Finite Group Databasebeta

← #13161399

Subgroups — #13161399

Order 1728.

1152 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1152 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
2#11723992C211yesmaximal, characteristic, complemented
3#11723439C211yesmaximal, characteristic, complemented
4#11723388C211yesmaximal, characteristic, complemented
5C32×#837C211yesmaximal, characteristic, complemented
6#11722046C211yesmaximal, characteristic, complemented
7#11722026C211yesmaximal, characteristic, complemented
8#10592524—31nomaximal
9#10591688—34nomaximal
10C33×#41C2211yescharacteristic
11C2×#6308C2211yescharacteristic, complemented
12C3×#3832—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3—21no—
15C3×#3828—21no—
16C3×#3828C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)—21no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19#88863—21no—
20#88861—21no—
21#88855C2211yescharacteristic, complemented
22C32×C4×Dic3C2211yescharacteristic, complemented
23C32×#208C2211yescharacteristic, complemented
24#88803—22no—
25#88800—21no—
26#88796—21no—
27#64366D311yescharacteristic
28C3×#1009—34no—
29#64160—31no—
30#64160—34no—
31#64160—64no—
32#64118—31no—
33#64084—31no—
34#63860—34no—
35#63831—34no—
36C32×#104—31no—
37#63650—31no—
38#63632—31no—
39C3×#837D314yescomplemented
40#63569—34no—
41#63558—34no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D412yes—
44C33×Q8D411yescharacteristic
45C33×D4—21no—
46C33×D4D411yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308—21no—
52C3×#638—22no—
53C3×#638—21no—
54C3×C4⋊(C3⋊D3)—22no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)D411yescharacteristic
58C3×C4×C3⋊D3—22no—
59C3×C4×C3⋊D3—21no—
60C3×#634D411yescharacteristic
61C2×C32×Dic3—21no—
62C3⋊(C3⋊(C3⋊C8))—21no—
63C32×C3⋊C8—21no—
64#5242—91no—
65#4825—94no—
66C32×#41—24no—
67C32×#41C2×D314yes—
68C32×#41C2×D311yescharacteristic
69#3832—21no—
70D4×C3⋊D3—22no—
71D4×C3⋊D3—21no—
72#3828—21no—
73#3828C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)—21no—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×#634—31no—
77C2×#634—34no—
78C2×#634—64no—
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—
86C32×M16—31no—
87C32×C42—31no—
88#3756—68no—
89#3756—64no—
90#3756—61no—
91#3754—68no—
92#3754—64no—
93#3754—61no—
94#3752—62no—
95#3748—31no—
96#3748—34no—
97#3748—64no—
98#3746—61no—
99#3744—61no—
100C3×C4×Dic3C2×D314yescomplemented
101C3×#208C2×D314yescomplemented
102(C2×D3)⋊Dic3—68no—
103#3719—64no—
104#3716—64no—
105C22×C33—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—42no—
110C2×C3×C3⋊D3—82no—
111C33×C4—21no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—21no—
116C32×Dic3—41no—
117C3×H(5,𝔽2)—91no—
118#1009—34no—
119#951—94no—
120#951—91no—
121#951—91no—
122#951—184no—
123#936—91no—
124#909—94no—
125#875—94no—
126C3×#104—34no—
127#837—91no—
128#837C3⋊D311yescharacteristic, complemented
129#833—91no—
130#823—94no—
131#805—94no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3#6712yes—
134C22×C3×D3—38no—
135C22×C3×D3—68no—
136C32×Q8—24no—
137C32×Q8D1214yes—
138C32×Q8#6711yescharacteristic
139C32×D4—21no—
140C32×D4—24no—
141C32×D4—24no—
142C32×D4—44no—
143C32×D4D1214yes—
144C32×D4#6711yescharacteristic
145C2×C32×C4—21no—
146C2×C32×C4—24no—
147C2×C32×C4—24no—
148C2×C32×C4—44no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C22×D311yescharacteristic
152#638—22no—
153#638—21no—
154C2×C3⋊Dic3—61no—
155C2×C3⋊Dic3—64no—
156C2×C3⋊Dic3—124no—
157C4⋊(C3⋊D3)—22no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)#6711yescharacteristic
161C4×C3⋊D3—22no—
162C4×C3⋊D3—21no—
163#634—68no—
164#634—64no—
165#634—61no—
166#634—61no—
167#634—64no—
168#634—124no—
169#634#6711yescharacteristic
170C3×#67—68no—
171C3×#67—64no—
172C2×C3×Dic3—24no—
173C3×D12—34no—
174C3×D12—68no—
175C3×D12—64no—
176C3×D12—64no—
177C3×C4×D3—68no—
178C3×C4×D3—64no—
179C3×Dic6—34no—
180C32×C8—61no—
181C3⋊(C3⋊C8)—68no—
182C3⋊(C3⋊C8)—64no—
183C3⋊(C3⋊C8)—61no—
184C3×C3⋊C8—24no—
185#455—271noSylow
186C2×C33—21no—
187C2×C33—41no—
188C2×C33#12011yescharacteristic
189C3×C3⋊D3—42no—
190C3×C3⋊D3—41no—
191C3×#41—24no—
192C3×#41—91no—
193C3×#41—181no—
194C3×#41—181no—
195C3×#41D3214yes—
196C3×#41C2×C3⋊D311yescharacteristic
197C2×C3×D4—91no—
198C2×C3×D4—181no—
199C2×C3×D4—182no—
200C2×C3×D4—181no—
201#239—64no—
202D3×D4—68no—
203D3×D4—64no—
204#235—34no—
205#235—64no—
206C2×D12—34no—
207C2×D12—64no—
208C2×Dic6—94no—
209C2×Dic6—91no—
210C2×Dic6—184no—
211C3×M16—34no—
212C3×C42—34no—
213#217—182no—
214#216—181no—
215#216—188no—
216#216—184no—
217#216—181no—
218#214—181no—
219#214—188no—
220#214—184no—
221#214—181no—
222(C2×D3)⋊C4—188no—
223C4×Dic3C2×C3⋊D311yescharacteristic, complemented
224#208—94no—
225#208—91no—
226#208—184no—
227#208C2×C3⋊D311yescharacteristic, complemented
228Dic12—184no—
229#204—184no—
230C22×C32—21no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—44no—
234C22×C32D3×D414yes—
235C22×C32C2×#6711yescharacteristic
236C2×C3⋊D3—22no—
237C2×C3⋊D3—21no—
238C2×C3⋊D3—42no—
239C2×C3⋊D3—82no—
240C2×C3×D3—64no—
241C2×C3×D3—68no—
242C2×C3×D3—128no—
243C2×C3×D3—248no—
244C32×C4—21no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—24no—
248C32×C4—44no—
249C32×C4—121no—
250C32×C4C2×D1214yes—
251C32×C4D3×D414yes—
252C32×C4C2×#6711yescharacteristic
253C32×C4C2×#6711yescharacteristic
254C3⋊Dic3—21no—
255C3⋊Dic3—128no—
256C3⋊Dic3—124no—
257C3⋊Dic3—121no—
258C3×Dic3—44no—
259C3×Dic3—64no—
260H(5,𝔽2)—91no—
261#137—271no—
262#137—271no—
263#124—271no—
264#104—31no—
265#104—271no—
266#100—271no—
267C33#45511yescharacteristic, complemented, Sylow
268C23×C3—92no—
269C23×C3—182no—
270C22×D3—38no—
271C22×D3—68no—
272C3×Q8—24no—
273C3×Q8—91no—
274C3×Q8#62614yes—
275C3×Q8C4⋊(C3⋊D3)11yescharacteristic
276C3×D4—24no—
277C3×D4—21no—
278C3×D4—24no—
279C3×D4—44no—
280C3×D4—91no—
281C3×D4—182no—
282C3×D4—182no—
283C3×D4—181no—
284C3×D4—181no—
285C3×D4—181no—
286C3×D4#62614yes—
287C3×D4C4⋊(C3⋊D3)11yescharacteristic
288C2×C3×C4—24no—
289C2×C3×C4—21no—
290C2×C3×C4—24no—
291C2×C3×C4—44no—
292C2×C3×C4—64no—
293C2×C3×C4—182no—
294C2×C3×C4—181no—
295C2×C3×C4C2×D3214yes—
296C2×C3×C4C22×C3⋊D311yescharacteristic
297#67—68no—
298#67—64no—
299C2×Dic3—21no—
300C2×Dic3—184no—
301C2×Dic3—181no—
302C2×Dic3—364no—
303D12—34no—
304D12—68no—
305D12—64no—
306D12—64no—
307C4×D3—68no—
308C4×D3—64no—
309Dic6—34no—
310Dic6—188no—
311Dic6—184no—
312Dic6—184no—
313Dic6—181no—
314Dic6—181no—
315Dic6—364no—
316C3×C8—64no—
317C3⋊C8—21no—
318C3⋊C8—188no—
319C3⋊C8—184no—
320C3⋊C8—181no—
321C2×C32—21no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—44no—
325C2×C32—41no—
326C2×C32—44no—
327C2×C32—84no—
328C2×C32#88214yes—
329C2×C32#95311yescharacteristic
330C3⋊D3—42no—
331C3⋊D3—41no—
332C3×D3—128no—
333C3×D3—124no—
334#41—91no—
335#41—181no—
336#41—181no—
337#41D3×C3⋊D311yescharacteristic
338C2×Q8—271no—
339C2×D4—91no—
340C2×D4—181no—
341C2×D4—182no—
342C2×D4—181no—
343Q16—541no—
344Q16—541no—
345SD16—541no—
346SD16—541no—
347M16—31no—
348M16—271no—
349C22⋊C4—542no—
350C42—31no—
351C22×C3—21no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—44no—
355C22×C3—181no—
356C22×C3—182no—
357C22×C3—362no—
358C22×C3—722no—
359C22×C3D3×#6714yes—
360C22×C3D4×C3⋊D311yescharacteristic
361C2×D3—64no—
362C2×D3—68no—
363C2×D3—128no—
364C2×D3—248no—
365C3×C4—21no—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—44no—
370C3×C4—124no—
371C3×C4—181no—
372C3×C4C2×#62614yes—
373C3×C4D3×#6714yes—
374C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
375C3×C4D4×C3⋊D311yescharacteristic
376Dic3—41no—
377Dic3—64no—
378Dic3—368no—
379Dic3—361no—
380Dic3—364no—
381C32—24no—
382C32#482514yescomplemented
383C32#524211yescharacteristic, complemented
384C23—92no—
385C23—182no—
386Q8—91no—
387Q8—541no—
388Q8—541no—
389Q8#629211yescharacteristic
390D4—21no—
391D4—91no—
392D4—182no—
393D4—182no—
394D4—181no—
395D4—181no—
396D4—181no—
397D4#629211yescharacteristic
398C2×C4—21no—
399C2×C4—61no—
400C2×C4—182no—
401C2×C4—181no—
402C2×C4—541no—
403C2×C4C2×D3×C3⋊D311yescharacteristic
404C8—61no—
405C8—541no—
406C2×C3—21no—
407C2×C3—24no—
408C2×C3—24no—
409C2×C3—44no—
410C2×C3—41no—
411C2×C3—44no—
412C2×C3—84no—
413C2×C3—362no—
414C2×C3—361no—
415C2×C3#6397714yes—
416C2×C3#6409111yescharacteristic
417D3—128no—
418D3—124no—
419C22—21no—
420C22—181no—
421C22—182no—
422C22—362no—
423C22—722no—
424C22C3⋊D3×#6711yescharacteristic
425C4—21no—
426C4—121no—
427C4—181no—
428C4—1081no—
429C4C2×#629211yescharacteristic
430C4C3⋊D3×#6711yescharacteristic
431C3—24no—
432C3#1059168814yescomplemented
433C3#1059252411yescharacteristic, complemented
434C2—21no—
435C2—41no—
436C2—362no—
437C2—361no—
438C2#1172363611yescharacteristic
subgroupG/H
centerC2#11723636
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33#120
prime radicalC3×#64366C2
hypercenter#41D3×C3⋊D3
hypocenterC33#455
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
2. #11723992C2semidirect
3. #11723439C2semidirect
4. #11723388C2semidirect
5. C32×#837C2semidirect
6. #11722046C2semidirect
7. #11722026C2semidirect
11. C2×#6308C22semidirect
21. #88855C22semidirect
22. C32×C4×Dic3C22semidirect
23. C32×#208C22semidirect
39. C3×#837D3semidirect
100. C3×C4×Dic3C2×D3semidirect
101. C3×#208C2×D3semidirect
128. #837C3⋊D3semidirect
223. C4×Dic3C2×C3⋊D3semidirect
227. #208C2×C3⋊D3semidirect
267. C33#455semidirect
382. C32#4825semidirect
383. C32#5242semidirect
432. C3#10591688semidirect
433. C3#10592524semidirect

Maximal subgroups

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

subgroupindex
C3×#643662
#117239922
#117234392
#117233882
C32×#8372
#117220462
#117220262
#105925243
#105916883