Finite Group Databasebeta

← #13161379

Subgroups — #13161379

Order 1728.

896 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723989C211yesmaximal, characteristic, complemented
2C3×#64158C211yesmaximal, characteristic, complemented
3C32×#949C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722059C211yesmaximal, characteristic, complemented
6#11722026C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592898—31nomaximal
9#10591678—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13#88855—21no—
14#88855C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))—21no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3755—21no—
18C3×#3753—21no—
19C3×#3748C2211yescharacteristic
20C32×#215—21no—
21C32×#213—21no—
22C32×#208C2211yescharacteristic
23#88789—21no—
24#88789—21no—
25C32×AGL(1,ℤ/8)—31no—
26#64158D311yescharacteristic, complemented
27#64157—31no—
28#64157—34no—
29#64157—64no—
30C2×#3753—31no—
31C3×#949—34no—
32C3×#949D314yescomplemented
33C2×#3713—34no—
34#63660—31no—
35#63650—31no—
36#63649—31no—
37#63578—34no—
38#63558—34no—
39#63558—34no—
40C22×C3×C3⋊D3—21no—
41C22×C32×D3—21no—
42C33×Q8D411yescharacteristic
43C33×D4—21no—
44C33×D4D411yescharacteristic
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C32×D12—21no—
50C32×D12—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
53C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#5199—94no—
57#5199—91no—
58C32×#41—24no—
59C32×#41C2×D314yes—
60C32×#41C2×D311yescharacteristic
61C2×C32×D4—31no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C3×D12—34no—
64C2×C3×D12C2×D314yes—
65C32×SD16—61no—
66C32×D8—61no—
67C32×M16—31no—
68#3755—21no—
69#3753—21no—
70#3753—61no—
71#3753—61no—
72#3748—31no—
73#3748—34no—
74#3748—61no—
75#3748—64no—
76#3748—64no—
77#3748—68no—
78#3748C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—34no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—64no—
83C2×C3⋊(C3⋊C8)—64no—
84C2×C3⋊(C3⋊C8)—68no—
85C3×#215—24no—
86C3×#215—64no—
87C3×#213—24no—
88C3×#213—64no—
89C3×#208—34no—
90C3×#208C2×D314yes—
91#3713—64no—
92#3713—64no—
93C22×C33—21no—
94C22×C33#4111yescharacteristic
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3—81no—
97C2×C32×D3—41no—
98C2×C32×D3—81no—
99C33×C4—21no—
100C33×C4C2×D411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3×AGL(1,ℤ/8)—34no—
103C3×AGL(1,ℤ/8)—91no—
104#949—34no—
105#949C3⋊D311yescharacteristic, complemented
106#948—94no—
107#948—91no—
108#948—184no—
109C2×#213—94no—
110C2×#213—91no—
111#833—94no—
112#833—91no—
113#823—94no—
114#823—91no—
115#822—94no—
116#822—91no—
117C23×C32—61no—
118C22×C3⋊D3—21no—
119C22×C3×D3—24no—
120C22×C3×D3—64no—
121C32×Q8—24no—
122C32×Q8#6714yes—
123C32×Q8#6711yescharacteristic
124C32×D4—28no—
125C32×D4—24no—
126C32×D4—21no—
127C32×D4—24no—
128C32×D4—61no—
129C32×D4—61no—
130C32×D4#6714yes—
131C32×D4#6711yescharacteristic
132C2×C32×C4—28no—
133C2×C32×C4—24no—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D311yescharacteristic
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C3×D12—24no—
141C3×D12—24no—
142C3×D12—64no—
143C3×D12—64no—
144C32×C8—61no—
145C3⋊(C3⋊C8)—21no—
146C3⋊(C3⋊C8)—31no—
147C3⋊(C3⋊C8)—31no—
148C3⋊(C3⋊C8)—34no—
149C3⋊(C3⋊C8)—34no—
150C3⋊(C3⋊C8)—64no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—68no—
153C3⋊(C3⋊C8)—64no—
154C3⋊(C3⋊C8)—61no—
155C3×C3⋊C8—24no—
156C3×C3⋊C8—64no—
157#472—271noSylow
158C2×C33—21no—
159C2×C33—41no—
160C2×C33#12111yescharacteristic
161C3×C3⋊D3—81no—
162C32×D3—81no—
163C3×#41—24no—
164C3×#41D3214yes—
165C3×#41C2×C3⋊D311yescharacteristic
166C2×C3×D4—34no—
167C2×C3×D4—91no—
168C2×D12—34no—
169C2×D12C2×C3⋊D311yescharacteristic
170C3×SD16—64no—
171C3×SD16—181no—
172C3×D8—64no—
173C3×D8—181no—
174C3×M16—34no—
175C3×M16—91no—
176#215—21no—
177#215—64no—
178#213—21no—
179#213—64no—
180#213—181no—
181#213—181no—
182#213—184no—
183#213—184no—
184#208—34no—
185#208—94no—
186#208—91no—
187#208—184no—
188#208—184no—
189#208—181no—
190#208—188no—
191#208C2×C3⋊D311yescharacteristic
192C2×C3⋊C8—94no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—184no—
195C2×C3⋊C8—184no—
196C2×C3⋊C8—181no—
197C2×C3⋊C8—188no—
198C22×C32—28no—
199C22×C32—24no—
200C22×C32—21no—
201C22×C32—24no—
202C22×C32—121no—
203C22×C32—241no—
204C22×C32#23714yes—
205C22×C32#23711yescharacteristic
206C2×C3⋊D3—41no—
207C2×C3⋊D3—81no—
208C2×C3×D3—44no—
209C2×C3×D3—84no—
210C2×C3×D3—124no—
211C2×C3×D3—244no—
212C32×C4—28no—
213C32×C4—24no—
214C32×C4—21no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4D3×D414yes—
218C32×C4D3×D411yescharacteristic
219C32×C4C2×#6714yes—
220C32×C4C2×#6711yescharacteristic
221AGL(1,ℤ/8)—31no—
222AGL(1,ℤ/8)—91no—
223C2×D8—271no—
224#131—271no—
225#108—271no—
226#100—271no—
227#100—271no—
228C33#47211yescharacteristic, complemented, Sylow
229C23×C3—64no—
230C23×C3—181no—
231C22×D3—21no—
232C22×D3—64no—
233C3×Q8—24no—
234C3×Q8#62514yes—
235C3×Q8#63811yescharacteristic
236C3×D4—28no—
237C3×D4—24no—
238C3×D4—21no—
239C3×D4—24no—
240C3×D4—64no—
241C3×D4—64no—
242C3×D4—181no—
243C3×D4—181no—
244C3×D4#62514yes—
245C3×D4#63811yescharacteristic
246C2×C3×C4—28no—
247C2×C3×C4—24no—
248C2×C3×C4—21no—
249C2×C3×C4—24no—
250C2×C3×C4C2×D3214yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252D12—21no—
253D12—21no—
254D12—64no—
255D12—64no—
256C3×C8—64no—
257C3×C8—181no—
258C3⋊C8—21no—
259C3⋊C8—64no—
260C3⋊C8—94no—
261C3⋊C8—94no—
262C3⋊C8—91no—
263C3⋊C8—91no—
264C3⋊C8—184no—
265C3⋊C8—184no—
266C3⋊C8—188no—
267C3⋊C8—184no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—44no—
273C2×C32—48no—
274C2×C32—44no—
275C2×C32—41no—
276C2×C32—241no—
277C2×C32#93914yes—
278C2×C32#93911yescharacteristic
279C3⋊D3—81no—
280C3×D3—84no—
281C3×D3—244no—
282#41D3×C3⋊D311yescharacteristic
283C2×D4—31no—
284C2×D4—91no—
285SD16—61no—
286SD16—181no—
287D8—61no—
288D8—181no—
289D8—541no—
290D8—541no—
291M16—31no—
292M16—91no—
293M16—271no—
294M16—541no—
295C2×C8—271no—
296C2×C8—541no—
297C22×C3—28no—
298C22×C3—21no—
299C22×C3—24no—
300C22×C3—24no—
301C22×C3—124no—
302C22×C3—244no—
303C22×C3—361no—
304C22×C3—721no—
305C22×C3#380514yes—
306C22×C3#383011yescharacteristic
307C2×D3—41no—
308C2×D3—81no—
309C2×D3—124no—
310C2×D3—244no—
311C3×C4—28no—
312C3×C4—21no—
313C3×C4—24no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4C2×#62514yes—
317C3×C4#381114yes—
318C3×C4D4×C3⋊D311yescharacteristic
319C3×C4C2×#63811yescharacteristic
320C32—24no—
321C32#519914yescomplemented
322C32#519911yescharacteristic, complemented
323C23—61no—
324C23—181no—
325Q8#629011yescharacteristic
326D4—21no—
327D4—61no—
328D4—61no—
329D4—181no—
330D4—181no—
331D4#629011yescharacteristic
332C2×C4—21no—
333C2×C4C2×D3×C3⋊D311yescharacteristic
334C8—61no—
335C8—181no—
336C8—271no—
337C8—271no—
338C8—541no—
339C2×C3—21no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—44no—
343C2×C3—48no—
344C2×C3—41no—
345C2×C3—44no—
346C2×C3—244no—
347C2×C3—721no—
348C2×C3#6398014yes—
349C2×C3#6414811yescharacteristic
350D3—81no—
351D3—244no—
352C22—21no—
353C22—121no—
354C22—241no—
355C22—361no—
356C22—721no—
357C22#8902911yescharacteristic
358C4—21no—
359C4C2×#629011yescharacteristic
360C4#8903911yescharacteristic
361C3—24no—
362C3#1059167814yescomplemented
363C3#1059289811yescharacteristic, complemented
364C2—21no—
365C2—41no—
366C2—241no—
367C2—721no—
368C2#1172363711yescharacteristic
subgroupG/H
centerC2#11723637
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33#121
hypercenter#41D3×C3⋊D3
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11723989C2semidirect
2. C3×#64158C2semidirect
3. C32×#949C2semidirect
4. C2×#88789C2semidirect
5. #11722059C2semidirect
6. #11722026C2semidirect
7. #11722025C2semidirect
26. #64158D3semidirect
32. C3×#949D3semidirect
105. #949C3⋊D3semidirect
228. C33#472semidirect
321. C32#5199semidirect
322. C32#5199semidirect
362. C3#10591678semidirect
363. C3#10592898semidirect

Maximal subgroups

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

subgroupindex
#117239892
C3×#641582
C32×#9492
C2×#887892
#117220592
#117220262
#117220252
#105928983
#105916783