Finite Group Databasebeta

← #13160953

Subgroups — #13160953

Order 1728.

980 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C3×#64147C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic, complemented
4C2×#88787C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722052C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592858—31nomaximal
9#10591451—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3757—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C32×#213—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88787—22no—
23C2×C32×D8—31no—
24#64147—31no—
25#64147D311yescharacteristic, complemented
26C2×#3743—31no—
27C3×#938—34no—
28C2×C3×#213D314yescomplemented
29#63908—34no—
30C2×#3711—34no—
31#63659—31no—
32#63659—31no—
33#63659—34no—
34#63659—64no—
35#63644—31no—
36#63573—34no—
37#63562—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C3×#638—42no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6290—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C32×C3⋊C8D412yescomplemented
55#5159—94no—
56#5039—91no—
57C2×C32×D4—24no—
58C2×C32×D4—31no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C2×#638—61no—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×C3×#67—64no—
65C2×C3×D12C2×D314yes—
66C2×C3×C4×D3—34no—
67C2×#625—64no—
68C32×D8—62no—
69C2×C32×C8—31no—
70#3757—21no—
71#3757—61no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79#3743—62no—
80C3×#217—64no—
81C3×#213—28no—
82C3×C4⋊Dic3—34no—
83C2×C3×C3⋊C8C2×D314yes—
84(C2×D3)⋊Dic3—64no—
85#3711—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C3×C3⋊Dic3—21no—
100C3×C3⋊Dic3—41no—
101C2×C3×D8—34no—
102C3×#121—91no—
103#938—34no—
104#938—94no—
105C2×#213C3⋊D311yescharacteristic, complemented
106#930—91no—
107C2×#208—91no—
108C2×ΓL(1,𝔽25)—94no—
109#832—94no—
110#832—94no—
111#832—91no—
112#832—184no—
113#817—94no—
114#809—91no—
115#808—91no—
116C23×C32—21no—
117C23×C32—24no—
118C23×C32—44no—
119C23×C32—61no—
120C22×C3⋊D3#6711yescharacteristic
121C22×C3×D3—24no—
122C22×C3×D3—34no—
123C32×D4—28no—
124C32×D4—21no—
125C32×D4—24no—
126C32×D4—21no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4—61no—
130C32×D4—61no—
131C2×C32×C4—24no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—42no—
135#638—122no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—61no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—124no—
140C2×C3⋊Dic3#6711yescharacteristic
141C4×C3⋊D3—21no—
142C4×C3⋊D3—21no—
143C3×#67—128no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—64no—
146C3×D12—24no—
147C3×D12—24no—
148C3×C4×D3—64no—
149C3×C4×D3—64no—
150#625—128no—
151C32×C8—32no—
152C3⋊(C3⋊C8)—61no—
153C3⋊(C3⋊C8)—64no—
154C3⋊(C3⋊C8)—124no—
155C3×C3⋊C8#6718yescomplemented
156#469—271noSylow
157C2×C33—81no—
158C2×C33#12111yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3×C3⋊D3—41no—
162C32×D3—81no—
163C2×C3×D4—24no—
164C2×C3×D4—34no—
165C2×C3×D4—181no—
166C2×C3×D4D3214yes—
167C2×C3×D4C2×C3⋊D311yescharacteristic
168C22×C3×C4—91no—
169C2×#67—64no—
170C2×#67—181no—
171C2×#67—184no—
172C2×D12C2×C3⋊D311yescharacteristic
173C2×C4×D3—34no—
174C3×D8—68no—
175C2×C3×C8—34no—
176C3×C4⋊C4—91no—
177C3×C22⋊C4—181no—
178#217—64no—
179#217—184no—
180#213—22no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—34no—
183C4⋊Dic3—94no—
184C4⋊Dic3—91no—
185C4⋊Dic3—184no—
186#208—182no—
187C2×C3⋊C8—94no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—184no—
190C2×C3⋊C8C2×C3⋊D311yescharacteristic
191ΓL(1,𝔽25)—188no—
192C22×C32—24no—
193C22×C32—48no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32—84no—
201C22×C32—84no—
202C22×C32—121no—
203C22×C32—121no—
204C22×C32—121no—
205C22×C32D3×D414yes—
206C22×C32C2×#6711yescharacteristic
207C2×C3⋊D3—21no—
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3×D3—44no—
211C2×C3×D3—44no—
212C2×C3×D3—44no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4D3×D411yescharacteristic
219C32×C4#23714yes—
220C32×C4#23711yescharacteristic
221C32×C4C2×#6714yes—
222C3⋊Dic3—21no—
223C3⋊Dic3—41no—
224C3⋊Dic3—121no—
225C3⋊Dic3—124no—
226C3⋊Dic3—244no—
227C3×Dic3—64no—
228C3×Dic3—124no—
229C2×D8—31no—
230C2×M16—271no—
231#121—91no—
232#121—271no—
233#106—271no—
234#102—271no—
235#102—271no—
236C33#46911yescharacteristic, complemented, Sylow
237C23×C3—24no—
238C23×C3—21no—
239C23×C3—44no—
240C23×C3—64no—
241C23×C3—91no—
242C22×D3—21no—
243C22×D3—34no—
244C3×D4—28no—
245C3×D4—24no—
246C3×D4—24no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—44no—
250C3×D4—64no—
251C3×D4—64no—
252C3×D4—362no—
253C2×C3×C4—24no—
254C2×C3×C4—91no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4C2×D3214yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260#67—128no—
261#67—362no—
262#67—368no—
263C2×Dic3—34no—
264C2×Dic3—64no—
265C2×Dic3—184no—
266C2×Dic3—181no—
267C2×Dic3—364no—
268D12—21no—
269D12—21no—
270C4×D3—64no—
271C4×D3—64no—
272C3×C8—38no—
273C3⋊C8—184no—
274C3⋊C8—181no—
275C3⋊C8—364no—
276C3⋊C8#63812yescomplemented
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—88no—
281C2×C32—84no—
282C2×C32—81no—
283C2×C32—241no—
284C2×C32#91114yes—
285C2×C32#91114yes—
286C2×C32#93211yescharacteristic
287C2×C32#93814yes—
288C2×C32#93911yescharacteristic
289C2×C32#94911yescharacteristic
290C3⋊D3—41no—
291C3×D3—84no—
292C3×D3—124no—
293C2×D4—31no—
294C2×D4—181no—
295C2×D4—541no—
296C2×D4D3×C3⋊D311yescharacteristic
297C22×C4—91no—
298D8—62no—
299M16—542no—
300C2×C8—31no—
301C2×C8—271no—
302C4⋊C4—91no—
303C4⋊C4—271no—
304C22⋊C4—181no—
305C22⋊C4—541no—
306C22×C3—24no—
307C22×C3—48no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—84no—
315C22×C3—84no—
316C22×C3—124no—
317C22×C3—124no—
318C22×C3—124no—
319C22×C3—181no—
320C22×C3—181no—
321C22×C3—181no—
322C22×C3D3×#6714yes—
323C22×C3D4×C3⋊D311yescharacteristic
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—64no—
328C2×D3—64no—
329C2×D3—64no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—181no—
333C3×C4—361no—
334C3×C4#380514yes—
335C3×C4D3×#6714yes—
336C3×C4#383011yescharacteristic
337C3×C4C2×#63811yescharacteristic
338Dic3—64no—
339Dic3—124no—
340Dic3—361no—
341Dic3—364no—
342Dic3—724no—
343C32—24no—
344C32#503911yescharacteristic, complemented
345C32#515914yescomplemented
346C23—21no—
347C23—61no—
348C23—91no—
349D4—21no—
350D4—21no—
351D4—61no—
352D4—61no—
353D4—362no—
354D4—1082no—
355C2×C4—91no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—541no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C8—32no—
362C8—541no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—88no—
367C2×C3—81no—
368C2×C3—84no—
369C2×C3—244no—
370C2×C3—361no—
371C2×C3#6392514yes—
372C2×C3#6393214yes—
373C2×C3#6397014yes—
374C2×C3#6412011yescharacteristic
375C2×C3#6412011yescharacteristic
376C2×C3#6414711yescharacteristic
377D3—81no—
378D3—124no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—181no—
386C22—181no—
387C22—181no—
388C22C3⋊D3×#6711yescharacteristic
389C4—181no—
390C4—361no—
391C4—1081no—
392C4#8902911yescharacteristic
393C4D3×#63811yescharacteristic
394C3—24no—
395C3#1059145114yescomplemented
396C3#1059285811yescharacteristic, complemented
397C2—81no—
398C2—241no—
399C2—361no—
400C2#1172354611yescharacteristic
401C2#1172356011yescharacteristic
402C2#1172363111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#469
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. C3×#64147C2semidirect
2. C2×C32×#213C2semidirect
3. #11723522C2semidirect
4. C2×#88787C2semidirect
5. #11722202C2semidirect
6. #11722052C2semidirect
7. #11722031C2semidirect
25. #64147D3semidirect
28. C2×C3×#213D3semidirect
54. C32×C3⋊C8D4semidirect
105. C2×#213C3⋊D3semidirect
155. C3×C3⋊C8#67semidirect
236. C33#469semidirect
276. C3⋊C8#638semidirect
344. C32#5039semidirect
345. C32#5159semidirect
395. C3#10591451semidirect
396. C3#10592858semidirect

Maximal subgroups

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

subgroupindex
C3×#641472
C2×C32×#2132
#117235222
C2×#887872
#117222022
#117220522
#117220312
#105928583
#105914513