Finite Group Databasebeta

← #13160958

Subgroups — #13160958

Order 1728.

1108 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1108 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×#938C211yesmaximal, characteristic, complemented
3#11723516C211yesmaximal, characteristic, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592737—31nomaximal
9#10591449—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6291—21no—
15#88860—22no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#217—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21#88804—21no—
22D3×C3⋊(C3⋊C8)—22no—
23C32×#121—31no—
24C2×#3753—31no—
25C2×#3753—34no—
26C2×#3753—64no—
27#64139—31no—
28C22×C3⋊(C3⋊C8)—31no—
29C3×#938D314yescomplemented
30#63909—34no—
31C2×D3×C3⋊C8—34no—
32#63659D311yescharacteristic, complemented
33#63636—31no—
34#63634—31no—
35C3×#832—34no—
36#63576—34no—
37#63564—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C32×D3D411yescharacteristic
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3—21no—
47C32×#67—42no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3—21no—
51C32×C4×D3—21no—
52#6291—42no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54C3×C3⋊(C3⋊C8)—21no—
55#5038—94no—
56#4775—91no—
57C2×C32×D4—24no—
58C2×C32×D4—61no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C22×C32×C4—31no—
62C2×#638—61no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—34no—
65C2×C4⋊(C3⋊D3)—64no—
66C2×C3×#67—24no—
67C2×C3×C4×D3C2×D314yes—
68C2×#626—64no—
69C32×C4⋊C4—31no—
70C32×C22⋊C4—61no—
71#3753—62no—
72#3753—68no—
73#3753—616no—
74#3752—61no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—62no—
79C2×C3⋊(C3⋊C8)—64no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#217—24no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3C2×D314yes—
84C2×C3×C3⋊C8—34no—
85#3722—64no—
86D3×C3⋊C8—68no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C32×D3—21no—
95C2×C32×D3—21no—
96C2×C32×D3D812yescomplemented
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3×C3⋊Dic3—41no—
100C32×Dic3—21no—
101C32×Dic3—41no—
102C3×#121—34no—
103#938C3⋊D311yescharacteristic, complemented
104C2×#213—94no—
105C2×#213—91no—
106C2×#213—184no—
107#930—94no—
108C22×C3⋊C8—94no—
109C2×D3×C8—91no—
110#895—91no—
111C3×#102—91no—
112#832—34no—
113#821—91no—
114#809—94no—
115#807—94no—
116#807—91no—
117C23×C32—21no—
118C23×C32—24no—
119C23×C32—31no—
120C23×C32—44no—
121C22×C3⋊D3—61no—
122C22×C3⋊D3—64no—
123C22×C3⋊D3—124no—
124C22×C3×D3#6714yes—
125C32×D4—28no—
126C32×D4—21no—
127C32×D4—24no—
128C32×D4—21no—
129C32×D4—24no—
130C32×D4—44no—
131C32×D4—122no—
132C2×C32×C4—24no—
133C2×C32×C4—31no—
134C2×C32×C4—61no—
135C2×C32×C4—61no—
136C2×C32×C4—61no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139#638—122no—
140C2×C3⋊Dic3—21no—
141C4⋊(C3⋊D3)—68no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—64no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)—64no—
146C4⋊(C3⋊D3)—124no—
147C3×#67—48no—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3#6714yes—
151C3×C4×D3—24no—
152C3×C4×D3—24no—
153#626—128no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—32no—
156C3⋊(C3⋊C8)—38no—
157C3⋊(C3⋊C8)—68no—
158C3×C3⋊C8—64no—
159#466—271noSylow
160C2×C33—81no—
161C2×C33#12111yescharacteristic
162C2×C33C2×D811yescharacteristic
163C2×C33#13511yescharacteristic
164C3⋊(C3⋊D3)—81no—
165C32×D3—22no—
166C2×C3×D4—24no—
167C2×C3×D4—64no—
168C2×C3×D4D3214yes—
169C2×C3×D4C2×C3⋊D311yescharacteristic
170C22×C3×C4—34no—
171C2×#67—21no—
172C2×#67—184no—
173C2×D12—94no—
174C2×D12—91no—
175C2×D12—184no—
176C2×D12—181no—
177C2×C4×D3C2×C3⋊D311yescharacteristic
178C2×C3×C8—91no—
179C3×C4⋊C4—34no—
180C3×C4⋊C4—91no—
181C3×C22⋊C4—64no—
182#217—21no—
183#213—188no—
184#213—182no—
185#213—1816no—
186(C2×D3)⋊C4—184no—
187(C2×D3)⋊C4—181no—
188C4⋊Dic3—34no—
189C4⋊Dic3C2×C3⋊D311yescharacteristic
190C2×C3⋊C8—34no—
191C2×C3⋊C8—94no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—184no—
194C2×C3⋊C8—188no—
195D3×C8—182no—
196C22×C32—24no—
197C22×C32—32no—
198C22×C32—48no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—44no—
203C22×C32—44no—
204C22×C32—44no—
205C22×C32—61no—
206C22×C32—61no—
207C22×C32—84no—
208C22×C32—84no—
209C22×C32D3×D411yescharacteristic
210C22×C32C2×#6714yes—
211C2×C3⋊D3—128no—
212C2×C3⋊D3—121no—
213C2×C3⋊D3—121no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—124no—
216C2×C3⋊D3—124no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—244no—
219C2×C3⋊D3—244no—
220C2×C3×D3—24no—
221C2×C3×D3—24no—
222C2×C3×D3#21318yescomplemented
223C32×C4—24no—
224C32×C4—24no—
225C32×C4—61no—
226C32×C4—121no—
227C32×C4#23511yescharacteristic
228C32×C4D3×D414yes—
229C32×C4D3×D411yescharacteristic
230C32×C4#23714yes—
231C3⋊Dic3—41no—
232C3×Dic3—24no—
233C3×Dic3—44no—
234C3×Dic3—124no—
235C2×D8—271no—
236C22×C8—271no—
237#121—31no—
238#121—271no—
239#107—271no—
240#102—91no—
241#102—271no—
242C33#46611yescharacteristic, complemented, Sylow
243C23×C3—24no—
244C23×C3—21no—
245C23×C3—34no—
246C23×C3—44no—
247C22×D3—184no—
248C22×D3—181no—
249C22×D3—364no—
250C22×D3#63811yescharacteristic
251C3×D4—28no—
252C3×D4—24no—
253C3×D4—24no—
254C3×D4—21no—
255C3×D4—21no—
256C3×D4—44no—
257C3×D4—128no—
258C2×C3×C4—24no—
259C2×C3×C4—34no—
260C2×C3×C4—64no—
261C2×C3×C4—64no—
262C2×C3×C4—64no—
263C2×C3×C4—181no—
264C2×C3×C4C2×D3214yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266#67—42no—
267#67—368no—
268C2×Dic3—21no—
269C2×Dic3—64no—
270C2×Dic3#63811yescharacteristic
271D12—188no—
272D12—184no—
273D12—184no—
274D12—181no—
275D12—181no—
276D12—364no—
277D12—362no—
278C4×D3—21no—
279C4×D3—21no—
280C3×C8—181no—
281C3⋊C8—64no—
282C3⋊C8—98no—
283C3⋊C8—92no—
284C3⋊C8—188no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—24no—
288C2×C32—62no—
289C2×C32—88no—
290C2×C32—81no—
291C2×C32—84no—
292C2×C32#88411yescharacteristic
293C2×C32D3×D811yescharacteristic
294C2×C32#91611yescharacteristic
295C2×C32C2×#21314yes—
296C2×C32#93914yes—
297C2×C32#95014yes—
298C3⋊D3—248no—
299C3⋊D3—241no—
300C3⋊D3—244no—
301C3×D3—28no—
302C2×D4—61no—
303C2×D4—271no—
304C2×D4—541no—
305C2×D4D3×C3⋊D311yescharacteristic
306C22×C4—31no—
307D8—542no—
308C2×C8—91no—
309C2×C8—271no—
310C2×C8—542no—
311C4⋊C4—31no—
312C4⋊C4—91no—
313C22⋊C4—61no—
314C22⋊C4—541no—
315C22×C3—24no—
316C22×C3—38no—
317C22×C3—48no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—44no—
322C22×C3—44no—
323C22×C3—44no—
324C22×C3—64no—
325C22×C3—64no—
326C22×C3—84no—
327C22×C3—84no—
328C22×C3D3×#6714yes—
329C22×C3C2×#63811yescharacteristic
330C2×D3—21no—
331C2×D3—21no—
332C2×D3—368no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—361no—
336C2×D3—364no—
337C2×D3—364no—
338C2×D3—364no—
339C2×D3—724no—
340C2×D3—724no—
341C2×D3#375312yescomplemented
342C3×C4—24no—
343C3×C4—24no—
344C3×C4—64no—
345C3×C4—124no—
346C3×C4—361no—
347C3×C4#380414yes—
348C3×C4#381114yes—
349C3×C4D4×C3⋊D311yescharacteristic
350C3×C4#383011yescharacteristic
351Dic3—21no—
352Dic3—41no—
353Dic3—124no—
354C32—24no—
355C32#477511yescharacteristic, complemented
356C32#503814yescomplemented
357C23—21no—
358C23—31no—
359C23—541no—
360D4—21no—
361D4—21no—
362D4—122no—
363D4—541no—
364D4—541no—
365D4—1082no—
366C2×C4—31no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—181no—
371C2×C4C2×D3×C3⋊D311yescharacteristic
372C8—181no—
373C8—272no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—68no—
378C2×C3—88no—
379C2×C3—84no—
380C2×C3—81no—
381C2×C3D3×#21314yes—
382C2×C3#6393514yes—
383C2×C3#6397114yes—
384C2×C3C2×#375311yescharacteristic
385C2×C3#6414811yescharacteristic
386C2×C3#6415911yescharacteristic
387D3—22no—
388D3—728no—
389D3—724no—
390D3—721no—
391C22—32no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—61no—
396C22—61no—
397C22—1081no—
398C22—1081no—
399C22—1081no—
400C22D3×#63811yescharacteristic
401C4—61no—
402C4—121no—
403C4—361no—
404C4#8902811yescharacteristic
405C4#8903911yescharacteristic
406C3—24no—
407C3#1059144914yescomplemented
408C3#1059273711yescharacteristic, complemented
409C2—62no—
410C2—81no—
411C2—2161no—
412C2D3×#375311yescharacteristic
413C2#1172355811yescharacteristic
414C2#1172361611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#466
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×#938C2semidirect
3. #11723516C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63659C2semidirect
6. #11722056C2semidirect
7. #11722035C2semidirect
29. C3×#938D3semidirect
32. #63659D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
96. C2×C32×D3D8semidirect
103. #938C3⋊D3semidirect
222. C2×C3×D3#213semidirect
242. C33#466semidirect
341. C2×D3#3753semidirect
355. C32#4775semidirect
356. C32#5038semidirect
407. C3#10591449semidirect
408. C3#10592737semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C32×#9382
#117235162
C2×D3×C3⋊(C3⋊C8)2
C3×#636592
#117220562
#117220352
#105927373
#105914493