Finite Group Databasebeta

← #13161368

Subgroups — #13161368

Order 1728.

872 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 152 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#11723991C211yesmaximal, characteristic
2C3×#64157C211yesmaximal, characteristic, complemented
3#11723357C211yesmaximal, characteristic, complemented
4C32×#837C211yesmaximal, characteristic, complemented
5#11722059C211yesmaximal, characteristic, complemented
6#11722048C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10592522—31nomaximal
9#10591663—34nomaximal
10C33×#41C2211yescharacteristic
11#89076C2211yescharacteristic
12#88863C411yescharacteristic, complemented
13#88862C411yescharacteristic, complemented
14#88861C411yescharacteristic, complemented
15#88860C411yescharacteristic, complemented
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3748—21no—
18C3×#3748C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4×Dic3C2211yescharacteristic
22C32×#208C2211yescharacteristic
23#88788—21no—
24#88785—21no—
25#64159—31no—
26#64159—34no—
27#64159—64no—
28#64157D311yescharacteristic, complemented
29#64116—31no—
30C3×#948—34no—
31#63817—34no—
32C32×#104—31no—
33#63649—31no—
34#63632—31no—
35#63630—31no—
36C3×#837D314yescomplemented
37#63578—34no—
38#63570—34no—
39#63554—34no—
40C33×Q8C2×C411yescharacteristic
41C33×D4—21no—
42C33×D4C2×C411yescharacteristic
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45#6312—21no—
46C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
47C4×C3⋊(C3⋊D3)—21no—
48#6308C2×C411yescharacteristic
49C2×C32×Dic3—21no—
50C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
51C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)D411yescharacteristic
54C3×C3⋊(C3⋊C8)D411yescharacteristic
55C32×C3⋊C8—21no—
56#5145—91no—
57#4823—94no—
58C32×#41—24no—
59C32×#41C2×D314yes—
60C32×#41C2×D311yescharacteristic
61#3828—31no—
62#3828—34no—
63#3828—64no—
64C32×M16—31no—
65C32×C42—31no—
66#3756—31no—
67#3756—34no—
68#3756—64no—
69#3755—31no—
70#3755—34no—
71#3755—64no—
72#3754—31no—
73#3754—34no—
74#3754—64no—
75#3753—31no—
76#3753—34no—
77#3753—64no—
78#3748—21no—
79#3748C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)—21no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—34no—
83C2×C3⋊(C3⋊C8)—64no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85#3743—61no—
86C8×C3⋊D3—61no—
87C3×C4×Dic3C2×D314yes—
88C3×#208—34no—
89C3×#208—64no—
90C3×#208C2×D314yes—
91C2×C3×C3⋊C8—34no—
92C2×C3×C3⋊C8—64no—
93#3712—64no—
94#3710—64no—
95C22×C33—21no—
96C22×C33#4111yescharacteristic
97C2×C3⋊(C3⋊D3)—21no—
98C33×C4—21no—
99C33×C4C22×C411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—21no—
102C32×Dic3—41no—
103C3×#131—91no—
104#950—94no—
105#950—91no—
106#950—184no—
107#948—34no—
108#907—94no—
109#901—91no—
110C3×#104—34no—
111#837C3⋊D311yescharacteristic, complemented
112#825—91no—
113#822—94no—
114#822—91no—
115#815—91no—
116#805—94no—
117#803—94no—
118C32×Q8—24no—
119C32×Q8C4×D314yes—
120C32×Q8C4×D311yescharacteristic
121C32×D4—21no—
122C32×D4—24no—
123C32×D4—24no—
124C32×D4—44no—
125C32×D4C4×D314yes—
126C32×D4C4×D311yescharacteristic
127C2×C32×C4—21no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—44no—
131C2×C32×C4—61no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—61no—
135#638—64no—
136#638—124no—
137C4⋊(C3⋊D3)—31no—
138C4⋊(C3⋊D3)—34no—
139C4⋊(C3⋊D3)—64no—
140C4×C3⋊D3—61no—
141C4×C3⋊D3—64no—
142C4×C3⋊D3—124no—
143#634—31no—
144#634—34no—
145#634—64no—
146C2×C3×Dic3—24no—
147C32×C8—61no—
148C3⋊(C3⋊C8)—21no—
149C3⋊(C3⋊C8)—31no—
150C3⋊(C3⋊C8)—31no—
151C3⋊(C3⋊C8)—34no—
152C3⋊(C3⋊C8)—34no—
153C3⋊(C3⋊C8)—64no—
154C3⋊(C3⋊C8)—64no—
155C3⋊(C3⋊C8)#6711yescharacteristic
156C3⋊(C3⋊C8)#6711yescharacteristic
157C3×C3⋊C8—24no—
158C3×C3⋊C8—34no—
159C3×C3⋊C8—34no—
160C3×C3⋊C8—64no—
161#444—271noSylow
162C2×C33—21no—
163C2×C33—41no—
164C2×C33C4×D411yescharacteristic
165C3⋊(C3⋊D3)—41no—
166C3×#41—24no—
167C3×#41D3214yes—
168C3×#41C2×C3⋊D311yescharacteristic
169#235—94no—
170#235—91no—
171#235—184no—
172C3×M16—34no—
173C3×M16—91no—
174C3×M16—181no—
175C2×C3×C8—91no—
176C2×C3×C8—181no—
177C3×C42—34no—
178#216—94no—
179#216—91no—
180#216—184no—
181#215—94no—
182#215—91no—
183#215—184no—
184#214—94no—
185#214—91no—
186#214—184no—
187#213—94no—
188#213—91no—
189#213—184no—
190C4×Dic3C2×C3⋊D311yescharacteristic
191#208—34no—
192#208—64no—
193#208C2×C3⋊D311yescharacteristic
194C2×C3⋊C8—34no—
195C2×C3⋊C8—64no—
196C2×C3⋊C8—94no—
197C2×C3⋊C8—91no—
198C2×C3⋊C8—184no—
199ΓL(1,𝔽25)—181no—
200ΓL(1,𝔽25)—184no—
201D3×C8—181no—
202D3×C8—184no—
203C22×C32—21no—
204C22×C32—24no—
205C22×C32—24no—
206C22×C32—44no—
207C22×C32#23511yescharacteristic
208C22×C32#23714yes—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—64no—
211C2×C3⋊D3—124no—
212C32×C4—21no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—44no—
217C32×C4—121no—
218C32×C4C2×C4×D314yes—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4D3×D414yes—
221C32×C4C2×#6711yescharacteristic
222C3⋊Dic3—61no—
223C3⋊Dic3—64no—
224C3⋊Dic3—124no—
225C3×Dic3—44no—
226#135—271no—
227#131—91no—
228#131—271no—
229#108—271no—
230#104—31no—
231#104—271no—
232#97—271no—
233C33#44411yescharacteristic, complemented, Sylow
234C3×Q8—24no—
235C3×Q8#62414yes—
236C3×Q8C4×C3⋊D311yescharacteristic
237C3×D4—24no—
238C3×D4—21no—
239C3×D4—24no—
240C3×D4—44no—
241C3×D4#62414yes—
242C3×D4C4×C3⋊D311yescharacteristic
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—24no—
246C2×C3×C4—44no—
247C2×C3×C4—64no—
248C2×C3×C4C2×D3214yes—
249C2×C3×C4C22×C3⋊D311yescharacteristic
250#67—184no—
251#67—181no—
252#67—364no—
253C2×Dic3—21no—
254D12—94no—
255D12—91no—
256D12—184no—
257C4×D3—184no—
258C4×D3—181no—
259C4×D3—364no—
260Dic6—94no—
261Dic6—91no—
262Dic6—184no—
263C3×C8—64no—
264C3×C8—91no—
265C3×C8—91no—
266C3×C8—181no—
267C3⋊C8—21no—
268C3⋊C8—34no—
269C3⋊C8—34no—
270C3⋊C8—64no—
271C3⋊C8—94no—
272C3⋊C8—94no—
273C3⋊C8—91no—
274C3⋊C8—91no—
275C3⋊C8—184no—
276C3⋊C8—184no—
277C2×C32—21no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—44no—
281C2×C32—41no—
282C2×C32—44no—
283C2×C32—84no—
284C2×C32#67⋊C414yes—
285C2×C32C4×#6711yescharacteristic
286C3⋊D3—121no—
287C3⋊D3—124no—
288C3⋊D3—244no—
289#41—271no—
290#41D3×C3⋊D311yescharacteristic
291Q16—271no—
292SD16—271no—
293SD16—271no—
294D8—271no—
295M16—31no—
296M16—91no—
297M16—181no—
298M16—541no—
299C2×C8—91no—
300C2×C8—181no—
301C2×C8—271no—
302C2×C8—541no—
303C42—31no—
304C22×C3—21no—
305C22×C3—24no—
306C22×C3—24no—
307C22×C3—44no—
308C22×C3#380414yes—
309C22×C3#383011yescharacteristic
310C2×D3—181no—
311C2×D3—184no—
312C2×D3—364no—
313C3×C4—21no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—44no—
318C3×C4—124no—
319C3×C4C2×#62414yes—
320C3×C4D3×#6714yes—
321C3×C4C2×C4×C3⋊D311yescharacteristic
322C3×C4D4×C3⋊D311yescharacteristic
323Dic3—41no—
324Dic3—181no—
325Dic3—184no—
326Dic3—364no—
327C32—24no—
328C32#482314yescomplemented
329C32#514511yescharacteristic, complemented
330Q8—271no—
331Q8#628911yescharacteristic
332D4—21no—
333D4—271no—
334D4—541no—
335D4#628911yescharacteristic
336C2×C4—21no—
337C2×C4—61no—
338C2×C4—541no—
339C2×C4C2×D3×C3⋊D311yescharacteristic
340C8—61no—
341C8—91no—
342C8—91no—
343C8—181no—
344C8—271no—
345C8—271no—
346C2×C3—21no—
347C2×C3—24no—
348C2×C3—24no—
349C2×C3—44no—
350C2×C3—41no—
351C2×C3—44no—
352C2×C3—84no—
353C2×C3#6395214yes—
354C2×C3#6409011yescharacteristic
355D3—361no—
356D3—364no—
357D3—724no—
358C22—21no—
359C22—541no—
360C22#8902811yescharacteristic
361C4—21no—
362C4—121no—
363C4—541no—
364C4C2×#628911yescharacteristic
365C4C3⋊D3×#6711yescharacteristic
366C3—24no—
367C3#1059166314yescomplemented
368C3#1059252211yescharacteristic, complemented
369C2—21no—
370C2—41no—
371C2—1081no—
372C2#1172362211yescharacteristic
subgroupG/H
centerC4C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C4×D4
prime radical#11723991C2
cosocleC2×C33×C4C23
hypercenter#41D3×C3⋊D3
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C3×#64157C2semidirect
3. #11723357C2semidirect
4. C32×#837C2semidirect
5. #11722059C2semidirect
6. #11722048C2semidirect
7. #11722019C2semidirect
12. #88863C4semidirect
13. #88862C4semidirect
14. #88861C4semidirect
15. #88860C4semidirect
28. #64157D3semidirect
36. C3×#837D3semidirect
111. #837C3⋊D3semidirect
233. C33#444semidirect
328. C32#4823semidirect
329. C32#5145semidirect
367. C3#10591663semidirect
368. C3#10592522semidirect

Maximal subgroups

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

subgroupindex
#117239912
C3×#641572
#117233572
C32×#8372
#117220592
#117220482
#117220192
#105925223
#105916633