Finite Group Databasebeta

← C3×#10588422 #13141646

Subgroups — C3×#10588422 #13141646

Order 1728.

532 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 102 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×#63710C211yesmaximal, characteristic
2C3×#63497C211yesmaximal, characteristic, complemented
3C3×#63448C212yesmaximal, complemented
4C3×C9×#99C211yesmaximal, characteristic, complemented
5C3×#63365C212yesmaximal, complemented
6C3×#4742—31nomaximal
7#10588422C311yesmaximal, characteristic, direct product factor
8C2×C3×D4×C9C2211yescharacteristic
9C2×C3×#611—21no—
10C2×C3×#611C2212yes—
11C22×C3×Dic9C2211yescharacteristic, complemented
12C3×#3699—22no—
13C3×D4×D9—22no—
14C3×#3697—21no—
15C3×C9×C22⋊C4C2212yescomplemented
16C3×#3676—21no—
17C3×#3676C2211yescharacteristic, complemented
18C3×(C2×D9)⋊C4—22no—
19C3×C4⋊Dic9—22no—
20C3×Dic9⋊C4—21no—
21C3×#1004—31no—
22C3×#935—31no—
23C3×#886—32no—
24#63710C2×C311yescharacteristic
25C32×#99D311yescharacteristic
26C3×#806—32no—
27#63497C2×C311yescharacteristic, complemented
28#63448C2×C312yescomplemented
29C9×#99—21no—
30C9×#99C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×C3×C9D411yescharacteristic
33C23×C3×C9C2311yescharacteristic
34C22×C3×D9—21no—
35C22×C3×D9D412yes—
36C3×D4×C9—22no—
37C2×C3×C4×C9—22no—
38C2×C3×C4×C9D411yescharacteristic
39C3×#611—22no—
40C3×#611—22no—
41C3×#611—22no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9—22no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D412yes—
47C3×D36—21no—
48C3×C4×D9—22no—
49C3×Dic18—21no—
50C3×#458—91no—
51#4742—31no—
52C2×C32×D4C2×D311yescharacteristic
53C2×C3×#67—32no—
54C2×C3×#67—61no—
55C22×C3×Dic3—31no—
56C3×#237—62no—
57C3×D3×D4—62no—
58C3×#235—61no—
59C32×C22⋊C4C2×D312yes—
60C3×#217—31no—
61C3×#217—61no—
62C3×(C2×D3)⋊C4—62no—
63C3×C4⋊Dic3—62no—
64C3×Dic3⋊C4—61no—
65C2×D4×C9—21no—
66C2×D4×C9C22×C311yescharacteristic
67C2×#611—21no—
68C2×#611C22×C312yes—
69C22×Dic9C22×C311yescharacteristic, complemented
70#3699—22no—
71D4×D9—22no—
72#3697—21no—
73C9×C22⋊C4—22no—
74C9×C22⋊C4C22×C312yescomplemented
75#3676—21no—
76#3676C22×C311yescharacteristic, complemented
77(C2×D9)⋊C4—22no—
78C4⋊Dic9—22no—
79Dic9⋊C4—21no—
80C22×C3×C9—21no—
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9C2×D411yescharacteristic
84C22×C3×C9C2×D412yes—
85C2×C3×D9—22no—
86C2×C3×D9—42no—
87C2×C3×D9—81no—
88C3×C4×C9—21no—
89C3×C4×C9—42no—
90C3×Dic9—22no—
91C3×Dic9—41no—
92C3×Dic9—41no—
93C3×H(5,𝔽2)—91no—
94#1004—31no—
95C3×#123—91no—
96C3×#123—92no—
97#935—31no—
98#886—32no—
99C3×#99—21no—
100C3×#99—92no—
101C3×#99D911yescharacteristic, complemented
102C3×#99C3×D311yescharacteristic
103#806—32no—
104C23×C32D1211yescharacteristic
105C23×C32C22×D311yescharacteristic
106C22×C3×D3—32no—
107C22×C3×D3—61no—
108C32×D4—22no—
109C2×C32×C4—22no—
110C2×C32×C4D1211yescharacteristic
111C3×#67—62no—
112C3×#67—62no—
113C3×#67—62no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—62no—
119C3×D12—61no—
120C3×C4×D3—62no—
121C3×Dic6—61no—
122C23×C9—21no—
123C23×C9—21no—
124C23×C9C3×D411yescharacteristic
125C23×C9C23×C311yescharacteristic
126C22×D9—21no—
127C22×D9C3×D412yes—
128D4×C9—22no—
129D4×C9—42no—
130C2×C4×C9—22no—
131C2×C4×C9—21no—
132C2×C4×C9—42no—
133C2×C4×C9C3×D411yescharacteristic
134#611—22no—
135#611—22no—
136#611—22no—
137C2×Dic9—21no—
138C2×Dic9—21no—
139C2×Dic9—21no—
140C2×Dic9—22no—
141C2×Dic9C3×D412yes—
142D36—21no—
143C4×D9—22no—
144Dic18—21no—
145#458—91noSylow
146C2×C3×C9—21no—
147C2×C3×C9—22no—
148C2×C3×C9—41no—
149C2×C3×C9#12011yescharacteristic
150C3×D9—42no—
151C3×#41—182no—
152C3×#41—181no—
153C2×C3×D4—21no—
154C2×C3×D4—92no—
155C2×C3×D4—182no—
156C2×C3×D4—181no—
157C2×C3×D4C2×D911yescharacteristic
158C2×C3×D4C2×C3×D311yescharacteristic
159C22×C3×C4—91no—
160C2×#67—32no—
161C2×#67—61no—
162C22×Dic3—31no—
163#237—62no—
164D3×D4—62no—
165#235—61no—
166C3×C4⋊C4—182no—
167C3×C4⋊C4—181no—
168C3×C22⋊C4—22no—
169C3×C22⋊C4—91no—
170C3×C22⋊C4—182no—
171C3×C22⋊C4—181no—
172C3×C22⋊C4C2×D912yescomplemented
173C3×C22⋊C4C2×C3×D312yes—
174#217—31no—
175#217—61no—
176(C2×D3)⋊C4—62no—
177C4⋊Dic3—62no—
178Dic3⋊C4—61no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32C2×D1211yescharacteristic
183C22×C32D3×D412yes—
184C2×C3×D3—62no—
185C2×C3×D3—122no—
186C2×C3×D3—241no—
187C32×C4—21no—
188C32×C4—42no—
189C3×Dic3—62no—
190C3×Dic3—121no—
191C3×Dic3—121no—
192C22×C9—21no—
193C22×C9—21no—
194C22×C9—22no—
195C22×C9—41no—
196C22×C9—41no—
197C22×C9—41no—
198C22×C9—81no—
199C22×C9—81no—
200C22×C9C2×C3×D411yescharacteristic
201C22×C9C2×C3×D412yes—
202C2×D9—22no—
203C2×D9—42no—
204C2×D9—81no—
205C4×C9—21no—
206C4×C9—41no—
207C4×C9—42no—
208C4×C9—82no—
209Dic9—22no—
210Dic9—41no—
211Dic9—41no—
212H(5,𝔽2)—91no—
213#123—91no—
214#123—92no—
215#99—92no—
216#99C3×D911yescharacteristic, complemented
217C3×C9#45811yescharacteristic, complemented, Sylow
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—92no—
221C23×C3—181no—
222C23×C3D3611yescharacteristic
223C23×C3C22×D911yescharacteristic
224C23×C3C3×D1211yescharacteristic
225C23×C3C22×C3×D311yescharacteristic
226C22×D3—32no—
227C22×D3—61no—
228C3×Q8—181no—
229C3×D4—22no—
230C3×D4—22no—
231C3×D4—42no—
232C3×D4—181no—
233C3×D4—182no—
234C3×D4—182no—
235C3×D4—182no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—42no—
240C2×C3×C4—92no—
241C2×C3×C4—182no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—182no—
246C2×C3×C4D3611yescharacteristic
247C2×C3×C4C3×D1211yescharacteristic
248#67—62no—
249#67—62no—
250#67—62no—
251C2×Dic3—32no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256D12—61no—
257C4×D3—62no—
258Dic6—61no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32#88211yescharacteristic
263C3×D3—122no—
264C2×C9—21no—
265C2×C9—22no—
266C2×C9—21no—
267C2×C9—42no—
268C2×C9—41no—
269C2×C9—41no—
270C2×C9—81no—
271C2×C9C3×#12011yescharacteristic
272D9—42no—
273#41—182no—
274#41—181no—
275C2×D4—92no—
276C2×D4—182no—
277C2×D4—181no—
278C2×D4C2×C3×D911yescharacteristic
279C22×C4—91no—
280C4⋊C4—182no—
281C4⋊C4—181no—
282C22⋊C4—91no—
283C22⋊C4—182no—
284C22⋊C4—181no—
285C22⋊C4C2×C3×D912yescomplemented
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—22no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—182no—
298C22×C3—362no—
299C22×C3—721no—
300C22×C3C2×D3611yescharacteristic
301C22×C3D4×D912yes—
302C22×C3C2×C3×D1211yescharacteristic
303C22×C3C3×D3×D412yes—
304C2×D3—62no—
305C2×D3—122no—
306C2×D3—241no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—41no—
310C3×C4—42no—
311C3×C4—42no—
312C3×C4—82no—
313C3×C4—182no—
314C3×C4—361no—
315C3×C4—361no—
316Dic3—62no—
317Dic3—121no—
318Dic3—121no—
319C32#474211yescharacteristic
320C9—21no—
321C9C3×#45811yescharacteristic, complemented
322C23—92no—
323C23—181no—
324C23C3×D3611yescharacteristic
325C23C22×C3×D911yescharacteristic
326Q8—181no—
327D4—22no—
328D4—181no—
329D4—182no—
330D4—182no—
331D4—182no—
332C2×C4—22no—
333C2×C4—92no—
334C2×C4—182no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—182no—
339C2×C4C3×D3611yescharacteristic
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—42no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—362no—
351C2×C3#6344411yescharacteristic
352C2×C3C3×#88211yescharacteristic
353D3—122no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—182no—
358C22—362no—
359C22—721no—
360C22C2×C3×D3611yescharacteristic
361C22C3×D4×D912yes—
362C4—21no—
363C4—42no—
364C4—182no—
365C4—361no—
366C4—361no—
367C3—21no—
368C3#1058842211yescharacteristic, direct product factor
369C3C3×#474211yescharacteristic
370C2—22no—
371C2—21no—
372C2—41no—
373C2—362no—
374C2C3×#6344411yescharacteristic
subgroupG/H
centerC2×C3#63444
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#99C2
socleC2×C32#882
prime radicalC3×#63710C2
hypercenterC3×#99D9
hypocenterC9C3×#458
LCSG ▷ C23×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. C3×#63497C2semidirect
3. C3×#63448C2semidirect
4. C3×C9×#99C2semidirect
5. C3×#63365C2semidirect
7. #10588422C3direct (N × H)
11. C22×C3×Dic9C22semidirect
15. C3×C9×C22⋊C4C22semidirect
17. C3×#3676C22semidirect
27. #63497C2×C3semidirect
28. #63448C2×C3semidirect
30. C9×#99C2×C3semidirect
31. #63365C2×C3semidirect
69. C22×Dic9C22×C3semidirect
74. C9×C22⋊C4C22×C3semidirect
76. #3676C22×C3semidirect
101. C3×#99D9semidirect
172. C3×C22⋊C4C2×D9semidirect
216. #99C3×D9semidirect
217. C3×C9#458semidirect
285. C22⋊C4C2×C3×D9semidirect
321. C9C3×#458semidirect
368. C3#10588422direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637102
C3×#634972
C3×#634482
C3×C9×#992
C3×#633652
C3×#47423
#105884223