Finite Group Databasebeta

← C3×#10596513 #13181032

Subgroups — C3×#10596513 #13181032

Order 1728.

996 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64306C212yesmaximal, complemented
2C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
3C3×#64242C214yesmaximal, complemented
4C3×#64230C214yesmaximal, complemented
5C3×D32⋊Q8C214yesmaximal, complemented
6#10596513C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2212yes—
8C2×C3×GO+(4,𝔽2)C2214yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C32×#235—22no—
11C3×#3811C2214yes—
12C3×#3805—22no—
13C3×#3802C2212yes—
14C3×C4×D32C2214yes—
15C3×#3798—22no—
16C3×#3797C2212yes—
17C3×#3796—22no—
18C3×C4⋊#157C2214yes—
19C3×#3773C2214yes—
20C3×D32⋊C4C2218yes—
21C3×D3×#41—62no—
22#64306C2×C312yescomplemented
23C2×C4⋊#157C2×C311yescharacteristic, complemented
24#64242C2×C314yescomplemented
25#64230C2×C314yescomplemented
26D32⋊Q8C2×C314yescomplemented
27C22×C3×C3⋊D3C2311yescharacteristic
28C2×C3×D32C2314yes—
29C2×C3×#157—42no—
30C2×C3×#157C2314yes—
31C3×GO+(4,𝔽2)—44no—
32C2×C33×C4D411yescharacteristic, complemented
33C2×C3×C3⋊Dic3D411yescharacteristic, complemented
34C3×C4×C3⋊D3D412yescomplemented
35C3×C4×C3⋊D3C2312yes—
36C32×#67—24no—
37C32×D12—22no—
38C32×C4×D3—24no—
39C32×Dic6—22no—
40C3×#627—22no—
41C3×#626—24no—
42C3×#625—22no—
43C3×#624C2314yes—
44C3×D3×Dic3—24no—
45C3×#537—91nomaximal
46C22×#157C22×C312yes—
47C2×GO+(4,𝔽2)C22×C314yes—
48C32×#41—62no—
49C2×C4×C3⋊D3C22×C311yescharacteristic
50C3×#239—62no—
51C3×D3×Q8—62no—
52C3×#237—62no—
53C3×#237—64no—
54C3×D3×D4—62no—
55C3×D3×D4—64no—
56C3×#235—22no—
57C3×#235—22no—
58C3×#235—42no—
59C3×#235—64no—
60C2×C3×C4×D3—64no—
61C2×C3×C4×D3—62no—
62#3811C22×C314yes—
63#3805—22no—
64#3802C22×C312yes—
65C4×D32C22×C314yes—
66#3798—22no—
67#3797C22×C312yes—
68#3796—22no—
69C4⋊#157C22×C314yes—
70#3773C22×C314yes—
71D32⋊C4C22×C318yes—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3C2×D411yescharacteristic
76C2×C3×C3⋊D3C2411yescharacteristic
77C2×C32×D3—24no—
78C3×D32—24no—
79C3×D32—24no—
80C3×#157—24no—
81C33×C4C2×D412yes—
82C3×C3⋊Dic3C2×D412yes—
83C32×Dic3—24no—
84C2×C3×#41—92no—
85D3×#41—62no—
86C3×C22⋊Q8—94no—
87C3×#121—94no—
88C3×#121—94no—
89C2×C3×C4⋊C4—91no—
90C22×C3⋊D3C23×C311yescharacteristic
91C22×C3×D3—64no—
92C22×C3×D3—62no—
93C2×D32C23×C314yes—
94C2×#157—42no—
95C2×#157C23×C314yes—
96GO+(4,𝔽2)—44no—
97C32×Q8—62no—
98C32×D4—62no—
99C32×D4—64no—
100C2×C32×C4—22no—
101C2×C32×C4—42no—
102C2×C32×C4—64no—
103C2×C32×C4C3×D411yescharacteristic, complemented
104C2×C3⋊Dic3C3×D411yescharacteristic, complemented
105C4×C3⋊D3C3×D412yescomplemented
106C4×C3⋊D3C23×C312yes—
107C3×#67—24no—
108C3×#67—24no—
109C3×#67—44no—
110C3×#67—64no—
111C3×#67—64no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—64no—
114C3×D12—22no—
115C3×D12—22no—
116C3×D12—42no—
117C3×D12—64no—
118C3×C4×D3—24no—
119C3×C4×D3—24no—
120C3×C4×D3—44no—
121C3×C4×D3—64no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×C4×D3—62no—
126C3×C4×D3—64no—
127C3×Dic6—22no—
128C3×Dic6—22no—
129C3×Dic6—42no—
130C3×Dic6—64no—
131#627—22no—
132#626—24no—
133#625—22no—
134#624C23×C314yes—
135D3×Dic3—24no—
136#537—91noSylow
137C2×C33—21no—
138C2×C33C22×D411yescharacteristic
139C3×C3⋊D3—21no—
140C3×C3⋊D3H(5,𝔽2)11yescharacteristic
141C3×C3⋊D32−1+411yescharacteristic
142C32×D3—44no—
143C3×#41—62no—
144C3×#41—62no—
145C3×#41—122no—
146C3×#41—182no—
147C3×#41—182no—
148C3×#41—182no—
149C2×C3×Q8—92no—
150C2×C3×D4—94no—
151C2×C3×D4—92no—
152C2×C3×D4—94no—
153C22×C3×C4—94no—
154C22×C3×C4—91no—
155C22×C3×C4—92no—
156#239—62no—
157D3×Q8—62no—
158#237—62no—
159#237—64no—
160D3×D4—62no—
161D3×D4—64no—
162#235—22no—
163#235—64no—
164C2×C4×D3—64no—
165C2×C4×D3—62no—
166C3×C4⋊C4—94no—
167C3×C4⋊C4—94no—
168C3×C22⋊C4—98no—
169C22×C32—22no—
170C22×C32—42no—
171C22×C32—64no—
172C22×C32C2×C3×D411yescharacteristic
173C2×C3⋊D3—21no—
174C2×C3⋊D3—21no—
175C2×C3⋊D3C2×C3×D411yescharacteristic
176C2×C3⋊D3C24×C311yescharacteristic
177C2×C3×D3—24no—
178C2×C3×D3—24no—
179C2×C3×D3—44no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—124no—
183C2×C3×D3—124no—
184C2×C3×D3—122no—
185C2×C3×D3—122no—
186D32—24no—
187D32—24no—
188#157—24no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—42no—
192C32×C4—42no—
193C32×C4—64no—
194C32×C4C2×C3×D412yes—
195C3⋊Dic3C2×C3×D412yes—
196C3×Dic3—24no—
197C3×Dic3—24no—
198C3×Dic3—44no—
199C3×Dic3—62no—
200C3×Dic3—62no—
201C2×#41—92no—
202C22⋊Q8—94no—
203#121—94no—
204#121—94no—
205C2×C4⋊C4—91no—
206C33#53711yescharacteristic, complemented, Sylow
207C23×C3—94no—
208C23×C3—91no—
209C22×D3—64no—
210C22×D3—62no—
211C3×Q8—62no—
212C3×Q8—62no—
213C3×Q8—122no—
214C3×Q8—182no—
215C3×D4—62no—
216C3×D4—64no—
217C3×D4—64no—
218C3×D4—62no—
219C3×D4—122no—
220C3×D4—124no—
221C3×D4—184no—
222C3×D4—182no—
223C3×D4—364no—
224C2×C3×C4—22no—
225C2×C3×C4—42no—
226C2×C3×C4—64no—
227C2×C3×C4—64no—
228C2×C3×C4—94no—
229C2×C3×C4—91no—
230C2×C3×C4—92no—
231C2×C3×C4—92no—
232C2×C3×C4—94no—
233C2×C3×C4—124no—
234C2×C3×C4—184no—
235C2×C3×C4—362no—
236C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
237#67—24no—
238#67—64no—
239#67—64no—
240C2×Dic3—62no—
241C2×Dic3—64no—
242D12—22no—
243D12—64no—
244C4×D3—24no—
245C4×D3—64no—
246C4×D3—62no—
247C4×D3—62no—
248C4×D3—62no—
249C4×D3—62no—
250C4×D3—64no—
251Dic6—22no—
252Dic6—64no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—42no—
256C2×C32—42no—
257C2×C32—82no—
258C2×C32—124no—
259C2×C32C22×C3×D411yescharacteristic
260C3⋊D3—21no—
261C3⋊D3C3×H(5,𝔽2)11yescharacteristic
262C3⋊D3C3×2−1+411yescharacteristic
263C3×D3—44no—
264C3×D3—44no—
265C3×D3—62no—
266C3×D3—62no—
267C3×D3—84no—
268C3×D3—122no—
269#41—62no—
270#41—182no—
271#41—182no—
272#41—182no—
273C2×Q8—92no—
274C2×D4—94no—
275C2×D4—92no—
276C2×D4—94no—
277C22×C4—94no—
278C22×C4—91no—
279C22×C4—92no—
280C4⋊C4—94no—
281C4⋊C4—94no—
282C22⋊C4—98no—
283C22×C3—22no—
284C22×C3—42no—
285C22×C3—64no—
286C22×C3—64no—
287C22×C3—91no—
288C22×C3—91no—
289C22×C3—124no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3—184no—
293C22×C3—184no—
294C22×C3C2×GO+(4,𝔽2)11yescharacteristic
295C2×D3—24no—
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—124no—
299C2×D3—124no—
300C2×D3—122no—
301C2×D3—122no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—64no—
307C3×C4—64no—
308C3×C4—92no—
309C3×C4—124no—
310C3×C4—184no—
311C3×C4C2×GO+(4,𝔽2)12yes—
312Dic3—24no—
313Dic3—62no—
314Dic3—62no—
315C32—22no—
316C32—42no—
317C32C3×#53711yescharacteristic, complemented
318C23—94no—
319C23—91no—
320Q8—62no—
321Q8—182no—
322D4—64no—
323D4—62no—
324D4—184no—
325D4—182no—
326D4—364no—
327C2×C4—64no—
328C2×C4—94no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—94no—
333C2×C4—184no—
334C2×C4—362no—
335C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—42no—
339C2×C3—42no—
340C2×C3—82no—
341C2×C3—91no—
342C2×C3—91no—
343C2×C3—124no—
344C2×C3—124no—
345C2×C3—181no—
346C2×C3—244no—
347C2×C3C22×GO+(4,𝔽2)11yescharacteristic
348D3—44no—
349D3—62no—
350D3—62no—
351D3—122no—
352C22—64no—
353C22—91no—
354C22—91no—
355C22—181no—
356C22—181no—
357C22—184no—
358C22—184no—
359C22C2×C3×GO+(4,𝔽2)11yescharacteristic
360C4—64no—
361C4—92no—
362C4—184no—
363C4C2×C3×GO+(4,𝔽2)12yes—
364C3—22no—
365C3—42no—
366C3#1059651311yescharacteristic, direct product factor
367C2—21no—
368C2—91no—
369C2—91no—
370C2—124no—
371C2—181no—
372C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#537
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64306C2semidirect
2. C2×C3×C4⋊#157C2semidirect
3. C3×#64242C2semidirect
4. C3×#64230C2semidirect
5. C3×D32⋊Q8C2semidirect
6. #10596513C3direct (N × H)
22. #64306C2×C3semidirect
23. C2×C4⋊#157C2×C3semidirect
24. #64242C2×C3semidirect
25. #64230C2×C3semidirect
26. D32⋊Q8C2×C3semidirect
32. C2×C33×C4D4semidirect
33. C2×C3×C3⋊Dic3D4semidirect
34. C3×C4×C3⋊D3D4semidirect
103. C2×C32×C4C3×D4semidirect
104. C2×C3⋊Dic3C3×D4semidirect
105. C4×C3⋊D3C3×D4semidirect
206. C33#537semidirect
236. C2×C3×C4GO+(4,𝔽2)semidirect
317. C32C3×#537semidirect
335. C2×C4C3×GO+(4,𝔽2)semidirect
366. C3#10596513direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C3×C4⋊#1572
C3×#642422
C3×#642302
C3×D32⋊Q82
#105965133
C3×#5379