Finite Group Databasebeta

← #13160916

Subgroups — #13160916

Order 1728.

1144 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 166 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×#940C211yesmaximal, characteristic, complemented
3#11723484C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592771—31nomaximal
9#10591429—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×#67—21no—
14C2×C32×D12C2211yescharacteristic
15C2×#6289—21no—
16#88860—22no—
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21#88804—21no—
22#88790—22no—
23C32×#127—31no—
24C2×#3753—31no—
25C2×#3753—31no—
26C2×#3753—34no—
27C2×#3753—64no—
28C4×C4⋊(C3⋊D3)—31no—
29C3×#940D314yescomplemented
30#63894—34no—
31C2×#3714—34no—
32#63659—31no—
33#63659D311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))—31no—
35C3×#832—34no—
36Dic3⋊(C3⋊C8)—34no—
37#63562—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))D412yescomplemented
46C2×C3×C3⋊Dic3—21no—
47C32×#67—42no—
48C32×#67—42no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C32×D12—21no—
52C32×D12—21no—
53#6289—42no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5072—94no—
57#4786—91no—
58C2×C32×D4—24no—
59C2×C32×D4—31no—
60C2×C32×D4—61no—
61C2×C32×D4—61no—
62C2×C32×D4C2×D314yes—
63C2×C32×D4C2×D311yescharacteristic
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—34no—
66C2×C4⋊(C3⋊D3)—64no—
67C2×C4×C3⋊D3—61no—
68C2×C3×#67—24no—
69C2×C3×#67—24no—
70C2×C3×D12C2×D314yes—
71C2×#624—64no—
72C32×C42—31no—
73#3753—62no—
74#3753—62no—
75#3753—68no—
76#3753—616no—
77#3752—61no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—34no—
81C2×C3⋊(C3⋊C8)—64no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C3×C4⋊Dic3—34no—
84C3×C4×Dic3C2×D314yes—
85C2×C3×C3⋊C8—34no—
86#3722—64no—
87#3714—68no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—22no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3⋊(C3⋊D3)—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C33×C4C2×D411yescharacteristic
99C33×C4#4111yescharacteristic
100C3×C3⋊Dic3—41no—
101C32×Dic3—21no—
102C32×Dic3D812yescomplemented
103C3×#127—34no—
104#940C3⋊D311yescharacteristic, complemented
105C2×#213—94no—
106C2×#213—94no—
107C2×#213—91no—
108C2×#213—184no—
109C2×D24—91no—
110#893—91no—
111C4×D12—94no—
112C3×#102—91no—
113#832—34no—
114#832—94no—
115Dic3⋊C8—91no—
116#809—91no—
117C4⋊(C3⋊C8)—94no—
118C23×C32—21no—
119C23×C32—24no—
120C23×C32—44no—
121C23×C32—61no—
122C22×C3⋊D3—61no—
123C22×C3⋊D3—64no—
124C22×C3⋊D3—124no—
125C22×C3×D3—24no—
126C32×D4—28no—
127C32×D4—21no—
128C32×D4—24no—
129C32×D4—21no—
130C32×D4—24no—
131C32×D4—44no—
132C32×D4—61no—
133C32×D4—61no—
134C32×D4—122no—
135C32×D4—122no—
136C2×C32×C4—24no—
137C2×C32×C4—31no—
138C2×C32×C4—31no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—21no—
142C4⋊(C3⋊D3)—32no—
143C4⋊(C3⋊D3)—38no—
144C4⋊(C3⋊D3)—68no—
145C4⋊(C3⋊D3)—61no—
146C4⋊(C3⋊D3)—64no—
147C4⋊(C3⋊D3)—124no—
148C4×C3⋊D3—122no—
149C3×#67—48no—
150C3×#67—48no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3#6714yes—
153C2×C3×Dic3#6714yes—
154C3×D12—24no—
155C3×D12—24no—
156#624—128no—
157C3⋊(C3⋊C8)—21no—
158C3⋊(C3⋊C8)—68no—
159C3⋊(C3⋊C8)—64no—
160C3⋊(C3⋊C8)—61no—
161C3×C3⋊C8—64no—
162#459—271noSylow
163C2×C33—81no—
164C2×C33#12111yescharacteristic
165C2×C33C2×D811yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C3⋊(C3⋊D3)—42no—
168C32×D3—81no—
169C2×C3×D4—24no—
170C2×C3×D4—34no—
171C2×C3×D4—64no—
172C2×C3×D4—64no—
173C2×C3×D4D3214yes—
174C2×C3×D4C2×C3⋊D311yescharacteristic
175C2×#67—21no—
176C2×#67—21no—
177C2×D12—94no—
178C2×D12—91no—
179C2×D12—184no—
180C2×D12C2×C3⋊D311yescharacteristic
181C2×C4×D3—184no—
182C2×C4×D3—181no—
183C2×C3×C8—91no—
184C3×C4⋊C4—91no—
185C3×C42—34no—
186#213—188no—
187#213—188no—
188#213—182no—
189#213—1816no—
190(C2×D3)⋊C4—184no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—34no—
193C4×Dic3C2×C3⋊D311yescharacteristic
194C2×C3⋊C8—34no—
195C2×C3⋊C8—94no—
196C2×C3⋊C8—91no—
197C2×C3⋊C8—184no—
198D24—182no—
199C22×C32—24no—
200C22×C32—48no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—44no—
205C22×C32—44no—
206C22×C32—44no—
207C22×C32—84no—
208C22×C32—84no—
209C22×C32—121no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×#6714yes—
214C2×C3⋊D3—62no—
215C2×C3⋊D3—68no—
216C2×C3⋊D3—128no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—121no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—124no—
221C2×C3⋊D3—244no—
222C2×C3⋊D3—244no—
223C2×C3×D3—44no—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C32×C4—24no—
227C32×C4—24no—
228C32×C4—32no—
229C32×C4—61no—
230C32×C4#23511yescharacteristic
231C32×C4D3×D414yes—
232C32×C4D3×D411yescharacteristic
233C32×C4#23714yes—
234C3⋊Dic3—41no—
235C3×Dic3—24no—
236C3×Dic3—124no—
237C3×Dic3#21318yescomplemented
238C2×D8—271no—
239C2×D8—271no—
240#127—31no—
241C4×D4—271no—
242C4⋊C8—271no—
243#102—91no—
244#102—271no—
245C33#45911yescharacteristic, complemented, Sylow
246C23×C3—24no—
247C23×C3—21no—
248C23×C3—44no—
249C23×C3—64no—
250C22×D3—21no—
251C22×D3—184no—
252C22×D3—181no—
253C22×D3—364no—
254C3×D4—28no—
255C3×D4—24no—
256C3×D4—24no—
257C3×D4—21no—
258C3×D4—21no—
259C3×D4—44no—
260C3×D4—64no—
261C3×D4—64no—
262C3×D4—128no—
263C3×D4—128no—
264C2×C3×C4—24no—
265C2×C3×C4—34no—
266C2×C3×C4—34no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3214yes—
269C2×C3×C4C22×C3⋊D311yescharacteristic
270#67—42no—
271#67—42no—
272C2×Dic3—64no—
273C2×Dic3#63811yescharacteristic
274C2×Dic3#63811yescharacteristic
275D12—21no—
276D12—21no—
277D12—98no—
278D12—92no—
279D12—188no—
280D12—184no—
281D12—181no—
282D12—364no—
283C4×D3—368no—
284C4×D3—362no—
285C3×C8—181no—
286C3⋊C8—64no—
287C3⋊C8—188no—
288C3⋊C8—184no—
289C3⋊C8—181no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32—88no—
294C2×C32—81no—
295C2×C32—84no—
296C2×C32—241no—
297C2×C32#88411yescharacteristic
298C2×C32D3×D811yescharacteristic
299C2×C32#91411yescharacteristic
300C2×C32C2×#21314yes—
301C2×C32#93214yes—
302C2×C32#93814yes—
303C3⋊D3—122no—
304C3⋊D3—128no—
305C3⋊D3—248no—
306C3×D3—84no—
307C2×D4—31no—
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—271no—
311C2×D4D3×C3⋊D311yescharacteristic
312C22×C4—541no—
313D8—542no—
314D8—542no—
315C2×C8—91no—
316C2×C8—271no—
317C4⋊C4—91no—
318C22⋊C4—541no—
319C42—31no—
320C22×C3—24no—
321C22×C3—48no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—44no—
326C22×C3—44no—
327C22×C3—44no—
328C22×C3—84no—
329C22×C3—84no—
330C22×C3—124no—
331C22×C3—124no—
332C22×C3—124no—
333C22×C3D3×#6714yes—
334C22×C3C2×#63811yescharacteristic
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—182no—
339C2×D3—188no—
340C2×D3—368no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—364no—
344C2×D3—364no—
345C2×D3—724no—
346C2×D3—724no—
347C3×C4—24no—
348C3×C4—24no—
349C3×C4—38no—
350C3×C4—64no—
351C3×C4—361no—
352C3×C4#380414yes—
353C3×C4#381114yes—
354C3×C4D4×C3⋊D311yescharacteristic
355C3×C4#383011yescharacteristic
356Dic3—21no—
357Dic3—124no—
358Dic3#375312yescomplemented
359C32—24no—
360C32#478611yescharacteristic, complemented
361C32#507214yescomplemented
362C23—21no—
363C23—61no—
364C23—541no—
365D4—21no—
366D4—21no—
367D4—61no—
368D4—61no—
369D4—122no—
370D4—122no—
371D4—272no—
372D4—541no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—181no—
376C2×C4—1082no—
377C2×C4C2×D3×C3⋊D311yescharacteristic
378C8—181no—
379C8—541no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—88no—
384C2×C3—81no—
385C2×C3—84no—
386C2×C3—244no—
387C2×C3D3×#21314yes—
388C2×C3#6393414yes—
389C2×C3#6395814yes—
390C2×C3C2×#375311yescharacteristic
391C2×C3#6414111yescharacteristic
392C2×C3#6414711yescharacteristic
393D3—81no—
394D3—362no—
395D3—368no—
396D3—728no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—542no—
404C22—1081no—
405C22—1081no—
406C22D3×#63811yescharacteristic
407C4—32no—
408C4—61no—
409C4—361no—
410C4#8902811yescharacteristic
411C4#8903911yescharacteristic
412C3—24no—
413C3#1059142914yescomplemented
414C3#1059277111yescharacteristic, complemented
415C2—81no—
416C2—241no—
417C2—1082no—
418C2D3×#375311yescharacteristic
419C2#1172355711yescharacteristic
420C2#1172359911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88860C2semidirect
2. C32×#940C2semidirect
3. #11723484C2semidirect
4. C2×#88790C2semidirect
5. C3×#63659C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722031C2semidirect
29. C3×#940D3semidirect
33. #63659D3semidirect
45. C4⋊(C3⋊(C3⋊D3))D4semidirect
102. C32×Dic3D8semidirect
104. #940C3⋊D3semidirect
237. C3×Dic3#213semidirect
245. C33#459semidirect
358. Dic3#3753semidirect
360. C32#4786semidirect
361. C32#5072semidirect
413. C3#10591429semidirect
414. C3#10592771semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C32×#9402
#117234842
C2×#887902
C3×#636592
Dic3⋊(C3⋊(C3⋊C8))2
#117220312
#105927713
#105914293