Finite Group Databasebeta

← #13154392

Subgroups — #13154392

Order 1728.

1364 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722676C211yesmaximal, characteristic, complemented
2#11722670C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#137C211yesmaximal, characteristic, complemented
4#11721871C211yesmaximal, characteristic, complemented
5C2×#88550C211yesmaximal, characteristic, complemented
6#11721807C212yesmaximal, complemented
7Q16×#272C212yesmaximal, complemented
8#11721802C212yesmaximal, complemented
9#11721801C212yesmaximal, complemented
10#11721788C211yesmaximal, characteristic, complemented
11#11721786C211yesmaximal, characteristic, complemented
12C3×#5779—34nomaximal
13#41×H(3,𝔽3)C2211yescharacteristic
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15#88748—21no—
16#88748C2212yes—
17Q8×#272—21no—
18Q8×#272C2212yes—
19Q8×#272C2212yes—
20#88744—22no—
21#88744C2212yes—
22#88740—22no—
23#88740—21no—
24#88740C2211yescharacteristic
25#88740C2211yescharacteristic
26C2×#6229—21no—
27C2×#6229C2211yescharacteristic
28Q16×H(3,𝔽3)C2212yes—
29SD16×H(3,𝔽3)C2212yes—
30M16×H(3,𝔽3)C2211yescharacteristic
31#88550C2212yes—
32#88550C2212yes—
33#88550C2211yescharacteristic
34#88549C2211yescharacteristic
35#88546C2211yescharacteristic
36#88545C2211yescharacteristic
37C2×#6180C2211yescharacteristic
38#88530C2212yes—
39#88528C2212yes—
40#88527C2212yes—
41C8×#272C2212yes—
42C3×#1010—34no—
43C3×#1007—34no—
44C32×#137D314yescomplemented
45C3×#950—34no—
46C2×C3×#216—34no—
47C3×#918—38no—
48C3×D3×Q16—38no—
49C3×#916—38no—
50C3×#915—38no—
51C3×#909—34no—
52C3×#907—34no—
53Q8×H(3,𝔽3)—21no—
54Q8×H(3,𝔽3)C2311yescharacteristic
55Q8×H(3,𝔽3)C2312yes—
56D4×H(3,𝔽3)—21no—
57D4×H(3,𝔽3)C2311yescharacteristic
58C2×C4×H(3,𝔽3)—21no—
59C2×C4×H(3,𝔽3)—21no—
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61#6235—21no—
62#6235—22no—
63#6235D412yes—
64C2×#1084—21no—
65C2×#1084—22no—
66C4⋊#272—22no—
67C4⋊#272D411yescharacteristic
68C4⋊#272C2311yescharacteristic
69C4×#272—22no—
70C4×#272—22no—
71C4×#272—21no—
72C4×#272—22no—
73C4×#272C2312yes—
74#6229—21no—
75#6229—22no—
76#6229—22no—
77#6229D411yescharacteristic
78#6229C2312yes—
79#6229C2311yescharacteristic
80C8×H(3,𝔽3)C2312yes—
81#6180C2311yescharacteristic
82#6180C2311yescharacteristic
83C3×#578—91no—
84#5779—34no—
85C32×#41C2×D314yes—
86C2×C32×Q8C2×D314yes—
87C3×#239—38no—
88C3×#239—64no—
89C3×D3×Q8—38no—
90C3×D3×Q8—38no—
91C3×D3×Q8—64no—
92C3×#237—38no—
93C3×#237—68no—
94C3×#235—34no—
95C3×#235—34no—
96C3×#235—64no—
97C3×#235—68no—
98C2×C3×Dic6—34no—
99C2×C3×Dic6—64no—
100C32×Q16C2×D318yes—
101C32×SD16C2×D318yes—
102C32×M16C2×D314yes—
103C3×#216—38no—
104C3×#216—38no—
105C3×#216—34no—
106C3×#215—34no—
107C3×#214—34no—
108C3×#213—34no—
109C2×C3×C3⋊C8—34no—
110C3×Dic12—38no—
111C3×#204—38no—
112C3×ΓL(1,𝔽25)—38no—
113C3×D3×C8—38no—
114C22×H(3,𝔽3)—21no—
115C22×H(3,𝔽3)C2×D411yescharacteristic
116C2×#272—21no—
117C2×#272C2×D412yes—
118C4×H(3,𝔽3)—21no—
119C4×H(3,𝔽3)—22no—
120C4×H(3,𝔽3)C2×D411yescharacteristic
121C4×H(3,𝔽3)C2411yescharacteristic
122#1084—21no—
123#1084—22no—
124#1084C2×D412yes—
125C3×2−1+4—91no—
126C3×2−1+4—91no—
127#1010—34no—
128#1007—34no—
129C3×#137—34no—
130C3×#137—92no—
131C3×#137—92no—
132C3×#137—91no—
133C3×#137C3⋊D311yescharacteristic
134C3×#135—91no—
135C3×#135—92no—
136C2×C3×Q16—91no—
137C2×C3×Q16—92no—
138C3×#131—91no—
139#950—34no—
140C2×#216—34no—
141#918—38no—
142D3×Q16—38no—
143#916—38no—
144#915—38no—
145#909—34no—
146#907—34no—
147C32×Q8—24no—
148C32×Q8C22×D314yes—
149C32×Q8C22×D318yes—
150C32×D4—24no—
151C32×D4C22×D314yes—
152C2×C32×C4—24no—
153C2×C32×C4—24no—
154C2×C32×C4C22×D314yes—
155C3×#67—38no—
156C3×#67—68no—
157C3×#67—64no—
158C2×C3×Dic3—68no—
159C2×C3×Dic3—64no—
160C3×D12—34no—
161C3×D12—34no—
162C3×D12—68no—
163C3×C4×D3—38no—
164C3×C4×D3—68no—
165C3×C4×D3—68no—
166C3×C4×D3—68no—
167C3×C4×D3—64no—
168C3×Dic6—34no—
169C3×Dic6—38no—
170C3×Dic6—34no—
171C3×Dic6—68no—
172C3×Dic6—68no—
173C3×Dic6—64no—
174C32×C8C22×D318yes—
175C3×C3⋊C8—34no—
176C3×C3⋊C8—34no—
177#578—91noSylow
178C2×H(3,𝔽3)—21no—
179C2×H(3,𝔽3)—41no—
180C2×H(3,𝔽3)C22×D411yescharacteristic
181#272—22no—
182#272—41no—
183C3×#41—34no—
184C3×#41—91no—
185C3×#41—92no—
186C3×#41—92no—
187C3×#41—91no—
188C3×#41—182no—
189C3×#41—181no—
190C3×#41—182no—
191C3×#41—181no—
192C3×#41C2×C3⋊D311yescharacteristic
193C2×C3×Q8—34no—
194C2×C3×Q8—92no—
195C2×C3×Q8—92no—
196C2×C3×Q8—91no—
197C2×C3×Q8—181no—
198C2×C3×Q8—181no—
199C2×C3×Q8C2×C3⋊D311yescharacteristic
200#239—38no—
201#239—64no—
202D3×Q8—38no—
203D3×Q8—38no—
204D3×Q8—64no—
205#237—38no—
206#237—68no—
207#235—34no—
208#235—34no—
209#235—64no—
210#235—68no—
211C2×Dic6—34no—
212C2×Dic6—64no—
213C3×Q16—38no—
214C3×Q16—91no—
215C3×Q16—92no—
216C3×Q16—92no—
217C3×Q16—92no—
218C3×Q16C2×C3⋊D312yes—
219C3×SD16—38no—
220C3×SD16—91no—
221C3×SD16—91no—
222C3×SD16—92no—
223C3×SD16C2×C3⋊D312yes—
224C3×D8—91no—
225C3×M16—34no—
226C3×M16—92no—
227C3×M16C2×C3⋊D311yescharacteristic
228C2×C3×C8—91no—
229C2×C3×C8—92no—
230#216—38no—
231#216—38no—
232#216—34no—
233#215—34no—
234#214—34no—
235#213—34no—
236C2×C3⋊C8—34no—
237Dic12—38no—
238#204—38no—
239ΓL(1,𝔽25)—38no—
240D3×C8—38no—
241C22×C32—24no—
242C22×C32D3×D414yes—
243C2×C3×D3—38no—
244C2×C3×D3—64no—
245C32×C4—24no—
246C32×C4—28no—
247C32×C4D3×D414yes—
248C32×C4C23×D314yes—
249C3×Dic3—38no—
250C3×Dic3—64no—
251C3×Dic3—68no—
2522−1+4—91no—
2532−1+4—91no—
254#137—92no—
255#137—92no—
256#137—91no—
257#137#27211yescharacteristic, complemented
258#135—91no—
259#135—92no—
260C2×Q16—91no—
261C2×Q16—92no—
262#131—91no—
263H(3,𝔽3)#57811yescharacteristic, complemented, Sylow
264C3×Q8—21no—
265C3×Q8—34no—
266C3×Q8—38no—
267C3×Q8—64no—
268C3×Q8—91no—
269C3×Q8—92no—
270C3×Q8—91no—
271C3×Q8—182no—
272C3×Q8—182no—
273C3×Q8—181no—
274C3×Q8C22×C3⋊D311yescharacteristic
275C3×Q8C22×C3⋊D312yes—
276C3×D4—21no—
277C3×D4—34no—
278C3×D4—64no—
279C3×D4—92no—
280C3×D4—91no—
281C3×D4—91no—
282C3×D4—182no—
283C3×D4—182no—
284C3×D4—181no—
285C3×D4C22×C3⋊D311yescharacteristic
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—34no—
289C2×C3×C4—64no—
290C2×C3×C4—64no—
291C2×C3×C4—92no—
292C2×C3×C4—182no—
293C2×C3×C4—182no—
294C2×C3×C4—182no—
295C2×C3×C4—182no—
296C2×C3×C4—181no—
297C2×C3×C4—181no—
298C2×C3×C4C22×C3⋊D311yescharacteristic
299#67—38no—
300#67—68no—
301#67—64no—
302C2×Dic3—68no—
303C2×Dic3—64no—
304D12—34no—
305D12—34no—
306D12—68no—
307C4×D3—38no—
308C4×D3—68no—
309C4×D3—64no—
310C4×D3—68no—
311C4×D3—68no—
312Dic6—34no—
313Dic6—38no—
314Dic6—34no—
315Dic6—68no—
316Dic6—68no—
317Dic6—64no—
318C3×C8—38no—
319C3×C8—91no—
320C3×C8—91no—
321C3×C8C22×C3⋊D312yes—
322C3⋊C8—34no—
323C3⋊C8—34no—
324C2×C32—24no—
325C2×C32—44no—
326C2×C32C2×D3×D414yes—
327C3×D3—68no—
328C3×D3—124no—
329#41—91no—
330#41—92no—
331#41—92no—
332#41—91no—
333#41—182no—
334#41—181no—
335#41—182no—
336#41—181no—
337#41C2×#27211yescharacteristic
338C2×Q8—92no—
339C2×Q8—92no—
340C2×Q8—91no—
341C2×Q8—181no—
342C2×Q8—181no—
343C2×Q8C2×#27211yescharacteristic
344Q16—91no—
345Q16—92no—
346Q16—92no—
347Q16—92no—
348Q16C2×#27212yes—
349SD16—91no—
350SD16—91no—
351SD16—92no—
352SD16C2×#27212yes—
353D8—91no—
354M16—92no—
355M16C2×#27211yescharacteristic
356C2×C8—91no—
357C2×C8—92no—
358C22×C3—21no—
359C22×C3—34no—
360C22×C3—64no—
361C22×C3—92no—
362C22×C3—181no—
363C22×C3D4×C3⋊D311yescharacteristic
364C2×D3—38no—
365C2×D3—64no—
366C3×C4—21no—
367C3×C4—22no—
368C3×C4—34no—
369C3×C4—34no—
370C3×C4—64no—
371C3×C4—68no—
372C3×C4—92no—
373C3×C4—181no—
374C3×C4—182no—
375C3×C4D4×C3⋊D311yescharacteristic
376C3×C4C23×C3⋊D311yescharacteristic
377Dic3—38no—
378Dic3—64no—
379Dic3—68no—
380C32#577914yescomplemented
381Q8—21no—
382Q8—91no—
383Q8—91no—
384Q8—92no—
385Q8—182no—
386Q8—182no—
387Q8—181no—
388Q8C22×#27211yescharacteristic
389Q8C22×#27212yes—
390D4—21no—
391D4—92no—
392D4—91no—
393D4—91no—
394D4—182no—
395D4—182no—
396D4—181no—
397D4C22×#27211yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—92no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4—182no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4C22×#27211yescharacteristic
408C8—91no—
409C8—91no—
410C8C22×#27212yes—
411C2×C3—21no—
412C2×C3—34no—
413C2×C3—41no—
414C2×C3—64no—
415C2×C3—124no—
416C2×C3—182no—
417C2×C3—361no—
418C2×C3C2×D4×C3⋊D311yescharacteristic
419D3—68no—
420D3—124no—
421C22—21no—
422C22—92no—
423C22—181no—
424C22D4×#27211yescharacteristic
425C4—21no—
426C4—22no—
427C4—92no—
428C4—181no—
429C4—182no—
430C4D4×#27211yescharacteristic
431C4C23×#27211yescharacteristic
432C3—34no—
433C3#1059625011yescharacteristic
434C2—21no—
435C2—41no—
436C2—182no—
437C2—361no—
438C2C2×D4×#27211yescharacteristic
subgroupG/H
center, socleC2×C3C2×D4×C3⋊D3
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C23×C3⋊D3
FittingH(3,𝔽3)×#137C2
hypercenterC3×#137C3⋊D3
hypocenterH(3,𝔽3)#578
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

Complements

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

NH ≅ G/Ntype
1. #11722676C2semidirect
2. #11722670C2semidirect
3. H(3,𝔽3)×#137C2semidirect
4. #11721871C2semidirect
5. C2×#88550C2semidirect
6. #11721807C2semidirect
7. Q16×#272C2semidirect
8. #11721802C2semidirect
9. #11721801C2semidirect
10. #11721788C2semidirect
11. #11721786C2semidirect
44. C32×#137D3semidirect
257. #137#272semidirect
263. H(3,𝔽3)#578semidirect
380. C32#5779semidirect

Maximal subgroups

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

subgroupindex
#117226762
#117226702
H(3,𝔽3)×#1372
#117218712
C2×#885502
#117218072
Q16×#2722
#117218022
#117218012
#117217882
#117217862
C3×#57793