Finite Group Databasebeta

← C3×#10591188 #13158368

Subgroups — C3×#10591188 #13158368

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C32×#924C212yesmaximal
3C2×C3×#3718C211yesmaximal, characteristic, complemented
4C3×#63576C211yesmaximal, characteristic, complemented
5C3×#63565C212yesmaximal, complemented
6C3×#5051—32nomaximal
7#10591188C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C2×C3×C3⋊(C3⋊C8)—22no—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×#217—22no—
14C32×C4⋊Dic3C2212yes—
15C32×Dic3⋊C4—22no—
16C3×#3718—22no—
17C32×C22⋊Q8—32no—
18C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
19C2×C3×#216—32no—
20C3×#924—22no—
21C3×#924D312yes—
22C3×#924C2×C312yes—
23C22×C3×C3⋊C8—32no—
24C22×C3×C3⋊C8—61no—
25C2×#3718C2×C311yescharacteristic, complemented
26C3×#835—32no—
27C3×#810—32no—
28C3×#807—32no—
29#63576C2×C311yescharacteristic, complemented
30#63565C2×C312yescomplemented
31C23×C33D411yescharacteristic
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C32×Dic3—22no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C3×C3⋊(C3⋊C8)—21no—
41C3×C3⋊(C3⋊C8)D412yes—
42C3×#467—91no—
43#5051—32no—
44C2×C32×Q8—32no—
45C22×C32×C4—21no—
46C22×C32×C4—22no—
47C22×C32×C4—41no—
48C22×C32×C4C2×D312yes—
49C22×C32×C4C22×C311yescharacteristic
50C2×C3×Dic6—22no—
51C2×C3×Dic6C2×D312yes—
52C2×C3×Dic6C22×C312yes—
53C32×C4⋊C4—32no—
54C32×C4⋊C4—62no—
55C32×C22⋊C4—62no—
56C2×C3⋊(C3⋊C8)—22no—
57C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
58C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
59C3×#217—24no—
60C3×#217—22no—
61C3×#217—22no—
62C3×#216—64no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C22×C312yes—
66C3×Dic3⋊C4—24no—
67C3×Dic3⋊C4—22no—
68C3×Dic3⋊C4—22no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—64no—
72C2×C3×C3⋊C8—61no—
73C2×C3×C3⋊C8—61no—
74C2×C3×C3⋊C8—64no—
75#3718—22no—
76C22×C33—21no—
77C22×C33—21no—
78C22×C33Q1612yescomplemented
79C22×C33C2×D411yescharacteristic
80C33×C4—21no—
81C33×C4C2×D411yescharacteristic
82C33×C4#4111yescharacteristic
83C32×Dic3—42no—
84C32×Dic3—42no—
85C2×C3×Q16—91no—
86C22×C3×C8—91no—
87C3×C22⋊Q8—32no—
88C3×C22⋊Q8—32no—
89C3×C22⋊Q8—62no—
90C2×#216—32no—
91#924C3×D312yes—
92C22×C3⋊C8—32no—
93C22×C3⋊C8—61no—
94C3×#107—91no—
95C3×#103—92no—
96#835—32no—
97#810—32no—
98#807—32no—
99C23×C32—21no—
100C23×C32—22no—
101C23×C32—41no—
102C23×C32#6712yes—
103C23×C32C3×D411yescharacteristic
104C32×Q8—62no—
105C32×Q8—62no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—42no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—42no—
122C2×C32×C4—62no—
123C2×C32×C4—62no—
124C2×C32×C4#6712yes—
125C2×C32×C4C3×D411yescharacteristic
126C2×C32×C4C22×D312yes—
127C2×C32×C4C23×C311yescharacteristic
128C2×C3×Dic3—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C3×Dic6—22no—
135C3×Dic6—22no—
136C3×Dic6—22no—
137C3×Dic6—22no—
138C3×Dic6—42no—
139C3×Dic6—42no—
140C3⋊(C3⋊C8)—21no—
141C3⋊(C3⋊C8)C3×D412yes—
142C3×C3⋊C8—34no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—62no—
146#467—91noSylow
147C2×C33—22no—
148C2×C33#12111yescharacteristic
149C2×C33C2×Q1611yescharacteristic
150C2×C33#13511yescharacteristic
151C2×C3×Q8—32no—
152C2×C3×Q8—32no—
153C2×C3×Q8—62no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—41no—
157C22×C3×C4D3211yescharacteristic
158C22×C3×C4C2×C3×D312yes—
159C2×Dic6C2×C3×D312yes—
160C3×Q16—182no—
161C2×C3×C8—91no—
162C2×C3×C8—91no—
163C2×C3×C8—182no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—62no—
167C3×C4⋊C4—64no—
168C3×C4⋊C4—62no—
169C3×C4⋊C4—62no—
170C3×C22⋊C4—64no—
171C3×C22⋊C4—62no—
172C3×C22⋊C4—62no—
173#217—22no—
174#216—64no—
175C4⋊Dic3C2×C3×D312yes—
176Dic3⋊C4—22no—
177C2×C3⋊C8—32no—
178C2×C3⋊C8—32no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—64no—
182C2×C3⋊C8—64no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—24no—
191C22×C32—21no—
192C22×C32—22no—
193C22×C32—42no—
194C22×C32—41no—
195C22×C32—42no—
196C22×C32—42no—
197C22×C32—42no—
198C22×C32—42no—
199C22×C32#21614yescomplemented
200C22×C32C3×Q1612yescomplemented
201C22×C32C2×#6712yes—
202C22×C32C2×C3×D411yescharacteristic
203C32×C4—21no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—42no—
213C32×C4—42no—
214C32×C4—122no—
215C32×C4—122no—
216C32×C4D3×D412yes—
217C32×C4#23712yes—
218C32×C4C2×C3×D411yescharacteristic
219C32×C4C3×#4111yescharacteristic
220C3×Dic3—44no—
221C3×Dic3—44no—
222C3×Dic3—42no—
223C3×Dic3—42no—
224C3×Dic3—42no—
225C3×Dic3—42no—
226C2×Q16—91no—
227C22×C8—91no—
228C22⋊Q8—32no—
229#107—91no—
230#103—92no—
231C33#46711yescharacteristic, complemented, Sylow
232C23×C3—21no—
233C23×C3—22no—
234C23×C3—41no—
235C23×C3#62511yescharacteristic
236C23×C3C3×#6712yes—
237C3×Q8—62no—
238C3×Q8—62no—
239C3×Q8—62no—
240C3×Q8—62no—
241C3×Q8—122no—
242C3×Q8—122no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—64no—
260C2×C3×C4—64no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4#62511yescharacteristic
266C2×C3×C4C3×#6712yes—
267C2×C3×C4C2×D3211yescharacteristic
268C2×C3×C4C22×C3×D312yes—
269C2×Dic3—22no—
270C2×Dic3—22no—
271Dic6—22no—
272Dic6—22no—
273C3×C8—92no—
274C3×C8—181no—
275C3⋊C8—34no—
276C3⋊C8—62no—
277C3⋊C8—62no—
278C3⋊C8—62no—
279C2×C32—22no—
280C2×C32—24no—
281C2×C32—24no—
282C2×C32—22no—
283C2×C32—22no—
284C2×C32—22no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—44no—
292C2×C32—44no—
293C2×C32#93912yes—
294C2×C32C2×#21612yes—
295C2×C32#95012yes—
296C2×C32C3×#12111yescharacteristic
297C2×C32C2×C3×Q1611yescharacteristic
298C2×C32C3×#13511yescharacteristic
299C2×Q8—32no—
300C22×C4C3×D3211yescharacteristic
301Q16—182no—
302C2×C8—91no—
303C2×C8—91no—
304C2×C8—182no—
305C4⋊C4—32no—
306C4⋊C4—62no—
307C22⋊C4—62no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—24no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—41no—
323C22×C3—42no—
324C22×C3#371812yescomplemented
325C22×C3C3×#21614yescomplemented
326C22×C3C2×#62511yescharacteristic
327C22×C3C2×C3×#6712yes—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—124no—
340C3×C4—124no—
341C3×C4—122no—
342C3×C4—122no—
343C3×C4—122no—
344C3×C4—122no—
345C3×C4#380511yescharacteristic
346C3×C4#381111yescharacteristic
347C3×C4C3×D3×D412yes—
348C3×C4C3×#23712yes—
349Dic3—42no—
350Dic3—42no—
351C32—22no—
352C32—21no—
353C32—41no—
354C32#505112yescomplemented
355C32C3×#46711yescharacteristic, complemented
356C23C3×#62511yescharacteristic
357Q8—62no—
358Q8—62no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4C3×#62511yescharacteristic
364C2×C4C2×C3×D3211yescharacteristic
365C8—92no—
366C8—181no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—22no—
370C2×C3—21no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—44no—
377C2×C3—44no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3#6382211yescharacteristic
382C2×C3C2×#371811yescharacteristic
383C2×C3#6398011yescharacteristic
384C2×C3C3×#93912yes—
385C2×C3C2×C3×#21612yes—
386C2×C3C3×#95012yes—
387C22—21no—
388C22—21no—
389C22C3×#371812yescomplemented
390C22C2×C3×#62511yescharacteristic
391C4—21no—
392C4—122no—
393C4—122no—
394C4C3×#380511yescharacteristic
395C4C3×#381111yescharacteristic
396C3—21no—
397C3—22no—
398C3—41no—
399C3#1059118811yescharacteristic, direct product factor
400C3C3×#505112yescomplemented
401C2—22no—
402C2C3×#6382211yescharacteristic
403C2C2×C3×#371811yescharacteristic
404C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3718C2semidirect
4. C3×#63576C2semidirect
5. C3×#63565C2semidirect
7. #10591188C3direct (N × H)
25. C2×#3718C2×C3semidirect
29. #63576C2×C3semidirect
30. #63565C2×C3semidirect
78. C22×C33Q16semidirect
199. C22×C32#216semidirect
200. C22×C32C3×Q16semidirect
231. C33#467semidirect
324. C22×C3#3718semidirect
325. C22×C3C3×#216semidirect
354. C32#5051semidirect
355. C32C3×#467semidirect
389. C22C3×#3718semidirect
399. C3#10591188direct (N × H)
400. C3C3×#5051semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C32×#9242
C2×C3×#37182
C3×#635762
C3×#635652
C3×#50513
#105911883