Finite Group Databasebeta

← #13155411

Subgroups — #13155411

Order 1728.

786 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 138 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
2C2×#88809C211yesmaximal, characteristic
3#11723404C211yesmaximal, characteristic
4#11722038C212yesmaximal, complemented
5C3×#63565C212yesmaximal, complemented
6#10591557—32nomaximal
7#10590602—31nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C2×C3×#627—21no—
11C2×C3×#624—21no—
12C3×#3797—22no—
13C3×#3797C414yescomplemented
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16C32×C4⋊Dic3C2211yescharacteristic
17#88809—22no—
18#88803—21no—
19Dic3×C3⋊Dic3—21no—
20C2×C3×D3×Q8—32no—
21C2×#3797D311yescharacteristic
22#64101—31no—
23#63841—32no—
24#63841—61no—
25C2×#3712—31no—
26C2×#3711—32no—
27#63662—32no—
28C3×#835—32no—
29C3×#810—32no—
30#63566—32no—
31#63566—32no—
32#63566—62no—
33#63565D312yescomplemented
34C22×C3×C3⋊D3D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C3⋊(C3⋊Dic3)—21no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3C2×C412yes—
39C3×C4×C3⋊D3D412yes—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—21no—
42C32×Dic6—22no—
43C32×Dic6C2×C414yes—
44C3×#627—24no—
45C3×#624—24no—
46C3×C3⋊(C3⋊C8)—22no—
47#5225—91no—
48#4802—92no—
49C2×C32×Q8—32no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
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×Dic6C2×D312yes—
60C2×#627—21no—
61C2×#624—21no—
62#3797—22no—
63#3797Dic314yescomplemented
64C32×C4⋊C4—31no—
65#3752—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—32no—
68C4⋊(C3⋊Dic3)—61no—
69C4⋊(C3⋊Dic3)—62no—
70C4⋊(C3⋊Dic3)—121no—
71C4×C3⋊Dic3—61no—
72C2×C3⋊(C3⋊C8)C2×D312yes—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3C2×D312yes—
75C2×C3×C3⋊C8—32no—
76C2×C3×C3⋊C8—32no—
77C2×C3×C3⋊C8—62no—
78(C2×D3)⋊Dic3—62no—
79(C2×D3)⋊Dic3—62no—
80Dic32—62no—
81Dic32—62no—
82#3712—62no—
83#3711—64no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3C22⋊C412yes—
88C33×C4C22×C411yescharacteristic
89C33×C4C2×D411yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3×C3⋊Dic3C22⋊C412yes—
92C32×Dic3—24no—
93C32×Dic3—41no—
94C22×C3×Q8—91no—
95C2×D3×Q8—32no—
96#926—91no—
97C2×#208—91no—
98C2×ΓL(1,𝔽25)—92no—
99#892—92no—
100#892—181no—
101C3×#103—92no—
102#835—32no—
103#835—92no—
104#835—92no—
105#835—182no—
106#816—92no—
107#810—32no—
108C22×C3⋊D3#6711yescharacteristic
109C22×C3×D3—32no—
110C22×C3×D3—61no—
111C32×Q8—34no—
112C32×Q8—62no—
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4—61no—
117C2×C32×C4—62no—
118C2×C32×C4C22×D312yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3—122no—
125C2×C3⋊Dic3#6711yescharacteristic
126C4×C3⋊D3C2×Dic312yes—
127C4×C3⋊D3#6712yes—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—61no—
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×Dic314yes—
147#627—24no—
148#624—24no—
149C3⋊(C3⋊C8)—22no—
150C3×C3⋊C8—62no—
151C3×C3⋊C8—62no—
152C3×C3⋊C8—122no—
153#419—271noSylow
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×Dic6D3212yes—
173C2×C3×C8—92no—
174C3×C4⋊C4—32no—
175C3×C4⋊C4—61no—
176#217—181no—
177(C2×D3)⋊C4—182no—
178(C2×D3)⋊C4—182no—
179C4⋊Dic3—92no—
180C4⋊Dic3—91no—
181C4⋊Dic3—181no—
182C4⋊Dic3—182no—
183C4⋊Dic3—361no—
184C4⋊Dic3D3211yescharacteristic
185C4×Dic3—181no—
186C4×Dic3—182no—
187C4×Dic3—182no—
188#208—182no—
189C2×C3⋊C8—32no—
190C2×C3⋊C8—32no—
191C2×C3⋊C8—62no—
192ΓL(1,𝔽25)—184no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—41no—
196C22×C32D3×D412yes—
197C22×C32C2×#6711yescharacteristic
198C2×C3⋊D3—21no—
199C2×C3⋊D3—21no—
200C2×C3⋊D3#21712yes—
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—64no—
214C32×C4—121no—
215C32×C4C2×C4×D312yes—
216C32×C4D3×D412yes—
217C32×C4C22×Dic311yescharacteristic
218C32×C4C2×#6711yescharacteristic
219C3⋊Dic3—121no—
220C3⋊Dic3—122no—
221C3⋊Dic3—122no—
222C3⋊Dic3—242no—
223C3⋊Dic3—242no—
224C3⋊Dic3#21712yes—
225C3×Dic3—24no—
226C3×Dic3—24no—
227C3×Dic3—34no—
228C3×Dic3—44no—
229C3×Dic3—42no—
230C3×Dic3—42no—
231C3×Dic3—62no—
232C22×Q8—91no—
233C2×M16—271no—
234#117—271no—
235#103—92no—
236#103—272no—
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—41no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—91no—
256C2×C3×C4—92no—
257C2×C3×C4—92no—
258C2×C3×C4—122no—
259C2×C3×C4—184no—
260C2×C3×C4C2×D3211yescharacteristic
261C2×C3×C4C2×D3212yes—
262C2×Dic3—22no—
263C2×Dic3—21no—
264C2×Dic3—32no—
265C2×Dic3—61no—
266C2×Dic3—182no—
267C2×Dic3—182no—
268C2×Dic3—181no—
269C2×Dic3—362no—
270C2×Dic3—362no—
271C4×D3—34no—
272C4×D3—34no—
273C4×D3—62no—
274C4×D3—62no—
275C4×D3—68no—
276Dic6—22no—
277Dic6—68no—
278Dic6D3×Dic314yes—
279C3×C8—182no—
280C3⋊C8—62no—
281C3⋊C8—62no—
282C3⋊C8—122no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32D3×C22⋊C412yes—
293C2×C32#91512yes—
294C2×C32#91812yes—
295C2×C32#94211yescharacteristic
296C2×C32#94211yescharacteristic
297C2×C32C2×#21711yescharacteristic
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—542no—
308C2×C8—92no—
309C4⋊C4—31no—
310C4⋊C4—271no—
311C22⋊C4—541no—
312C42—541no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—41no—
316C22×C3—92no—
317C22×C3—181no—
318C22×C3—181no—
319C22×C3D3×#6712yes—
320C22×C3#381111yescharacteristic
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—41no—
332C3×C4—41no—
333C3×C4—64no—
334C3×C4—64no—
335C3×C4—92no—
336C3×C4—124no—
337C3×C4—122no—
338C3×C4—122no—
339C3×C4C4×D3211yescharacteristic
340C3×C4#380211yescharacteristic
341C3×C4C2×D3×Dic312yes—
342C3×C4D3×#6712yes—
343Dic3—24no—
344Dic3—34no—
345Dic3—41no—
346Dic3—62no—
347Dic3—362no—
348Dic3—361no—
349Dic3—362no—
350Dic3—722no—
351Dic3—722no—
352C32—22no—
353C32—21no—
354C32—41no—
355C32#480212yescomplemented
356C32#522511yescharacteristic, complemented
357C23—91no—
358Q8—34no—
359Q8—62no—
360Q8—184no—
361C2×C4—62no—
362C2×C4—61no—
363C2×C4—91no—
364C2×C4—92no—
365C2×C4—92no—
366C2×C4—184no—
367C2×C4—541no—
368C2×C4D3311yescharacteristic
369C8—182no—
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×C3#6394612yes—
384C2×C3#6394812yes—
385C2×C3D3×#21712yes—
386D3—64no—
387D3—122no—
388C22—92no—
389C22—181no—
390C22—181no—
391C22#8895211yescharacteristic
392C4—64no—
393C4—92no—
394C4—121no—
395C4—1081no—
396C4D32×Dic311yescharacteristic
397C4#8895311yescharacteristic
398C3—21no—
399C3—22no—
400C3—41no—
401C3#1059060211yescharacteristic, complemented
402C3#1059155712yescomplemented
403C2—182no—
404C2#1172288911yescharacteristic
405C2#1172289011yescharacteristic
406C2#1172290211yescharacteristic
subgroupG/H
centerC22#88952
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722038C2semidirect
5. C3×#63565C2semidirect
13. C3×#3797C4semidirect
33. #63565D3semidirect
63. #3797Dic3semidirect
237. C33#419semidirect
355. C32#4802semidirect
356. C32#5225semidirect
401. C3#10590602semidirect
402. C3#10591557semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×#888092
#117234042
#117220382
C3×#635652
#105915573
#105906023