Finite Group Databasebeta

← #13155970

Subgroups — #13155970

Order 1728.

780 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 110 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
1#11723697C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3714C212yesmaximal, complemented
4#11722034C212yesmaximal, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6#10591486—31nomaximal
7#10590753—32nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C2×C32×D12C2211yescharacteristic
10C2×#6295—21no—
11C2×#6295—21no—
12C2×D3×C3⋊Dic3—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C32×C3⋊C8C2212yes—
16(C2×D3)⋊(C3⋊Dic3)—21no—
17C3×#3714—24no—
18D4×C3⋊Dic3—31no—
19C2×C3×#213—32no—
20C2×C3×D24—32no—
21#63914—31no—
22#63911—32no—
23Dic3×D12—32no—
24Dic3×D12—61no—
25C2×#3714D312yescomplemented
26#63648—32no—
27C3×Dic3⋊C8—32no—
28#63572D311yescharacteristic, complemented
29#63563—32no—
30#63563—32no—
31#63563—62no—
32C22×C3×C3⋊D3—22no—
33C22×C32×D3—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4⋊(C3⋊D3)—22no—
39C3×C4⋊(C3⋊D3)—22no—
40C32×D12—21no—
41C32×D12D412yescomplemented
42#6295—42no—
43#6295—42no—
44D3×C3⋊Dic3—42no—
45C32×C3⋊C8—22no—
46#5150—92no—
47#4843—91no—
48C2×C32×D4—31no—
49C22×C3⋊Dic3—61no—
50C2×C4⋊(C3⋊D3)C2×D312yes—
51C2×C3×D12—21no—
52C2×C3×D12—32no—
53C2×C3×D12—32no—
54C2×C3×D12—62no—
55C2×C3×D12C2×D312yes—
56C2×#626—62no—
57C2×#626—62no—
58C2×#625—61no—
59C2×#625—61no—
60C2×D3×Dic3—62no—
61C2×D3×Dic3—62no—
62C2×C32×C8—32no—
63#3757—61no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—32no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)—62no—
68C4⋊(C3⋊Dic3)—121no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C3×#213—64no—
71C3×C4×Dic3—32no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—22no—
74C2×C3×C3⋊C8C2×D312yes—
75C2×C3×C3⋊C8C2×D312yes—
76C3×D24—64no—
77(C2×D3)⋊Dic3—62no—
78(C2×D3)⋊Dic3—62no—
79#3714—24no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—42no—
82C2×C3×C3⋊D3—42no—
83C2×C3×C3⋊D3—42no—
84C2×C32×D3—22no—
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C33×C4C2×D411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—41no—
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3D812yescomplemented
92C2×C3×D8—92no—
93#940—92no—
94D4×Dic3—92no—
95D4×Dic3—181no—
96C2×#213—32no—
97C2×D24—32no—
98#874—91no—
99C4×D12—91no—
100C3×C4⋊C8—91no—
101#832—92no—
102#821—92no—
103#821—92no—
104#821—182no—
105Dic3⋊C8—32no—
106C23×C32—61no—
107C22×C3⋊D3—22no—
108C22×C3×D3—22no—
109C22×C3×D3—22no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—122no—
113C32×D4—32no—
114C32×D4—61no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D312yes—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3—122no—
125C2×C3⋊Dic3—122no—
126C2×C3⋊Dic3D1211yescharacteristic
127C2×C3⋊Dic3D1211yescharacteristic
128C4⋊(C3⋊D3)—22no—
129C4⋊(C3⋊D3)—22no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C3×D12—22no—
135C3×D12—22no—
136C3×D12—22no—
137C3×D12—64no—
138C3×D12—62no—
139C3×D12—62no—
140C3×D12—62no—
141C3×D12—62no—
142C3×D12—122no—
143C3×D12#6714yescomplemented
144#626—124no—
145#626—124no—
146#625—122no—
147#625—122no—
148D3×Dic3—124no—
149D3×Dic3—124no—
150C32×C8—62no—
151C3×C3⋊C8—22no—
152C3×C3⋊C8—22no—
153C3×C3⋊C8—42no—
154#459—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×D811yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3×C3⋊D3—82no—
159C32×D3—42no—
160C2×C3×D4—32no—
161C2×C3×D4—61no—
162C2×C3×D4—92no—
163C2×#67—182no—
164C2×#67—182no—
165C22×Dic3—182no—
166C22×Dic3—182no—
167C2×D12—32no—
168C2×D12—32no—
169C2×D12—62no—
170C2×D12—181no—
171C2×D12—181no—
172C2×D12D3211yescharacteristic
173C2×C4×D3—181no—
174C3×D8—184no—
175C2×C3×C8—32no—
176C2×C3×C8—32no—
177C2×C3×C8—62no—
178C3×C42—91no—
179#217—182no—
180#217—182no—
181#213—64no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—91no—
184C4⋊Dic3—92no—
185C4⋊Dic3—181no—
186C4⋊Dic3—182no—
187C4⋊Dic3—361no—
188C4×Dic3—32no—
189C4×Dic3—61no—
190C2×C3⋊C8D3212yes—
191D24—64no—
192C22×C32—22no—
193C22×C32—21no—
194C22×C32—41no—
195C22×C32—62no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32C2×D1211yescharacteristic
199C22×C32D3×D412yes—
200C2×C3⋊D3—42no—
201C2×C3⋊D3—42no—
202C2×C3⋊D3—42no—
203C2×C3×D3—24no—
204C2×C3×D3—24no—
205C2×C3×D3—42no—
206C2×C3×D3—42no—
207C2×C3×D3—42no—
208C2×C3×D3—42no—
209C2×C3×D3—124no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—242no—
217C2×C3×D3—242no—
218C32×C4—22no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—22no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4D3×D411yescharacteristic
225C32×C4#23712yes—
226C32×C4#23911yescharacteristic
227C32×C4C2×#6712yes—
228C3⋊Dic3—21no—
229C3⋊Dic3—121no—
230C3⋊Dic3—122no—
231C3⋊Dic3—122no—
232C3⋊Dic3—242no—
233C3⋊Dic3—242no—
234C3⋊Dic3D2412yescomplemented
235C3×Dic3—34no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—121no—
239C2×D8—92no—
240#127—271no—
241C4×D4—271no—
242C4⋊C8—91no—
243#102—272no—
244C33#45911yescharacteristic, complemented, Sylow
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—182no—
248C22×D3—21no—
249C22×D3—62no—
250C22×D3—62no—
251C22×D3—122no—
252C3×D4—34no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—182no—
257C3×D4—182no—
258C2×C3×C4—21no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4C2×D3212yes—
264C2×C3×C4C2×D3211yescharacteristic
265#67—364no—
266#67—364no—
267C2×Dic3—32no—
268C2×Dic3—32no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—182no—
272C2×Dic3—182no—
273C2×Dic3—181no—
274C2×Dic3—362no—
275C2×Dic3—362no—
276C2×Dic3—364no—
277C2×Dic3—364no—
278D12—21no—
279D12—64no—
280D12—62no—
281D12—62no—
282D12—62no—
283D12—62no—
284D12—122no—
285D12—362no—
286D12—362no—
287D12#62512yescomplemented
288C4×D3—362no—
289C3×C8—62no—
290C3×C8—62no—
291C3×C8—122no—
292C3⋊C8—22no—
293C2×C32—22no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—122no—
303C2×C32#89511yescharacteristic
304C2×C32C2×D2411yescharacteristic
305C2×C32#90811yescharacteristic
306C2×C32D3×D812yes—
307C2×C32#91112yes—
308C2×C32#93912yes—
309C3⋊D3—82no—
310C3×D3—44no—
311C3×D3—44no—
312C3×D3—244no—
313C3×D3—242no—
314C3×D3—242no—
315C2×D4—31no—
316C2×D4—92no—
317C2×D4—541no—
318C2×D4—541no—
319C22×C4—541no—
320D8—184no—
321C2×C8—32no—
322C4⋊C4—271no—
323C22⋊C4—541no—
324C42—91no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—41no—
328C22×C3—64no—
329C22×C3—64no—
330C22×C3—122no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3—122no—
334C22×C3—362no—
335C22×C3—362no—
336C22×C3—362no—
337C22×C3D3×D1212yes—
338C22×C3#381111yescharacteristic
339C2×D3—22no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—124no—
343C2×D3—122no—
344C2×D3—122no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—122no—
348C2×D3—122no—
349C2×D3—242no—
350C2×D3—242no—
351C3×C4—22no—
352C3×C4—21no—
353C3×C4—22no—
354C3×C4—21no—
355C3×C4—41no—
356C3×C4—41no—
357C3×C4—92no—
358C3×C4—181no—
359C3×C4#379612yes—
360C3×C4#380511yescharacteristic
361C3×C4C2×#62511yescharacteristic
362C3×C4D3×#6712yes—
363Dic3—34no—
364Dic3—62no—
365Dic3—62no—
366Dic3—121no—
367Dic3—362no—
368Dic3—362no—
369Dic3—361no—
370Dic3—722no—
371Dic3—722no—
372C32—22no—
373C32—21no—
374C32—41no—
375C32#484311yescharacteristic, complemented
376C32#515012yescomplemented
377C23—61no—
378C23—182no—
379D4—32no—
380D4—61no—
381D4—182no—
382D4—182no—
383D4—1082no—
384D4—1082no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—541no—
388C2×C4—1082no—
389C2×C4D3311yescharacteristic
390C8—62no—
391C2×C3—21no—
392C2×C3—22no—
393C2×C3—21no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—124no—
401C2×C3—124no—
402C2×C3—722no—
403C2×C3D3×D2412yes—
404C2×C3#6379512yes—
405C2×C3#6388612yes—
406C2×C3#6392611yescharacteristic
407C2×C3#6392711yescharacteristic
408C2×C3#6398011yescharacteristic
409D3—42no—
410D3—244no—
411D3—242no—
412D3—242no—
413C22—62no—
414C22—121no—
415C22—121no—
416C22—362no—
417C22—362no—
418C22—362no—
419C22#8895511yescharacteristic
420C4—92no—
421C4—181no—
422C4—1081no—
423C4D3×#62511yescharacteristic
424C4#8896411yescharacteristic
425C3—22no—
426C3—21no—
427C3—41no—
428C3#1059075312yescomplemented
429C3#1059148611yescharacteristic, complemented
430C2—122no—
431C2—722no—
432C2#1172283311yescharacteristic
433C2#1172288011yescharacteristic
434C2#1172295511yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723697C2semidirect
2. D12×C3⋊Dic3C2semidirect
3. C2×C3×#3714C2semidirect
4. #11722034C2semidirect
5. C3×#63572C2semidirect
25. C2×#3714D3semidirect
28. #63572D3semidirect
41. C32×D12D4semidirect
91. C3×C3⋊Dic3D8semidirect
143. C3×D12#67semidirect
234. C3⋊Dic3D24semidirect
244. C33#459semidirect
287. D12#625semidirect
375. C32#4843semidirect
376. C32#5150semidirect
428. C3#10590753semidirect
429. C3#10591486semidirect

Maximal subgroups

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

subgroupindex
#117236972
D12×C3⋊Dic32
C2×C3×#37142
#117220342
C3×#635722
#105914863
#105907533