Finite Group Databasebeta

← #13144303

Subgroups — #13144303

Order 1728.

1573 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic
2C22×C4×#272C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88552C211yesmaximal, characteristic, complemented
5C2×#88542C211yesmaximal, characteristic, complemented
6#11720574C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4956—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
19C2×C4×#272—22no—
20C2×C4×#272—22no—
21C2×C4×#272—22no—
22C22⋊C4×H(3,𝔽3)—22no—
23#88552—22no—
24#88542—22no—
25C22×C3×#67—34no—
26C22×C3×C4×D3—34no—
27C2×C32×C22⋊C4D314yescomplemented
28C2×C3×#217—34no—
29C2×C3×(C2×D3)⋊C4—34no—
30C3×C3⋊#95—34no—
31C3×C3⋊#95—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×C412yescomplemented
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272—21no—
44C22×#272C2×C412yes—
45C22×#272D412yes—
46C22×#272D412yes—
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×#1084—21no—
60C2×#1084—21no—
61C2×#1084C2×C412yescomplemented
62C2×#1084D412yescomplemented
63C4×#272—28no—
64C3×#386—91no—
65#4956—34no—
66C24×C32C2×D314yes—
67C23×C3×D3—34no—
68C22×C32×C4C2×D314yes—
69C22×C32×C4C2×D314yes—
70C2×C3×#67—316no—
71C2×C3×#67—68no—
72C2×C3×#67—68no—
73C22×C3×Dic3—34no—
74C22×C3×Dic3—34no—
75C22×C3×Dic3—34no—
76C2×C3×C4×D3—68no—
77C2×C3×C4×D3—68no—
78C2×C3×C4×D3—68no—
79C32×C22⋊C4—28no—
80C3×#217—68no—
81C3×(C2×D3)⋊C4—68no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—22no—
84C22×H(3,𝔽3)—22no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)C22×C411yescharacteristic
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C22×H(3,𝔽3)C2×D411yescharacteristic
94C22×H(3,𝔽3)#4111yescharacteristic
95C22×H(3,𝔽3)#4111yescharacteristic
96C2×#272—22no—
97C2×#272—22no—
98C2×#272—22no—
99C2×#272—22no—
100C2×#272—22no—
101C2×#272—22no—
102C2×#272C22⋊C414yescomplemented
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)—41no—
105#1084—22no—
106#1084—22no—
107#1084—41no—
108C22×C3×D4—91no—
109C23×C3×C4—91no—
110C22×#67—34no—
111C22×C4×D3—34no—
112C2×C3×C22⋊C4—34no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4C3⋊D311yescharacteristic
116C2×#217—34no—
117C2×(C2×D3)⋊C4—34no—
118C3×#95—91no—
119C3×#95—91no—
120C3⋊#95—34no—
121C3⋊#95—34no—
122C23×C32—24no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32—24no—
128C23×C32C4×D318yescomplemented
129C23×C32C22×D314yes—
130C22×C3×D3—38no—
131C22×C3×D3—38no—
132C22×C3×D3—38no—
133C22×C3×D3—64no—
134C22×C3×D3—64no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4#6718yescomplemented
143C3×#67—632no—
144C2×C3×Dic3—38no—
145C2×C3×Dic3—38no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—64no—
153C3×C4×D3—632no—
154#386—91noSylow
155C2×H(3,𝔽3)—42no—
156C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
157C2×H(3,𝔽3)C4×D411yescharacteristic
158C2×H(3,𝔽3)C4×D411yescharacteristic
159C2×H(3,𝔽3)#12011yescharacteristic
160C2×H(3,𝔽3)#12111yescharacteristic
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)#12311yescharacteristic
163#272—24no—
164C24×C3—34no—
165C24×C3—91no—
166C24×C3C2×C3⋊D311yescharacteristic
167C23×D3—34no—
168C2×C3×D4—94no—
169C2×C3×D4—182no—
170C2×C3×D4—182no—
171C22×C3×C4—34no—
172C22×C3×C4—34no—
173C22×C3×C4—91no—
174C22×C3×C4—91no—
175C22×C3×C4—91no—
176C22×C3×C4—182no—
177C22×C3×C4—182no—
178C22×C3×C4—182no—
179C22×C3×C4C2×C3⋊D311yescharacteristic
180C22×C3×C4C2×C3⋊D311yescharacteristic
181C2×#67—316no—
182C2×#67—68no—
183C2×#67—68no—
184C22×Dic3—34no—
185C22×Dic3—34no—
186C22×Dic3—34no—
187C2×C4×D3—68no—
188C2×C4×D3—68no—
189C2×C4×D3—68no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—68no—
192C3×C22⋊C4—182no—
193C3×C22⋊C4—182no—
194#217—68no—
195(C2×D3)⋊C4—68no—
196C22×C32—28no—
197C22×C32—28no—
198C22×C32—28no—
199C22×C32—44no—
200C22×C32—44no—
201C22×C32—44no—
202C22×C32—44no—
203C22×C32C2×C4×D314yes—
204C22×C32#23514yes—
205C22×C32D3×D414yes—
206C22×C32D3×D414yes—
207C22×C32D3×D414yes—
208C22×C32#23714yes—
209C22×C32C2×#6714yes—
210C2×C3×D3—316no—
211C2×C3×D3—68no—
212C2×C3×D3—68no—
213C2×C3×D3—68no—
214C2×C3×D3—68no—
215C2×C3×D3—68no—
216C2×C3×D3—68no—
217C32×C4—28no—
218C32×C4—44no—
219C3×Dic3—68no—
220C3×Dic3—68no—
221C3×Dic3—124no—
222C22×D4—91no—
223C23×C4—91no—
224C2×C22⋊C4—91no—
225C2×C22⋊C4—91no—
226C2×C22⋊C4#27211yescharacteristic, complemented
227#95—91no—
228#95—91no—
229H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—38no—
237C23×C3—34no—
238C23×C3—64no—
239C23×C3—64no—
240C23×C3—64no—
241C23×C3—64no—
242C23×C3—64no—
243C23×C3—64no—
244C23×C3—92no—
245C23×C3—92no—
246C23×C3—92no—
247C23×C3—181no—
248C23×C3—181no—
249C23×C3—181no—
250C23×C3—181no—
251C23×C3C4×C3⋊D312yes—
252C23×C3C22×C3⋊D311yescharacteristic
253C22×D3—38no—
254C22×D3—38no—
255C22×D3—38no—
256C22×D3—64no—
257C22×D3—64no—
258C22×D3—64no—
259C22×D3—64no—
260C3×D4—188no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—38no—
267C2×C3×C4—64no—
268C2×C3×C4—64no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4—64no—
272C2×C3×C4—92no—
273C2×C3×C4—92no—
274C2×C3×C4—188no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4#63812yes—
283#67—632no—
284C2×Dic3—38no—
285C2×Dic3—38no—
286C2×Dic3—64no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3—64no—
290C2×Dic3—64no—
291C2×Dic3—64no—
292C2×Dic3—64no—
293C4×D3—632no—
294C2×C32—48no—
295C2×C32D3×C22⋊C414yes—
296C2×C32#67⋊C414yes—
297C2×C32#88314yes—
298C2×C32#88414yes—
299C2×C32C4×#6714yes—
300C2×C32#93714yes—
301C2×C32#93814yes—
302C3×D3—616no—
303C24—91no—
304C24C2×#27211yescharacteristic
305C2×D4—94no—
306C2×D4—182no—
307C2×D4—182no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—91no—
311C22×C4—182no—
312C22×C4—182no—
313C22×C4—182no—
314C22×C4C2×#27211yescharacteristic
315C22×C4C2×#27211yescharacteristic
316C22⋊C4—22no—
317C22⋊C4—182no—
318C22⋊C4—182no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—34no—
323C22×C3—34no—
324C22×C3—34no—
325C22×C3—34no—
326C22×C3—34no—
327C22×C3—34no—
328C22×C3—34no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—68no—
334C22×C3—68no—
335C22×C3—68no—
336C22×C3—94no—
337C22×C3—124no—
338C22×C3—124no—
339C22×C3—124no—
340C22×C3—124no—
341C22×C3—182no—
342C22×C3—182no—
343C22×C3—182no—
344C22×C3—182no—
345C22×C3—182no—
346C22×C3—182no—
347C22×C3C2×C4×C3⋊D311yescharacteristic
348C22×C3#382811yescharacteristic
349C22×C3D4×C3⋊D311yescharacteristic
350C22×C3D4×C3⋊D311yescharacteristic
351C22×C3D4×C3⋊D311yescharacteristic
352C22×C3#383011yescharacteristic
353C22×C3C2×#63811yescharacteristic
354C2×D3—316no—
355C2×D3—68no—
356C2×D3—68no—
357C2×D3—68no—
358C2×D3—68no—
359C2×D3—68no—
360C2×D3—68no—
361C3×C4—22no—
362C3×C4—41no—
363C3×C4—68no—
364C3×C4—124no—
365C3×C4—182no—
366C3×C4—182no—
367C3×C4—361no—
368Dic3—68no—
369Dic3—68no—
370Dic3—124no—
371C32#495614yescomplemented
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—92no—
379C23—92no—
380C23—92no—
381C23—181no—
382C23—181no—
383C23—181no—
384C23—181no—
385C23C4×#27212yescomplemented
386C23C22×#27211yescharacteristic
387D4—188no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C4—188no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4—181no—
402C2×C4—181no—
403C2×C4#623512yescomplemented
404C2×C3—34no—
405C2×C3—34no—
406C2×C3—34no—
407C2×C3—34no—
408C2×C3—34no—
409C2×C3—34no—
410C2×C3—34no—
411C2×C3—42no—
412C2×C3—128no—
413C2×C3—184no—
414C2×C3C22⋊C4×C3⋊D311yescharacteristic
415C2×C3#6409011yescharacteristic
416C2×C3#6409211yescharacteristic
417C2×C3#6409311yescharacteristic
418C2×C3C4×#63811yescharacteristic
419C2×C3#6414611yescharacteristic
420C2×C3#6414711yescharacteristic
421D3—616no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—94no—
430C22—182no—
431C22—182no—
432C22—182no—
433C22—182no—
434C22—182no—
435C22—182no—
436C22C2×C4×#27211yescharacteristic
437C22#8874011yescharacteristic
438C22D4×#27211yescharacteristic
439C22D4×#27211yescharacteristic
440C22D4×#27211yescharacteristic
441C22#8874411yescharacteristic
442C22C2×#623511yescharacteristic
443C4—22no—
444C4—41no—
445C4—182no—
446C4—182no—
447C4—361no—
448C3—34no—
449C3#1059265511yescharacteristic
450C2—42no—
451C2—184no—
452C2C22⋊C4×#27211yescharacteristic
453C2#1172173711yescharacteristic
454C2#1172173911yescharacteristic
455C2#1172174011yescharacteristic
456C2C4×#623511yescharacteristic
457C2#1172184711yescharacteristic
458C2#1172184811yescharacteristic
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)#386
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. C22×C4×#272C2semidirect
3. C2×C22⋊C4×H(3,𝔽3)C2semidirect
4. C2×#88552C2semidirect
5. C2×#88542C2semidirect
6. #11720574C2semidirect
7. #11720574C2semidirect
15. C2×#6235C4semidirect
27. C2×C32×C22⋊C4D3semidirect
38. C23×H(3,𝔽3)C2×C4semidirect
52. C2×C4×H(3,𝔽3)D4semidirect
61. C2×#1084C2×C4semidirect
62. C2×#1084D4semidirect
102. C2×#272C22⋊C4semidirect
128. C23×C32C4×D3semidirect
142. C2×C32×C4#67semidirect
226. C2×C22⋊C4#272semidirect
229. H(3,𝔽3)#386semidirect
371. C32#4956semidirect
385. C23C4×#272semidirect
403. 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
C22×C4×#2722
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×#885422
#117205742
C3×#49563