Finite Group Databasebeta

← C3×#10590602 #13157619

Subgroups — C3×#10590602 #13157619

Order 1728.

786 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C2×C3×#3797C211yesmaximal, characteristic
2C3×#64101C211yesmaximal, characteristic
3C2×C3×#3712C211yesmaximal, characteristic
4C32×#810C212yesmaximal, complemented
5C3×#63566C212yesmaximal, complemented
6C3×#4802—32nomaximal
7#10590602C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C2×C3×#627—21no—
11C2×C3×#624—21no—
12C3×#3797—22no—
13C3×#3797C414yescomplemented
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3—21no—
18C2×C32×C3⋊C8C2212yes—
19C3×#3712—22no—
20C2×C3×D3×Q8—32no—
21C2×#3797C2×C311yescharacteristic
22#64101C2×C311yescharacteristic
23C2×C3×ΓL(1,𝔽25)—32no—
24C3×#892—32no—
25C3×#892—61no—
26C2×#3712C2×C311yescharacteristic
27C32×#103—32no—
28C3×#835—32no—
29C3×#816—32no—
30C3×#810—22no—
31C3×#810D312yescomplemented
32C3×#810C2×C312yescomplemented
33#63566C2×C312yescomplemented
34C22×C3×C3⋊D3D411yescharacteristic
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×C4×C3⋊D3C2×C412yes—
40C3×C4×C3⋊D3D412yes—
41C2×C32×Dic3—22no—
42C32×Dic6—22no—
43C32×Dic6C2×C414yes—
44C3×#627—24no—
45C3×#624—24no—
46C32×C3⋊C8—22no—
47C3×#419—91no—
48#4802—32no—
49C2×C32×Q8—32no—
50C2×C4×C3⋊D3C22×C311yescharacteristic
51C3×D3×Q8—38no—
52C3×D3×Q8—64no—
53C2×C3×C4×D3—32no—
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—61no—
56C2×C3×Dic6—22no—
57C2×C3×Dic6—62no—
58C2×C3×Dic6C2×D312yes—
59C2×C3×Dic6C22×C312yes—
60C2×#627—21no—
61C2×#624—21no—
62#3797—22no—
63#3797C3×C414yescomplemented
64C2×C32×C8—32no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—22no—
67C32×C4⋊C4—41no—
68C32×C4⋊C4C2×D312yes—
69C32×C4⋊C4C22×C311yescharacteristic
70#3752—21no—
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72C4×C3⋊Dic3—21no—
73C3×(C2×D3)⋊C4—62no—
74C3×(C2×D3)⋊C4—62no—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—61no—
77C3×C4×Dic3—62no—
78C3×C4×Dic3—62no—
79C2×C3×C3⋊C8—22no—
80C2×C3×C3⋊C8C2×D312yes—
81C2×C3×C3⋊C8C22×C312yes—
82C3×ΓL(1,𝔽25)—64no—
83#3712—22no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3C22⋊C412yes—
88C33×C4—41no—
89C33×C4C22×C411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C3×C3⋊Dic3—41no—
92C3×C3⋊Dic3C22⋊C412yes—
93C32×Dic3—24no—
94C22×C3×Q8—91no—
95C2×D3×Q8—32no—
96C2×C3×M16—91no—
97C3×#117—91no—
98C2×ΓL(1,𝔽25)—32no—
99#892—32no—
100#892—61no—
101C3×#103—32no—
102C3×#103—32no—
103C3×#103—62no—
104C3×#103—92no—
105#835—32no—
106#816—32no—
107#810C3×D312yescomplemented
108C22×C3⋊D3C3×D411yescharacteristic
109C22×C3×D3—32no—
110C22×C3×D3—61no—
111C32×Q8—34no—
112C32×Q8—62no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—41no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—62no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C23×C311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3C3×D411yescharacteristic
126C4×C3⋊D3C2×C3×C412yes—
127C4×C3⋊D3C3×D412yes—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—42no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C3×C4×D3—34no—
136C3×C4×D3—34no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139C3×C4×D3—68no—
140C3×Dic6—22no—
141C3×Dic6—22no—
142C3×Dic6—24no—
143C3×Dic6—42no—
144C3×Dic6—68no—
145C3×Dic6C4×D314yes—
146C3×Dic6C2×C3×C414yes—
147#627—24no—
148#624—24no—
149C32×C8—62no—
150C3×C3⋊C8—22no—
151C3×C3⋊C8—22no—
152C3×C3⋊C8—42no—
153#419—91noSylow
154C2×C33C2×C22⋊C411yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C33#13711yescharacteristic
157C3×C3⋊D3—22no—
158C2×C3×Q8—32no—
159C2×C3×Q8—32no—
160C2×C3×Q8—62no—
161C2×C3×Q8—94no—
162C2×C3×Q8—181no—
163C2×C3×Q8—182no—
164C22×C3×C4—91no—
165C22×C3×C4—181no—
166D3×Q8—38no—
167D3×Q8—64no—
168C2×C4×D3—32no—
169C2×C4×D3—61no—
170C2×C4×D3—62no—
171C2×Dic6—62no—
172C2×Dic6C2×C3×D312yes—
173C3×M16—182no—
174C2×C3×C8—32no—
175C2×C3×C8—32no—
176C2×C3×C8—62no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—41no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4D3211yescharacteristic
182C3×C4⋊C4C2×C3×D312yes—
183C3×C22⋊C4—181no—
184C3×C42—181no—
185(C2×D3)⋊C4—62no—
186(C2×D3)⋊C4—62no—
187C4⋊Dic3—32no—
188C4⋊Dic3—61no—
189C4×Dic3—62no—
190C4×Dic3—62no—
191C2×C3⋊C8C2×C3×D312yes—
192ΓL(1,𝔽25)—64no—
193C22×C32—21no—
194C22×C32—22no—
195C22×C32—41no—
196C22×C32D3×D412yes—
197C22×C32C2×C3×D411yescharacteristic
198C2×C3⋊D3—21no—
199C2×C3⋊D3—21no—
200C2×C3⋊D3C3×C22⋊C412yes—
201C2×C3×D3—34no—
202C2×C3×D3—62no—
203C2×C3×D3—62no—
204C2×C3×D3—62no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4—41no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4—42no—
215C32×C4—42no—
216C32×C4—64no—
217C32×C4—82no—
218C32×C4—82no—
219C32×C4C2×C4×D312yes—
220C32×C4D3×D412yes—
221C32×C4C22×C3×C411yescharacteristic
222C32×C4C2×C3×D411yescharacteristic
223C3⋊Dic3—41no—
224C3⋊Dic3C3×C22⋊C412yes—
225C3×Dic3—24no—
226C3×Dic3—24no—
227C3×Dic3—34no—
228C3×Dic3—44no—
229C3×Dic3—62no—
230C3×Dic3—122no—
231C3×Dic3—122no—
232C22×Q8—91no—
233C2×M16—91no—
234#117—91no—
235#103—32no—
236#103—92no—
237C33#41911yescharacteristic, complemented, Sylow
238C23×C3—91no—
239C22×D3—32no—
240C22×D3—61no—
241C3×Q8—34no—
242C3×Q8—34no—
243C3×Q8—64no—
244C3×Q8—62no—
245C3×Q8—62no—
246C3×Q8—122no—
247C3×Q8—184no—
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—41no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—91no—
259C2×C3×C4—92no—
260C2×C3×C4—92no—
261C2×C3×C4—122no—
262C2×C3×C4—184no—
263C2×C3×C4—181no—
264C2×C3×C4C2×D3211yescharacteristic
265C2×C3×C4C22×C3×D312yes—
266C2×Dic3—22no—
267C2×Dic3—32no—
268C2×Dic3—61no—
269C2×Dic3—62no—
270C2×Dic3—62no—
271C4×D3—34no—
272C4×D3—34no—
273C4×D3—62no—
274C4×D3—62no—
275C4×D3—68no—
276Dic6—22no—
277Dic6—68no—
278Dic6C3×C4×D314yes—
279C3×C8—62no—
280C3×C8—62no—
281C3×C8—122no—
282C3⋊C8—22no—
283C2×C32—22no—
284C2×C32—22no—
285C2×C32—22no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32D3×C22⋊C412yes—
293C2×C32#91512yes—
294C2×C32#91812yes—
295C2×C32C2×C3×C22⋊C411yescharacteristic
296C2×C32C3×#13711yescharacteristic
297C2×C32C3×#13711yescharacteristic
298C3⋊D3—22no—
299C3×D3—64no—
300C3×D3—122no—
301C2×Q8—32no—
302C2×Q8—94no—
303C2×Q8—181no—
304C2×Q8—182no—
305C22×C4—91no—
306C22×C4—181no—
307M16—182no—
308C2×C8—32no—
309C4⋊C4—91no—
310C4⋊C4C3×D3211yescharacteristic
311C22⋊C4—181no—
312C42—181no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—41no—
316C22×C3—92no—
317C22×C3—181no—
318C22×C3—181no—
319C22×C3#381111yescharacteristic
320C22×C3C3×D3×D412yes—
321C2×D3—34no—
322C2×D3—62no—
323C2×D3—62no—
324C2×D3—62no—
325C2×D3—121no—
326C2×D3—121no—
327C3×C4—21no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—21no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—64no—
337C3×C4—64no—
338C3×C4—82no—
339C3×C4—82no—
340C3×C4—92no—
341C3×C4—124no—
342C3×C4—361no—
343C3×C4C4×D3211yescharacteristic
344C3×C4#380211yescharacteristic
345C3×C4C2×C3×C4×D312yes—
346C3×C4C3×D3×D412yes—
347Dic3—24no—
348Dic3—34no—
349Dic3—62no—
350Dic3—122no—
351Dic3—122no—
352C32—21no—
353C32—22no—
354C32—41no—
355C32#480212yescomplemented
356C32C3×#41911yescharacteristic, complemented
357C23—91no—
358Q8—34no—
359Q8—62no—
360Q8—184no—
361C2×C4—21no—
362C2×C4—62no—
363C2×C4—91no—
364C2×C4—92no—
365C2×C4—92no—
366C2×C4—184no—
367C2×C4—181no—
368C2×C4C2×C3×D3211yescharacteristic
369C8—62no—
370C2×C3—21no—
371C2×C3—22no—
372C2×C3—21no—
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—22no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—182no—
380C2×C3#6392111yescharacteristic
381C2×C3#6393111yescharacteristic
382C2×C3#6394511yescharacteristic
383C2×C3C3×D3×C22⋊C412yes—
384C2×C3C3×#91512yes—
385C2×C3C3×#91812yes—
386D3—64no—
387D3—122no—
388C22—92no—
389C22—181no—
390C22—181no—
391C22C3×#381111yescharacteristic
392C4—41no—
393C4—64no—
394C4—92no—
395C4—361no—
396C4C3×C4×D3211yescharacteristic
397C4C3×#380211yescharacteristic
398C3—21no—
399C3—22no—
400C3—41no—
401C3#1059060211yescharacteristic, direct product factor
402C3C3×#480212yescomplemented
403C2—182no—
404C2C3×#6392111yescharacteristic
405C2C3×#6393111yescharacteristic
406C2C3×#6394511yescharacteristic
subgroupG/H
centerC22×C3#3811
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#419
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C32×#810C2semidirect
5. C3×#63566C2semidirect
7. #10590602C3direct (N × H)
13. C3×#3797C4semidirect
31. C3×#810D3semidirect
32. C3×#810C2×C3semidirect
33. #63566C2×C3semidirect
63. #3797C3×C4semidirect
107. #810C3×D3semidirect
237. C33#419semidirect
355. C32#4802semidirect
356. C32C3×#419semidirect
401. C3#10590602direct (N × H)
402. C3C3×#4802semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C3×#641012
C2×C3×#37122
C32×#8102
C3×#635662
C3×#48023
#105906023