Finite Group Databasebeta

← #13143731

Subgroups — #13143731

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×C4⋊#272C211yesmaximal, characteristic
2C22×C4×#272C211yesmaximal, characteristic, complemented
3C2×#88542C211yesmaximal, characteristic, complemented
4C2×#88542C211yesmaximal, characteristic, complemented
5C2×#88542C211yesmaximal, characteristic, complemented
6H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4665—34nomaximal
9C23×#272C2211yescharacteristic
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C2×C4⋊#272—22no—
17C2×C4⋊#272—22no—
18C2×C4⋊#272C414yescomplemented
19C2×C4×#272—22no—
20C2×C4×#272—22no—
21C2×C4×#272—22no—
22#88542—22no—
23#88542—22no—
24#88542—22no—
25C22×C3×D12—34no—
26C22×C3×C4×D3—34no—
27C2×C3×(C2×D3)⋊C4—34no—
28C2×C3×(C2×D3)⋊C4—34no—
29C2×C3×(C2×D3)⋊C4—34no—
30C32×#95D314yescomplemented
31C3×C3⋊#95—34no—
32C23×H(3,𝔽3)C2311yescharacteristic
33C22×#272—21no—
34C22×#272—21no—
35C22×#272—21no—
36C22×#272—21no—
37C22×#272—21no—
38C22×#272—21no—
39C22×#272—21no—
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272C2×C412yes—
44C22×#272C2×C412yescomplemented
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)—21no—
53C2×C4×H(3,𝔽3)—21no—
54C2×C4×H(3,𝔽3)C2×C412yescomplemented
55C2×C4×H(3,𝔽3)D412yescomplemented
56C2×#1084—21no—
57C2×#1084—21no—
58C2×#1084—21no—
59C2×#1084—21no—
60C2×#1084—21no—
61C2×#1084D412yescomplemented
62C4⋊#272—28no—
63C4×#272—28no—
64C3×#386—91no—
65#4665—34no—
66C23×C3×D3—34no—
67C23×C3×D3—34no—
68C22×C32×C4C2×D314yes—
69C22×C32×C4C2×D314yes—
70C22×C32×C4C2×D314yes—
71C22×C3×Dic3—34no—
72C22×C3×Dic3—34no—
73C2×C3×D12—316no—
74C2×C3×D12—68no—
75C2×C3×D12—68no—
76C2×C3×C4×D3—68no—
77C2×C3×C4×D3—68no—
78C2×C3×C4×D3—68no—
79C3×(C2×D3)⋊C4—68no—
80C3×(C2×D3)⋊C4—68no—
81C3×(C2×D3)⋊C4—68no—
82C22×H(3,𝔽3)C22×C411yescharacteristic
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C22×H(3,𝔽3)#4111yescharacteristic
88C22×H(3,𝔽3)#4111yescharacteristic
89C2×#272—22no—
90C2×#272—22no—
91C2×#272—22no—
92C2×#272—22no—
93C2×#272—22no—
94C2×#272—22no—
95C2×#272—22no—
96C2×#272—22no—
97C2×#272—22no—
98C2×#272—41no—
99C2×#272—41no—
100C2×#272—41no—
101C2×#272—41no—
102C2×#272C22⋊C414yescomplemented
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)—22no—
105C4×H(3,𝔽3)—41no—
106#1084—22no—
107#1084—41no—
108C22×C3×D4—91no—
109C23×C3×C4—91no—
110C22×D12—34no—
111C22×C4×D3—34no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×(C2×D3)⋊C4—34no—
116C2×(C2×D3)⋊C4—34no—
117C2×(C2×D3)⋊C4—34no—
118C3×#95—34no—
119C3×#95—91no—
120C3×#95C3⋊D311yescharacteristic
121C3⋊#95—34no—
122C23×C32C22×D314yes—
123C22×C3×D3—38no—
124C22×C3×D3—38no—
125C22×C3×D3—38no—
126C22×C3×D3—38no—
127C22×C3×D3—64no—
128C22×C3×D3—64no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—64no—
132C22×C3×D3—64no—
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—24no—
143C2×C32×C4—24no—
144C2×C32×C4C4×D318yescomplemented
145C2×C32×C4D1218yescomplemented
146C2×C3×Dic3—38no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—64no—
152C3×D12—632no—
153C3×C4×D3—632no—
154#386—91noSylow
155C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
156C2×H(3,𝔽3)C4×D411yescharacteristic
157C2×H(3,𝔽3)C4×D411yescharacteristic
158C2×H(3,𝔽3)#12011yescharacteristic
159C2×H(3,𝔽3)#12111yescharacteristic
160C2×H(3,𝔽3)#12111yescharacteristic
161C2×H(3,𝔽3)#12311yescharacteristic
162#272—24no—
163#272—42no—
164C24×C3—91no—
165C24×C3—91no—
166C23×D3—34no—
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—34no—
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
181C22×C3×C4C2×C3⋊D311yescharacteristic
182C22×Dic3—34no—
183C22×Dic3—34no—
184C2×D12—316no—
185C2×D12—68no—
186C2×D12—68no—
187C2×C4×D3—68no—
188C2×C4×D3—68no—
189C2×C4×D3—68no—
190C3×C22⋊C4—182no—
191C3×C22⋊C4—182no—
192C3×C22⋊C4—182no—
193(C2×D3)⋊C4—68no—
194(C2×D3)⋊C4—68no—
195(C2×D3)⋊C4—68no—
196C22×C32C2×C4×D314yes—
197C22×C32C2×D1214yes—
198C22×C32#23514yes—
199C22×C32D3×D414yes—
200C22×C32D3×D414yes—
201C22×C32D3×D414yes—
202C22×C32#23914yes—
203C2×C3×D3—316no—
204C2×C3×D3—68no—
205C2×C3×D3—68no—
206C2×C3×D3—68no—
207C2×C3×D3—68no—
208C2×C3×D3—68no—
209C2×C3×D3—68no—
210C2×C3×D3—68no—
211C2×C3×D3—68no—
212C2×C3×D3—68no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C2×C3×D3—124no—
216C2×C3×D3—124no—
217C32×C4—28no—
218C32×C4—28no—
219C32×C4—44no—
220C3×Dic3—68no—
221C3×Dic3—124no—
222C22×D4—91no—
223C23×C4—91no—
224C2×C22⋊C4—91no—
225C2×C22⋊C4—91no—
226C2×C22⋊C4—91no—
227#95—91no—
228#95#27211yescharacteristic, complemented
229H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
230C23×C3—34no—
231C23×C3—92no—
232C23×C3—92no—
233C23×C3—92no—
234C23×C3—92no—
235C23×C3—181no—
236C23×C3—181no—
237C23×C3—181no—
238C23×C3—181no—
239C23×C3—181no—
240C23×C3—181no—
241C23×C3—181no—
242C23×C3—181no—
243C23×C3—181no—
244C23×C3—181no—
245C23×C3C22×C3⋊D311yescharacteristic
246C22×D3—38no—
247C22×D3—38no—
248C22×D3—38no—
249C22×D3—38no—
250C22×D3—64no—
251C22×D3—64no—
252C22×D3—64no—
253C22×D3—64no—
254C22×D3—64no—
255C22×D3—64no—
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—21no—
267C2×C3×C4—21no—
268C2×C3×C4—38no—
269C2×C3×C4—38no—
270C2×C3×C4—64no—
271C2×C3×C4—64no—
272C2×C3×C4—64no—
273C2×C3×C4—64no—
274C2×C3×C4—64no—
275C2×C3×C4—64no—
276C2×C3×C4—64no—
277C2×C3×C4—92no—
278C2×C3×C4—188no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4C4×C3⋊D312yes—
285C2×C3×C4C4⋊(C3⋊D3)12yes—
286C2×Dic3—38no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3—64no—
290C2×Dic3—64no—
291C2×Dic3—64no—
292D12—632no—
293C4×D3—632no—
294C2×C32C4×D1214yes—
295C2×C32D3×C22⋊C414yes—
296C2×C32#88214yes—
297C2×C32#88414yes—
298C2×C32#89314yes—
299C2×C32#89414yes—
300C2×C32#89514yes—
301C3×D3—616no—
302C3×D3—128no—
303C24—91no—
304C24—91no—
305C2×D4—94no—
306C2×D4—182no—
307C2×D4—182no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—182no—
311C22×C4—182no—
312C22×C4—182no—
313C22×C4C2×#27211yescharacteristic
314C22×C4C2×#27211yescharacteristic
315C22×C4C2×#27211yescharacteristic
316C22⋊C4—182no—
317C22⋊C4—182no—
318C22⋊C4—182no—
319C22×C3—34no—
320C22×C3—34no—
321C22×C3—34no—
322C22×C3—34no—
323C22×C3—34no—
324C22×C3—34no—
325C22×C3—34no—
326C22×C3—94no—
327C22×C3—182no—
328C22×C3—182no—
329C22×C3—182no—
330C22×C3—182no—
331C22×C3—182no—
332C22×C3—182no—
333C22×C3—182no—
334C22×C3—182no—
335C22×C3—182no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3C2×C4×C3⋊D311yescharacteristic
341C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
342C22×C3#382811yescharacteristic
343C22×C3D4×C3⋊D311yescharacteristic
344C22×C3D4×C3⋊D311yescharacteristic
345C22×C3D4×C3⋊D311yescharacteristic
346C22×C3#383211yescharacteristic
347C2×D3—316no—
348C2×D3—68no—
349C2×D3—68no—
350C2×D3—68no—
351C2×D3—68no—
352C2×D3—68no—
353C2×D3—68no—
354C2×D3—68no—
355C2×D3—68no—
356C2×D3—68no—
357C2×D3—124no—
358C2×D3—124no—
359C2×D3—124no—
360C2×D3—124no—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—41no—
364C3×C4—68no—
365C3×C4—68no—
366C3×C4—124no—
367C3×C4—182no—
368C3×C4—361no—
369Dic3—68no—
370Dic3—124no—
371C32#466514yescomplemented
372C23—92no—
373C23—92no—
374C23—92no—
375C23—92no—
376C23—181no—
377C23—181no—
378C23—181no—
379C23—181no—
380C23—181no—
381C23—181no—
382C23—181no—
383C23—181no—
384C23—181no—
385C23—181no—
386C23C22×#27211yescharacteristic
387D4—188no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—92no—
396C2×C4—188no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4—181no—
402C2×C4C4×#27212yescomplemented
403C2×C4C4⋊#27212yescomplemented
404C2×C3—34no—
405C2×C3—34no—
406C2×C3—34no—
407C2×C3—34no—
408C2×C3—34no—
409C2×C3—34no—
410C2×C3—34no—
411C2×C3—184no—
412C2×C3—362no—
413C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
414C2×C3C22⋊C4×C3⋊D311yescharacteristic
415C2×C3#6409111yescharacteristic
416C2×C3#6409311yescharacteristic
417C2×C3#6410211yescharacteristic
418C2×C3#6410311yescharacteristic
419C2×C3#6410411yescharacteristic
420D3—616no—
421D3—128no—
422C22—94no—
423C22—182no—
424C22—182no—
425C22—182no—
426C22—182no—
427C22—182no—
428C22—182no—
429C22—182no—
430C22—182no—
431C22—182no—
432C22—361no—
433C22—361no—
434C22—361no—
435C22—361no—
436C22C2×C4×#27211yescharacteristic
437C22C2×C4⋊#27211yescharacteristic
438C22#8874011yescharacteristic
439C22D4×#27211yescharacteristic
440C22D4×#27211yescharacteristic
441C22D4×#27211yescharacteristic
442C22#8874811yescharacteristic
443C4—22no—
444C4—22no—
445C4—41no—
446C4—182no—
447C4—361no—
448C3—34no—
449C3#1059236411yescharacteristic
450C2—184no—
451C2—362no—
452C2C4×C4⋊#27211yescharacteristic
453C2C22⋊C4×#27211yescharacteristic
454C2#1172173811yescharacteristic
455C2#1172174011yescharacteristic
456C2#1172176111yescharacteristic
457C2#1172176211yescharacteristic
458C2#1172176311yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#95C2
prime radicalC22×C4⋊#272C2
cosocleC23×H(3,𝔽3)C23
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#386
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
2. C22×C4×#272C2semidirect
3. C2×#88542C2semidirect
4. C2×#88542C2semidirect
5. C2×#88542C2semidirect
6. H(3,𝔽3)×#95C2semidirect
7. #11720574C2semidirect
18. C2×C4⋊#272C4semidirect
30. C32×#95D3semidirect
44. C22×#272C2×C4semidirect
54. C2×C4×H(3,𝔽3)C2×C4semidirect
55. C2×C4×H(3,𝔽3)D4semidirect
61. C2×#1084D4semidirect
102. C2×#272C22⋊C4semidirect
144. C2×C32×C4C4×D3semidirect
145. C2×C32×C4D12semidirect
228. #95#272semidirect
229. H(3,𝔽3)#386semidirect
371. C32#4665semidirect
402. C2×C4C4×#272semidirect
403. C2×C4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C22×C4×#2722
C2×#885422
H(3,𝔽3)×#952
#117205742
C3×#46653