Finite Group Databasebeta

← C3×#10591621 #13158650

Subgroups — C3×#10591621 #13158650

Order 1728.

826 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C3×#64156C211yesmaximal, characteristic, complemented
2C2×C32×#215C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4C2×C3×#3714C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#5176—32nomaximal
7#10591621C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C32×D12C2212yes—
11C2×C3×#625—22no—
12C3×#3752—22no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×#215—24no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3714—24no—
17C2×C32×SD16—32no—
18#64156C2×C311yescharacteristic, complemented
19C3×#947—32no—
20C3×#947—61no—
21C2×C3×#215—22no—
22C2×C3×#215D312yescomplemented
23C2×C3×#215C2×C312yescomplemented
24C3×#940—32no—
25C2×C3×#213—32no—
26C2×C3×D24—32no—
27#63914C2×C311yescharacteristic, complemented
28C2×#3714C2×C312yescomplemented
29C3×#815—32no—
30#63555C2×C311yescharacteristic, complemented
31C22×C3×C3⋊D3—21no—
32C22×C32×D3—22no—
33C33×Q8—22no—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3D412yes—
37C3×C4⋊(C3⋊D3)—22no—
38C32×D12—22no—
39C32×D12—22no—
40C3×#625—44no—
41C32×C3⋊C8D414yescomplemented
42C3×#496—91no—
43#5176—32no—
44C2×C32×Q8—21no—
45C2×C32×Q8—22no—
46C2×C32×Q8—41no—
47C2×C32×Q8C2×D312yes—
48C2×C32×Q8C22×C311yescharacteristic
49C2×C32×D4—32no—
50C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
51C2×C3×#67—64no—
52C2×C3×D12—22no—
53C2×C3×D12—32no—
54C2×C3×D12—61no—
55C2×C3×D12C2×D312yes—
56C2×C3×D12C22×C312yes—
57C2×#625—22no—
58C32×SD16—64no—
59C2×C32×C8—32no—
60#3752—22no—
61C4×C3⋊Dic3C22×C311yescharacteristic
62C3×#215—28no—
63C3×#215—24no—
64C3×#215—24no—
65C3×#213—64no—
66C3×(C2×D3)⋊C4—64no—
67C3×(C2×D3)⋊C4—64no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C22×C312yes—
73C3×D24—64no—
74#3714—24no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—42no—
78C2×C32×D3—42no—
79C2×C32×D3—44no—
80C33×C4—41no—
81C33×C4C2×D412yes—
82C3×C3⋊Dic3—22no—
83C2×C3×SD16—32no—
84C2×C3×SD16—32no—
85C2×C3×SD16—62no—
86C2×C3×D8—92no—
87C3×#127—91no—
88C3×#124—91no—
89#947—32no—
90#947—61no—
91C2×#215C3×D312yescomplemented
92#940—32no—
93C2×#213—32no—
94C2×D24—32no—
95C3×#97—91no—
96#815—32no—
97C23×C32—62no—
98C22×C3⋊D3—21no—
99C22×C3×D3—22no—
100C22×C3×D3—22no—
101C22×C3×D3—42no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C32×Q8—22no—
105C32×Q8—24no—
106C32×Q8—22no—
107C32×Q8—22no—
108C32×Q8—42no—
109C32×Q8—42no—
110C32×Q8—44no—
111C32×D4—62no—
112C32×D4—62no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—41no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4C22×D312yes—
122C2×C32×C4C23×C311yescharacteristic
123C2×C3⋊Dic3C3×D412yes—
124C4⋊(C3⋊D3)—22no—
125C3×#67—128no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—62no—
128C3×D12—24no—
129C3×D12—22no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12—42no—
134C3×D12—62no—
135C3×D12—62no—
136C3×D12—122no—
137#625—44no—
138C32×C8—34no—
139C3×C3⋊C8—24no—
140C3×C3⋊C8#6714yescomplemented
141C3×C3⋊C8C3×D414yescomplemented
142#496—91noSylow
143C2×C33#12711yescharacteristic
144C2×C33AGL(1,ℤ/8)12yes—
145C3×C3⋊D3—81no—
146C32×D3—82no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—41no—
150C2×C3×Q8D3211yescharacteristic
151C2×C3×Q8C2×C3×D312yes—
152C2×C3×D4—32no—
153C2×C3×D4—32no—
154C2×C3×D4—62no—
155C2×C3×D4—91no—
156C2×C3×D4—182no—
157C2×#67—64no—
158C2×D12—32no—
159C2×D12—61no—
160C2×D12C2×C3×D312yes—
161C3×SD16—64no—
162C3×SD16—64no—
163C3×SD16—68no—
164C3×D8—184no—
165C2×C3×C8—32no—
166C2×C3×C8—32no—
167C2×C3×C8—62no—
168C3×C22⋊C4—182no—
169C3×C42—91no—
170#215—24no—
171#213—64no—
172(C2×D3)⋊C4—64no—
173(C2×D3)⋊C4—64no—
174C4×Dic3—32no—
175C4×Dic3—61no—
176C2×C3⋊C8C2×C3×D312yes—
177D24—64no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—41no—
181C22×C32—122no—
182C22×C32—124no—
183C22×C32D3×D412yes—
184C22×C32C2×C3×D411yescharacteristic
185C2×C3⋊D3—41no—
186C2×C3⋊D3—42no—
187C2×C3×D3—44no—
188C2×C3×D3—44no—
189C2×C3×D3—42no—
190C2×C3×D3—42no—
191C2×C3×D3—44no—
192C2×C3×D3—84no—
193C2×C3×D3—122no—
194C2×C3×D3—124no—
195C2×C3×D3—124no—
196C2×C3×D3—122no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—42no—
201C32×C4—42no—
202C32×C4—44no—
203C32×C4—44no—
204C32×C4—42no—
205C32×C4—41no—
206C32×C4D3×D412yes—
207C32×C4C2×#6712yes—
208C32×C4C2×C3×D412yes—
209C3⋊Dic3—22no—
210C3×Dic3—64no—
211C3×Dic3—122no—
212C2×SD16—32no—
213C2×D8—92no—
214#127—91no—
215#124—91no—
216#97—91no—
217C33#49611yescharacteristic, complemented, Sylow
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—122no—
221C23×C3—181no—
222C22×D3—22no—
223C22×D3—62no—
224C22×D3—62no—
225C3×Q8—24no—
226C3×Q8—22no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—42no—
230C3×Q8—42no—
231C3×Q8—44no—
232C3×D4—64no—
233C3×D4—62no—
234C3×D4—62no—
235C3×D4—62no—
236C3×D4—62no—
237C3×D4—122no—
238C3×D4—182no—
239C3×D4—364no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—92no—
249C2×C3×C4C2×D3211yescharacteristic
250C2×C3×C4C22×C3×D312yes—
251#67—128no—
252C2×Dic3—34no—
253C2×Dic3—62no—
254D12—22no—
255D12—22no—
256D12—62no—
257D12—62no—
258D12—122no—
259C3×C8—34no—
260C3×C8—34no—
261C3×C8—64no—
262C3⋊C8C3×#6714yescomplemented
263C2×C32—24no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—41no—
268C2×C32—42no—
269C2×C32—242no—
270C2×C32#91414yes—
271C2×C32#94012yes—
272C2×C32C3×#12711yescharacteristic
273C2×C32C3×AGL(1,ℤ/8)12yes—
274C3⋊D3—81no—
275C3×D3—84no—
276C3×D3—82no—
277C3×D3—82no—
278C3×D3—242no—
279C3×D3—242no—
280C2×Q8C3×D3211yescharacteristic
281C2×D4—32no—
282C2×D4—91no—
283C2×D4—182no—
284SD16—64no—
285D8—184no—
286C2×C8—32no—
287C22⋊C4—182no—
288C42—91no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—41no—
292C22×C3—124no—
293C22×C3—124no—
294C22×C3—122no—
295C22×C3—124no—
296C22×C3—122no—
297C22×C3—244no—
298C22×C3—362no—
299C22×C3—361no—
300C22×C3#381111yescharacteristic
301C22×C3C3×D3×D412yes—
302C2×D3—44no—
303C2×D3—42no—
304C2×D3—122no—
305C2×D3—124no—
306C2×D3—124no—
307C2×D3—122no—
308C3×C4—22no—
309C3×C4—22no—
310C3×C4—22no—
311C3×C4—42no—
312C3×C4—44no—
313C3×C4—44no—
314C3×C4—42no—
315C3×C4—41no—
316C3×C4—42no—
317C3×C4—182no—
318C3×C4D3×#6712yes—
319C3×C4C3×D3×D412yes—
320C3×C4C2×C3×#6712yes—
321Dic3—64no—
322Dic3—122no—
323C32—22no—
324C32—21no—
325C32—41no—
326C32#517612yescomplemented
327C32C3×#49611yescharacteristic, complemented
328C23—62no—
329C23—181no—
330Q8—22no—
331D4—62no—
332D4—62no—
333D4—182no—
334D4—364no—
335C2×C4—21no—
336C2×C4—92no—
337C2×C4C2×C3×D3211yescharacteristic
338C8—34no—
339C2×C3—22no—
340C2×C3—24no—
341C2×C3—21no—
342C2×C3—22no—
343C2×C3—42no—
344C2×C3—41no—
345C2×C3—244no—
346C2×C3—242no—
347C2×C3—242no—
348C2×C3—721no—
349C2×C3#6394112yes—
350C2×C3#6397911yescharacteristic
351C2×C3C3×#91414yes—
352C2×C3C3×#94012yes—
353D3—82no—
354D3—242no—
355D3—242no—
356C22—124no—
357C22—122no—
358C22—362no—
359C22—361no—
360C22C3×#381111yescharacteristic
361C4—41no—
362C4—182no—
363C4C3×D3×#6712yes—
364C3—21no—
365C3—22no—
366C3—41no—
367C3#1059162111yescharacteristic, direct product factor
368C3C3×#517612yescomplemented
369C2—242no—
370C2—721no—
371C2C3×#6394112yes—
372C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C2×C32×#215C2semidirect
3. C3×#63914C2semidirect
4. C2×C3×#3714C2semidirect
5. C3×#63555C2semidirect
7. #10591621C3direct (N × H)
18. #64156C2×C3semidirect
22. C2×C3×#215D3semidirect
23. C2×C3×#215C2×C3semidirect
27. #63914C2×C3semidirect
28. C2×#3714C2×C3semidirect
30. #63555C2×C3semidirect
41. C32×C3⋊C8D4semidirect
91. C2×#215C3×D3semidirect
140. C3×C3⋊C8#67semidirect
141. C3×C3⋊C8C3×D4semidirect
217. C33#496semidirect
262. C3⋊C8C3×#67semidirect
326. C32#5176semidirect
327. C32C3×#496semidirect
367. C3#10591621direct (N × H)
368. C3C3×#5176semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C32×#2152
C3×#639142
C2×C3×#37142
C3×#635552
C3×#51763
#105916213