Finite Group Databasebeta

← #13155402

Subgroups — #13155402

Order 1728.

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

normal subgroups only

This group has 950 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×#3802C211yesmaximal, characteristic
2#11723667C211yesmaximal, characteristic, complemented
3C2×#88787C211yesmaximal, characteristic, complemented
4#11722031C212yesmaximal, complemented
5C3×#63563C212yesmaximal, complemented
6#10591421—32nomaximal
7#10590941—31nomaximal
8C22×C3×D32—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C2212yes—
11C2×C3×#625—21no—
12C3×#3802—22no—
13C3×#3802C414yescomplemented
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
15C3×C4⋊(C3⋊Dic3)C2212yescomplemented
16C2×C32×C3⋊C8C2211yescharacteristic, complemented
17#88814—21no—
18#88813—21no—
19#88787—22no—
20C2×C3×D3×D4—32no—
21C2×#3802D311yescharacteristic
22C2×#3743—31no—
23#63849—31no—
24#63841—32no—
25C2×#3711—32no—
26C2×#3711—61no—
27#63659—32no—
28C3×#832—32no—
29C3×#821—32no—
30#63563D312yescomplemented
31#63562—32no—
32#63562—32no—
33#63562—62no—
34C22×C3×C3⋊D3D411yescharacteristic
35C22×C32×D3—22no—
36C2×C3×D32—24no—
37C2×C3×D32—44no—
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4×C3⋊D3C2×C412yes—
42C3×C4×C3⋊D3D412yes—
43C32×D12—22no—
44C32×D12C2×C414yes—
45C3×#625—24no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C32×C3⋊C8—21no—
48#5215—91no—
49#4770—92no—
50C23×C3×D3—62no—
51C22×D32—21no—
52C2×C32×D4—32no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×#67—62no—
55C3×D3×D4—38no—
56C3×D3×D4—64no—
57C2×C3×D12—22no—
58C2×C3×D12C2×D312yes—
59C2×C3×D12C2×D312yes—
60C2×C3×C4×D3—32no—
61C2×C3×C4×D3—61no—
62C2×#625—21no—
63#3802—22no—
64#3802Dic314yescomplemented
65C2×C32×C8—31no—
66C4⋊(C3⋊Dic3)C2×D312yescomplemented
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—32no—
69C2×C3⋊(C3⋊C8)—61no—
70C2×C3⋊(C3⋊C8)—62no—
71C2×C3⋊(C3⋊C8)—121no—
72#3743—62no—
73C3×C4⋊Dic3—32no—
74C3×C4⋊Dic3—32no—
75C3×C4⋊Dic3—62no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8C2×D312yescomplemented
78#3722—61no—
79(C2×D3)⋊Dic3—62no—
80Dic32—62no—
81Dic32—61no—
82#3711—64no—
83#3711—64no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3C22⋊C412yes—
87C2×C32×D3—24no—
88C2×C32×D3—42no—
89C2×C32×D3—42no—
90C3×D32—48no—
91C33×C4C22×C411yescharacteristic
92C33×C4C2×D411yescharacteristic
93C3×C3⋊Dic3—42no—
94C3×C3⋊Dic3C22⋊C412yes—
95C22×C3×D4—91no—
96C2×D3×D4—32no—
97#926—91no—
98C2×#208—91no—
99C2×ΓL(1,𝔽25)—92no—
100C2×ΓL(1,𝔽25)—181no—
101#892—92no—
102C3×#102—92no—
103#832—32no—
104#832—92no—
105#832—92no—
106#832—182no—
107#821—32no—
108#809—92no—
109C23×C32—62no—
110C22×C3⋊D3#6711yescharacteristic
111C22×C3×D3—22no—
112C22×C3×D3—22no—
113C22×C3×D3—32no—
114C22×C3×D3—42no—
115C22×C3×D3—61no—
116C22×C3×D3—68no—
117C22×C3×D3—124no—
118C22×C3×D3—124no—
119C2×D32—24no—
120C2×D32—44no—
121C32×D4—34no—
122C32×D4—62no—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D312yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3#6711yescharacteristic
130C4×C3⋊D3C2×Dic312yes—
131C4×C3⋊D3#6712yes—
132C3×#67—68no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—122no—
138C3×D12—22no—
139C3×D12—22no—
140C3×D12—24no—
141C3×D12—42no—
142C3×D12C4×D314yes—
143C3×D12C2×Dic314yes—
144C3×C4×D3—34no—
145C3×C4×D3—34no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148#625—24no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—61no—
151C3⋊(C3⋊C8)—62no—
152C3⋊(C3⋊C8)—62no—
153C3⋊(C3⋊C8)—122no—
154C3⋊(C3⋊C8)—122no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—22no—
157#418—271noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33AGL(1,ℤ/8)12yes—
160C3×C3⋊D3—22no—
161C32×D3—44no—
162C24×C3—181no—
163C23×D3—62no—
164C2×C3×D4—32no—
165C2×C3×D4—32no—
166C2×C3×D4—62no—
167C2×C3×D4—94no—
168C2×C3×D4—182no—
169C2×C3×D4—181no—
170C22×C3×C4—91no—
171C2×#67—62no—
172D3×D4—38no—
173D3×D4—64no—
174C2×D12D3212yes—
175C2×C4×D3—32no—
176C2×C4×D3—61no—
177C2×C3×C8—32no—
178C2×C3×C8—61no—
179C3×C4⋊C4—92no—
180#217—181no—
181(C2×D3)⋊C4—182no—
182C4⋊Dic3—32no—
183C4⋊Dic3—32no—
184C4⋊Dic3—62no—
185C4×Dic3—182no—
186C4×Dic3—181no—
187#208—182no—
188C2×C3⋊C8—92no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—181no—
191C2×C3⋊C8—182no—
192C2×C3⋊C8—361no—
193C2×C3⋊C8D3211yescharacteristic, complemented
194ΓL(1,𝔽25)—184no—
195ΓL(1,𝔽25)—184no—
196C22×C32—21no—
197C22×C32—22no—
198C22×C32—41no—
199C22×C32—64no—
200C22×C32—122no—
201C22×C32—122no—
202C22×C32D3×D412yes—
203C22×C32C2×#6711yescharacteristic
204C2×C3⋊D3—22no—
205C2×C3⋊D3#21712yes—
206C2×C3×D3—24no—
207C2×C3×D3—24no—
208C2×C3×D3—34no—
209C2×C3×D3—44no—
210C2×C3×D3—42no—
211C2×C3×D3—42no—
212C2×C3×D3—42no—
213C2×C3×D3—42no—
214C2×C3×D3—62no—
215C2×C3×D3—62no—
216C2×C3×D3—62no—
217C2×C3×D3—82no—
218C2×C3×D3—82no—
219C2×C3×D3—122no—
220C2×C3×D3—1216no—
221D32—48no—
222C32×C4—21no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—21no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4C2×C4×D312yes—
229C32×C4D3×D412yes—
230C32×C4C22×Dic311yescharacteristic
231C32×C4C2×#6711yescharacteristic
232C3⋊Dic3—42no—
233C3⋊Dic3#21712yes—
234C3×Dic3—34no—
235C3×Dic3—62no—
236C3×Dic3—122no—
237C3×Dic3—122no—
238C3×Dic3—242no—
239C22×D4—91no—
240C2×M16—271no—
241#117—271no—
242#102—92no—
243#102—272no—
244C33#41811yescharacteristic, complemented, Sylow
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—91no—
248C23×C3—122no—
249C23×C3—184no—
250C23×C3—364no—
251C22×D3—22no—
252C22×D3—32no—
253C22×D3—61no—
254C22×D3—68no—
255C22×D3—124no—
256C22×D3—124no—
257C3×D4—34no—
258C3×D4—34no—
259C3×D4—64no—
260C3×D4—62no—
261C3×D4—62no—
262C3×D4—122no—
263C3×D4—184no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—91no—
268C2×C3×C4—92no—
269C2×C3×C4—92no—
270C2×C3×C4—182no—
271C2×C3×C4C2×D3211yescharacteristic
272C2×C3×C4C2×D3212yes—
273#67—68no—
274C2×Dic3—32no—
275C2×Dic3—61no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—122no—
279D12—22no—
280D12D3×Dic314yes—
281C4×D3—34no—
282C4×D3—34no—
283C4×D3—62no—
284C4×D3—62no—
285C3×C8—62no—
286C3×C8—62no—
287C3⋊C8—21no—
288C3⋊C8—182no—
289C3⋊C8—182no—
290C3⋊C8—181no—
291C3⋊C8—362no—
292C3⋊C8—362no—
293C2×C32—22no—
294C2×C32—21no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—22no—
298C2×C32—41no—
299C2×C32—42no—
300C2×C32—124no—
301C2×C32D3×C22⋊C412yes—
302C2×C32#91112yes—
303C2×C32#91412yes—
304C2×C32#93212yes—
305C2×C32C2×#21711yescharacteristic
306C3⋊D3—22no—
307C3×D3—44no—
308C3×D3—44no—
309C3×D3—64no—
310C3×D3—84no—
311C3×D3—122no—
312C24—181no—
313C2×D4—32no—
314C2×D4—94no—
315C2×D4—182no—
316C2×D4—181no—
317C22×C4—91no—
318M16—542no—
319C2×C8—31no—
320C2×C8—271no—
321C4⋊C4—92no—
322C22⋊C4—541no—
323C42—541no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—41no—
327C22×C3—64no—
328C22×C3—64no—
329C22×C3—92no—
330C22×C3—124no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3—122no—
334C22×C3—122no—
335C22×C3—182no—
336C22×C3—242no—
337C22×C3—242no—
338C22×C3—368no—
339C22×C3#380211yescharacteristic
340C22×C3D3×#6712yes—
341C2×D3—24no—
342C2×D3—34no—
343C2×D3—42no—
344C2×D3—42no—
345C2×D3—62no—
346C2×D3—62no—
347C2×D3—62no—
348C2×D3—122no—
349C2×D3—1216no—
350C3×C4—21no—
351C3×C4—22no—
352C3×C4—21no—
353C3×C4—22no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—92no—
357C3×C4—362no—
358C3×C4C4×D3211yescharacteristic
359C3×C4C2×D3×Dic312yes—
360C3×C4D3×#6712yes—
361C3×C4#381111yescharacteristic
362Dic3—34no—
363Dic3—62no—
364Dic3—122no—
365Dic3—122no—
366Dic3—242no—
367C32—22no—
368C32—21no—
369C32—41no—
370C32#477012yescomplemented
371C32#521511yescharacteristic, complemented
372C23—62no—
373C23—91no—
374C23—184no—
375C23—364no—
376D4—34no—
377D4—62no—
378D4—184no—
379C2×C4—91no—
380C2×C4—92no—
381C2×C4—92no—
382C2×C4—182no—
383C2×C4D3311yescharacteristic
384C8—61no—
385C8—541no—
386C2×C3—22no—
387C2×C3—22no—
388C2×C3—22no—
389C2×C3—21no—
390C2×C3—22no—
391C2×C3—42no—
392C2×C3—41no—
393C2×C3—124no—
394C2×C3—124no—
395C2×C3—182no—
396C2×C3—244no—
397C2×C3#6379812yes—
398C2×C3#6392111yescharacteristic
399C2×C3#6393212yes—
400C2×C3#6393412yes—
401C2×C3D3×#21712yes—
402D3—44no—
403D3—64no—
404D3—122no—
405C22—64no—
406C22—92no—
407C22—122no—
408C22—122no—
409C22—182no—
410C22—368no—
411C22#8895311yescharacteristic
412C4—92no—
413C4—362no—
414C4D32×Dic311yescharacteristic
415C4#8895211yescharacteristic
416C3—21no—
417C3—22no—
418C3—41no—
419C3#1059094111yescharacteristic, complemented
420C3#1059142112yescomplemented
421C2—124no—
422C2—182no—
423C2#1172287512yes—
424C2#1172290211yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723667C2semidirect
3. C2×#88787C2semidirect
4. #11722031C2semidirect
5. C3×#63563C2semidirect
13. C3×#3802C4semidirect
14. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
15. C3×C4⋊(C3⋊Dic3)C22semidirect
16. C2×C32×C3⋊C8C22semidirect
30. #63563D3semidirect
64. #3802Dic3semidirect
66. C4⋊(C3⋊Dic3)C2×D3semidirect
77. C2×C3×C3⋊C8C2×D3semidirect
193. C2×C3⋊C8D32semidirect
244. C33#418semidirect
370. C32#4770semidirect
371. C32#5215semidirect
419. C3#10590941semidirect
420. C3#10591421semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
#117236672
C2×#887872
#117220312
C3×#635632
#105914213
#105909413