Finite Group Databasebeta

← #13144289

Subgroups — #13144289

Order 1728.

1419 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic
2C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#88552C211yesmaximal, characteristic, complemented
4C2×#88542C211yesmaximal, characteristic, complemented
5C2×#88542C211yesmaximal, characteristic, complemented
6C2×#88540C211yesmaximal, characteristic, complemented
7C2×C4×#1084C211yesmaximal, characteristic, complemented
8C3×#4957—34nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6235—22no—
14C2×#6235—22no—
15C2×#6235C414yescomplemented
16C22×#1084C2211yescharacteristic
17C22×#1084C2211yescharacteristic
18C22×#1084C2211yescharacteristic
19C22⋊C4×H(3,𝔽3)—22no—
20#88552—22no—
21#88542—22no—
22#88542—22no—
23#88540—22no—
24C4×#1084—22no—
25C22×C3×#67—34no—
26C2×C32×C22⋊C4D314yescomplemented
27C2×C3×#217—34no—
28C2×C3×(C2×D3)⋊C4—34no—
29C2×C3×(C2×D3)⋊C4—34no—
30C2×C3×Dic3⋊C4—34no—
31C2×C3×C4×Dic3—34no—
32C23×H(3,𝔽3)—21no—
33C23×H(3,𝔽3)—21no—
34C23×H(3,𝔽3)—21no—
35C23×H(3,𝔽3)—21no—
36C23×H(3,𝔽3)—21no—
37C23×H(3,𝔽3)—21no—
38C23×H(3,𝔽3)C2×C412yes—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272—21no—
44C22×#272—21no—
45C22×#272—21no—
46C22×#272C2×C412yes—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)—21no—
51C2×C4×H(3,𝔽3)—21no—
52C2×C4×H(3,𝔽3)D412yescomplemented
53#6235—28no—
54C2×#1084—21no—
55C2×#1084—21no—
56C2×#1084—21no—
57C2×#1084—21no—
58C2×#1084—21no—
59C2×#1084C2×C412yes—
60C2×#1084D412yes—
61C2×#1084D412yes—
62C2×#1084D412yescomplemented
63C3×#390—91no—
64#4957—34no—
65C24×C32C2×D314yes—
66C23×C3×D3—34no—
67C22×C32×C4C2×D314yes—
68C22×C32×C4C2×D314yes—
69C2×C3×#67—316no—
70C2×C3×#67—68no—
71C2×C3×#67—68no—
72C22×C3×Dic3—34no—
73C22×C3×Dic3—34no—
74C22×C3×Dic3—34no—
75C32×C22⋊C4—28no—
76C3×#217—68no—
77C3×(C2×D3)⋊C4—68no—
78C3×(C2×D3)⋊C4—68no—
79C3×Dic3⋊C4—68no—
80C3×C4×Dic3—68no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—22no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C22×C411yescharacteristic
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C22×H(3,𝔽3)#4111yescharacteristic
94C22×H(3,𝔽3)#4111yescharacteristic
95C2×#272—22no—
96C2×#272—22no—
97C2×#272—22no—
98C2×#272—41no—
99C2×#272—41no—
100C2×#272—41no—
101C2×#272—41no—
102C4×H(3,𝔽3)—22no—
103C4×H(3,𝔽3)—41no—
104#1084—22no—
105#1084—41no—
106#1084C22⋊C414yescomplemented
107C22×C3×D4—91no—
108C22×#67—34no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C22⋊C4—34no—
111C2×C3×C22⋊C4—91no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4C3⋊D311yescharacteristic
115C2×C3×C42—91no—
116C2×#217—34no—
117C2×(C2×D3)⋊C4—34no—
118C2×(C2×D3)⋊C4—34no—
119C2×Dic3⋊C4—34no—
120C2×C4×Dic3—34no—
121C23×C32—24no—
122C23×C32—24no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32C4×D318yes—
128C23×C32C22×D314yes—
129C22×C3×D3—38no—
130C22×C3×D3—64no—
131C22×C3×D3—64no—
132C22×C3×D3—64no—
133C22×C3×D3—64no—
134C22×C3×D3—64no—
135C22×C3×D3—64no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4#6718yescomplemented
142C3×#67—632no—
143C2×C3×Dic3—38no—
144C2×C3×Dic3—38no—
145C2×C3×Dic3—38no—
146C2×C3×Dic3—38no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—64no—
152#390—91noSylow
153C2×H(3,𝔽3)—42no—
154C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
155C2×H(3,𝔽3)C4×D411yescharacteristic
156C2×H(3,𝔽3)C4×D411yescharacteristic
157C2×H(3,𝔽3)#12111yescharacteristic
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)#12411yescharacteristic
160C2×H(3,𝔽3)#12711yescharacteristic
161#272—42no—
162C24×C3—34no—
163C24×C3—91no—
164C24×C3C2×C3⋊D311yescharacteristic
165C23×D3—34no—
166C2×C3×D4—94no—
167C2×C3×D4—182no—
168C2×C3×D4—182no—
169C22×C3×C4—34no—
170C22×C3×C4—34no—
171C22×C3×C4—91no—
172C22×C3×C4—91no—
173C22×C3×C4—91no—
174C22×C3×C4C2×C3⋊D311yescharacteristic
175C22×C3×C4C2×C3⋊D311yescharacteristic
176C2×#67—316no—
177C2×#67—68no—
178C2×#67—68no—
179C22×Dic3—34no—
180C22×Dic3—34no—
181C22×Dic3—34no—
182C3×C4⋊C4—182no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—68no—
185C3×C22⋊C4—182no—
186C3×C22⋊C4—182no—
187C3×C22⋊C4—182no—
188C3×C42—182no—
189#217—68no—
190(C2×D3)⋊C4—68no—
191(C2×D3)⋊C4—68no—
192Dic3⋊C4—68no—
193C4×Dic3—68no—
194C22×C32—28no—
195C22×C32—28no—
196C22×C32—28no—
197C22×C32—44no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32—44no—
201C22×C32C2×C4×D314yes—
202C22×C32#23514yes—
203C22×C32D3×D414yes—
204C22×C32D3×D414yes—
205C22×C32D3×D414yes—
206C22×C32#23714yes—
207C22×C32C2×#6714yes—
208C2×C3×D3—68no—
209C2×C3×D3—68no—
210C2×C3×D3—68no—
211C2×C3×D3—124no—
212C2×C3×D3—124no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C32×C4—28no—
216C32×C4—44no—
217C3×Dic3—316no—
218C3×Dic3—68no—
219C3×Dic3—124no—
220C22×D4—91no—
221C2×C4⋊C4—91no—
222C2×C22⋊C4—91no—
223C2×C22⋊C4—91no—
224C2×C22⋊C4—91no—
225C2×C22⋊C4#27211yescharacteristic, complemented
226C2×C42—91no—
227H(3,𝔽3)#39011yescharacteristic, complemented, Sylow
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—38no—
235C23×C3—34no—
236C23×C3—64no—
237C23×C3—64no—
238C23×C3—64no—
239C23×C3—64no—
240C23×C3—64no—
241C23×C3—64no—
242C23×C3—92no—
243C23×C3—181no—
244C23×C3—181no—
245C23×C3—181no—
246C23×C3—181no—
247C23×C3—181no—
248C23×C3—181no—
249C23×C3C4×C3⋊D312yes—
250C23×C3C22×C3⋊D311yescharacteristic
251C22×D3—38no—
252C22×D3—64no—
253C22×D3—64no—
254C22×D3—64no—
255C22×D3—64no—
256C22×D3—64no—
257C22×D3—64no—
258C3×D4—188no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—38no—
265C2×C3×C4—64no—
266C2×C3×C4—64no—
267C2×C3×C4—64no—
268C2×C3×C4—64no—
269C2×C3×C4—64no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—92no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4#63812yes—
280#67—632no—
281C2×Dic3—38no—
282C2×Dic3—38no—
283C2×Dic3—38no—
284C2×Dic3—38no—
285C2×Dic3—64no—
286C2×Dic3—64no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3—64no—
290C2×C32—48no—
291C2×C32D3×C22⋊C414yes—
292C2×C32#67⋊C414yes—
293C2×C32#88414yes—
294C2×C32#88514yes—
295C2×C32C4×#6714yes—
296C2×C32#93914yes—
297C2×C32#94014yes—
298C3×D3—128no—
299C24—91no—
300C24C2×#27211yescharacteristic
301C2×D4—94no—
302C2×D4—182no—
303C2×D4—182no—
304C22×C4—91no—
305C22×C4—91no—
306C22×C4—91no—
307C22×C4C2×#27211yescharacteristic
308C22×C4C2×#27211yescharacteristic
309C4⋊C4—182no—
310C22⋊C4—22no—
311C22⋊C4—182no—
312C22⋊C4—182no—
313C22⋊C4—182no—
314C42—182no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—34no—
319C22×C3—34no—
320C22×C3—34no—
321C22×C3—34no—
322C22×C3—34no—
323C22×C3—34no—
324C22×C3—34no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—68no—
330C22×C3—68no—
331C22×C3—68no—
332C22×C3—124no—
333C22×C3—124no—
334C22×C3—124no—
335C22×C3—124no—
336C22×C3—182no—
337C22×C3—182no—
338C22×C3—182no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3C2×C4×C3⋊D311yescharacteristic
344C22×C3#382811yescharacteristic
345C22×C3D4×C3⋊D311yescharacteristic
346C22×C3D4×C3⋊D311yescharacteristic
347C22×C3D4×C3⋊D311yescharacteristic
348C22×C3#383011yescharacteristic
349C22×C3C2×#63811yescharacteristic
350C2×D3—68no—
351C2×D3—68no—
352C2×D3—68no—
353C2×D3—124no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—124no—
357C3×C4—22no—
358C3×C4—41no—
359C3×C4—68no—
360C3×C4—94no—
361C3×C4—124no—
362C3×C4—182no—
363C3×C4—361no—
364Dic3—316no—
365Dic3—68no—
366Dic3—124no—
367C32#495714yescomplemented
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—92no—
375C23—181no—
376C23—181no—
377C23—181no—
378C23—181no—
379C23—181no—
380C23—181no—
381C23C4×#27212yes—
382C23C22×#27211yescharacteristic
383D4—188no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—92no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—92no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4#623512yescomplemented
399C2×C3—34no—
400C2×C3—34no—
401C2×C3—34no—
402C2×C3—34no—
403C2×C3—34no—
404C2×C3—34no—
405C2×C3—34no—
406C2×C3—42no—
407C2×C3—128no—
408C2×C3—362no—
409C2×C3C22⋊C4×C3⋊D311yescharacteristic
410C2×C3#6409011yescharacteristic
411C2×C3#6409311yescharacteristic
412C2×C3#6409411yescharacteristic
413C2×C3C4×#63811yescharacteristic
414C2×C3#6414811yescharacteristic
415C2×C3#6414911yescharacteristic
416D3—128no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—182no—
425C22—182no—
426C22—182no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22C2×C4×#27211yescharacteristic
432C22#8874011yescharacteristic
433C22D4×#27211yescharacteristic
434C22D4×#27211yescharacteristic
435C22D4×#27211yescharacteristic
436C22#8874411yescharacteristic
437C22C2×#623511yescharacteristic
438C4—22no—
439C4—41no—
440C4—94no—
441C4—182no—
442C4—361no—
443C3—34no—
444C3#1059265611yescharacteristic
445C2—42no—
446C2—362no—
447C2C22⋊C4×#27211yescharacteristic
448C2#1172173711yescharacteristic
449C2#1172174011yescharacteristic
450C2#1172174111yescharacteristic
451C2C4×#623511yescharacteristic
452C2#1172184911yescharacteristic
453C2#1172185011yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC22×#6235C2
cosocleC23×H(3,𝔽3)C23
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#390
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×H(3,𝔽3)C2semidirect
3. C2×#88552C2semidirect
4. C2×#88542C2semidirect
5. C2×#88542C2semidirect
6. C2×#88540C2semidirect
7. C2×C4×#1084C2semidirect
15. C2×#6235C4semidirect
26. C2×C32×C22⋊C4D3semidirect
52. C2×C4×H(3,𝔽3)D4semidirect
62. C2×#1084D4semidirect
106. #1084C22⋊C4semidirect
141. C2×C32×C4#67semidirect
225. C2×C22⋊C4#272semidirect
227. H(3,𝔽3)#390semidirect
367. C32#4957semidirect
398. C2×C4#6235semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×#885422
C2×#885402
C2×C4×#10842
C3×#49573