Finite Group Databasebeta

← #13161285

Subgroups — #13161285

Order 1728.

864 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 130 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#11723983C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3#11723378C211yesmaximal, characteristic
4#11723356C211yesmaximal, characteristic, complemented
5C3×#63663C211yesmaximal, characteristic, complemented
6#11722058C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592876—31nomaximal
9#10591542—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11#89076C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13#88863—21no—
14#88862—21no—
15#88855C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C3×#3748C2211yescharacteristic
18C32×#215—21no—
19C32×#215—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21#88794—21no—
22#88790—21no—
23#88788—21no—
24#88785—21no—
25C2×C32×SD16—31no—
26#64151—31no—
27#64151—34no—
28#64151—64no—
29#64141—31no—
30#64110—31no—
31C2×C3×#215D314yescomplemented
32#63830—34no—
33#63817—34no—
34#63663D311yescharacteristic, complemented
35#63660—31no—
36#63646—31no—
37C3×#836—34no—
38#63577—34no—
39#63558—34no—
40C22×C32×D3—21no—
41C33×Q8—21no—
42C33×Q8—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45#6312—21no—
46C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
47C4×C3⋊(C3⋊D3)—21no—
48#6308D411yescharacteristic
49C32×D12—21no—
50C32×D12—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8D411yescharacteristic
55C32×C3⋊C8D411yescharacteristic
56#5177—94no—
57#4754—91no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×D4—31no—
62#3828—31no—
63#3828—34no—
64#3828—64no—
65C2×C3×D12C2×D314yes—
66C32×SD16—61no—
67C32×SD16—61no—
68C2×C32×C8—31no—
69#3756—61no—
70#3756—64no—
71#3756—68no—
72#3755—61no—
73#3755—64no—
74#3755—68no—
75#3754—61no—
76#3753—61no—
77#3748—31no—
78#3748—34no—
79#3748—64no—
80#3748C2×D311yescharacteristic
81#3748C2×D311yescharacteristic
82#3743—61no—
83C8×C3⋊D3—61no—
84C3×#215—24no—
85C3×#215—24no—
86C3×#208—34no—
87C3×#208—34no—
88C2×C3×C3⋊C8C2×D314yes—
89#3716—64no—
90#3714—64no—
91#3712—64no—
92#3710—64no—
93C22×C33#4111yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C32×D3—41no—
96C2×C32×D3—81no—
97C33×C4—41no—
98C33×C4C2×D411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3⋊(C3⋊Dic3)—21no—
101C2×C3×SD16—34no—
102#942—94no—
103#942—91no—
104#942—184no—
105C2×#215C3⋊D311yescharacteristic, complemented
106#932—94no—
107#908—91no—
108#907—91no—
109#901—94no—
110C3×#101—91no—
111#836—34no—
112#833—94no—
113#823—91no—
114#822—91no—
115#819—94no—
116C23×C32—61no—
117C22×C3×D3—24no—
118C32×Q8—28no—
119C32×Q8—24no—
120C32×Q8—21no—
121C32×Q8—21no—
122C32×Q8—24no—
123C32×Q8—44no—
124C32×D4—61no—
125C32×D4—61no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—44no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132#638—61no—
133#638—64no—
134#638—124no—
135C4⋊(C3⋊D3)—31no—
136C4⋊(C3⋊D3)—34no—
137C4⋊(C3⋊D3)—64no—
138C4×C3⋊D3—61no—
139C4×C3⋊D3—64no—
140C4×C3⋊D3—124no—
141#634—31no—
142#634—34no—
143#634—64no—
144C3×D12—24no—
145C3×D12—24no—
146C32×C8—31no—
147C32×C8—31no—
148C3⋊(C3⋊C8)—21no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—68no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—61no—
153C3×C3⋊C8—64no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8#6714yes—
156C3×C3⋊C8#6714yes—
157#471—271noSylow
158C2×C33—21no—
159C2×C33#12111yescharacteristic
160C3⋊(C3⋊D3)—41no—
161C32×D3—81no—
162C2×C3×Q8—24no—
163C2×C3×Q8D3214yes—
164C2×C3×Q8C2×C3⋊D311yescharacteristic
165C2×C3×D4—34no—
166#235—94no—
167#235—91no—
168#235—184no—
169C2×D12C2×C3⋊D311yescharacteristic
170C3×SD16—64no—
171C3×SD16—64no—
172C3×M16—91no—
173C3×M16—91no—
174C2×C3×C8—34no—
175#216—184no—
176#216—181no—
177#216—188no—
178#215—21no—
179#215—21no—
180#215—184no—
181#215—181no—
182#215—188no—
183#214—184no—
184#213—184no—
185#208—34no—
186#208—34no—
187#208—94no—
188#208—91no—
189#208—184no—
190C2×C3⋊C8C2×C3⋊D311yescharacteristic
191D24—181no—
192#204—181no—
193ΓL(1,𝔽25)—184no—
194ΓL(1,𝔽25)—181no—
195D3×C8—184no—
196D3×C8—181no—
197C22×C32—24no—
198C22×C32—121no—
199C22×C32—241no—
200C22×C32#23511yescharacteristic
201C22×C32#23714yes—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—64no—
204C2×C3⋊D3—124no—
205C2×C3×D3—44no—
206C2×C3×D3—84no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—48no—
210C32×C4—41no—
211C32×C4—44no—
212C32×C4D3×D414yes—
213C32×C4D3×D411yescharacteristic
214C32×C4D3×D411yescharacteristic
215C32×C4C2×#6714yes—
216C3⋊Dic3—61no—
217C3⋊Dic3—64no—
218C3⋊Dic3—124no—
219#137—271no—
220AGL(1,ℤ/8)—271no—
221C2×SD16—31no—
222#131—271no—
223#108—271no—
224#101—91no—
225#100—271no—
226C33#47111yescharacteristic, complemented, Sylow
227C23×C3—64no—
228C22×D3—21no—
229C3×Q8—28no—
230C3×Q8—24no—
231C3×Q8—24no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—44no—
235C3×D4—64no—
236C3×D4—64no—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—44no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243#67—184no—
244#67—181no—
245#67—364no—
246D12—21no—
247D12—21no—
248D12—94no—
249D12—91no—
250D12—184no—
251C4×D3—184no—
252C4×D3—181no—
253C4×D3—364no—
254Dic6—94no—
255Dic6—91no—
256Dic6—184no—
257C3×C8—34no—
258C3×C8—34no—
259C3×C8—181no—
260C3×C8—181no—
261C3⋊C8—64no—
262C3⋊C8—64no—
263C3⋊C8—188no—
264C3⋊C8—184no—
265C3⋊C8—181no—
266C3⋊C8#63811yescharacteristic
267C3⋊C8#63811yescharacteristic
268C2×C32—21no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—44no—
272C2×C32—241no—
273C2×C32#88411yescharacteristic
274C2×C32#93814yes—
275C3⋊D3—121no—
276C3⋊D3—124no—
277C3⋊D3—244no—
278C3×D3—84no—
279#41—271no—
280C2×Q8D3×C3⋊D311yescharacteristic
281C2×D4—31no—
282Q16—541no—
283SD16—61no—
284SD16—61no—
285SD16—541no—
286SD16—541no—
287D8—541no—
288M16—91no—
289M16—91no—
290M16—271no—
291M16—541no—
292C2×C8—31no—
293C2×C8—541no—
294C22×C3—24no—
295C22×C3—124no—
296C22×C3—244no—
297C22×C3#380414yes—
298C22×C3#383011yescharacteristic
299C2×D3—41no—
300C2×D3—81no—
301C2×D3—181no—
302C2×D3—184no—
303C2×D3—364no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—48no—
307C3×C4—41no—
308C3×C4—44no—
309C3×C4D3×#6714yes—
310C3×C4#381114yes—
311C3×C4D4×C3⋊D311yescharacteristic
312C3×C4C2×#63811yescharacteristic
313Dic3—181no—
314Dic3—184no—
315Dic3—364no—
316C32—24no—
317C32#475411yescharacteristic, complemented
318C32#517714yescomplemented
319C23—61no—
320Q8—21no—
321Q8—21no—
322Q8—271no—
323D4—61no—
324D4—61no—
325D4—271no—
326D4—541no—
327C2×C4—21no—
328C2×C4—541no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—31no—
331C8—31no—
332C8—181no—
333C8—181no—
334C8—541no—
335C2×C3—24no—
336C2×C3—21no—
337C2×C3—24no—
338C2×C3—44no—
339C2×C3—244no—
340C2×C3#6395814yes—
341C2×C3#6414711yescharacteristic
342D3—81no—
343D3—364no—
344D3—361no—
345D3—724no—
346C22—121no—
347C22—241no—
348C22—541no—
349C22#8902811yescharacteristic
350C4—41no—
351C4—541no—
352C4D3×#63811yescharacteristic
353C4#8903911yescharacteristic
354C3—24no—
355C3#1059154214yescomplemented
356C3#1059287611yescharacteristic, complemented
357C2—21no—
358C2—241no—
359C2—1081no—
360C2#1172359911yescharacteristic
subgroupG/H
centerC2#11723599
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#121
prime radical#11723378C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#471
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11723983C2semidirect
2. C2×C32×#215C2semidirect
4. #11723356C2semidirect
5. C3×#63663C2semidirect
6. #11722058C2semidirect
7. #11722025C2semidirect
31. C2×C3×#215D3semidirect
34. #63663D3semidirect
105. C2×#215C3⋊D3semidirect
226. C33#471semidirect
317. C32#4754semidirect
318. C32#5177semidirect
355. C3#10591542semidirect
356. C3#10592876semidirect

Maximal subgroups

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

subgroupindex
#117239832
C2×C32×#2152
#117233782
#117233562
C3×#636632
#117220582
#117220252
#105928763
#105915423