Finite Group Databasebeta

← #13161149

Subgroups — #13161149

Order 1728.

992 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 150 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×#88862C211yesmaximal, characteristic
2C3×#64153C211yesmaximal, characteristic, complemented
3#11723494C211yesmaximal, characteristic
4C2×#88797C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592517—31nomaximal
9#10591565—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×#6289—21no—
14#88862—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3750—21no—
17C3×#3750—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88804—21no—
22#88797—22no—
23#64153D311yescharacteristic, complemented
24C2×#3755—31no—
25C2×#3755—34no—
26C2×#3755—64no—
27C2×#3744—31no—
28#64102—31no—
29C3×#944—34no—
30#63894—34no—
31C2×#3717—34no—
32C32×#103—31no—
33#63641—31no—
34#63637—31no—
35C3×#835D314yescomplemented
36Dic3⋊(C3⋊C8)—34no—
37#63565—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))D412yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×#634—21no—
49C3×#634—21no—
50C2×C32×Dic3—21no—
51#6289—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8—21no—
54#5083—91no—
55#4818—94no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—34no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C4×C3⋊D3—61no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—34no—
65C2×#624—64no—
66C2×C32×C8—31no—
67C32×C4⋊C4—31no—
68#3755—62no—
69#3755—68no—
70#3755—616no—
71#3752—61no—
72#3750—21no—
73#3750—21no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3744—62no—
79C3×C4⋊Dic3C2×D314yes—
80C3×Dic3⋊C4—64no—
81C3×Dic3⋊C4—64no—
82C3×C4×Dic3—34no—
83C2×C3×C3⋊C8C2×D314yes—
84#3722—64no—
85#3717—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—22no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C33×C4—41no—
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—41no—
95C3×C3⋊Dic3SD1612yescomplemented
96C32×Dic3—41no—
97C3×C4⋊Q8—91no—
98#944—34no—
99C2×#215—94no—
100C2×#215—91no—
101C2×#215—91no—
102C2×#215—184no—
103C2×#204—94no—
104#893—94no—
105C4×D12—91no—
106C3×#103—34no—
107#835—91no—
108#835C3⋊D311yescharacteristic, complemented
109Dic3⋊C8—94no—
110#810—94no—
111C4⋊(C3⋊C8)—91no—
112C22×C3⋊D3—61no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—124no—
115C32×Q8—28no—
116C32×Q8—24no—
117C32×Q8—21no—
118C32×Q8—21no—
119C32×Q8—24no—
120C32×Q8—44no—
121C2×C32×C4—21no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—44no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3#6711yescharacteristic
130C2×C3⋊Dic3#6711yescharacteristic
131C4⋊(C3⋊D3)—32no—
132C4⋊(C3⋊D3)—38no—
133C4⋊(C3⋊D3)—68no—
134C4⋊(C3⋊D3)—61no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)—124no—
137C4×C3⋊D3—122no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—64no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146#624—128no—
147C32×C8—61no—
148C3⋊(C3⋊C8)—68no—
149C3⋊(C3⋊C8)—64no—
150C3⋊(C3⋊C8)—61no—
151C3×C3⋊C8—24no—
152#461—271noSylow
153C2×C33#12111yescharacteristic
154C2×C33C2×SD1611yescharacteristic
155C2×C33#13711yescharacteristic
156C3⋊(C3⋊D3)—42no—
157C2×C3×Q8—24no—
158C2×C3×Q8—91no—
159C2×C3×Q8D3214yes—
160C2×C3×Q8C2×C3⋊D311yescharacteristic
161C2×D12—94no—
162C2×D12—91no—
163C2×D12—184no—
164C2×C4×D3—181no—
165C2×C4×D3—184no—
166C2×Dic6—34no—
167C2×C3×C8—34no—
168C3×C4⋊C4—34no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4—181no—
171C3×C42—91no—
172#215—182no—
173#215—188no—
174#215—182no—
175#215—1816no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—184no—
178C4⋊Dic3C2×C3⋊D311yescharacteristic
179Dic3⋊C4—64no—
180Dic3⋊C4—64no—
181C4×Dic3—34no—
182C2×C3⋊C8—94no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—184no—
185C2×C3⋊C8C2×C3⋊D311yescharacteristic
186#204—188no—
187C22×C32—24no—
188C22×C32D3×D414yes—
189C22×C32C2×#6711yescharacteristic
190C2×C3⋊D3—62no—
191C2×C3⋊D3—68no—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—124no—
197C2×C3⋊D3—244no—
198C2×C3⋊D3—244no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—44no—
203C32×C4—41no—
204C32×C4—121no—
205C32×C4#23514yes—
206C32×C4D3×D414yes—
207C32×C4D3×D411yescharacteristic
208C32×C4#23711yescharacteristic
209C3⋊Dic3—21no—
210C3⋊Dic3—41no—
211C3⋊Dic3#21512yescomplemented
212C3×Dic3—38no—
213C3×Dic3—44no—
214C3×Dic3—64no—
215C3×Dic3—124no—
216C2×SD16—271no—
217C2×SD16—271no—
218C4⋊Q8—91no—
219C4×D4—271no—
220C4⋊C8—271no—
221#103—31no—
222#103—271no—
223C33#46111yescharacteristic, complemented, Sylow
224C22×D3—184no—
225C22×D3—181no—
226C22×D3—364no—
227C3×Q8—28no—
228C3×Q8—24no—
229C3×Q8—24no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—44no—
233C3×Q8—181no—
234C3×Q8—181no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—44no—
239C2×C3×C4—64no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—34no—
247C2×Dic3—34no—
248C2×Dic3—64no—
249D12—98no—
250D12—92no—
251D12—188no—
252D12—184no—
253D12—181no—
254D12—364no—
255C4×D3—368no—
256C4×D3—362no—
257Dic6—64no—
258Dic6—64no—
259C3×C8—64no—
260C3⋊C8—21no—
261C3⋊C8—188no—
262C3⋊C8—184no—
263C3⋊C8—181no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32#88414yes—
268C2×C32D3×SD1614yes—
269C2×C32#91814yes—
270C2×C32#93811yescharacteristic
271C2×C32C2×#21511yescharacteristic
272C2×C32#94211yescharacteristic
273C3⋊D3—122no—
274C3⋊D3—128no—
275C3⋊D3—248no—
276C2×Q8—91no—
277C2×Q8D3×C3⋊D311yescharacteristic
278C2×D4—271no—
279C22×C4—541no—
280SD16—542no—
281SD16—542no—
282C2×C8—31no—
283C2×C8—271no—
284C4⋊C4—31no—
285C4⋊C4—181no—
286C4⋊C4—181no—
287C22⋊C4—541no—
288C42—91no—
289C22×C3—24no—
290C22×C3D3×#6714yes—
291C22×C3D4×C3⋊D311yescharacteristic
292C2×D3—182no—
293C2×D3—188no—
294C2×D3—368no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—364no—
298C2×D3—364no—
299C2×D3—724no—
300C2×D3—724no—
301C3×C4—24no—
302C3×C4—24no—
303C3×C4—48no—
304C3×C4—41no—
305C3×C4—44no—
306C3×C4—92no—
307C3×C4—124no—
308C3×C4—181no—
309C3×C4—361no—
310C3×C4#380414yes—
311C3×C4#381114yes—
312C3×C4#382811yescharacteristic
313C3×C4D4×C3⋊D311yescharacteristic
314Dic3—38no—
315Dic3—41no—
316Dic3—64no—
317Dic3—124no—
318C32—24no—
319C32#481814yescomplemented
320C32#508311yescharacteristic, complemented
321C23—541no—
322Q8—21no—
323Q8—21no—
324Q8—181no—
325Q8—181no—
326D4—272no—
327D4—541no—
328C2×C4—21no—
329C2×C4—61no—
330C2×C4—91no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—1082no—
334C2×C4C2×D3×C3⋊D311yescharacteristic
335C8—61no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3D3×#21514yes—
341C2×C3#6394814yes—
342C2×C3#6395814yes—
343C2×C3#6409311yescharacteristic
344C2×C3SD16×C3⋊D311yescharacteristic
345C2×C3#6412711yescharacteristic
346D3—362no—
347D3—368no—
348D3—728no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22C3⋊D3×#6711yescharacteristic
353C4—41no—
354C4—92no—
355C4—121no—
356C4—181no—
357C4—361no—
358C4#8903111yescharacteristic
359C4#8903911yescharacteristic
360C3—24no—
361C3#1059156514yescomplemented
362C3#1059251711yescharacteristic, complemented
363C2—1082no—
364C2C3⋊D3×#21511yescharacteristic
365C2#1172358611yescharacteristic
366C2#1172362311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#461
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. C3×#64153C2semidirect
5. C32×#835C2semidirect
6. #11722051C2semidirect
7. #11722039C2semidirect
23. #64153D3semidirect
35. C3×#835D3semidirect
95. C3×C3⋊Dic3SD16semidirect
108. #835C3⋊D3semidirect
211. C3⋊Dic3#215semidirect
223. C33#461semidirect
319. C32#4818semidirect
320. C32#5083semidirect
361. C3#10591565semidirect
362. C3#10592517semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×#641532
#117234942
C2×#887972
C32×#8352
#117220512
#117220392
#105925173
#105915653