Dic6semidirect
213. C33#476semidirect
245. D12#627semidirect
321. C32#4761semidirect
322. C32#5084semidirect
365. C3#10590465semidirect
366. C3#10591495semidirect

Maximal subgroups

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

subgroupindex
D12×C3⋊Dic32
C3×#639192
#117220782
#117220542
C3×#635622
#105914953
#105904653