Finite Group Databasebeta

← C3×#10590565 #13157558

Subgroups — C3×#10590565 #13157558

Order 1728.

684 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 124 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×#64102C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic, complemented
3C2×C3×#3717C212yesmaximal
4C32×#810C212yesmaximal, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#4818—32nomaximal
7#10590565C311yesmaximal, characteristic, direct product factor
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×C4×C3⋊D3—21no—
10C2×C32×Dic6C2212yes—
11C33×C4⋊C4C2211yescharacteristic
12C3×#3752—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—21no—
16C3×#3724—21no—
17C3×#3717—24no—
18#64102C2×C311yescharacteristic
19C3×#944—32no—
20C2×C3×#215—32no—
21C2×C3×#204—32no—
22C3×#893—32no—
23C3×#893—61no—
24#63873C2×C311yescharacteristic, complemented
25C2×#3717C2×C312yes—
26C32×#103—32no—
27C3×Dic3⋊C8—32no—
28C3×#810—22no—
29C3×#810D312yescomplemented
30C3×#810C2×C312yescomplemented
31#63572C2×C311yescharacteristic, complemented
32C22×C3×C3⋊D3—21no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×C4⋊(C3⋊D3)—21no—
38C3×C4⋊(C3⋊D3)D412yes—
39C3×C4×C3⋊D3—42no—
40C2×C32×Dic3—22no—
41C32×Dic6—22no—
42C32×Dic6—22no—
43C32×C3⋊C8—22no—
44C3×#461—91no—
45#4818—32no—
46C2×C32×Q8—32no—
47C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
48C2×C4×C3⋊D3—21no—
49C2×C3×D12—32no—
50C2×C3×D12—61no—
51C2×C3×C4×D3—62no—
52C2×C3×C4×D3—62no—
53C2×C3×Dic6—22no—
54C2×C3×Dic6C2×D312yes—
55C2×C3×Dic6C22×C312yes—
56C2×C32×C8—32no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—22no—
59C32×C4⋊C4—41no—
60C32×C4⋊C4C2×D312yes—
61C32×C4⋊C4C22×C311yescharacteristic
62#3752—21no—
63C4×C3⋊Dic3C22×C311yescharacteristic
64C3×#215—64no—
65C3×(C2×D3)⋊C4—62no—
66C3×(C2×D3)⋊C4—62no—
67C3×Dic3⋊C4—62no—
68C3×Dic3⋊C4—62no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—61no—
71C2×C3×C3⋊C8—22no—
72C2×C3×C3⋊C8C2×D312yes—
73C2×C3×C3⋊C8C22×C312yes—
74C3×#204—64no—
75#3724—21no—
76#3724—21no—
77#3717—24no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—22no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C33×C4—41no—
83C33×C4C2×D411yescharacteristic
84C33×C4#4111yescharacteristic
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3SD1612yescomplemented
87C32×Dic3—42no—
88C2×C3×SD16—92no—
89C3×C4⋊Q8—91no—
90C3×C4×D4—91no—
91#944—32no—
92C2×#215—32no—
93C2×#204—32no—
94#893—32no—
95#893—61no—
96C3×C4⋊C8—91no—
97C3×#103—32no—
98C3×#103—32no—
99C3×#103—62no—
100Dic3⋊C8—32no—
101#810C3×D312yescomplemented
102C22×C3⋊D3—21no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C32×Q8—62no—
106C32×Q8—62no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—41no—
113C2×C32×C4—42no—
114C2×C32×C4—42no—
115C2×C32×C4—62no—
116C2×C32×C4C22×D312yes—
117C2×C32×C4C23×C311yescharacteristic
118C2×C3⋊Dic3C3×D411yescharacteristic
119C2×C3⋊Dic3C3×D411yescharacteristic
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)C3×D412yes—
122C4×C3⋊D3—42no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—42no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×D12—34no—
131C3×D12—62no—
132C3×D12—62no—
133C3×D12—62no—
134C3×C4×D3—124no—
135C3×C4×D3—124no—
136C3×Dic6—24no—
137C3×Dic6—22no—
138C3×Dic6—22no—
139C3×Dic6—22no—
140C3×Dic6—22no—
141C3×Dic6—42no—
142C32×C8—62no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—22no—
145C3×C3⋊C8—42no—
146#461—91noSylow
147C2×C33#12111yescharacteristic
148C2×C33C2×SD1611yescharacteristic
149C2×C33#13711yescharacteristic
150C3×C3⋊D3—42no—
151C2×C3×Q8—32no—
152C2×C3×Q8—32no—
153C2×C3×Q8—62no—
154C2×C3×D4—91no—
155C22×C3×C4—181no—
156C2×D12—32no—
157C2×D12—61no—
158C2×C4×D3—62no—
159C2×C4×D3—62no—
160C2×Dic6C2×C3×D312yes—
161C3×SD16—184no—
162C2×C3×C8—32no—
163C2×C3×C8—32no—
164C2×C3×C8—62no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—181no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4D3211yescharacteristic
171C3×C4⋊C4C2×C3×D312yes—
172C3×C22⋊C4—181no—
173C3×C42—91no—
174#215—64no—
175(C2×D3)⋊C4—62no—
176(C2×D3)⋊C4—62no—
177Dic3⋊C4—62no—
178Dic3⋊C4—62no—
179C4×Dic3—32no—
180C4×Dic3—61no—
181C2×C3⋊C8C2×C3×D312yes—
182#204—64no—
183C22×C32—21no—
184C22×C32—22no—
185C22×C32—41no—
186C22×C32D3×D412yes—
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3—22no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3×D3—64no—
192C2×C3×D3—64no—
193C2×C3×D3—122no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4—41no—
204C32×C4—42no—
205C32×C4—42no—
206C32×C4—82no—
207C32×C4—82no—
208C32×C4—122no—
209C32×C4#23512yes—
210C32×C4D3×D412yes—
211C32×C4C2×C3×D411yescharacteristic
212C32×C4C3×#4111yescharacteristic
213C3⋊Dic3—21no—
214C3⋊Dic3C3×SD1612yescomplemented
215C3×Dic3—34no—
216C3×Dic3—44no—
217C3×Dic3—42no—
218C3×Dic3—42no—
219C3×Dic3—62no—
220C3×Dic3—62no—
221C3×Dic3—121no—
222C2×SD16—92no—
223C4⋊Q8—91no—
224C4×D4—91no—
225C4⋊C8—91no—
226#103—32no—
227C33#46111yescharacteristic, complemented, Sylow
228C23×C3—181no—
229C22×D3—62no—
230C22×D3—62no—
231C3×Q8—64no—
232C3×Q8—62no—
233C3×Q8—62no—
234C3×Q8—62no—
235C3×Q8—62no—
236C3×Q8—122no—
237C3×D4—92no—
238C3×D4—181no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4—122no—
252C2×C3×C4—362no—
253C2×C3×C4C2×D3211yescharacteristic
254C2×C3×C4C22×C3×D312yes—
255C2×Dic3—22no—
256C2×Dic3—32no—
257C2×Dic3—32no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260D12—34no—
261D12—62no—
262D12—62no—
263D12—62no—
264C4×D3—124no—
265C4×D3—124no—
266Dic6—22no—
267Dic6—22no—
268C3×C8—62no—
269C3×C8—62no—
270C3×C8—122no—
271C3⋊C8—22no—
272C2×C32—22no—
273C2×C32—22no—
274C2×C32—22no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32#88412yes—
282C2×C32D3×SD1612yes—
283C2×C32#91812yes—
284C2×C32C3×#12111yescharacteristic
285C2×C32C2×C3×SD1611yescharacteristic
286C2×C32C3×#13711yescharacteristic
287C3⋊D3—42no—
288C3×D3—124no—
289C3×D3—124no—
290C2×Q8—32no—
291C2×D4—91no—
292C22×C4—181no—
293SD16—184no—
294C2×C8—32no—
295C4⋊C4—181no—
296C4⋊C4—181no—
297C4⋊C4C3×D3211yescharacteristic
298C22⋊C4—181no—
299C42—91no—
300C22×C3—21no—
301C22×C3—22no—
302C22×C3—41no—
303C22×C3—182no—
304C22×C3—361no—
305C22×C3—361no—
306C22×C3#381111yescharacteristic
307C22×C3C3×D3×D412yes—
308C2×D3—64no—
309C2×D3—64no—
310C2×D3—122no—
311C2×D3—122no—
312C2×D3—122no—
313C2×D3—122no—
314C3×C4—21no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—42no—
320C3×C4—41no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—82no—
324C3×C4—82no—
325C3×C4—92no—
326C3×C4—124no—
327C3×C4—122no—
328C3×C4—122no—
329C3×C4—181no—
330C3×C4#379811yescharacteristic
331C3×C4#380211yescharacteristic
332C3×C4C3×#23512yes—
333C3×C4C3×D3×D412yes—
334Dic3—34no—
335Dic3—42no—
336Dic3—62no—
337Dic3—62no—
338Dic3—121no—
339C32—21no—
340C32—22no—
341C32—41no—
342C32#481812yescomplemented
343C32C3×#46111yescharacteristic, complemented
344C23—181no—
345Q8—62no—
346Q8—62no—
347D4—92no—
348D4—181no—
349C2×C4—21no—
350C2×C4—62no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—362no—
354C2×C4C2×C3×D3211yescharacteristic
355C8—62no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—22no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—362no—
366C2×C3#6391211yescharacteristic
367C2×C3#6393011yescharacteristic
368C2×C3#6394511yescharacteristic
369C2×C3C3×#88412yes—
370C2×C3C3×D3×SD1612yes—
371C2×C3C3×#91812yes—
372D3—124no—
373D3—124no—
374C22—182no—
375C22—361no—
376C22—361no—
377C22C3×#381111yescharacteristic
378C4—41no—
379C4—92no—
380C4—122no—
381C4—181no—
382C4C3×#379811yescharacteristic
383C4C3×#380211yescharacteristic
384C3—21no—
385C3—22no—
386C3—41no—
387C3#1059056511yescharacteristic, direct product factor
388C3C3×#481812yescomplemented
389C2—362no—
390C2C3×#6391211yescharacteristic
391C2C3×#6393011yescharacteristic
392C2C3×#6394511yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63873C2semidirect
4. C32×#810C2semidirect
5. C3×#63572C2semidirect
7. #10590565C3direct (N × H)
24. #63873C2×C3semidirect
29. C3×#810D3semidirect
30. C3×#810C2×C3semidirect
31. #63572C2×C3semidirect
86. C3×C3⋊Dic3SD16semidirect
101. #810C3×D3semidirect
214. C3⋊Dic3C3×SD16semidirect
227. C33#461semidirect
342. C32#4818semidirect
343. C32C3×#461semidirect
387. C3#10590565direct (N × H)
388. C3C3×#4818semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
C3×#638732
C2×C3×#37172
C32×#8102
C3×#635722
C3×#48183
#105905653