Finite Group Databasebeta

← #13143732

Subgroups — #13143732

Order 1728.

1287 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88542C211yesmaximal, characteristic
3C2×#88542C211yesmaximal, characteristic
4C2×C4⋊#1084C211yesmaximal, characteristic, complemented
5C2×#88540C211yesmaximal, characteristic, complemented
6H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4668—34nomaximal
9C23×#272C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C2×C4×#272—22no—
17C2×C4×#272—22no—
18C2×C4×#272—22no—
19#88542—22no—
20#88542C414yescomplemented
21C4⋊#1084—22no—
22#88540—22no—
23C22×C3×C4×D3—34no—
24C2×C3×(C2×D3)⋊C4—34no—
25C2×C3×(C2×D3)⋊C4—34no—
26C2×C3×C4⋊Dic3—34no—
27C2×C3×Dic3⋊C4—34no—
28C32×#95D314yescomplemented
29C3×C3⋊#95—34no—
30C23×H(3,𝔽3)C2311yescharacteristic
31C22×#272—21no—
32C22×#272—21no—
33C22×#272—21no—
34C22×#272—21no—
35C22×#272C2×C412yes—
36C22×#272D412yes—
37C22×#272Q812yes—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2×C412yescomplemented
46C2×C4×H(3,𝔽3)D412yes—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084—21no—
53C2×#1084—21no—
54C2×#1084C2×C412yescomplemented
55C2×#1084D412yes—
56C4×#272—28no—
57C3×#385—91no—
58#4668—34no—
59C23×C3×D3—34no—
60C22×C32×C4C2×D314yes—
61C22×C32×C4C2×D314yes—
62C22×C32×C4C2×D314yes—
63C22×C3×Dic3—34no—
64C22×C3×Dic3—34no—
65C22×C3×Dic3—34no—
66C2×C3×C4×D3—68no—
67C2×C3×C4×D3—68no—
68C2×C3×C4×D3—68no—
69C3×(C2×D3)⋊C4—316no—
70C3×(C2×D3)⋊C4—68no—
71C3×C4⋊Dic3—68no—
72C3×Dic3⋊C4—68no—
73C22×H(3,𝔽3)C22×C411yescharacteristic
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C22×H(3,𝔽3)C2×Q811yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4111yescharacteristic
80C2×#272—22no—
81C2×#272—22no—
82C2×#272—22no—
83C2×#272—22no—
84C2×#272—22no—
85C2×#272—22no—
86C2×#272C4⋊C414yescomplemented
87C4×H(3,𝔽3)—22no—
88C4×H(3,𝔽3)—22no—
89C4×H(3,𝔽3)—41no—
90#1084—22no—
91#1084—22no—
92#1084—41no—
93C23×C3×C4—91no—
94C22×C4×D3—34no—
95C2×C3×C4⋊C4—91no—
96C2×C3×C4⋊C4—91no—
97C2×C3×C22⋊C4—91no—
98C2×C3×C22⋊C4—91no—
99C2×(C2×D3)⋊C4—34no—
100C2×(C2×D3)⋊C4—34no—
101C2×C4⋊Dic3—34no—
102C2×Dic3⋊C4—34no—
103C3×#95—34no—
104C3×#95—91no—
105C3×#95C3⋊D311yescharacteristic
106C3⋊#95—34no—
107C23×C32C22×D314yes—
108C22×C3×D3—38no—
109C22×C3×D3—38no—
110C22×C3×D3—38no—
111C22×C3×D3—64no—
112C22×C3×D3—64no—
113C22×C3×D3—64no—
114C22×C3×D3—64no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4C4×D318yescomplemented
123C2×C32×C4D1218yes—
124C2×C3×Dic3—38no—
125C2×C3×Dic3—38no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—64no—
133C3×C4×D3—632no—
134#385—91noSylow
135C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
136C2×H(3,𝔽3)C4×D411yescharacteristic
137C2×H(3,𝔽3)C4×D411yescharacteristic
138C2×H(3,𝔽3)#12011yescharacteristic
139C2×H(3,𝔽3)C22⋊Q811yescharacteristic
140C2×H(3,𝔽3)C22⋊Q811yescharacteristic
141C2×H(3,𝔽3)#12311yescharacteristic
142#272—24no—
143C24×C3—91no—
144C23×D3—34no—
145C22×C3×C4—34no—
146C22×C3×C4—34no—
147C22×C3×C4—34no—
148C22×C3×C4—91no—
149C22×C3×C4—91no—
150C22×C3×C4—91no—
151C22×C3×C4—182no—
152C22×C3×C4—182no—
153C22×C3×C4—182no—
154C22×C3×C4C2×C3⋊D311yescharacteristic
155C22×C3×C4C2×C3⋊D311yescharacteristic
156C22×C3×C4C2×C3⋊D311yescharacteristic
157C22×Dic3—34no—
158C22×Dic3—34no—
159C22×Dic3—34no—
160C2×C4×D3—68no—
161C2×C4×D3—68no—
162C2×C4×D3—68no—
163C3×C4⋊C4—182no—
164C3×C4⋊C4—182no—
165C3×C22⋊C4—94no—
166C3×C22⋊C4—182no—
167(C2×D3)⋊C4—316no—
168(C2×D3)⋊C4—68no—
169C4⋊Dic3—68no—
170Dic3⋊C4—68no—
171C22×C32C2×C4×D314yes—
172C22×C32C2×D1214yes—
173C22×C32#23514yes—
174C22×C32D3×D414yes—
175C22×C32D3×D414yes—
176C22×C32#23714yes—
177C22×C32D3×Q814yes—
178C2×C3×D3—316no—
179C2×C3×D3—68no—
180C2×C3×D3—68no—
181C2×C3×D3—68no—
182C2×C3×D3—68no—
183C2×C3×D3—68no—
184C2×C3×D3—68no—
185C32×C4—28no—
186C32×C4—28no—
187C32×C4—44no—
188C3×Dic3—68no—
189C3×Dic3—68no—
190C3×Dic3—124no—
191C23×C4—91no—
192C2×C4⋊C4—91no—
193C2×C4⋊C4—91no—
194C2×C22⋊C4—91no—
195C2×C22⋊C4—91no—
196#95—91no—
197#95#27211yescharacteristic, complemented
198H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
199C23×C3—34no—
200C23×C3—92no—
201C23×C3—92no—
202C23×C3—92no—
203C23×C3—181no—
204C23×C3—181no—
205C23×C3—181no—
206C23×C3—181no—
207C23×C3C22×C3⋊D311yescharacteristic
208C22×D3—38no—
209C22×D3—38no—
210C22×D3—38no—
211C22×D3—64no—
212C22×D3—64no—
213C22×D3—64no—
214C22×D3—64no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—38no—
223C2×C3×C4—38no—
224C2×C3×C4—64no—
225C2×C3×C4—64no—
226C2×C3×C4—64no—
227C2×C3×C4—64no—
228C2×C3×C4—64no—
229C2×C3×C4—64no—
230C2×C3×C4—64no—
231C2×C3×C4—92no—
232C2×C3×C4—92no—
233C2×C3×C4—188no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4C4×C3⋊D312yes—
242C2×C3×C4C4⋊(C3⋊D3)12yes—
243C2×Dic3—38no—
244C2×Dic3—38no—
245C2×Dic3—64no—
246C2×Dic3—64no—
247C2×Dic3—64no—
248C2×Dic3—64no—
249C2×Dic3—64no—
250C2×Dic3—64no—
251C2×Dic3—64no—
252C4×D3—632no—
253C2×C32C4×D1214yes—
254C2×C32#67⋊C414yes—
255C2×C32#88214yes—
256C2×C32#88314yes—
257C2×C32D3×C4⋊C414yes—
258C2×C32#89614yes—
259C2×C32#89714yes—
260C3×D3—616no—
261C24—91no—
262C22×C4—91no—
263C22×C4—91no—
264C22×C4—91no—
265C22×C4—182no—
266C22×C4—182no—
267C22×C4—182no—
268C22×C4C2×#27211yescharacteristic
269C22×C4C2×#27211yescharacteristic
270C22×C4C2×#27211yescharacteristic
271C4⋊C4—182no—
272C4⋊C4—182no—
273C22⋊C4—94no—
274C22⋊C4—182no—
275C22×C3—34no—
276C22×C3—34no—
277C22×C3—34no—
278C22×C3—34no—
279C22×C3—34no—
280C22×C3—34no—
281C22×C3—34no—
282C22×C3—94no—
283C22×C3—182no—
284C22×C3—182no—
285C22×C3—182no—
286C22×C3—182no—
287C22×C3—182no—
288C22×C3—182no—
289C22×C3C2×C4×C3⋊D311yescharacteristic
290C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
291C22×C3#382811yescharacteristic
292C22×C3D4×C3⋊D311yescharacteristic
293C22×C3D4×C3⋊D311yescharacteristic
294C22×C3#383011yescharacteristic
295C22×C3Q8×C3⋊D311yescharacteristic
296C2×D3—316no—
297C2×D3—68no—
298C2×D3—68no—
299C2×D3—68no—
300C2×D3—68no—
301C2×D3—68no—
302C2×D3—68no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—41no—
306C3×C4—68no—
307C3×C4—68no—
308C3×C4—124no—
309C3×C4—182no—
310C3×C4—182no—
311C3×C4—361no—
312Dic3—68no—
313Dic3—68no—
314Dic3—124no—
315C32#466814yescomplemented
316C23—92no—
317C23—92no—
318C23—92no—
319C23—181no—
320C23—181no—
321C23—181no—
322C23—181no—
323C23C22×#27211yescharacteristic
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—92no—
332C2×C4—92no—
333C2×C4—188no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4C4×#27212yescomplemented
342C2×C4C4⋊#27212yes—
343C2×C3—34no—
344C2×C3—34no—
345C2×C3—34no—
346C2×C3—34no—
347C2×C3—34no—
348C2×C3—34no—
349C2×C3—34no—
350C2×C3—184no—
351C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
352C2×C3#6409011yescharacteristic
353C2×C3#6409111yescharacteristic
354C2×C3#6409211yescharacteristic
355C2×C3C4⋊C4×C3⋊D311yescharacteristic
356C2×C3#6410511yescharacteristic
357C2×C3#6410611yescharacteristic
358D3—616no—
359C22—94no—
360C22—182no—
361C22—182no—
362C22—182no—
363C22—182no—
364C22—182no—
365C22—182no—
366C22C2×C4×#27211yescharacteristic
367C22C2×C4⋊#27211yescharacteristic
368C22#8874011yescharacteristic
369C22D4×#27211yescharacteristic
370C22D4×#27211yescharacteristic
371C22#8874411yescharacteristic
372C22Q8×#27211yescharacteristic
373C4—22no—
374C4—22no—
375C4—41no—
376C4—182no—
377C4—182no—
378C4—361no—
379C3—34no—
380C3#1059236711yescharacteristic
381C2—184no—
382C2C4×C4⋊#27211yescharacteristic
383C2#1172173711yescharacteristic
384C2#1172173811yescharacteristic
385C2#1172173911yescharacteristic
386C2C4⋊C4×#27211yescharacteristic
387C2#1172176411yescharacteristic
388C2#1172176511yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#95C2
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#385
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
4. C2×C4⋊#1084C2semidirect
5. C2×#88540C2semidirect
6. H(3,𝔽3)×#95C2semidirect
7. #11720574C2semidirect
20. #88542C4semidirect
28. C32×#95D3semidirect
45. C2×C4×H(3,𝔽3)C2×C4semidirect
54. C2×#1084C2×C4semidirect
86. C2×#272C4⋊C4semidirect
122. C2×C32×C4C4×D3semidirect
197. #95#272semidirect
198. H(3,𝔽3)#385semidirect
315. C32#4668semidirect
341. 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
C2×#885422
C2×C4⋊#10842
C2×#885402
H(3,𝔽3)×#952
#117205742
C3×#46683