Finite Group Databasebeta

← #13161094

Subgroups — #13161094

Order 1728.

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

normal subgroups only

This group has 952 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
1C2×#88860C211yesmaximal, characteristic, complemented
2C32×#932C211yesmaximal, characteristic, complemented
3#11723367C211yesmaximal, characteristic, complemented
4#11723350C211yesmaximal, characteristic, complemented
5C3×#63660C211yesmaximal, characteristic, complemented
6#11722059C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592898—31nomaximal
9#10591508—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C32×#235C2211yescharacteristic
13#88860—21no—
14#88860—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3748C2211yescharacteristic
17C3×#3748C2211yescharacteristic
18C32×#214—21no—
19C32×#213—21no—
20C32×#208C2211yescharacteristic
21#88795—21no—
22#88790—21no—
23#88786—21no—
24D3×C3⋊(C3⋊C8)—21no—
25C32×AGL(1,ℤ/8)—31no—
26#64158—31no—
27#64157—31no—
28C2×#3753—31no—
29C2×#3753—34no—
30C2×#3753—64no—
31C3×#932D314yescomplemented
32#63825—34no—
33#63815—34no—
34#63660D311yescharacteristic, complemented
35#63650—31no—
36#63649—31no—
37C3×#833—34no—
38#63578—34no—
39#63559—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)—21no—
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4C2311yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C32×#67—21no—
48C32×D12D411yescharacteristic
49C32×C4×D3—21no—
50C32×Dic6D411yescharacteristic
51C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
53C3×C3⋊(C3⋊C8)—21no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#5199—94no—
57#4749—91no—
58C32×#41—31no—
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C3×#235C2×D314yes—
66C32×SD16—61no—
67C32×D8—61no—
68C32×M16—31no—
69#3755—61no—
70#3753—61no—
71#3753—61no—
72#3753—61no—
73#3753—64no—
74#3753—64no—
75#3753—68no—
76#3753—68no—
77#3748—61no—
78#3748C2×D311yescharacteristic
79#3748C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—64no—
84C3×#214—24no—
85C3×#213—24no—
86C3×#208—34no—
87C3×#208—34no—
88C3×#208C2×D314yes—
89#3717—64no—
90#3714—64no—
91#3711—64no—
92D3×C3⋊C8—64no—
93C22×C33—41no—
94C22×C33—81no—
95C22×C33#4111yescharacteristic
96C2×C3⋊(C3⋊D3)—41no—
97C2×C3⋊(C3⋊D3)—81no—
98C2×C32×D3—21no—
99C33×C4C2×D411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C32×Dic3—21no—
102C3×AGL(1,ℤ/8)—34no—
103#949—94no—
104#948—94no—
105#932C3⋊D311yescharacteristic, complemented
106C2×#213—94no—
107C2×#213—91no—
108C2×#213—184no—
109#908—91no—
110#907—91no—
111C3×#100—91no—
112#833—34no—
113#823—94no—
114#823—91no—
115#822—94no—
116#822—91no—
117C23×C32—21no—
118C23×C32—24no—
119C23×C32—44no—
120C22×C3⋊D3—61no—
121C22×C3⋊D3—64no—
122C22×C3⋊D3—124no—
123C32×Q8—31no—
124C32×D4—28no—
125C32×D4—24no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—24no—
129C32×D4—31no—
130C32×D4—44no—
131C32×D4—61no—
132C2×C32×C4—24no—
133C2×C32×C4—61no—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136C4⋊(C3⋊D3)—68no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—124no—
142C3×#67—24no—
143C3×D12#6714yes—
144C3×C4×D3—24no—
145C3×Dic6#6714yes—
146C32×C8—61no—
147C3⋊(C3⋊C8)—21no—
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—24no—
156C3×C3⋊C8—64no—
157C3×C3⋊C8—64no—
158#472—271noSylow
159C2×C33—21no—
160C2×C33—81no—
161C2×C33#12111yescharacteristic
162C3⋊(C3⋊D3)—81no—
163C32×D3—41no—
164C3×#41—34no—
165C2×C3×D4—24no—
166C2×C3×D4D3214yes—
167C2×C3×D4C2×C3⋊D311yescharacteristic
168#235C2×C3⋊D311yescharacteristic
169C2×D12—94no—
170C2×D12—91no—
171C2×D12—184no—
172C3×SD16—64no—
173C3×D8—64no—
174C3×M16—34no—
175C3×M16—91no—
176C3×M16—91no—
177#215—184no—
178#214—21no—
179#213—21no—
180#213—184no—
181#213—184no—
182#213—184no—
183#213—181no—
184#213—181no—
185#213—188no—
186#213—188no—
187#208—34no—
188#208—34no—
189#208—184no—
190#208C2×C3⋊D311yescharacteristic
191C2×C3⋊C8—94no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—184no—
194C2×C3⋊C8—184no—
195D24—181no—
196#204—181no—
197ΓL(1,𝔽25)—181no—
198D3×C8—181no—
199C22×C32—24no—
200C22×C32—48no—
201C22×C32—41no—
202C22×C32—44no—
203C22×C32—61no—
204C22×C32—81no—
205C22×C32—84no—
206C22×C32—88no—
207C22×C32#23511yescharacteristic
208C22×C32#23714yes—
209C2×C3⋊D3—128no—
210C2×C3⋊D3—121no—
211C2×C3⋊D3—124no—
212C2×C3⋊D3—248no—
213C2×C3⋊D3—244no—
214C2×C3⋊D3—241no—
215C2×C3×D3—24no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—61no—
219C32×C4D3×D414yes—
220C32×C4D3×D411yescharacteristic
221C32×C4D3×D411yescharacteristic
222C32×C4C2×#6714yes—
223C3×Dic3—24no—
224AGL(1,ℤ/8)—31no—
225AGL(1,ℤ/8)—271no—
226C2×D8—271no—
227#131—271no—
228#108—271no—
229#100—91no—
230#100—271no—
231C33#47211yescharacteristic, complemented, Sylow
232C23×C3—24no—
233C23×C3—21no—
234C23×C3—44no—
235C22×D3—184no—
236C22×D3—181no—
237C22×D3—364no—
238C3×Q8—34no—
239C3×D4—28no—
240C3×D4—24no—
241C3×D4—24no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—34no—
245C3×D4—44no—
246C3×D4—64no—
247C2×C3×C4—24no—
248C2×C3×C4—64no—
249C2×C3×C4C2×D3214yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251#67—21no—
252D12—188no—
253D12—184no—
254D12—184no—
255D12—181no—
256D12—181no—
257D12—364no—
258D12#63811yescharacteristic
259C4×D3—21no—
260Dic6#63811yescharacteristic
261C3×C8—64no—
262C3×C8—181no—
263C3×C8—181no—
264C3⋊C8—21no—
265C3⋊C8—64no—
266C3⋊C8—64no—
267C3⋊C8—94no—
268C3⋊C8—94no—
269C3⋊C8—91no—
270C3⋊C8—91no—
271C3⋊C8—184no—
272C3⋊C8—184no—
273C2×C32—28no—
274C2×C32—24no—
275C2×C32—21no—
276C2×C32—24no—
277C2×C32—88no—
278C2×C32—84no—
279C2×C32—81no—
280C2×C32—121no—
281C2×C32#88411yescharacteristic
282C2×C32#93914yes—
283C3⋊D3—248no—
284C3⋊D3—244no—
285C3⋊D3—241no—
286C3×D3—44no—
287#41—31no—
288C2×D4—271no—
289C2×D4D3×C3⋊D311yescharacteristic
290SD16—61no—
291SD16—541no—
292D8—61no—
293D8—541no—
294D8—541no—
295D8—541no—
296M16—31no—
297M16—91no—
298M16—91no—
299M16—541no—
300C2×C8—271no—
301C2×C8—541no—
302C22×C3—24no—
303C22×C3—48no—
304C22×C3—41no—
305C22×C3—44no—
306C22×C3—64no—
307C22×C3—84no—
308C22×C3—81no—
309C22×C3—88no—
310C22×C3#380414yes—
311C22×C3#383011yescharacteristic
312C2×D3—21no—
313C2×D3—368no—
314C2×D3—361no—
315C2×D3—364no—
316C2×D3—728no—
317C2×D3—724no—
318C2×D3—721no—
319C3×C4—24no—
320C3×C4—24no—
321C3×C4—64no—
322C3×C4D3×#6714yes—
323C3×C4#381114yes—
324C3×C4D4×C3⋊D311yescharacteristic
325C3×C4C2×#63811yescharacteristic
326Dic3—21no—
327C32—24no—
328C32#474911yescharacteristic, complemented
329C32#519914yescomplemented
330C23—21no—
331C23—541no—
332Q8—31no—
333D4—21no—
334D4—21no—
335D4—31no—
336D4—61no—
337D4—541no—
338D4—541no—
339C2×C4—61no—
340C2×C4C2×D3×C3⋊D311yescharacteristic
341C8—61no—
342C8—181no—
343C8—181no—
344C8—271no—
345C8—271no—
346C2×C3—28no—
347C2×C3—24no—
348C2×C3—21no—
349C2×C3—24no—
350C2×C3—88no—
351C2×C3—84no—
352C2×C3—81no—
353C2×C3—124no—
354C2×C3#6397114yes—
355C2×C3#6414811yescharacteristic
356D3—41no—
357D3—728no—
358D3—724no—
359D3—721no—
360C22—41no—
361C22—61no—
362C22—81no—
363C22—1081no—
364C22—2161no—
365C22#8902811yescharacteristic
366C4—61no—
367C4D3×#63811yescharacteristic
368C4#8903911yescharacteristic
369C3—24no—
370C3#1059150814yescomplemented
371C3#1059289811yescharacteristic, complemented
372C2—21no—
373C2—81no—
374C2—121no—
375C2—2161no—
376C2#1172361611yescharacteristic
subgroupG/H
centerC2#11723616
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#121
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88860C2semidirect
2. C32×#932C2semidirect
3. #11723367C2semidirect
4. #11723350C2semidirect
5. C3×#63660C2semidirect
6. #11722059C2semidirect
7. #11722027C2semidirect
31. C3×#932D3semidirect
34. #63660D3semidirect
105. #932C3⋊D3semidirect
231. C33#472semidirect
328. C32#4749semidirect
329. C32#5199semidirect
370. C3#10591508semidirect
371. C3#10592898semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C32×#9322
#117233672
#117233502
C3×#636602
#117220592
#117220272
#105928983
#105915083