Finite Group Databasebeta

← #13147783

Subgroups — #13147783

Order 1728.

1186 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88649C212yesmaximal, complemented
2C4×C4⋊#272C211yesmaximal, characteristic, complemented
3#11721385C211yesmaximal, characteristic, complemented
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5#11721159C212yesmaximal, complemented
6#11721158C212yesmaximal, complemented
7#11721132C212yesmaximal, complemented
8#11721127C214yesmaximal, complemented
9#10593906—31nomaximal
10#10593902—31nomaximal
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
15C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
16C2×#634⋊C3C2211yescharacteristic
17C2×#6200C2214yes—
18C2×#6200C2214yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
20#88649—28no—
21C42×H(3,𝔽3)C2211yescharacteristic
22#88542C2212yes—
23C4⋊#1084C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27#88450C2214yes—
28C2×#3799—32no—
29C2×#3795—32no—
30#64083D311yescharacteristic, complemented
31C4×#634D311yescharacteristic, complemented
32C3×#874—31no—
33C3×C4×D12—31no—
34C3×C4×D12—31no—
35C3×C4×D12—61no—
36C3×C4×Dic6—31no—
37#63914—32no—
38#63911—32no—
39#63910—34no—
40#63909—32no—
41#63908—32no—
42#63898—32no—
43#63894—32no—
44#63885—34no—
45C22×#272C2312yes—
46C22×(C3⋊D3)⋊C3C2314yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×#1084C2312yes—
51C4⋊#272—22no—
52C4×#272—24no—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55C4⋊((C3⋊D3)⋊C3)—24no—
56C4⋊((C3⋊D3)⋊C3)—24no—
57C4⋊((C3⋊D3)⋊C3)—42no—
58#634⋊C3D414yescomplemented
59#6200—28no—
60#6200—44no—
61((C3⋊D3)⋊C3)⋊C4—28no—
62#5718—91no—
63#5577—91no—
64#5553—91no—
65C2×C4⋊(C3⋊D3)C2×D312yes—
66C2×C4⋊(C3⋊D3)C2×D312yes—
67C2×C4⋊(C3⋊D3)C2×D312yes—
68C2×#634C2×D311yescharacteristic
69C2×C3×D12—32no—
70C2×C3×D12—32no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—32no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—32no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—62no—
78C2×C3×Dic6—31no—
79C2×#626—34no—
80C2×#626—34no—
81C2×#625—34no—
82C2×#625—34no—
83C2×#624—34no—
84C2×D3×Dic3—34no—
85#3799—68no—
86#3795—68no—
87C32×C42—21no—
88C32×C42C2×D311yescharacteristic
89C32×C42C2×D311yescharacteristic
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91#3750C2×D312yes—
92C4×C3⋊Dic3C2×D312yes—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—62no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—61no—
100C3×Dic3⋊C4—32no—
101C3×C4×Dic3—32no—
102#3722—34no—
103(C2×D3)⋊Dic3—34no—
104C22×H(3,𝔽3)C2411yescharacteristic
105C2×#272—22no—
106C2×#272—22no—
107C2×#272—22no—
108C2×(C3⋊D3)⋊C3—24no—
109C2×(C3⋊D3)⋊C3—24no—
110C2×(C3⋊D3)⋊C3—24no—
111C4×H(3,𝔽3)—21no—
112C4×H(3,𝔽3)C2×D412yes—
113C4×H(3,𝔽3)#4112yescomplemented
114#1084—22no—
115(C3⋊Dic3)⋊C3—22no—
116(C3⋊Dic3)⋊C3C2×D414yes—
117C2×#239—92no—
118C2×#237—92no—
119C2×#235—92no—
120C3×#127—91no—
121C3×C4×Q8—91no—
122C3×C4×D4—91no—
123#940—92no—
124#940—92no—
125#939—94no—
126#938—92no—
127D4×Dic3—92no—
128#930—92no—
129#895—94no—
130#895—92no—
131#893—92no—
132#884—94no—
133#874—31no—
134#874—91no—
135#874—92no—
136C4×D12—31no—
137C4×D12—31no—
138C4×D12—61no—
139C4×D12—92no—
140C4×Dic6—31no—
141C4×Dic6—91no—
142C22×C3⋊D3C22×D314yes—
143C22×C3×D3—34no—
144C22×C3×D3—32no—
145C22×C3×D3—32no—
146C22×C3×D3—62no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C4⋊(C3⋊D3)—24no—
159C4⋊(C3⋊D3)—24no—
160C4⋊(C3⋊D3)—42no—
161#634D1214yescomplemented
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—62no—
167C3×D12—64no—
168C3×D12—64no—
169C3×D12—62no—
170C3×D12—64no—
171C3×D12—62no—
172C3×D12—122no—
173C3×C4×D3—64no—
174C3×C4×D3—64no—
175C3×C4×D3—124no—
176C3×Dic6—34no—
177#626—68no—
178#626—124no—
179#625—68no—
180#625—124no—
181#624—68no—
182D3×Dic3—68no—
183#550—271noSylow
184C2×H(3,𝔽3)C22×D411yescharacteristic
185C2×H(3,𝔽3)C2×#4111yescharacteristic
186C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
187#272—42no—
188(C3⋊D3)⋊C3—44no—
189C2×C3×Q8—91no—
190C2×C3×D4—92no—
191C2×C3×D4—92no—
192C2×C3×D4—92no—
193C2×C3×D4—91no—
194C22×C3×C4—92no—
195C2×#67—94no—
196C2×#67—94no—
197C2×#67—94no—
198C2×#67—94no—
199C22×Dic3—94no—
200#239—188no—
201#237—188no—
202#235—188no—
203C2×D12—31no—
204C2×D12—31no—
205C2×D12—32no—
206C2×D12—32no—
207C2×D12—32no—
208C2×D12—61no—
209C2×D12—92no—
210C2×D12—92no—
211C2×D12—92no—
212C2×D12—94no—
213C2×D12—94no—
214C2×C4×D3—32no—
215C2×C4×D3—32no—
216C2×C4×D3—62no—
217C2×C4×D3—94no—
218C2×C4×D3—94no—
219C2×Dic6—31no—
220C2×Dic6—91no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4—91no—
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—32no—
233(C2×D3)⋊C4—62no—
234(C2×D3)⋊C4—94no—
235(C2×D3)⋊C4—94no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—91no—
241Dic3⋊C4—32no—
242Dic3⋊C4—92no—
243C4×Dic3—32no—
244C4×Dic3—92no—
245C22×C32—21no—
246C22×C32C23×D311yescharacteristic
247C22×C32C23×D311yescharacteristic
248C2×C3⋊D3—24no—
249C2×C3⋊D3—24no—
250C2×C3⋊D3—24no—
251C2×C3×D3—64no—
252C2×C3×D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C2×C3×D3—64no—
258C2×C3×D3—62no—
259C2×C3×D3—64no—
260C2×C3×D3—62no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—41no—
268C32×C4C2×D1212yes—
269C32×C4D3×D412yes—
270C32×C4#23712yescomplemented
271C32×C4#23912yescomplemented
272C3⋊Dic3—22no—
273C3⋊Dic3C2×D1214yes—
274C3×Dic3—34no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—122no—
279C2×#41—272no—
280#127—91no—
281#127—272no—
282#121—274no—
283#121—272no—
284C4×Q8—91no—
285C4×D4—91no—
286C4×D4—272no—
287H(3,𝔽3)#55011yescharacteristic, complemented, Sylow
288C23×C3—94no—
289C23×C3—92no—
290C22×D3—32no—
291C22×D3—32no—
292C22×D3—34no—
293C22×D3—62no—
294C22×D3—94no—
295C3×Q8—94no—
296C3×D4—184no—
297C3×D4—184no—
298C3×D4—182no—
299C3×D4—362no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—184no—
313C2×C3×C4C2×D3211yescharacteristic
314C2×C3×C4C2×D3211yescharacteristic
315C2×C3×C4C2×D3211yescharacteristic
316#67—188no—
317#67—188no—
318#67—364no—
319#67—364no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—62no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—188no—
328D12—62no—
329D12—62no—
330D12—64no—
331D12—64no—
332D12—64no—
333D12—122no—
334D12—184no—
335D12—184no—
336D12—188no—
337D12—362no—
338D12—364no—
339C4×D3—64no—
340C4×D3—64no—
341C4×D3—124no—
342C4×D3—188no—
343C4×D3—188no—
344Dic6—34no—
345Dic6—94no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32C22×D1211yescharacteristic
350C2×C32C2×D3×D411yescharacteristic
351C2×C32C2×#23711yescharacteristic
352C2×C32#100411yescharacteristic
353C2×C32C2×#23911yescharacteristic
354C2×C32#100911yescharacteristic
355C3⋊D3—44no—
356C3×D3—124no—
357C3×D3—122no—
358C3×D3—124no—
359C3×D3—122no—
360#41—548no—
361C2×Q8—91no—
362C2×D4—91no—
363C2×D4—92no—
364C2×D4—92no—
365C2×D4—92no—
366C2×D4—274no—
367C2×D4—274no—
368C22×C4—92no—
369C22×C4—274no—
370C4⋊C4—91no—
371C4⋊C4—92no—
372C4⋊C4—91no—
373C22⋊C4—92no—
374C22⋊C4—274no—
375C42—92no—
376C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—61no—
380C22×C3—184no—
381C22×C3—184no—
382C22×C3—184no—
383C22×C3—182no—
384C22×C3—182no—
385C22×C3—182no—
386C22×C3C22×D3211yescharacteristic
387C2×D3—62no—
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—62no—
392C2×D3—62no—
393C2×D3—64no—
394C2×D3—64no—
395C2×D3—64no—
396C2×D3—64no—
397C2×D3—122no—
398C2×D3—122no—
399C2×D3—184no—
400C2×D3—184no—
401C2×D3—184no—
402C3×C4—21no—
403C3×C4—32no—
404C3×C4—32no—
405C3×C4—32no—
406C3×C4—32no—
407C3×C4—61no—
408C3×C4—61no—
409C3×C4—62no—
410C3×C4—62no—
411C3×C4—94no—
412C3×C4—121no—
413C3×C4—182no—
414C3×C4—182no—
415C3×C4#379612yes—
416C3×C4D3×D1212yes—
417Dic3—34no—
418Dic3—62no—
419Dic3—62no—
420Dic3—62no—
421Dic3—94no—
422Dic3—122no—
423Dic3—182no—
424C32—21no—
425C32#557711yescharacteristic, complemented
426C32#571811yescharacteristic, complemented
427C23—92no—
428C23—94no—
429Q8—94no—
430D4—182no—
431D4—184no—
432D4—184no—
433D4—362no—
434D4—548no—
435D4—1084no—
436C2×C4—92no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—184no—
440C2×C4—548no—
441C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
442C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
443C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—61no—
451C2×C3—61no—
452C2×C3—61no—
453C2×C3—364no—
454C2×C3—362no—
455C2×C3C2×#379611yescharacteristic
456C2×C3C2×D3×D1211yescharacteristic
457C2×C3#6430711yescharacteristic
458D3—122no—
459D3—122no—
460D3—124no—
461D3—124no—
462D3—364no—
463C22—182no—
464C22—184no—
465C22—182no—
466C22—184no—
467C22—184no—
468C22—182no—
469C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
470C4—21no—
471C4—94no—
472C4—182no—
473C4—182no—
474C4#8864912yescomplemented
475C4#8865412yes—
476C3—31no—
477C3—31no—
478C3—61no—
479C3#1059390011yescharacteristic
480C2—362no—
481C2—364no—
482C2C2×#8864911yescharacteristic
483C2C2×#8865411yescharacteristic
484C2#1172249211yescharacteristic
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×#88649C2semidirect
2. C4×C4⋊#272C2semidirect
3. #11721385C2semidirect
4. C4×#634⋊C3C2semidirect
5. #11721159C2semidirect
6. #11721158C2semidirect
7. #11721132C2semidirect
8. #11721127C2semidirect
30. #64083D3semidirect
31. C4×#634D3semidirect
58. #634⋊C3D4semidirect
113. C4×H(3,𝔽3)#41semidirect
161. #634D12semidirect
270. C32×C4#237semidirect
271. C32×C4#239semidirect
287. H(3,𝔽3)#550semidirect
425. C32#5577semidirect
426. C32#5718semidirect
474. C4#88649semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
C4×C4⋊#2722
#117213852
C4×#634⋊C32
#117211592
#117211582
#117211322
#117211272
#105939063
#105939023