Finite Group Databasebeta

← C3×#10590402 #13157418

Subgroups — C3×#10590402 #13157418

Order 1728.

778 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 116 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×#64298C211yesmaximal, characteristic, complemented
2C3×#64084C211yesmaximal, characteristic, complemented
3C3×#63825C212yesmaximal, complemented
4C32×#805C212yesmaximal, complemented
5C3×#63561C211yesmaximal, characteristic, complemented
6C3×#4827—32nomaximal
7#10590402C311yesmaximal, characteristic, direct product factor
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C32×#235C2212yes—
11C3×#3805—22no—
12C3×#3795—22no—
13C3×D3×Dic6—22no—
14C33×C42C2211yescharacteristic
15C3×#3752—22no—
16C32×#208C2212yes—
17C3×#3717—22no—
18C3×#3714—22no—
19C3×#1010—32no—
20#64298C2×C311yescharacteristic, complemented
21#64084C2×C311yescharacteristic, complemented
22C3×#949—32no—
23C3×#908—32no—
24C3×#875—32no—
25C3×#875—61no—
26#63825C2×C312yescomplemented
27C32×#104—32no—
28C3×#824—32no—
29C3×#805—22no—
30C3×#805D312yescomplemented
31C3×#805C2×C312yescomplemented
32#63561C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3—21no—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3D412yes—
37C3×C4⋊(C3⋊D3)—22no—
38C3×#634—22no—
39C32×#67—22no—
40C32×D12D412yes—
41C32×C4×D3—22no—
42C32×Dic6D412yes—
43C3×#627—22no—
44C3×#625—22no—
45C3×D3×Dic3—24no—
46C32×C3⋊C8—22no—
47C3×#454—91no—
48#4827—32no—
49C32×#41—32no—
50C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
51C2×#634C22×C311yescharacteristic
52C3×D3×Q8—62no—
53C3×#237—62no—
54C3×#237—64no—
55C3×#235—22no—
56C3×#235—62no—
57C3×#235C2×D312yes—
58C3×#235C22×C312yes—
59C2×C3×D12—32no—
60C2×C3×D12—61no—
61C2×C3×Dic6—32no—
62C2×C3×Dic6—62no—
63C2×C3×Dic6—61no—
64#3805—22no—
65#3795—22no—
66D3×Dic6—22no—
67C32×M16—32no—
68C32×C42—21no—
69C32×C42—22no—
70C32×C42—41no—
71C32×C42C2×D312yes—
72C32×C42C22×C311yescharacteristic
73#3752—22no—
74C3×#215—62no—
75C3×#213—62no—
76C3×(C2×D3)⋊C4—64no—
77C3×(C2×D3)⋊C4—64no—
78C3×#208—22no—
79C3×#208C2×D312yes—
80C3×#208C22×C312yes—
81C3×D24—62no—
82C3×#204—62no—
83#3717—22no—
84#3714—22no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—42no—
87C2×C3×C3⋊D3—41no—
88C2×C32×D3—22no—
89C33×C4—41no—
90C33×C4C2×D412yes—
91C3×C3⋊Dic3—22no—
92C32×Dic3—22no—
93C3×2−1+4—91no—
94#1010—32no—
95C3×AGL(1,ℤ/8)—92no—
96C3×#124—91no—
97#949—32no—
98#908—32no—
99#875—32no—
100#875—61no—
101C3×#104—32no—
102C3×#104—32no—
103C3×#104—62no—
104C3×#101—91no—
105#824—32no—
106#805C3×D312yescomplemented
107C22×C3⋊D3—21no—
108C22×C3×D3—62no—
109C22×C3×D3—62no—
110C32×Q8—32no—
111C32×D4—32no—
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×C4—62no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C23×C311yescharacteristic
124C2×C3⋊Dic3C3×D412yes—
125C4⋊(C3⋊D3)—22no—
126#634—22no—
127C3×#67—22no—
128C3×#67—22no—
129C3×#67—42no—
130C3×#67—64no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—64no—
134C3×D12—22no—
135C3×D12—62no—
136C3×D12—62no—
137C3×D12—122no—
138C3×D12D1212yes—
139C3×D12C3×D412yes—
140C3×C4×D3—22no—
141C3×C4×D3—22no—
142C3×C4×D3—42no—
143C3×C4×D3—64no—
144C3×Dic6—22no—
145C3×Dic6—64no—
146C3×Dic6—62no—
147C3×Dic6—62no—
148C3×Dic6—122no—
149C3×Dic6D1212yes—
150C3×Dic6C3×D412yes—
151#627—22no—
152#625—22no—
153D3×Dic3—24no—
154C32×C8—62no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—42no—
158#454—91noSylow
159C2×C33—21no—
160C2×C33#12011yescharacteristic
161C3×C3⋊D3—81no—
162C32×D3—42no—
163C3×#41—32no—
164C3×#41—32no—
165C3×#41—62no—
166C3×#41—182no—
167C3×#41—182no—
168C2×C3×Q8—91no—
169C2×C3×Q8—182no—
170C2×C3×D4—91no—
171D3×Q8—62no—
172#237—62no—
173#237—64no—
174#235—62no—
175#235C2×C3×D312yes—
176C2×D12—32no—
177C2×D12—61no—
178C2×Dic6—32no—
179C2×Dic6—61no—
180C2×Dic6—62no—
181C3×SD16—182no—
182C3×D8—182no—
183C3×M16—32no—
184C3×M16—32no—
185C3×M16—62no—
186C3×C22⋊C4—182no—
187C3×C42—21no—
188C3×C42—22no—
189C3×C42—41no—
190C3×C42D3211yescharacteristic
191C3×C42C2×C3×D312yes—
192#215—62no—
193#213—62no—
194(C2×D3)⋊C4—64no—
195(C2×D3)⋊C4—64no—
196#208C2×C3×D312yes—
197D24—62no—
198#204—62no—
199C22×C32—21no—
200C22×C32—22no—
201C22×C32—41no—
202C22×C32—62no—
203C22×C32D3×D412yes—
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—42no—
206C2×C3⋊D3—41no—
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—42no—
210C2×C3×D3—122no—
211C2×C3×D3—124no—
212C2×C3×D3—124no—
213C2×C3×D3—122no—
214C32×C4—22no—
215C32×C4—22no—
216C32×C4—22no—
217C32×C4—42no—
218C32×C4—41no—
219C32×C4—42no—
220C32×C4—42no—
221C32×C4—44no—
222C32×C4—44no—
223C32×C4—62no—
224C32×C4C2×D1212yes—
225C32×C4D3×D412yes—
226C32×C4C2×C3×D412yes—
227C3⋊Dic3—22no—
228C3×Dic3—22no—
229C3×Dic3—22no—
230C3×Dic3—42no—
231C3×Dic3—64no—
232C3×Dic3—122no—
2332−1+4—91no—
234AGL(1,ℤ/8)—92no—
235#124—91no—
236#104—32no—
237#101—91no—
238C33#45411yescharacteristic, complemented, Sylow
239C23×C3—181no—
240C22×D3—62no—
241C22×D3—62no—
242C3×Q8—32no—
243C3×Q8—32no—
244C3×Q8—62no—
245C3×Q8—182no—
246C3×Q8—182no—
247C3×D4—32no—
248C3×D4—32no—
249C3×D4—62no—
250C3×D4—62no—
251C3×D4—62no—
252C3×D4—122no—
253C3×D4—182no—
254C3×D4—182no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—92no—
266C2×C3×C4—122no—
267C2×C3×C4—184no—
268C2×C3×C4C2×D3211yescharacteristic
269C2×C3×C4C22×C3×D312yes—
270#67—22no—
271#67—64no—
272C2×Dic3—34no—
273C2×Dic3—62no—
274C2×Dic3—64no—
275D12—62no—
276D12—62no—
277D12—122no—
278D12C3×D1212yes—
279C4×D3—22no—
280C4×D3—64no—
281Dic6—64no—
282Dic6—62no—
283Dic6—62no—
284Dic6—122no—
285Dic6C3×D1212yes—
286C3×C8—62no—
287C3×C8—62no—
288C3×C8—122no—
289C3⋊C8—22no—
290C2×C32—21no—
291C2×C32—22no—
292C2×C32—22no—
293C2×C32—22no—
294C2×C32—21no—
295C2×C32—41no—
296C2×C32—42no—
297C2×C32—42no—
298C2×C32—122no—
299C2×C32#88212yes—
300C2×C32C3×#12011yescharacteristic
301C3⋊D3—81no—
302C3×D3—42no—
303C3×D3—42no—
304C3×D3—82no—
305C3×D3—242no—
306C3×D3—242no—
307#41—32no—
308#41—182no—
309#41—182no—
310C2×Q8—91no—
311C2×Q8—182no—
312C2×D4—91no—
313SD16—182no—
314D8—182no—
315M16—32no—
316C22⋊C4—182no—
317C42C3×D3211yescharacteristic
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—41no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3—122no—
324C22×C3—362no—
325C22×C3—361no—
326C22×C3#381111yescharacteristic
327C22×C3C3×D3×D412yes—
328C2×D3—22no—
329C2×D3—122no—
330C2×D3—124no—
331C2×D3—124no—
332C2×D3—122no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337C3×C4—42no—
338C3×C4—41no—
339C3×C4—44no—
340C3×C4—44no—
341C3×C4—42no—
342C3×C4—62no—
343C3×C4—62no—
344C3×C4—122no—
345C3×C4—182no—
346C3×C4D3×D1212yes—
347C3×C4C2×C3×D1212yes—
348C3×C4C3×D3×D412yes—
349Dic3—22no—
350Dic3—64no—
351Dic3—122no—
352C32—22no—
353C32—21no—
354C32—41no—
355C32#482712yescomplemented
356C32C3×#45411yescharacteristic, complemented
357C23—181no—
358Q8—32no—
359Q8—182no—
360Q8—182no—
361D4—32no—
362D4—62no—
363D4—182no—
364D4—182no—
365C2×C4—21no—
366C2×C4—62no—
367C2×C4—92no—
368C2×C4—184no—
369C2×C4C2×C3×D3211yescharacteristic
370C8—62no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—22no—
376C2×C3—42no—
377C2×C3—42no—
378C2×C3—41no—
379C2×C3—122no—
380C2×C3—122no—
381C2×C3—242no—
382C2×C3—721no—
383C2×C3#6392311yescharacteristic
384C2×C3C3×#88212yes—
385D3—42no—
386D3—242no—
387D3—242no—
388C22—62no—
389C22—362no—
390C22—361no—
391C22C3×#381111yescharacteristic
392C4—41no—
393C4—62no—
394C4—182no—
395C4C3×D3×D1212yes—
396C3—21no—
397C3—22no—
398C3—41no—
399C3#1059040211yescharacteristic, direct product factor
400C3C3×#482712yescomplemented
401C2—21no—
402C2—122no—
403C2—721no—
404C2C3×#6392311yescharacteristic
subgroupG/H
centerC2×C3#63923
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33#120
hypercenterC3×C42D32
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64298C2semidirect
2. C3×#64084C2semidirect
3. C3×#63825C2semidirect
4. C32×#805C2semidirect
5. C3×#63561C2semidirect
7. #10590402C3direct (N × H)
20. #64298C2×C3semidirect
21. #64084C2×C3semidirect
26. #63825C2×C3semidirect
30. C3×#805D3semidirect
31. C3×#805C2×C3semidirect
32. #63561C2×C3semidirect
106. #805C3×D3semidirect
238. C33#454semidirect
355. C32#4827semidirect
356. C32C3×#454semidirect
399. C3#10590402direct (N × H)
400. C3C3×#4827semidirect

Maximal subgroups

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

subgroupindex
C3×#642982
C3×#640842
C3×#638252
C32×#8052
C3×#635612
C3×#48273
#105904023