Finite Group Databasebeta

← #13161296

Subgroups — #13161296

Order 1728.

976 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723988C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3#11723410C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic
8#10592497—31nomaximal
9#10591579—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13#88859—21no—
14#88859—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3756—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22#88797—22no—
23#64156—31no—
24#64156—34no—
25#64156—64no—
26C2×#3756D311yescharacteristic, complemented
27C2×#3744—31no—
28#64096—31no—
29C2×C3×#216—34no—
30#63843—34no—
31C2×#3717—34no—
32C32×#103—31no—
33#63641—31no—
34#63636—31no—
35C3×#835D314yescomplemented
36#63572—34no—
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×#634—21no—
49C3×#634D412yes—
50C2×C32×Dic3—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C32×C3⋊C8—21no—
53#5185—91no—
54#4798—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—34no—
63C2×C32×C8—31no—
64C32×C4⋊C4—31no—
65#3756—22no—
66#3752—61no—
67#3752—61no—
68#3752—64no—
69#3752—64no—
70#3752—68no—
71#3752—68no—
72#3750—61no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—34no—
75C4×C3⋊Dic3—61no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3744—62no—
79C3×#216—68no—
80C3×C4⋊Dic3C2×D314yes—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—64no—
84Dic32—64no—
85#3717—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
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)—21no—
95C3×C3⋊Dic3—22no—
96C32×Dic3—41no—
97C2×C3×Q16—91no—
98#947—94no—
99#947—91no—
100#947—184no—
101Q8×Dic3—91no—
102C2×#216—34no—
103C2×#215—91no—
104C2×#204—94no—
105#887—94no—
106C3×#103—34no—
107#835C3⋊D311yescharacteristic, complemented
108#821—91no—
109Dic3⋊C8—94no—
110Dic3⋊C8—91no—
111#809—94no—
112C22×C3⋊D3—68no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—61no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—24no—
119C32×Q8—44no—
120C32×Q8—44no—
121C2×C32×C4—28no—
122C2×C32×C4—24no—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—34no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—64no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C4⋊(C3⋊D3)—124no—
141#634—21no—
142#634#6712yes—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C3×Dic6—38no—
146C3×Dic6—64no—
147C32×C8—61no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—24no—
150C3×C3⋊C8—64no—
151#464—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13511yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C2×C3×Q8—24no—
157C2×C3×Q8—91no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×D12—94no—
161C2×D12—91no—
162C2×D12—184no—
163C2×Dic6—34no—
164C3×Q16—182no—
165C2×C3×C8—34no—
166C2×C3×C8—91no—
167C3×C4⋊C4—34no—
168#216—68no—
169#215—182no—
170(C2×D3)⋊C4—184no—
171(C2×D3)⋊C4—184no—
172(C2×D3)⋊C4—181no—
173(C2×D3)⋊C4—181no—
174(C2×D3)⋊C4—188no—
175(C2×D3)⋊C4—188no—
176C4⋊Dic3—181no—
177C4⋊Dic3C2×C3⋊D311yescharacteristic
178Dic3⋊C4—184no—
179C4×Dic3—94no—
180C4×Dic3—91no—
181C4×Dic3—184no—
182C4×Dic3—181no—
183C4×Dic3—184no—
184C2×C3⋊C8—34no—
185C2×C3⋊C8C2×C3⋊D311yescharacteristic
186#204—188no—
187C22×C32—24no—
188C22×C32D3×D414yes—
189C22×C32D3×D411yescharacteristic
190C2×C3⋊D3—128no—
191C2×C3⋊D3—128no—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—124no—
194C2×C3⋊D3—124no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—121no—
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×C4#23711yescharacteristic
208C32×C4C2×#6711yescharacteristic
209C3⋊Dic3—22no—
210C3⋊Dic3—61no—
211C3⋊Dic3—61no—
212C3⋊Dic3—64no—
213C3⋊Dic3—64no—
214C3⋊Dic3—124no—
215C3⋊Dic3—124no—
216C3×Dic3—44no—
217C3×Dic3—68no—
218C2×Q16—91no—
219C2×SD16—271no—
220#124—271no—
221C4×Q8—271no—
222C4⋊C8—271no—
223#103—31no—
224#102—271no—
225C33#46411yescharacteristic, complemented, Sylow
226C22×D3—188no—
227C22×D3—184no—
228C22×D3—181no—
229C3×Q8—24no—
230C3×Q8—24no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—44no—
234C3×Q8—44no—
235C3×Q8—92no—
236C3×Q8—181no—
237C2×C3×C4—28no—
238C2×C3×C4—24no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—64no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—64no—
247C2×Dic3—94no—
248C2×Dic3—94no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—184no—
252C2×Dic3—184no—
253D12—184no—
254D12—184no—
255D12—181no—
256D12—181no—
257D12—364no—
258D12—364no—
259Dic6—38no—
260Dic6—64no—
261C3×C8—64no—
262C3×C8—181no—
263C3⋊C8—21no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#88414yes—
269C2×C32#91614yes—
270C2×C32#91814yes—
271C2×C32#91811yescharacteristic
272C2×C32#91911yescharacteristic
273C2×C32#93911yescharacteristic
274C3⋊D3—248no—
275C3⋊D3—241no—
276C3⋊D3—244no—
277C2×Q8—91no—
278C2×Q8D3×C3⋊D311yescharacteristic
279C2×D4—271no—
280Q16—182no—
281SD16—542no—
282C2×C8—31no—
283C2×C8—91no—
284C4⋊C4—31no—
285C4⋊C4—541no—
286C22⋊C4—541no—
287C22⋊C4—541no—
288C42—271no—
289C42—541no—
290C22×C3—24no—
291C22×C3#381114yes—
292C22×C3D4×C3⋊D311yescharacteristic
293C2×D3—368no—
294C2×D3—368no—
295C2×D3—368no—
296C2×D3—361no—
297C2×D3—361no—
298C2×D3—361no—
299C2×D3—364no—
300C2×D3—364no—
301C2×D3—364no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—48no—
305C3×C4—41no—
306C3×C4—44no—
307C3×C4—124no—
308C3×C4—182no—
309C3×C4#380414yes—
310C3×C4D3×#6714yes—
311C3×C4#382811yescharacteristic
312C3×C4D4×C3⋊D311yescharacteristic
313Dic3—41no—
314Dic3—68no—
315Dic3—181no—
316Dic3—181no—
317Dic3—184no—
318Dic3—184no—
319Dic3—364no—
320Dic3—364no—
321C32—24no—
322C32#479814yescomplemented
323C32#518511yescharacteristic, complemented
324C23—541no—
325Q8—21no—
326Q8—21no—
327Q8—92no—
328Q8—181no—
329D4—541no—
330D4—541no—
331C2×C4—21no—
332C2×C4—61no—
333C2×C4—181no—
334C2×C4—271no—
335C2×C4—271no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—61no—
338C8—181no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3#6394514yes—
343C2×C3#6395014yes—
344C2×C3#6397114yes—
345C2×C3#6409311yescharacteristic
346C2×C3#6412511yescharacteristic
347C2×C3#6412711yescharacteristic
348D3—728no—
349D3—724no—
350D3—721no—
351C22—1081no—
352C22—1081no—
353C22—1081no—
354C22#8903911yescharacteristic
355C4—41no—
356C4—121no—
357C4—182no—
358C4—541no—
359C4—541no—
360C4#8903111yescharacteristic
361C4C3⋊D3×#6711yescharacteristic
362C3—24no—
363C3#1059157914yescomplemented
364C3#1059249711yescharacteristic, complemented
365C2—2161no—
366C2#1172357811yescharacteristic
367C2#1172358711yescharacteristic
368C2#1172362411yescharacteristic
subgroupG/H
centerC22#89039
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#464
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
3. #11723410C2semidirect
5. C32×#835C2semidirect
6. #11722050C2semidirect
26. C2×#3756D3semidirect
35. C3×#835D3semidirect
107. #835C3⋊D3semidirect
225. C33#464semidirect
322. C32#4798semidirect
323. C32#5185semidirect
363. C3#10591579semidirect
364. C3#10592497semidirect

Maximal subgroups

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

subgroupindex
#117239882
C2×C3×#37562
#117234102
C2×#887972
C32×#8352
#117220502
#117220352
#105924973
#105915793