Finite Group Databasebeta

← #13161389

Subgroups — #13161389

Order 1728.

1184 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 148 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×#64366C211yesmaximal, characteristic, complemented
2C32×#949C211yesmaximal, characteristic, complemented
3#11723529C211yesmaximal, characteristic, complemented
4#11723364C211yesmaximal, characteristic, complemented
5#11722207C211yesmaximal, characteristic, complemented
6#11722045C211yesmaximal, characteristic, complemented
7#11722026C211yesmaximal, characteristic, complemented
8#10592897—31nomaximal
9#10591665—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832—21no—
12C3×D4×C3⋊D3—22no—
13C3×D4×C3⋊D3—21no—
14C3×#3828—21no—
15C3×#3828C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)—21no—
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C32×D12C2211yescharacteristic, complemented
19C2×#6290—22no—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21#88855C2211yescharacteristic, complemented
22C32×#215—21no—
23C32×#213—21no—
24C32×#208C2211yescharacteristic, complemented
25#88792—21no—
26#88789—21no—
27#64366D311yescharacteristic, complemented
28C3×#1009—34no—
29C32×AGL(1,ℤ/8)—31no—
30#64149—31no—
31#64141—31no—
32C3×#949D314yescomplemented
33#63914—34no—
34#63823—34no—
35#63664—31no—
36#63664—34no—
37#63664—64no—
38#63652—31no—
39#63650—31no—
40#63568—34no—
41#63558—34no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D412yes—
44C22×C32×D3—21no—
45C33×Q8D411yescharacteristic, complemented
46C33×D4—21no—
47C33×D4D411yescharacteristic, complemented
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51C3×#638—22no—
52C3×#638—21no—
53C3×C4⋊(C3⋊D3)—22no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
57C3×C4×C3⋊D3—22no—
58C3×C4×C3⋊D3—21no—
59C3×#634D411yescharacteristic, complemented
60C32×D12—21no—
61C32×D12—21no—
62#6290—44no—
63C3⋊(C3⋊(C3⋊C8))—21no—
64C32×C3⋊C8—21no—
65#5241—91no—
66#5198—94no—
67C32×#41—24no—
68C32×#41C2×D314yes—
69C32×#41C2×D311yescharacteristic
70C2×C32×D4—31no—
71C2×#638—62no—
72#3832—21no—
73D4×C3⋊D3—22no—
74D4×C3⋊D3—21no—
75#3828—21no—
76#3828C2×D311yescharacteristic
77C2×C4⋊(C3⋊D3)—21no—
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
79C3×#239—64no—
80C3×D3×D4—68no—
81C3×D3×D4—64no—
82C3×#235—34no—
83C3×#235—64no—
84C2×C3×D12—34no—
85C2×C3×D12—64no—
86C2×C3×D12C2×D314yescomplemented
87C2×#625—68no—
88C32×SD16—61no—
89C32×D8—61no—
90C32×M16—31no—
91#3754—61no—
92#3753—61no—
93C4×C3⋊Dic3—31no—
94C4×C3⋊Dic3—34no—
95C4×C3⋊Dic3—64no—
96#3748—31no—
97#3748—34no—
98#3748—64no—
99C3×#215—24no—
100C3×#213—24no—
101C3×#208C2×D314yescomplemented
102#3715—64no—
103#3713—64no—
104C22×C33—21no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—42no—
109C2×C3×C3⋊D3—82no—
110C2×C32×D3—42no—
111C2×C32×D3—41no—
112C33×C4—21no—
113C33×C4C2×D411yescharacteristic
114C33×C4C2×D411yescharacteristic
115C3⋊(C3⋊Dic3)—22no—
116C3×C3⋊Dic3—21no—
117C3×H(5,𝔽2)—91no—
118#1009—34no—
119C3×AGL(1,ℤ/8)—34no—
120#949—91no—
121#949C3⋊D311yescharacteristic, complemented
122#940—94no—
123#940—91no—
124#932—94no—
125#837—94no—
126#837—91no—
127#837—91no—
128#837—184no—
129#833—91no—
130#825—94no—
131#823—94no—
132C23×C32—61no—
133C22×C3⋊D3—22no—
134C22×C3⋊D3#6712yes—
135C22×C3×D3—24no—
136C22×C3×D3—38no—
137C22×C3×D3—68no—
138C32×Q8—24no—
139C32×Q8#6714yescomplemented
140C32×Q8#6711yescharacteristic, complemented
141C32×D4—21no—
142C32×D4—24no—
143C32×D4—24no—
144C32×D4—44no—
145C32×D4—61no—
146C32×D4—61no—
147C32×D4#6714yescomplemented
148C32×D4#6711yescharacteristic, complemented
149C2×C32×C4—21no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—44no—
153C2×C32×C4C22×D314yes—
154C2×C32×C4C22×D311yescharacteristic
155#638—22no—
156#638—21no—
157#638—124no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—64no—
160C2×C3⋊Dic3—124no—
161C4⋊(C3⋊D3)—22no—
162C4⋊(C3⋊D3)—21no—
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)#6711yescharacteristic, complemented
165C4×C3⋊D3—22no—
166C4×C3⋊D3—21no—
167#634#6711yescharacteristic, complemented
168C3×#67—68no—
169C3×#67—64no—
170C3×D12—24no—
171C3×D12—24no—
172C3×D12—34no—
173C3×D12—68no—
174C3×D12—64no—
175C3×D12—64no—
176C3×C4×D3—68no—
177C3×C4×D3—64no—
178C3×Dic6—34no—
179#625—1216no—
180C32×C8—61no—
181C3⋊(C3⋊C8)—61no—
182C3⋊(C3⋊C8)—64no—
183C3⋊(C3⋊C8)—124no—
184C3×C3⋊C8—24no—
185#453—271noSylow
186C2×C33—21no—
187C2×C33—41no—
188C2×C33#12011yescharacteristic
189C3×C3⋊D3—42no—
190C3×C3⋊D3—41no—
191C32×D3—81no—
192C3×#41—24no—
193C3×#41—91no—
194C3×#41—181no—
195C3×#41—181no—
196C3×#41D3214yes—
197C3×#41C2×C3⋊D311yescharacteristic
198C2×C3×D4—34no—
199C2×C3×D4—91no—
200C2×C3×D4—181no—
201C2×C3×D4—182no—
202C2×C3×D4—181no—
203C2×#67—182no—
204C2×#67—188no—
205#239—64no—
206D3×D4—68no—
207D3×D4—64no—
208#235—34no—
209#235—64no—
210C2×D12—34no—
211C2×D12—64no—
212C2×D12C2×C3⋊D311yescharacteristic, complemented
213C3×SD16—64no—
214C3×D8—64no—
215C3×M16—34no—
216#215—21no—
217#215—181no—
218#214—184no—
219#213—21no—
220#213—181no—
221#213—184no—
222C4×Dic3—94no—
223C4×Dic3—91no—
224C4×Dic3—184no—
225#208—94no—
226#208—91no—
227#208—184no—
228#208C2×C3⋊D311yescharacteristic, complemented
229C22×C32—21no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—44no—
233C22×C32—122no—
234C22×C32—121no—
235C22×C32D3×D414yes—
236C22×C32C2×#6711yescharacteristic
237C2×C3⋊D3—22no—
238C2×C3⋊D3—21no—
239C2×C3⋊D3—42no—
240C2×C3⋊D3—82no—
241C2×C3×D3—48no—
242C2×C3×D3—44no—
243C2×C3×D3—64no—
244C2×C3×D3—68no—
245C2×C3×D3—128no—
246C2×C3×D3—248no—
247C32×C4—21no—
248C32×C4—24no—
249C32×C4—24no—
250C32×C4—24no—
251C32×C4—44no—
252C32×C4D3×D414yes—
253C32×C4C2×#6714yes—
254C32×C4C2×#6711yescharacteristic
255C32×C4C2×#6711yescharacteristic
256C3⋊Dic3—21no—
257C3⋊Dic3—62no—
258C3⋊Dic3—68no—
259C3⋊Dic3—128no—
260C3×Dic3—64no—
261H(5,𝔽2)—91no—
262AGL(1,ℤ/8)—31no—
263AGL(1,ℤ/8)—271no—
264#127—271no—
265#104—271no—
266#104—271no—
267#100—271no—
268C33#45311yescharacteristic, complemented, Sylow
269C23×C3—64no—
270C23×C3—92no—
271C23×C3—182no—
272C22×D3—21no—
273C22×D3—38no—
274C22×D3—68no—
275C3×Q8—24no—
276C3×Q8—91no—
277C3×Q8#62514yescomplemented
278C3×Q8#63811yescharacteristic, complemented
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—24no—
282C3×D4—44no—
283C3×D4—64no—
284C3×D4—64no—
285C3×D4—91no—
286C3×D4—182no—
287C3×D4—182no—
288C3×D4—181no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4#62514yescomplemented
292C3×D4#63811yescharacteristic, complemented
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—24no—
296C2×C3×C4—44no—
297C2×C3×C4—182no—
298C2×C3×C4—181no—
299C2×C3×C4C2×D3214yes—
300C2×C3×C4C22×C3⋊D311yescharacteristic
301#67—68no—
302#67—64no—
303#67—364no—
304#67—3616no—
305C2×Dic3—184no—
306C2×Dic3—181no—
307C2×Dic3—364no—
308D12—21no—
309D12—21no—
310D12—34no—
311D12—68no—
312D12—64no—
313D12—64no—
314C4×D3—68no—
315C4×D3—64no—
316Dic6—34no—
317C3×C8—64no—
318C3⋊C8—21no—
319C3⋊C8—184no—
320C3⋊C8—181no—
321C3⋊C8—364no—
322C2×C32—21no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—44no—
326C2×C32—41no—
327C2×C32—44no—
328C2×C32—84no—
329C2×C32—241no—
330C2×C32#93714yes—
331C2×C32#95311yescharacteristic
332C3⋊D3—42no—
333C3⋊D3—41no—
334C3×D3—84no—
335C3×D3—128no—
336C3×D3—124no—
337#41—91no—
338#41—181no—
339#41—181no—
340#41D3×C3⋊D311yescharacteristic
341C2×D4—31no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—182no—
345C2×D4—181no—
346C2×D4—542no—
347SD16—61no—
348SD16—541no—
349D8—61no—
350D8—541no—
351M16—31no—
352M16—271no—
353C42—271no—
354C22×C3—21no—
355C22×C3—24no—
356C22×C3—24no—
357C22×C3—44no—
358C22×C3—128no—
359C22×C3—124no—
360C22×C3—181no—
361C22×C3—182no—
362C22×C3—362no—
363C22×C3—722no—
364C22×C3D3×#6714yes—
365C22×C3D4×C3⋊D311yescharacteristic
366C2×D3—42no—
367C2×D3—41no—
368C2×D3—64no—
369C2×D3—68no—
370C2×D3—128no—
371C2×D3—248no—
372C3×C4—21no—
373C3×C4—24no—
374C3×C4—24no—
375C3×C4—24no—
376C3×C4—44no—
377C3×C4—181no—
378C3×C4C2×#62514yes—
379C3×C4D3×#6714yes—
380C3×C4D4×C3⋊D311yescharacteristic
381C3×C4C2×#63811yescharacteristic
382Dic3—64no—
383Dic3—182no—
384Dic3—188no—
385Dic3—368no—
386C32—24no—
387C32#519814yescomplemented
388C32#524111yescharacteristic, complemented
389C23—61no—
390C23—92no—
391C23—182no—
392Q8—91no—
393Q8#629011yescharacteristic, complemented
394D4—21no—
395D4—61no—
396D4—61no—
397D4—91no—
398D4—182no—
399D4—182no—
400D4—181no—
401D4—181no—
402D4—181no—
403D4—1084no—
404D4#629011yescharacteristic, complemented
405C2×C4—21no—
406C2×C4—182no—
407C2×C4—181no—
408C2×C4—541no—
409C2×C4C2×D3×C3⋊D311yescharacteristic
410C8—61no—
411C8—541no—
412C2×C3—21no—
413C2×C3—24no—
414C2×C3—24no—
415C2×C3—44no—
416C2×C3—41no—
417C2×C3—44no—
418C2×C3—84no—
419C2×C3—244no—
420C2×C3—362no—
421C2×C3—361no—
422C2×C3#6397614yes—
423C2×C3#6414611yescharacteristic
424D3—81no—
425D3—128no—
426D3—124no—
427C22—21no—
428C22—122no—
429C22—121no—
430C22—181no—
431C22—182no—
432C22—362no—
433C22—722no—
434C22C3⋊D3×#6711yescharacteristic
435C4—21no—
436C4—181no—
437C4—542no—
438C4C2×#629011yescharacteristic
439C4C3⋊D3×#6711yescharacteristic
440C3—24no—
441C3#1059166514yescomplemented
442C3#1059289711yescharacteristic, complemented
443C2—21no—
444C2—41no—
445C2—241no—
446C2—362no—
447C2—361no—
448C2#1172363411yescharacteristic
subgroupG/H
centerC2#11723634
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33#120
hypercenter#41D3×C3⋊D3
hypocenterC33#453
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. C3×#64366C2semidirect
2. C32×#949C2semidirect
3. #11723529C2semidirect
4. #11723364C2semidirect
5. #11722207C2semidirect
6. #11722045C2semidirect
7. #11722026C2semidirect
18. C2×C32×D12C22semidirect
20. C4×C3⋊(C3⋊Dic3)C22semidirect
21. #88855C22semidirect
24. C32×#208C22semidirect
27. #64366D3semidirect
32. C3×#949D3semidirect
45. C33×Q8D4semidirect
47. C33×D4D4semidirect
56. C3×C4⋊(C3⋊D3)D4semidirect
59. C3×#634D4semidirect
86. C2×C3×D12C2×D3semidirect
101. C3×#208C2×D3semidirect
121. #949C3⋊D3semidirect
139. C32×Q8#67semidirect
140. C32×Q8#67semidirect
147. C32×D4#67semidirect
148. C32×D4#67semidirect
164. C4⋊(C3⋊D3)#67semidirect
167. #634#67semidirect
212. C2×D12C2×C3⋊D3semidirect
228. #208C2×C3⋊D3semidirect
268. C33#453semidirect
277. C3×Q8#625semidirect
278. C3×Q8#638semidirect
291. C3×D4#625semidirect
292. C3×D4#638semidirect
387. C32#5198semidirect
388. C32#5241semidirect
393. Q8#6290semidirect
404. D4#6290semidirect
441. C3#10591665semidirect
442. C3#10592897semidirect

Maximal subgroups

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

subgroupindex
C3×#643662
C32×#9492
#117235292
#117233642
#117222072
#117220452
#117220262
#105928973
#105916653