Finite Group Databasebeta

← C3×#10593501 #13166875

Subgroups — C3×#10593501 #13166875

Order 1728.

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

normal subgroups only

This group has 410 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×#63982C211yesmaximal, characteristic
2C3×#63957C211yesmaximal, characteristic
3C3×C22⋊#622C211yesmaximal, characteristic
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63740C211yesmaximal, characteristic, complemented
8#10593501C311yesmaximal, characteristic, direct product factor
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C32×#217—21no—
14C32×#217—21no—
15C3×#3724C2211yescharacteristic
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×Dic32—21no—
20C3×#878—61no—
21C3×#877—61no—
22#63982C2×C311yescharacteristic
23#63957C2×C311yescharacteristic
24C22⋊#622C2×C311yescharacteristic
25#63743C2×C311yescharacteristic, complemented
26#63742C2×C311yescharacteristic, complemented
27#63740C2×C311yescharacteristic, complemented
28#63740C2×C311yescharacteristic, complemented
29C23×C33D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C3×#627—21no—
39C3×#627—21no—
40C3×#622—21no—
41C3×#622—21no—
42C3×#485—91nomaximal
43C22×C3⋊Dic3C22×C311yescharacteristic
44C22×C3×Dic3—61no—
45C22×C3×Dic3—61no—
46C2×C3×Dic6—61no—
47C2×#627C22×C311yescharacteristic
48C2×#622C22×C311yescharacteristic
49C2×#622C22×C311yescharacteristic
50C32×C22⋊C4—61no—
51C32×C22⋊C4—61no—
52C3×#217—21no—
53C3×#217—21no—
54C3×#217—21no—
55C3×#217—21no—
56C3×#217—41no—
57C3×#217—41no—
58C3×C4⋊Dic3—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×C4×Dic3—61no—
64C3×C4×Dic3—61no—
65#3724C22×C311yescharacteristic
66#3724C22×C311yescharacteristic
67#3724C22×C311yescharacteristic
68Dic3⋊Dic3—21no—
69Dic32—21no—
70C22×C33—21no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33C2×D411yescharacteristic
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3#4111yescharacteristic
76C3×C3⋊Dic3#4111yescharacteristic
77C32×Dic3—41no—
78C32×Dic3—41no—
79C32×Dic3—41no—
80C32×Dic3—41no—
81C3×C22⋊Q8—91no—
82C3×#117—91no—
83#878—61no—
84#877—61no—
85C3×#107—91no—
86C3×#106—91no—
87C3×#103—91no—
88C3×#103—91no—
89C3×C22⋊C8—91no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—41no—
93C23×C32—41no—
94C23×C32C3×D411yescharacteristic
95C2×C32×C4—61no—
96C2×C32×C4—61no—
97C2×C32×C4—61no—
98C2×C32×C4—61no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3C3×D411yescharacteristic
102C2×C3⋊Dic3C23×C311yescharacteristic
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—41no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—41no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—121no—
122C2×C3×Dic3—121no—
123C3×Dic6—121no—
124C3×Dic6—121no—
125#627—21no—
126#627—21no—
127#622—21no—
128#622—21no—
129#485—91noSylow
130C2×C33—41no—
131C2×C33#12311yescharacteristic
132C2×C33#13511yescharacteristic
133C2×C33#13711yescharacteristic
134C2×C3×Q8—91no—
135C22×C3×C4—91no—
136C22×Dic3—61no—
137C22×Dic3—61no—
138C2×Dic6—61no—
139C2×C3×C8—91no—
140C2×C3×C8—91no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—181no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—61no—
147C3×C22⋊C4—61no—
148C3×C22⋊C4—61no—
149C3×C22⋊C4—121no—
150C3×C22⋊C4—121no—
151C3×C42—181no—
152#217—21no—
153#217—21no—
154C4⋊Dic3—61no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—61no—
158Dic3⋊C4—61no—
159C4×Dic3—61no—
160C4×Dic3—61no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32—81no—
177C22×C32—81no—
178C22×C32—81no—
179C22×C32—81no—
180C22×C32C2×C3×D411yescharacteristic
181C32×C4—121no—
182C32×C4—121no—
183C32×C4—121no—
184C32×C4—121no—
185C3⋊Dic3—21no—
186C3⋊Dic3C3×#4111yescharacteristic
187C3⋊Dic3C3×#4111yescharacteristic
188C3×Dic3—41no—
189C3×Dic3—41no—
190C3×Dic3—41no—
191C3×Dic3—41no—
192C3×Dic3—41no—
193C3×Dic3—41no—
194C3×Dic3—41no—
195C3×Dic3—41no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—62no—
201C3×Dic3—81no—
202C3×Dic3—81no—
203C3×Dic3—81no—
204C3×Dic3—81no—
205C3×Dic3—121no—
206C22⋊Q8—91no—
207#117—91no—
208#107—91no—
209#106—91no—
210#103—91no—
211#103—91no—
212C22⋊C8—91no—
213C33#48511yescharacteristic, complemented, Sylow
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3GO+(4,𝔽2)11yescharacteristic
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—121no—
232C2×C3×C4—121no—
233C2×C3×C4—121no—
234C2×C3×C4—121no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—121no—
248C2×Dic3—121no—
249Dic6—121no—
250Dic6—121no—
251C3×C8—181no—
252C3×C8—181no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32—82no—
269C2×C32—82no—
270C2×C32C3×#12311yescharacteristic
271C2×C32C3×#13511yescharacteristic
272C2×C32C3×#13711yescharacteristic
273C2×Q8—91no—
274C22×C4—91no—
275C2×C8—91no—
276C2×C8—91no—
277C4⋊C4—91no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—181no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C42—181no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3C2×GO+(4,𝔽2)11yescharacteristic
304C3×C4—91no—
305C3×C4—91no—
306C3×C4—121no—
307C3×C4—121no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—121no—
312C3×C4—121no—
313C3×C4—121no—
314C3×C4—181no—
315C3×C4—241no—
316C3×C4—241no—
317C3×C4—241no—
318C3×C4—241no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323Dic3—61no—
324Dic3—61no—
325Dic3—61no—
326Dic3—61no—
327Dic3—62no—
328Dic3—121no—
329C32—21no—
330C32—21no—
331C32—41no—
332C32—41no—
333C32C3×#48511yescharacteristic, complemented
334C23C3×GO+(4,𝔽2)11yescharacteristic
335Q8—181no—
336Q8—181no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C8—181no—
346C8—181no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—42no—
354C2×C3—42no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—82no—
363C2×C3—82no—
364C2×C3#6423311yescharacteristic
365C2×C3#6423711yescharacteristic
366C2×C3#6423911yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22C2×C3×GO+(4,𝔽2)11yescharacteristic
371C4—91no—
372C4—91no—
373C4—121no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4—181no—
378C3—21no—
379C3—21no—
380C3—41no—
381C3—41no—
382C3#1059350111yescharacteristic, direct product factor
383C2—41no—
384C2C3×#6423311yescharacteristic
385C2C3×#6423711yescharacteristic
386C2C3×#6423911yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#485
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C3×#63743C2semidirect
5. C3×#63742C2semidirect
6. C3×#63740C2semidirect
7. C3×#63740C2semidirect
8. #10593501C3direct (N × H)
25. #63743C2×C3semidirect
26. #63742C2×C3semidirect
27. #63740C2×C3semidirect
28. #63740C2×C3semidirect
213. C33#485semidirect
333. C32C3×#485semidirect
382. C3#10593501direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639822
C3×#639572
C3×C22⋊#6222
C3×#637432
C3×#637422
C3×#637402
#105935013
C3×#4859