Finite Group Databasebeta

← #13147853

Subgroups — #13147853

Order 1728.

1200 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88652C212yesmaximal, complemented
2#11721720C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721158C212yesmaximal, complemented
6#11721157C214yesmaximal, complemented
7#11721156C212yesmaximal, complemented
8#11721149C212yesmaximal, complemented
9#10593906—31nomaximal
10#10593900—31nomaximal
11C2×C4⋊#272C2212yes—
12C2×C4⋊#272C2212yes—
13C2×C4⋊#272C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×#634⋊C3C2211yescharacteristic
17C2×#6200C2214yes—
18C2×#6200C2214yes—
19C2×#6199C2214yes—
20#88652—28no—
21C42×H(3,𝔽3)C2211yescharacteristic
22#3752⋊C3C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27#88451C2214yes—
28C2×#3796—32no—
29C2×#3795—32no—
30C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
31C4×#634D311yescharacteristic, complemented
32C3×#874—31no—
33C3×#874—31no—
34C3×#874—61no—
35C3×C4×D12—31no—
36C3×C4×Dic6—31no—
37#63916—32no—
38#63914—32no—
39#63911—32no—
40#63908—32no—
41#63898—32no—
42Dic3×D12—32no—
43#63886—34no—
44#63885—34no—
45C22×#272C2314yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C4⋊#272—24no—
51C4⋊#272—24no—
52C4⋊#272—42no—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C4⋊((C3⋊D3)⋊C3)—22no—
57C4×(C3⋊D3)⋊C3—24no—
58#634⋊C3D414yescomplemented
59#6200—28no—
60#6200—44no—
61#6199—28no—
62#5718—91no—
63#5577—91no—
64#5553—91no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3C2×D312yes—
67C2×#634C2×D311yescharacteristic
68C2×C3×D12—32no—
69C2×C3×D12—32no—
70C2×C3×D12—31no—
71C2×C3×D12—32no—
72C2×C3×D12—32no—
73C2×C3×D12—32no—
74C2×C3×D12—32no—
75C2×C3×D12—62no—
76C2×C3×D12—62no—
77C2×C3×D12—62no—
78C2×C3×C4×D3—32no—
79C2×C3×Dic6—31no—
80C2×#626—34no—
81C2×#626—34no—
82C2×#625—34no—
83C2×#625—34no—
84C2×D3×Dic3—34no—
85C2×D3×Dic3—34no—
86#3796—68no—
87#3795—68no—
88C32×C42—21no—
89C32×C42C2×D311yescharacteristic
90C32×C42C2×D311yescharacteristic
91#3752C2×D312yes—
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94#3750C2×D312yes—
95C4×C3⋊Dic3C2×D312yes—
96C3×(C2×D3)⋊C4—32no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×Dic3⋊C4—32no—
100C3×C4×Dic3—32no—
101(C2×D3)⋊Dic3—34no—
102(C2×D3)⋊Dic3—34no—
103C22×H(3,𝔽3)C2411yescharacteristic
104C2×#272—24no—
105C2×#272—24no—
106C2×#272—24no—
107C2×(C3⋊D3)⋊C3—22no—
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—22no—
110C4×H(3,𝔽3)—21no—
111C4×H(3,𝔽3)C2×D412yes—
112C4×H(3,𝔽3)#4112yescomplemented
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3C2×D414yes—
116C2×#239—92no—
117C2×#237—92no—
118C2×#235—92no—
119C3×#127—91no—
120C3×C4×Q8—91no—
121C3×C4×D4—91no—
122#940—92no—
123#940—92no—
124#939—94no—
125#938—92no—
126D4×Dic3—92no—
127#930—92no—
128#895—94no—
129#895—92no—
130#893—92no—
131#884—94no—
132#874—31no—
133#874—31no—
134#874—61no—
135#874—92no—
136C4×D12—31no—
137C4×D12—91no—
138C4×D12—92no—
139C4×Dic6—31no—
140C4×Dic6—91no—
141C22×C3⋊D3C22×D312yes—
142C22×C3×D3—32no—
143C22×C3×D3—34no—
144C22×C3×D3—34no—
145C22×C3×D3—64no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3C22×D312yes—
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C4⋊(C3⋊D3)—22no—
159C4×C3⋊D3—24no—
160#634D1214yescomplemented
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C3×D12—64no—
165C3×D12—64no—
166C3×D12—64no—
167C3×D12—62no—
168C3×D12—64no—
169C3×D12—124no—
170C3×D12—124no—
171C3×D12—122no—
172C3×D12—122no—
173C3×D12—124no—
174C3×C4×D3—64no—
175C3×Dic6—34no—
176#626—68no—
177#626—124no—
178#625—68no—
179#625—124no—
180D3×Dic3—68no—
181D3×Dic3—68no—
182#550—271noSylow
183C2×H(3,𝔽3)C22×D411yescharacteristic
184C2×H(3,𝔽3)C2×#4111yescharacteristic
185C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
186#272—44no—
187(C3⋊D3)⋊C3—42no—
188C2×C3×Q8—91no—
189C2×C3×D4—91no—
190C2×C3×D4—92no—
191C2×C3×D4—92no—
192C2×C3×D4—92no—
193C22×C3×C4—92no—
194C2×#67—94no—
195C2×#67—94no—
196C2×#67—94no—
197C2×#67—94no—
198C22×Dic3—94no—
199#239—188no—
200#237—188no—
201#235—188no—
202C2×D12—32no—
203C2×D12—32no—
204C2×D12—32no—
205C2×D12—32no—
206C2×D12—31no—
207C2×D12—32no—
208C2×D12—32no—
209C2×D12—62no—
210C2×D12—62no—
211C2×D12—62no—
212C2×D12—91no—
213C2×D12—94no—
214C2×D12—94no—
215C2×C4×D3—32no—
216C2×C4×D3—92no—
217C2×C4×D3—94no—
218C2×C4×D3—94no—
219C2×Dic6—31no—
220C2×Dic6—91no—
221C3×C4⋊C4—91no—
222C3×C4⋊C4—92no—
223C3×C4⋊C4—91no—
224C3×C22⋊C4—92no—
225C3×C42—31no—
226C3×C42—31no—
227C3×C42—61no—
228C3×C42—92no—
229C3×C42D3211yescharacteristic
230#217—94no—
231(C2×D3)⋊C4—32no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—94no—
234(C2×D3)⋊C4—94no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—91no—
238C4⋊Dic3—91no—
239Dic3⋊C4—32no—
240Dic3⋊C4—92no—
241C4×Dic3—32no—
242C4×Dic3—92no—
243C22×C32—21no—
244C22×C32C23×D311yescharacteristic
245C22×C32C23×D311yescharacteristic
246C2×C3⋊D3—22no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—22no—
249C2×C3×D3—62no—
250C2×C3×D3—64no—
251C2×C3×D3—64no—
252C2×C3×D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—62no—
256C2×C3×D3—64no—
257C2×C3×D3—64no—
258C2×C3×D3—124no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C32×C4—22no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—22no—
265C32×C4—22no—
266C32×C4C2×D1212yes—
267C32×C4#23512yescomplemented
268C32×C4D3×D412yes—
269C32×C4#23912yescomplemented
270C3⋊Dic3—22no—
271C3⋊Dic3—22no—
272C3⋊Dic3C2×D1214yes—
273C3×Dic3—34no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C2×#41—272no—
277#127—91no—
278#127—272no—
279#121—274no—
280#121—272no—
281C4×Q8—91no—
282C4×D4—91no—
283C4×D4—272no—
284H(3,𝔽3)#55011yescharacteristic, complemented, Sylow
285C23×C3—92no—
286C23×C3—94no—
287C22×D3—34no—
288C22×D3—34no—
289C22×D3—32no—
290C22×D3—64no—
291C22×D3—92no—
292C3×Q8—94no—
293C3×D4—182no—
294C3×D4—184no—
295C3×D4—184no—
296C3×D4—362no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—92no—
307C2×C3×C4—92no—
308C2×C3×C4—92no—
309C2×C3×C4—184no—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313#67—188no—
314#67—188no—
315#67—364no—
316#67—364no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—188no—
324D12—64no—
325D12—64no—
326D12—64no—
327D12—64no—
328D12—62no—
329D12—124no—
330D12—124no—
331D12—122no—
332D12—122no—
333D12—124no—
334D12—182no—
335D12—188no—
336D12—364no—
337C4×D3—64no—
338C4×D3—184no—
339C4×D3—188no—
340C4×D3—188no—
341Dic6—34no—
342Dic6—94no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32C22×D1211yescharacteristic
347C2×C32C2×#23511yescharacteristic
348C2×C32C2×D3×D411yescharacteristic
349C2×C32C2×#23911yescharacteristic
350C2×C32#100911yescharacteristic
351C2×C32#100911yescharacteristic
352C3⋊D3—42no—
353C3×D3—124no—
354C3×D3—122no—
355C3×D3—124no—
356C3×D3—244no—
357#41—548no—
358C2×Q8—91no—
359C2×D4—91no—
360C2×D4—92no—
361C2×D4—92no—
362C2×D4—92no—
363C2×D4—274no—
364C2×D4—274no—
365C22×C4—92no—
366C22×C4—274no—
367C4⋊C4—91no—
368C4⋊C4—92no—
369C4⋊C4—91no—
370C22⋊C4—92no—
371C22⋊C4—274no—
372C42—92no—
373C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—61no—
377C22×C3—182no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—184no—
381C22×C3—184no—
382C22×C3—184no—
383C22×C3C22×D3211yescharacteristic
384C2×D3—64no—
385C2×D3—64no—
386C2×D3—64no—
387C2×D3—64no—
388C2×D3—64no—
389C2×D3—64no—
390C2×D3—62no—
391C2×D3—62no—
392C2×D3—62no—
393C2×D3—124no—
394C2×D3—124no—
395C2×D3—124no—
396C2×D3—182no—
397C2×D3—182no—
398C2×D3—182no—
399C3×C4—21no—
400C3×C4—32no—
401C3×C4—32no—
402C3×C4—32no—
403C3×C4—32no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—62no—
407C3×C4—62no—
408C3×C4—62no—
409C3×C4—94no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4#379912yes—
413C3×C4D3×D1212yes—
414Dic3—34no—
415Dic3—62no—
416Dic3—62no—
417Dic3—94no—
418Dic3—182no—
419Dic3—182no—
420C32—21no—
421C32#555311yescharacteristic, complemented
422C32#557711yescharacteristic, complemented
423C23—92no—
424C23—94no—
425Q8—94no—
426D4—184no—
427D4—182no—
428D4—184no—
429D4—362no—
430D4—548no—
431D4—1084no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—184no—
436C2×C4—548no—
437C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
438C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—61no—
447C2×C3—61no—
448C2×C3—61no—
449C2×C3—362no—
450C2×C3—364no—
451C2×C3C2×#379911yescharacteristic
452C2×C3C2×D3×D1211yescharacteristic
453C2×C3#6430811yescharacteristic
454D3—124no—
455D3—124no—
456D3—122no—
457D3—244no—
458D3—362no—
459C22—184no—
460C22—182no—
461C22—182no—
462C22—184no—
463C22—184no—
464C22—182no—
465C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
466C4—21no—
467C4—94no—
468C4—182no—
469C4—182no—
470C4#8865212yescomplemented
471C4#8865412yes—
472C3—31no—
473C3—31no—
474C3—61no—
475C3#1059390211yescharacteristic
476C2—364no—
477C2—362no—
478C2C2×#8865211yescharacteristic
479C2C2×#8865411yescharacteristic
480C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#550
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88652C2semidirect
2. #11721720C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C4×#634⋊C3C2semidirect
5. #11721158C2semidirect
6. #11721157C2semidirect
7. #11721156C2semidirect
8. #11721149C2semidirect
30. C4×C4⋊(C3⋊D3)D3semidirect
31. C4×#634D3semidirect
58. #634⋊C3D4semidirect
112. C4×H(3,𝔽3)#41semidirect
160. #634D12semidirect
267. C32×C4#235semidirect
269. C32×C4#239semidirect
284. H(3,𝔽3)#550semidirect
421. C32#5553semidirect
422. C32#5577semidirect
470. C4#88652semidirect

Maximal subgroups

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

subgroupindex
C2×#886522
#117217202
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#117211582
#117211572
#117211562
#117211492
#105939063
#105939003