Finite Group Databasebeta

← C3×#10593492 #13166859

Subgroups — C3×#10593492 #13166859

Order 1728.

522 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 70 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×#63980C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic, complemented
3C3×C22⋊#622C211yesmaximal, characteristic, complemented
4C3×#63743C212yesmaximal, complemented
5C3×#63739C212yesmaximal, complemented
6#10593492C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×#67—21no—
9C22×C32×Dic3—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3×#622C2212yes—
12C3×#3724—21no—
13C3×#3724C2212yes—
14C3×#3724C2211yescharacteristic
15C3×(C2×D3)⋊Dic3—21no—
16C3×#939—61no—
17C2×C3×Dic3⋊C4—61no—
18#63980C2×C311yescharacteristic
19C2×#3724C2×C311yescharacteristic, complemented
20C22⋊#622C2×C311yescharacteristic, complemented
21#63743C2×C312yescomplemented
22#63739C2×C312yescomplemented
23C23×C33D411yescharacteristic
24C22×C32×D3—21no—
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C32×#67—42no—
29C2×C32×Dic3—21no—
30C2×C32×Dic3—22no—
31C2×C32×Dic3—42no—
32C3×#625—22no—
33C3×#622—22no—
34C3×#480—91nomaximal
35C2×C32×D4—61no—
36C22×C32×C4—61no—
37C22×C3⋊Dic3C22×C311yescharacteristic
38C2×C3×#67—21no—
39C2×C3×#67—21no—
40C2×C3×#67—41no—
41C2×C3×#67—61no—
42C22×C3×Dic3—21no—
43C22×C3×Dic3—21no—
44C22×C3×Dic3—41no—
45C22×C3×Dic3—61no—
46C22×C3×Dic3—61no—
47C2×#625C22×C311yescharacteristic
48C2×#622C22×C312yes—
49C3×#217—61no—
50C3×(C2×D3)⋊C4—61no—
51C3×Dic3⋊C4—62no—
52C3×Dic3⋊C4—61no—
53C3×Dic3⋊C4—62no—
54#3724—21no—
55#3724C22×C312yes—
56#3724C22×C311yescharacteristic
57(C2×D3)⋊Dic3—21no—
58C22×C33—21no—
59C22×C33—21no—
60C22×C33D812yes—
61C22×C33C2×D411yescharacteristic
62C2×C32×D3—41no—
63C2×C32×D3—41no—
64C2×C32×D3—41no—
65C3×C3⋊Dic3—21no—
66C3×C3⋊Dic3#4112yes—
67C32×Dic3—42no—
68C32×Dic3—41no—
69C3×#121—91no—
70C2×C3×C4⋊C4—91no—
71#939—61no—
72C2×Dic3⋊C4—61no—
73C3×#107—92no—
74C3×#102—92no—
75C3×C22⋊C8—91no—
76C23×C32—21no—
77C23×C32—21no—
78C23×C32—41no—
79C23×C32—41no—
80C23×C32—61no—
81C23×C32C3×D411yescharacteristic
82C22×C3×D3—21no—
83C22×C3×D3—21no—
84C22×C3×D3—41no—
85C32×D4—122no—
86C2×C32×C4—62no—
87C2×C32×C4—61no—
88C2×C32×C4—122no—
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3C3×D411yescharacteristic
91C2×C3⋊Dic3C23×C311yescharacteristic
92C3×#67—42no—
93C3×#67—44no—
94C3×#67—42no—
95C3×#67—122no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—42no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—42no—
103C2×C3×Dic3—44no—
104C2×C3×Dic3—42no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—122no—
111#625—22no—
112#622—22no—
113#480—91noSylow
114C2×C33—22no—
115C2×C33#12311yescharacteristic
116C2×C33C2×D811yescharacteristic
117C2×C33#13711yescharacteristic
118C32×D3—81no—
119C2×C3×D4—61no—
120C2×C3×D4—61no—
121C2×C3×D4—91no—
122C2×C3×D4—121no—
123C22×C3×C4—61no—
124C22×C3×C4—61no—
125C22×C3×C4—91no—
126C22×C3×C4—121no—
127C2×#67—21no—
128C2×#67—61no—
129C22×Dic3—21no—
130C22×Dic3—61no—
131C22×Dic3—61no—
132C2×C3×C8—92no—
133C3×C4⋊C4—92no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—181no—
136C3×C22⋊C4—181no—
137#217—61no—
138(C2×D3)⋊C4—61no—
139Dic3⋊C4—62no—
140Dic3⋊C4—61no—
141Dic3⋊C4—62no—
142C22×C32—22no—
143C22×C32—22no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—22no—
150C22×C32—42no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32—42no—
154C22×C32—41no—
155C22×C32—41no—
156C22×C32—42no—
157C22×C32—42no—
158C22×C32—81no—
159C22×C32—81no—
160C22×C32—121no—
161C22×C32—121no—
162C22×C32—121no—
163C22×C32C3×D812yes—
164C22×C32C2×C3×D411yescharacteristic
165C2×C3×D3—41no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C2×C3×D3—42no—
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—41no—
172C2×C3×D3—81no—
173C2×C3×D3—81no—
174C32×C4—122no—
175C32×C4—121no—
176C3⋊Dic3—21no—
177C3⋊Dic3C3×#4112yes—
178C3×Dic3—42no—
179C3×Dic3—41no—
180C3×Dic3—44no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—41no—
184C3×Dic3—62no—
185C3×Dic3—62no—
186C3×Dic3—62no—
187C3×Dic3—121no—
188#121—91no—
189C2×C4⋊C4—91no—
190#107—92no—
191#102—92no—
192C22⋊C8—91no—
193C33#48011yescharacteristic, complemented, Sylow
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—61no—
199C23×C3—61no—
200C23×C3—121no—
201C23×C3GO+(4,𝔽2)11yescharacteristic
202C22×D3—21no—
203C3×D4—122no—
204C3×D4—124no—
205C3×D4—122no—
206C3×D4—182no—
207C2×C3×C4—62no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—62no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—122no—
214C2×C3×C4—121no—
215C2×C3×C4—122no—
216C2×C3×C4—124no—
217C2×C3×C4—122no—
218C2×C3×C4—182no—
219#67—42no—
220#67—122no—
221C2×Dic3—22no—
222C2×Dic3—21no—
223C2×Dic3—42no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—64no—
229C2×Dic3—122no—
230C3×C8—182no—
231C2×C32—24no—
232C2×C32—22no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—42no—
246C2×C32—44no—
247C2×C32—82no—
248C2×C32—241no—
249C2×C32C3×#12311yescharacteristic
250C2×C32C2×C3×D811yescharacteristic
251C2×C32C3×#13711yescharacteristic
252C3×D3—81no—
253C3×D3—82no—
254C3×D3—81no—
255C2×D4—61no—
256C2×D4—91no—
257C22×C4—61no—
258C22×C4—91no—
259C2×C8—92no—
260C4⋊C4—92no—
261C4⋊C4—91no—
262C4⋊C4—181no—
263C22⋊C4—181no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—42no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3—121no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3—122no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3—241no—
290C22×C3—241no—
291C22×C3#377412yes—
292C22×C3C2×GO+(4,𝔽2)11yescharacteristic
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C3×C4—92no—
297C3×C4—122no—
298C3×C4—121no—
299C3×C4—124no—
300C3×C4—122no—
301C3×C4—122no—
302C3×C4—121no—
303C3×C4—181no—
304Dic3—42no—
305Dic3—41no—
306Dic3—62no—
307Dic3—62no—
308Dic3—62no—
309Dic3—121no—
310C32—21no—
311C32—21no—
312C32—41no—
313C32—41no—
314C32C3×#48011yescharacteristic, complemented
315C23—61no—
316C23C3×GO+(4,𝔽2)11yescharacteristic
317D4—122no—
318D4—182no—
319C2×C4—61no—
320C2×C4—62no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—122no—
324C2×C4—182no—
325C8—182no—
326C2×C3—24no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—42no—
335C2×C3—44no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—82no—
343C2×C3—241no—
344C2×C3—242no—
345C2×C3—241no—
346C2×C3#6423311yescharacteristic
347C2×C3C2×#377411yescharacteristic
348C2×C3#6423911yescharacteristic
349D3—81no—
350C22—21no—
351C22—21no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22C3×#377412yes—
356C22C2×C3×GO+(4,𝔽2)11yescharacteristic
357C4—92no—
358C4—122no—
359C4—121no—
360C4—181no—
361C3—21no—
362C3—21no—
363C3—41no—
364C3—41no—
365C3#1059349211yescharacteristic, direct product factor
366C2—22no—
367C2—241no—
368C2C3×#6423311yescharacteristic
369C2C2×C3×#377411yescharacteristic
370C2C3×#6423911yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC3×#63980C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#480
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
2. C2×C3×#3724C2semidirect
3. C3×C22⋊#622C2semidirect
4. C3×#63743C2semidirect
5. C3×#63739C2semidirect
6. #10593492C3direct (N × H)
19. C2×#3724C2×C3semidirect
20. C22⋊#622C2×C3semidirect
21. #63743C2×C3semidirect
22. #63739C2×C3semidirect
193. C33#480semidirect
314. C32C3×#480semidirect
365. C3#10593492direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639802
C2×C3×#37242
C3×C22⋊#6222
C3×#637432
C3×#637392
#105934923
C3×#4809