Finite Group Databasebeta

← C3×#10591497 #13158509

Subgroups — C3×#10591497 #13158509

Order 1728.

764 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 136 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×D4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×#214C212yesmaximal
3C3×#63873C211yesmaximal, characteristic, complemented
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×#63566C212yesmaximal, complemented
6C3×#5162—32nomaximal
7#10591497C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C32×Dic6C2212yes—
11C3×#3757—21no—
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×#214—24no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17C3×#3724—21no—
18C2×C32×SD16—32no—
19D4×C3⋊Dic3C2×C311yescharacteristic
20C3×#944—32no—
21C3×D4×Dic3—32no—
22C3×D4×Dic3—61no—
23C2×C3×#214—22no—
24C2×C3×#214D312yes—
25C2×C3×#214C2×C312yes—
26#63873C2×C311yescharacteristic, complemented
27C3×#835—32no—
28C3×#816—32no—
29C3×Dic3⋊C8—32no—
30#63572C2×C311yescharacteristic, complemented
31#63566C2×C312yescomplemented
32C23×C33—21no—
33C33×D4—21no—
34C33×D4D412yes—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—42no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3D411yescharacteristic
40C2×C32×Dic3—22no—
41C32×Dic6—22no—
42C32×Dic6—22no—
43C32×C3⋊C8—22no—
44C3×#461—91no—
45#5162—32no—
46C2×C32×Q8—32no—
47C2×C32×D4—21no—
48C2×C32×D4—22no—
49C2×C32×D4—41no—
50C2×C32×D4C2×D312yes—
51C2×C32×D4C22×C311yescharacteristic
52C22×C3⋊Dic3—21no—
53C22×C3×Dic3—62no—
54C22×C3×Dic3—62no—
55C2×C3×Dic6—22no—
56C2×C3×Dic6C2×D312yes—
57C2×C3×Dic6C22×C312yes—
58C32×SD16—64no—
59C2×C32×C8—32no—
60#3757—21no—
61C4⋊(C3⋊Dic3)C22×C311yescharacteristic
62C4×C3⋊Dic3C22×C311yescharacteristic
63C3×#217—62no—
64C3×#217—62no—
65C3×#214—28no—
66C3×#214—24no—
67C3×#214—24no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—62no—
71C3×Dic3⋊C4—62no—
72C3×C4×Dic3—32no—
73C3×C4×Dic3—61no—
74C2×C3×C3⋊C8—22no—
75C2×C3×C3⋊C8C2×D312yes—
76C2×C3×C3⋊C8C22×C312yes—
77#3724—21no—
78#3724—21no—
79C22×C33—22no—
80C22×C33—41no—
81C22×C33—41no—
82C22×C33C2×D411yescharacteristic
83C33×C4C2×D411yescharacteristic
84C33×C4#4111yescharacteristic
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3SD1612yescomplemented
88C32×Dic3—42no—
89C2×C3×SD16—32no—
90C2×C3×SD16—32no—
91C2×C3×SD16—62no—
92C3×C4⋊Q8—91no—
93C3×C4×D4—91no—
94#944—32no—
95D4×Dic3—32no—
96D4×Dic3—61no—
97C2×#214C3×D312yes—
98C3×C4⋊C8—91no—
99C3×#103—92no—
100#835—32no—
101#816—32no—
102Dic3⋊C8—32no—
103C23×C32—21no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—42no—
107C23×C32—42no—
108C32×Q8—62no—
109C32×Q8—62no—
110C32×D4—21no—
111C32×D4—22no—
112C32×D4—22no—
113C32×D4—22no—
114C32×D4—24no—
115C32×D4—42no—
116C32×D4—42no—
117C32×D4—42no—
118C32×D4#6714yes—
119C32×D4C3×D412yes—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4—41no—
123C2×C32×C4—62no—
124C2×C32×C4C22×D312yes—
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—42no—
128C2×C3⋊Dic3C3×D411yescharacteristic
129C2×C3⋊Dic3C3×D411yescharacteristic
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—42no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—124no—
140C2×C3×Dic3—124no—
141C3×Dic6—24no—
142C3×Dic6—22no—
143C3×Dic6—22no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—42no—
147C32×C8—62no—
148C3×C3⋊C8—24no—
149C3×C3⋊C8—22no—
150C3×C3⋊C8—22no—
151#461—91noSylow
152C2×C33—42no—
153C2×C33#12111yescharacteristic
154C2×C33C2×SD1611yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C3×Q8—32no—
157C2×C3×Q8—32no—
158C2×C3×Q8—62no—
159C2×C3×D4—21no—
160C2×C3×D4—22no—
161C2×C3×D4—41no—
162C2×C3×D4D3211yescharacteristic
163C2×C3×D4C2×C3×D312yes—
164C22×C3×C4—181no—
165C22×Dic3—62no—
166C22×Dic3—62no—
167C2×Dic6C2×C3×D312yes—
168C3×SD16—68no—
169C3×SD16—64no—
170C3×SD16—64no—
171C2×C3×C8—32no—
172C2×C3×C8—32no—
173C2×C3×C8—62no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—181no—
176C3×C4⋊C4—181no—
177C3×C22⋊C4—181no—
178C3×C42—91no—
179#217—62no—
180#217—62no—
181#214—24no—
182C4⋊Dic3—32no—
183C4⋊Dic3—61no—
184Dic3⋊C4—62no—
185Dic3⋊C4—62no—
186C4×Dic3—32no—
187C4×Dic3—61no—
188C2×C3⋊C8C2×C3×D312yes—
189C22×C32—22no—
190C22×C32—24no—
191C22×C32—24no—
192C22×C32—21no—
193C22×C32—22no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—42no—
200C22×C32—42no—
201C22×C32—42no—
202C22×C32—42no—
203C22×C32—82no—
204C22×C32—82no—
205C22×C32—82no—
206C22×C32—82no—
207C22×C32D3×D412yes—
208C22×C32C2×C3×D411yescharacteristic
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—122no—
216C32×C4#23712yes—
217C32×C4C2×#6712yes—
218C32×C4C2×C3×D411yescharacteristic
219C32×C4C3×#4111yescharacteristic
220C3⋊Dic3—21no—
221C3⋊Dic3—41no—
222C3⋊Dic3C3×SD1612yescomplemented
223C3×Dic3—34no—
224C3×Dic3—44no—
225C3×Dic3—42no—
226C3×Dic3—42no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—121no—
230C3×Dic3—122no—
231C3×Dic3—122no—
232C2×SD16—32no—
233C4⋊Q8—91no—
234C4×D4—91no—
235C4⋊C8—91no—
236#103—92no—
237C33#46111yescharacteristic, complemented, Sylow
238C23×C3—22no—
239C23×C3—21no—
240C23×C3—22no—
241C23×C3—42no—
242C23×C3—42no—
243C3×Q8—64no—
244C3×Q8—62no—
245C3×Q8—62no—
246C3×Q8—62no—
247C3×Q8—62no—
248C3×Q8—122no—
249C3×D4—22no—
250C3×D4—21no—
251C3×D4—22no—
252C3×D4—22no—
253C3×D4—24no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4#62512yes—
258C3×D4C3×#6714yes—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—41no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—91no—
265C2×C3×C4—91no—
266C2×C3×C4—122no—
267C2×C3×C4—181no—
268C2×C3×C4—362no—
269C2×C3×C4C2×D3211yescharacteristic
270C2×C3×C4C22×C3×D312yes—
271C2×Dic3—22no—
272C2×Dic3—32no—
273C2×Dic3—32no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—124no—
279C2×Dic3—124no—
280Dic6—22no—
281Dic6—22no—
282C3×C8—64no—
283C3×C8—62no—
284C3×C8—62no—
285C3⋊C8—22no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32—42no—
296C2×C32—44no—
297C2×C32—44no—
298C2×C32—84no—
299C2×C32—84no—
300C2×C32D3×SD1612yes—
301C2×C32#91512yes—
302C2×C32#93912yes—
303C2×C32C3×#12111yescharacteristic
304C2×C32C2×C3×SD1611yescharacteristic
305C2×C32C3×#13711yescharacteristic
306C2×Q8—32no—
307C2×D4C3×D3211yescharacteristic
308C22×C4—181no—
309SD16—64no—
310C2×C8—32no—
311C4⋊C4—91no—
312C4⋊C4—181no—
313C4⋊C4—181no—
314C22⋊C4—181no—
315C42—91no—
316C22×C3—24no—
317C22×C3—24no—
318C22×C3—22no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—44no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—82no—
331C22×C3—82no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3#381111yescharacteristic
335C22×C3C3×D3×D412yes—
336C3×C4—21no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—21no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4—92no—
343C3×C4—124no—
344C3×C4—122no—
345C3×C4—122no—
346C3×C4—181no—
347C3×C4—361no—
348C3×C4#380511yescharacteristic
349C3×C4C2×#62511yescharacteristic
350C3×C4C3×#23712yes—
351C3×C4C2×C3×#6712yes—
352Dic3—34no—
353Dic3—42no—
354Dic3—62no—
355Dic3—62no—
356Dic3—121no—
357Dic3—122no—
358Dic3—122no—
359C32—22no—
360C32—21no—
361C32—41no—
362C32#516212yescomplemented
363C32C3×#46111yescharacteristic, complemented
364C23—21no—
365Q8—62no—
366Q8—62no—
367D4—21no—
368D4C3×#62512yes—
369C2×C4—62no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—181no—
373C2×C4—362no—
374C2×C4C2×C3×D3211yescharacteristic
375C8—62no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—21no—
381C2×C3—22no—
382C2×C3—44no—
383C2×C3—44no—
384C2×C3—42no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—84no—
389C2×C3—84no—
390C2×C3#6393011yescharacteristic
391C2×C3#6393111yescharacteristic
392C2×C3#6398011yescharacteristic
393C2×C3C3×D3×SD1612yes—
394C2×C3C3×#91512yes—
395C2×C3C3×#93912yes—
396C22—22no—
397C22—41no—
398C22—41no—
399C22C3×#381111yescharacteristic
400C4—92no—
401C4—122no—
402C4—181no—
403C4—361no—
404C4C3×#380511yescharacteristic
405C4C2×C3×#62511yescharacteristic
406C3—21no—
407C3—22no—
408C3—41no—
409C3#1059149711yescharacteristic, direct product factor
410C3C3×#516212yescomplemented
411C2—42no—
412C2C3×#6393011yescharacteristic
413C2C3×#6393111yescharacteristic
414C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
3. C3×#63873C2semidirect
4. C3×#63572C2semidirect
5. C3×#63566C2semidirect
7. #10591497C3direct (N × H)
26. #63873C2×C3semidirect
30. #63572C2×C3semidirect
31. #63566C2×C3semidirect
87. C3×C3⋊Dic3SD16semidirect
222. C3⋊Dic3C3×SD16semidirect
237. C33#461semidirect
362. C32#5162semidirect
363. C32C3×#461semidirect
409. C3#10591497direct (N × H)
410. C3C3×#5162semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C32×#2142
C3×#638732
C3×#635722
C3×#635662
C3×#51623
#105914973