Finite Group Databasebeta

← #13161212

Subgroups — #13161212

Order 1728.

1056 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 200 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×C32×D3×Q8C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3#11723425C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
8#10592927—31nomaximal
9#10591594—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C32×D3×Q8—22no—
13C32×D3×Q8—22no—
14C2×C32×C4×D3—21no—
15C2×C32×C4×D3C2211yescharacteristic
16C2×C32×Dic6—21no—
17C2×C32×Dic6C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×#3756—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88806—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88798—22no—
25C22×C32×Q8—31no—
26C2×C3×D3×Q8D314yes—
27C2×#3756—31no—
28C2×#3756D311yescharacteristic, complemented
29#64133—31no—
30C2×C3×#216—34no—
31#63856—34no—
32C2×#3718—34no—
33#63662—31no—
34#63662—31no—
35#63662—34no—
36#63662—64no—
37#63657—31no—
38#63566—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C32×D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8D412yes—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3D411yescharacteristic
51C32×C4×D3—24no—
52C32×C4×D3—21no—
53C32×C4×D3—21no—
54C32×Dic6—24no—
55C32×Dic6—21no—
56C32×Dic6D412yes—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5228—94no—
60#5186—91no—
61C2×C32×Q8—24no—
62C2×C32×Q8—31no—
63C2×C32×Q8—61no—
64C2×C32×Q8—62no—
65C2×C32×Q8—62no—
66C2×C32×Q8C2×D314yes—
67C2×C32×Q8C2×D311yescharacteristic
68C22×C32×C4—31no—
69C22×C32×C4—61no—
70C2×#634C2×D311yescharacteristic
71C3×D3×Q8—28no—
72C3×D3×Q8—28no—
73C2×C3×C4×D3—24no—
74C2×C3×C4×D3C2×D314yes—
75C2×C3×Dic6—24no—
76C2×C3×Dic6—34no—
77C2×C3×Dic6C2×D314yes—
78#3757—61no—
79#3756—22no—
80#3756—62no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)—64no—
84#3750—61no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—34no—
87C2×C3⋊(C3⋊C8)—64no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×#216—68no—
90C2×C3×C3⋊C8—34no—
91#3724—64no—
92(C2×D3)⋊Dic3—64no—
93#3718—68no—
94C22×C33C2×D411yescharacteristic
95C2×C32×D3—21no—
96C2×C32×D3—21no—
97C2×C32×D3Q1612yescomplemented
98C33×C4—22no—
99C33×C4C2×D411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—41no—
102C3×C3⋊Dic3—41no—
103C32×Dic3—21no—
104C32×Dic3—22no—
105C22×C3×Q8—34no—
106C2×D3×Q8C3⋊D311yescharacteristic
107C2×C3×Q16—91no—
108#946—91no—
109C2×#216—34no—
110C2×#216—94no—
111C2×#216—91no—
112#924—94no—
113#835—94no—
114#835—94no—
115#835—91no—
116#835—184no—
117#830—94no—
118(C2×D3)⋊C8—91no—
119#816—91no—
120C23×C32—31no—
121C22×C3×D3#6714yes—
122C32×Q8—21no—
123C32×Q8—28no—
124C32×Q8—24no—
125C32×Q8—32no—
126C32×Q8—44no—
127C32×Q8—64no—
128C32×Q8—61no—
129C32×Q8#6718yes—
130C32×Q8#6712yes—
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—31no—
135C2×C32×C4—44no—
136C2×C32×C4—64no—
137C2×C32×C4—61no—
138C2×C32×C4—61no—
139C2×C32×C4—61no—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—124no—
146#634—21no—
147#634—21no—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3#6714yes—
151C3×C4×D3—216no—
152C3×C4×D3—24no—
153C3×C4×D3—24no—
154C3×Dic6—216no—
155C3×Dic6—24no—
156C3×Dic6—64no—
157C3×Dic6—64no—
158C3×Dic6#6718yes—
159C3⋊(C3⋊C8)—21no—
160C3⋊(C3⋊C8)—61no—
161C3⋊(C3⋊C8)—64no—
162C3⋊(C3⋊C8)—124no—
163C3×C3⋊C8—64no—
164#451—271noSylow
165C2×C33#12011yescharacteristic
166C2×C33C2×Q1611yescharacteristic
167C2×C33#13711yescharacteristic
168C32×D3—22no—
169C2×C3×Q8—24no—
170C2×C3×Q8—34no—
171C2×C3×Q8—64no—
172C2×C3×Q8—68no—
173C2×C3×Q8—68no—
174C2×C3×Q8—91no—
175C2×C3×Q8D3214yes—
176C2×C3×Q8C2×C3⋊D311yescharacteristic
177C22×C3×C4—34no—
178C22×C3×C4—64no—
179D3×Q8—22no—
180D3×Q8—22no—
181C2×C4×D3—21no—
182C2×C4×D3C2×C3⋊D311yescharacteristic
183C2×Dic6—21no—
184C2×Dic6—34no—
185C2×Dic6C2×C3⋊D311yescharacteristic
186C3×Q16—182no—
187C2×C3×C8—91no—
188#217—184no—
189#216—68no—
190#216—188no—
191#216—182no—
192(C2×D3)⋊C4—181no—
193C4⋊Dic3—94no—
194C4⋊Dic3—91no—
195C4⋊Dic3—184no—
196Dic3⋊C4—184no—
197Dic3⋊C4—181no—
198C2×C3⋊C8—34no—
199C2×C3⋊C8—94no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—184no—
202C22×C32—24no—
203C22×C32—32no—
204C22×C32—61no—
205C22×C32—61no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×#6714yes—
208C2×C3×D3—24no—
209C2×C3×D3—24no—
210C2×C3×D3#21618yescomplemented
211C32×C4—22no—
212C32×C4—24no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—48no—
216C32×C4—61no—
217C32×C4—62no—
218C32×C4D3×D411yescharacteristic
219C32×C4C2×#6714yes—
220C32×C4C2×#6714yes—
221C32×C4C2×#6711yescharacteristic
222C3⋊Dic3—41no—
223C3⋊Dic3—121no—
224C3⋊Dic3—124no—
225C3⋊Dic3—244no—
226C3×Dic3—24no—
227C3×Dic3—28no—
228C3×Dic3—124no—
229C22×Q8—31no—
230C2×Q16—91no—
231C2×Q16—271no—
232C22⋊Q8—271no—
233#103—271no—
234#103—271no—
235C22⋊C8—271no—
236C33#45111yescharacteristic, complemented, Sylow
237C23×C3—34no—
238C22×D3#63811yescharacteristic
239C3×Q8—24no—
240C3×Q8—21no—
241C3×Q8—28no—
242C3×Q8—38no—
243C3×Q8—44no—
244C3×Q8—616no—
245C3×Q8—64no—
246C3×Q8—181no—
247C3×Q8—181no—
248C3×Q8#62518yes—
249C3×Q8#63812yes—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—34no—
254C2×C3×C4—44no—
255C2×C3×C4—616no—
256C2×C3×C4—64no—
257C2×C3×C4—64no—
258C2×C3×C4—64no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—64no—
264C2×Dic3—184no—
265C2×Dic3—181no—
266C2×Dic3—364no—
267C2×Dic3#63811yescharacteristic
268C4×D3—24no—
269C4×D3—21no—
270C4×D3—21no—
271Dic6—24no—
272Dic6—21no—
273Dic6—64no—
274Dic6—64no—
275Dic6#63812yes—
276C3×C8—181no—
277C3⋊C8—64no—
278C3⋊C8—184no—
279C3⋊C8—181no—
280C3⋊C8—364no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—62no—
285C2×C32D3×Q1611yescharacteristic
286C2×C32#91811yescharacteristic
287C2×C32#93711yescharacteristic
288C2×C32#94214yes—
289C2×C32C2×#21614yes—
290C2×C32#95314yes—
291C3×D3—28no—
292C2×Q8—31no—
293C2×Q8—61no—
294C2×Q8—62no—
295C2×Q8—62no—
296C2×Q8—91no—
297C2×Q8D3×C3⋊D311yescharacteristic
298C22×C4—31no—
299C22×C4—61no—
300Q16—182no—
301Q16—542no—
302C2×C8—91no—
303C2×C8—271no—
304C4⋊C4—271no—
305C4⋊C4—541no—
306C22⋊C4—541no—
307C22×C3—24no—
308C22×C3—38no—
309C22×C3—64no—
310C22×C3—64no—
311C22×C3D3×#6714yes—
312C22×C3C2×#63811yescharacteristic
313C2×D3—21no—
314C2×D3—21no—
315C2×D3#375612yescomplemented
316C3×C4—22no—
317C3×C4—28no—
318C3×C4—24no—
319C3×C4—24no—
320C3×C4—48no—
321C3×C4—64no—
322C3×C4—68no—
323C3×C4—361no—
324C3×C4C2×#62514yes—
325C3×C4D3×#6714yes—
326C3×C4C2×#63811yescharacteristic
327C3×C4C2×#63811yescharacteristic
328Dic3—21no—
329Dic3—22no—
330Dic3—124no—
331Dic3—361no—
332Dic3—364no—
333Dic3—724no—
334C32—24no—
335C32#518611yescharacteristic, complemented
336C32#522814yescomplemented
337C23—31no—
338Q8—21no—
339Q8—32no—
340Q8—64no—
341Q8—61no—
342Q8—181no—
343Q8—181no—
344Q8#629012yes—
345C2×C4—21no—
346C2×C4—31no—
347C2×C4—64no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—181no—
352C2×C4—541no—
353C2×C4C2×D3×C3⋊D311yescharacteristic
354C8—181no—
355C8—541no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—68no—
360C2×C3D3×#21614yes—
361C2×C3#6394814yes—
362C2×C3#6397614yes—
363C2×C3#6415111yescharacteristic
364C2×C3C2×#375611yescharacteristic
365C2×C3#6416211yescharacteristic
366D3—22no—
367C22—32no—
368C22—61no—
369C22—61no—
370C22D3×#63811yescharacteristic
371C4—22no—
372C4—61no—
373C4—62no—
374C4—361no—
375C4—1081no—
376C4C2×#629011yescharacteristic
377C4D3×#63811yescharacteristic
378C3—24no—
379C3#1059159414yescomplemented
380C3#1059292711yescharacteristic, complemented
381C2—62no—
382C2D3×#375611yescharacteristic
383C2#1172358111yescharacteristic
384C2#1172362911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
4. C2×#88798C2semidirect
5. #11722205C2semidirect
6. #11722038C2semidirect
28. C2×#3756D3semidirect
97. C2×C32×D3Q16semidirect
210. C2×C3×D3#216semidirect
236. C33#451semidirect
315. C2×D3#3756semidirect
335. C32#5186semidirect
336. C32#5228semidirect
379. C3#10591594semidirect
380. C3#10592927semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×Q82
C2×C3×#37562
#117234252
C2×#887982
#117222052
#117220382
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929273
#105915943