Finite Group Databasebeta

← #13161348

Subgroups — #13161348

Order 1728.

1152 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 134 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
1#11724580C211yesmaximal, characteristic
2#11723983C211yesmaximal, characteristic, complemented
3C3×#64156C211yesmaximal, characteristic, complemented
4C32×#942C211yesmaximal, characteristic, complemented
5#11722047C211yesmaximal, characteristic, complemented
6#11722044C211yesmaximal, characteristic, complemented
7#11722026C211yesmaximal, characteristic, complemented
8#10592901—31nomaximal
9#10591659—34nomaximal
10C2×C33×Q8C2211yescharacteristic, complemented
11#89076C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C32×#235C2211yescharacteristic
14C3⋊D3×#67—22no—
15#89026—21no—
16D3×C4⋊(C3⋊D3)—21no—
17#89021—21no—
18#89017—21no—
19#88863—21no—
20#88862—21no—
21#88855C2211yescharacteristic, complemented
22C3×#3752—22no—
23C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
24C32×#216—21no—
25C32×#215—21no—
26C32×#208C2211yescharacteristic, complemented
27#64366—31no—
28#64307—34no—
29C32×#137—31no—
30#64156D311yescharacteristic, complemented
31#64151—31no—
32#64151—34no—
33#64151—64no—
34C3×#947—34no—
35C3×#942D314yescomplemented
36#63664—31no—
37#63652—31no—
38#63650—31no—
39#63570—34no—
40#63569—34no—
41#63558—34no—
42C22×C3×C3⋊D3D412yes—
43C2×D3×C3⋊D3—22no—
44C33×Q8—21no—
45C33×Q8—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48#6312—21no—
49C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
50C4×C3⋊(C3⋊D3)—21no—
51#6308D411yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C32×#67—21no—
56C32×D12D411yescharacteristic
57C32×C4×D3—21no—
58C32×Dic6D411yescharacteristic
59#6292—22no—
60#6290—22no—
61Dic3×C3⋊D3—22no—
62C3⋊(C3⋊(C3⋊C8))—21no—
63C32×C3⋊C8—21no—
64#5202—94no—
65#5061—91no—
66C32×#41—31no—
67C2×C32×Q8—24no—
68C2×C32×Q8C2×D314yescomplemented
69C2×C32×Q8C2×D311yescharacteristic, complemented
70#3832—61no—
71D4×C3⋊D3—61no—
72D4×C3⋊D3—62no—
73#3828—31no—
74#3828—34no—
75#3828—61no—
76#3828—64no—
77C2×C4⋊(C3⋊D3)—61no—
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
79C3×#235C2×D314yes—
80C2×C3×D12—34no—
81D3×#67—68no—
82#3802—64no—
83D3×D12—64no—
84#3796—64no—
85#3795—64no—
86C32×Q16—61no—
87C32×SD16—61no—
88C32×M16—31no—
89#3756—61no—
90#3756—64no—
91#3756—68no—
92#3755—61no—
93#3755—64no—
94#3755—68no—
95#3752—22no—
96C4×C3⋊Dic3C2×D311yescharacteristic, complemented
97#3748—31no—
98#3748—34no—
99#3748—64no—
100C3×#216—24no—
101C3×#215—24no—
102C3×(C2×D3)⋊C4—68no—
103C3×C4×Dic3—34no—
104C3×#208C2×D314yescomplemented
105C22×C33C2×D411yescharacteristic
106C2×C3⋊(C3⋊D3)—21no—
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—42no—
109C2×C32×D3—21no—
110D3×C3⋊D3—82no—
111C33×C4—41no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3⋊(C3⋊Dic3)—21no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—21no—
117#1009—94no—
118#1004—91no—
119C3×#137—34no—
120C3×#124—91no—
121#947—34no—
122#942—94no—
123#942—91no—
124#942—184no—
125#942C3⋊D311yescharacteristic, complemented
126#837—94no—
127#833—91no—
128#825—94no—
129#823—94no—
130#805—91no—
131#805—91no—
132C22×C3⋊D3—62no—
133C22×C3⋊D3#6712yes—
134C22×C3×D3—38no—
135C2×D32—68no—
136C32×Q8—28no—
137C32×Q8—24no—
138C32×Q8—21no—
139C32×Q8—21no—
140C32×Q8—24no—
141C32×Q8—31no—
142C32×Q8—44no—
143C32×D4—31no—
144C32×D4—61no—
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—62no—
153#638—61no—
154#638—64no—
155#638—124no—
156C2×C3⋊Dic3—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159C4⋊(C3⋊D3)—31no—
160C4⋊(C3⋊D3)—34no—
161C4⋊(C3⋊D3)—64no—
162C4⋊(C3⋊D3)—62no—
163C4×C3⋊D3—62no—
164C4×C3⋊D3—61no—
165C4×C3⋊D3—64no—
166C4×C3⋊D3—124no—
167#634—31no—
168#634—34no—
169#634—64no—
170C3×#67—24no—
171C2×C3×Dic3—64no—
172C3×D12—64no—
173C3×D12—64no—
174C3×D12#6714yes—
175C3×C4×D3—24no—
176C3×Dic6#6714yes—
177#626—68no—
178#625—68no—
179D3×Dic3—68no—
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#12011yescharacteristic
188C3⋊(C3⋊D3)—41no—
189C3×C3⋊D3—42no—
190C32×D3—41no—
191C3×#41—34no—
192C2×C3×Q8—24no—
193C2×C3×Q8D3214yescomplemented
194C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
195C2×C3×D4—91no—
196C2×#67—182no—
197#239—184no—
198#237—181no—
199#237—181no—
200D3×D4—181no—
201D3×D4—184no—
202D3×D4—188no—
203D3×D4—181no—
204#235—94no—
205#235—91no—
206#235—184no—
207#235—184no—
208#235C2×C3⋊D311yescharacteristic
209C2×D12—34no—
210C2×D12—184no—
211C3×Q16—64no—
212C3×SD16—64no—
213C3×M16—34no—
214C3×C22⋊C4—182no—
215C3×C42—91no—
216#216—21no—
217#216—184no—
218#216—181no—
219#216—188no—
220#215—21no—
221#215—184no—
222#215—181no—
223#215—188no—
224(C2×D3)⋊C4—68no—
225C4×Dic3—34no—
226#208—94no—
227#208—91no—
228#208—184no—
229#208C2×C3⋊D311yescharacteristic, complemented
230C22×C32—24no—
231C22×C32—61no—
232C22×C32D3×D414yes—
233C22×C32C2×#6711yescharacteristic
234C2×C3⋊D3—22no—
235C2×C3⋊D3—42no—
236C2×C3⋊D3—61no—
237C2×C3⋊D3—64no—
238C2×C3⋊D3—124no—
239C2×C3⋊D3—242no—
240C2×C3×D3—24no—
241C2×C3×D3—68no—
242C2×C3×D3—128no—
243D32—248no—
244C32×C4—24no—
245C32×C4—24no—
246C32×C4—48no—
247C32×C4—44no—
248C32×C4—41no—
249C32×C4—61no—
250C32×C4D3×D414yes—
251C32×C4D3×D411yescharacteristic
252C32×C4D3×D411yescharacteristic
253C32×C4C2×#6714yes—
254C3⋊Dic3—41no—
255C3⋊Dic3—61no—
256C3⋊Dic3—64no—
257C3⋊Dic3—124no—
258C3×Dic3—24no—
259C3×Dic3—124no—
260H(5,𝔽2)—271no—
261#137—31no—
262#137—271no—
263#124—91no—
264#104—271no—
265#104—271no—
266#100—271no—
267C33#45511yescharacteristic, complemented, Sylow
268C23×C3—92no—
269C22×D3—38no—
270C22×D3—182no—
271C22×D3—188no—
272C3×Q8—28no—
273C3×Q8—24no—
274C3×Q8—24no—
275C3×Q8—21no—
276C3×Q8—21no—
277C3×Q8—34no—
278C3×Q8—44no—
279C3×D4—34no—
280C3×D4—64no—
281C3×D4—181no—
282C3×D4—181no—
283C2×C3×C4—24no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—44no—
287C2×C3×C4—64no—
288C2×C3×C4—181no—
289C2×C3×C4C2×D3214yes—
290C2×C3×C4C22×C3⋊D311yescharacteristic
291#67—21no—
292#67—182no—
293#67—182no—
294#67—188no—
295#67—184no—
296#67—181no—
297#67—364no—
298C2×Dic3—64no—
299C2×Dic3—182no—
300D12—64no—
301D12—64no—
302D12—94no—
303D12—91no—
304D12—184no—
305D12—188no—
306D12#63811yescharacteristic
307C4×D3—21no—
308C4×D3—188no—
309C4×D3—184no—
310C4×D3—181no—
311C4×D3—364no—
312Dic6—94no—
313Dic6—91no—
314Dic6—184no—
315Dic6#63811yescharacteristic
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—121no—
326C2×C32#93714yes—
327C2×C32#93711yescharacteristic
328C3⋊D3—42no—
329C3⋊D3—121no—
330C3⋊D3—124no—
331C3⋊D3—244no—
332C3×D3—44no—
333C3×D3—128no—
334#41—31no—
335#41—271no—
336#41—541no—
337#41—541no—
338C2×Q8D3×C3⋊D311yescharacteristic, complemented
339C2×D4—91no—
340C2×D4—541no—
341C2×D4—541no—
342C2×D4—542no—
343Q16—61no—
344Q16—541no—
345SD16—61no—
346SD16—541no—
347M16—31no—
348M16—271no—
349C22⋊C4—182no—
350C42—91no—
351C22×C3—24no—
352C22×C3—64no—
353C22×C3—182no—
354C22×C3—362no—
355C22×C3D3×#6714yes—
356C22×C3D4×C3⋊D311yescharacteristic
357C2×D3—21no—
358C2×D3—68no—
359C2×D3—128no—
360C2×D3—181no—
361C2×D3—184no—
362C2×D3—364no—
363C2×D3—722no—
364C2×D3—728no—
365C3×C4—24no—
366C3×C4—24no—
367C3×C4—48no—
368C3×C4—41no—
369C3×C4—44no—
370C3×C4—64no—
371C3×C4—361no—
372C3×C4D3×#6714yes—
373C3×C4#381114yes—
374C3×C4D4×C3⋊D311yescharacteristic
375C3×C4C2×#63811yescharacteristic
376Dic3—21no—
377Dic3—124no—
378Dic3—181no—
379Dic3—184no—
380Dic3—364no—
381C32—24no—
382C32#506111yescharacteristic, complemented
383C32#520214yescomplemented
384C23—92no—
385C23—542no—
386Q8—21no—
387Q8—21no—
388Q8—31no—
389Q8—271no—
390D4—31no—
391D4—61no—
392D4—181no—
393D4—181no—
394D4—271no—
395D4—542no—
396D4—542no—
397D4—541no—
398C2×C4—21no—
399C2×C4—61no—
400C2×C4—181no—
401C2×C4—542no—
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—124no—
411C2×C3—362no—
412C2×C3#6398314yes—
413C2×C3#6414611yescharacteristic
414D3—41no—
415D3—128no—
416D3—361no—
417D3—364no—
418D3—724no—
419C22—61no—
420C22—182no—
421C22—362no—
422C22—541no—
423C22—2162no—
424C22C3⋊D3×#6711yescharacteristic
425C4—41no—
426C4—61no—
427C4—361no—
428C4—541no—
429C4D3×#63811yescharacteristic
430C4#8903911yescharacteristic
431C3—24no—
432C3#1059165914yescomplemented
433C3#1059290111yescharacteristic, complemented
434C2—21no—
435C2—121no—
436C2—362no—
437C2—1081no—
438C2#1172364311yescharacteristic
subgroupG/H
centerC2#11723643
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
prime radical#11724580C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723983C2semidirect
3. C3×#64156C2semidirect
4. C32×#942C2semidirect
5. #11722047C2semidirect
6. #11722044C2semidirect
7. #11722026C2semidirect
10. C2×C33×Q8C22semidirect
21. #88855C22semidirect
23. C3×C4×C3⋊Dic3C22semidirect
26. C32×#208C22semidirect
30. #64156D3semidirect
35. C3×#942D3semidirect
68. C2×C32×Q8C2×D3semidirect
69. C2×C32×Q8C2×D3semidirect
96. C4×C3⋊Dic3C2×D3semidirect
104. C3×#208C2×D3semidirect
125. #942C3⋊D3semidirect
193. C2×C3×Q8D32semidirect
194. C2×C3×Q8C2×C3⋊D3semidirect
229. #208C2×C3⋊D3semidirect
267. C33#455semidirect
338. C2×Q8D3×C3⋊D3semidirect
382. C32#5061semidirect
383. C32#5202semidirect
432. C3#10591659semidirect
433. C3#10592901semidirect

Maximal subgroups

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

subgroupindex
#117245802
#117239832
C3×#641562
C32×#9422
#117220472
#117220442
#117220262
#105929013
#105916593