Finite Group Databasebeta

← C3×#10593452 #13166821

Subgroups — C3×#10593452 #13166821

Order 1728.

564 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 62 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
2C3×#64239C212yesmaximal, complemented
3C3×#63840C211yesmaximal, characteristic, complemented
4C3×#63780C211yesmaximal, characteristic, complemented
5C3×#63741C212yesmaximal, complemented
6#10593452C311yesmaximal, characteristic, direct product factor
7C2×C3×#634C2211yescharacteristic
8C32×#235—21no—
9C2×C3×#627C2211yescharacteristic
10C3×#3805—21no—
11C3×#3805C2212yes—
12C3×#3795—22no—
13C3×D3×Dic6—22no—
14C3×#3792C2212yes—
15C3×#3776—22no—
16C3×#3775—22no—
17C32×Dic3⋊C4—21no—
18C3×Dic3⋊Dic3—21no—
19C3×Dic32C2211yescharacteristic
20C3×#1010—61no—
21#64298C2×C311yescharacteristic
22#64239C2×C312yescomplemented
23C3×#888—61no—
24#63840C2×C311yescharacteristic, complemented
25#63780C2×C311yescharacteristic, complemented
26#63741C2×C312yescomplemented
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×#634—22no—
31C32×#67—22no—
32C2×C32×Dic3—21no—
33C2×C32×Dic3—21no—
34C32×D12—21no—
35C32×C4×D3—22no—
36C32×Dic6—21no—
37C3×#627—21no—
38C3×#627—22no—
39C3×#627D412yes—
40C3×#625—21no—
41C3×#625D412yes—
42C3×D3×Dic3—22no—
43C3×D3×Dic3—22no—
44C3×#622—22no—
45C3×#456—91nomaximal
46C32×#41—61no—
47C2×#634C22×C311yescharacteristic
48C3×D3×Q8—62no—
49C3×#237—62no—
50C3×#237—62no—
51C3×#237—62no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—41no—
55C3×#235—62no—
56C2×C3×Dic6—62no—
57C2×C3×Dic6—61no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—61no—
60C2×#627C22×C311yescharacteristic
61#3805—21no—
62#3805C22×C312yes—
63#3795—22no—
64D3×Dic6—22no—
65#3792C22×C312yes—
66#3776—22no—
67#3775—22no—
68C32×C4⋊C4—61no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—21no—
72C3×Dic3⋊C4—41no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—61no—
75Dic3⋊Dic3—21no—
76Dic32C22×C311yescharacteristic
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—22no—
79C33×C4—21no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3C2×D412yes—
82C32×Dic3—22no—
83C32×Dic3—22no—
84C32×Dic3—41no—
85C3×2−1+4—91no—
86#1010—61no—
87C3×#137—92no—
88C3×C4⋊Q8—91no—
89#888—61no—
90C3×#104—92no—
91C3×#101—91no—
92C32×Q8—61no—
93C32×D4—61no—
94C32×D4—62no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4—62no—
100C2×C32×C4—61no—
101C2×C32×C4—61no—
102C2×C32×C4C3×D411yescharacteristic
103C2×C3⋊Dic3C3×D411yescharacteristic
104C2×C3⋊Dic3C23×C311yescharacteristic
105#634—22no—
106C3×#67—22no—
107C3×#67—22no—
108C3×#67—42no—
109C3×#67—62no—
110C3×#67—62no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C3×D12—21no—
124C3×D12—21no—
125C3×D12—41no—
126C3×C4×D3—22no—
127C3×C4×D3—22no—
128C3×C4×D3—42no—
129C3×C4×D3—62no—
130C3×C4×D3—62no—
131C3×Dic6—21no—
132C3×Dic6—21no—
133C3×Dic6—41no—
134C3×Dic6—62no—
135C3×Dic6—62no—
136C3×Dic6—62no—
137C3×Dic6—62no—
138C3×Dic6—122no—
139C3×Dic6—122no—
140#627—21no—
141#627—22no—
142#627C3×D412yes—
143#625—21no—
144#625C3×D412yes—
145D3×Dic3—22no—
146D3×Dic3—22no—
147#622—22no—
148#456—91noSylow
149C2×C33—21no—
150C2×C33#12011yescharacteristic
151C32×D3—42no—
152C3×#41—61no—
153C3×#41—61no—
154C3×#41—92no—
155C3×#41—121no—
156C3×#41—182no—
157C3×#41—181no—
158C2×C3×Q8—91no—
159C2×C3×Q8—91no—
160C2×C3×Q8—182no—
161D3×Q8—62no—
162#237—62no—
163#237—62no—
164#237—62no—
165#235—21no—
166#235—62no—
167C2×Dic6—61no—
168C2×Dic6—61no—
169C2×Dic6—61no—
170C2×Dic6—62no—
171C3×Q16—182no—
172C3×SD16—182no—
173C3×M16—92no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—121no—
177C3×C4⋊C4—181no—
178C3×C42—91no—
179C4⋊Dic3—61no—
180Dic3⋊C4—21no—
181Dic3⋊C4—61no—
182C4×Dic3—61no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—62no—
188C22×C32C2×C3×D411yescharacteristic
189C2×C3×D3—22no—
190C2×C3×D3—22no—
191C2×C3×D3—42no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4—41no—
198C32×C4—62no—
199C32×C4—62no—
200C32×C4—81no—
201C32×C4—121no—
202C3⋊Dic3—21no—
203C3⋊Dic3C2×C3×D412yes—
204C3×Dic3—22no—
205C3×Dic3—22no—
206C3×Dic3—22no—
207C3×Dic3—22no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C3×Dic3—41no—
211C3×Dic3—41no—
212C3×Dic3—62no—
213C3×Dic3—62no—
214C3×Dic3—62no—
215C3×Dic3—81no—
216C3×Dic3—121no—
2172−1+4—91no—
218#137—92no—
219C4⋊Q8—91no—
220#104—92no—
221#101—91no—
222C33#45611yescharacteristic, complemented, Sylow
223C3×Q8—61no—
224C3×Q8—61no—
225C3×Q8—92no—
226C3×Q8—121no—
227C3×Q8—182no—
228C3×Q8—181no—
229C3×Q8—182no—
230C3×D4—61no—
231C3×D4—61no—
232C3×D4—62no—
233C3×D4—62no—
234C3×D4—92no—
235C3×D4—121no—
236C3×D4—122no—
237C3×D4—181no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—91no—
249C2×C3×C4—91no—
250C2×C3×C4—122no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×C3×C4GO+(4,𝔽2)11yescharacteristic
256#67—22no—
257#67—62no—
258#67—62no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267D12—21no—
268C4×D3—22no—
269C4×D3—62no—
270C4×D3—62no—
271Dic6—21no—
272Dic6—62no—
273Dic6—62no—
274Dic6—62no—
275Dic6—62no—
276Dic6—122no—
277Dic6—122no—
278C3×C8—182no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—81no—
290C2×C32—122no—
291C2×C32C3×#12011yescharacteristic
292C3×D3—42no—
293C3×D3—42no—
294C3×D3—82no—
295#41—61no—
296#41—92no—
297#41—182no—
298#41—181no—
299C2×Q8—91no—
300C2×Q8—91no—
301C2×Q8—182no—
302Q16—182no—
303SD16—182no—
304M16—92no—
305C4⋊C4—61no—
306C4⋊C4—181no—
307C42—91no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—122no—
315C22×C3C2×GO+(4,𝔽2)11yescharacteristic
316C2×D3—22no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—41no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—62no—
324C3×C4—62no—
325C3×C4—62no—
326C3×C4—62no—
327C3×C4—81no—
328C3×C4—92no—
329C3×C4—122no—
330C3×C4—122no—
331C3×C4—121no—
332C3×C4—121no—
333C3×C4—181no—
334C3×C4—241no—
335Dic3—22no—
336Dic3—22no—
337Dic3—41no—
338Dic3—62no—
339Dic3—62no—
340Dic3—62no—
341Dic3—121no—
342C32—21no—
343C32—21no—
344C32—41no—
345C32—41no—
346C32C3×#45611yescharacteristic, complemented
347Q8—61no—
348Q8—92no—
349Q8—182no—
350Q8—181no—
351Q8—182no—
352D4—61no—
353D4—62no—
354D4—92no—
355D4—181no—
356C2×C4—62no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—91no—
360C2×C4—91no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4C3×GO+(4,𝔽2)11yescharacteristic
364C8—182no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—81no—
376C2×C3—122no—
377C2×C3—122no—
378C2×C3—242no—
379C2×C3#6424111yescharacteristic
380D3—42no—
381C22—62no—
382C22C2×C3×GO+(4,𝔽2)11yescharacteristic
383C4—21no—
384C4—62no—
385C4—62no—
386C4—92no—
387C4—121no—
388C4—181no—
389C3—21no—
390C3—21no—
391C3—41no—
392C3—41no—
393C3#1059345211yescharacteristic, direct product factor
394C2—21no—
395C2—122no—
396C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33#120
prime radicalC3×#64298C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#456
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64239C2semidirect
3. C3×#63840C2semidirect
4. C3×#63780C2semidirect
5. C3×#63741C2semidirect
6. #10593452C3direct (N × H)
22. #64239C2×C3semidirect
24. #63840C2×C3semidirect
25. #63780C2×C3semidirect
26. #63741C2×C3semidirect
222. C33#456semidirect
346. C32C3×#456semidirect
393. C3#10593452direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642982
C3×#642392
C3×#638402
C3×#637802
C3×#637412
#105934523
C3×#4569