Finite Group Databasebeta

← #13144300

Subgroups — #13144300

Order 1728.

1045 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88552C211yesmaximal, characteristic
3C2×#88552C211yesmaximal, characteristic
4C2×#88540C211yesmaximal, characteristic, complemented
5C2×C4×#1084C211yesmaximal, characteristic, complemented
6#11720574C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4945—34nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22⋊C4×H(3,𝔽3)—22no—
17#88552—22no—
18#88552C414yescomplemented
19#88540—22no—
20C4×#1084—22no—
21C2×C32×C22⋊C4D314yes—
22C2×C3×#217—34no—
23C2×C3×#217—34no—
24C2×C3×Dic3⋊C4—34no—
25C2×C3×C4×Dic3—34no—
26C3×C3⋊#95—34no—
27C3×C3⋊#95—34no—
28C23×H(3,𝔽3)—21no—
29C23×H(3,𝔽3)—21no—
30C23×H(3,𝔽3)—21no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)—21no—
33C23×H(3,𝔽3)—21no—
34C23×H(3,𝔽3)C2×C412yes—
35C23×H(3,𝔽3)C2311yescharacteristic
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)D412yes—
42C2×#1084—21no—
43C2×#1084—21no—
44C2×#1084—21no—
45C2×#1084—21no—
46C2×#1084—21no—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084C2×C412yes—
52C2×#1084C2×C412yes—
53C2×#1084D412yes—
54C3×#388—91no—
55#4945—34no—
56C24×C32C2×D314yes—
57C22×C32×C4C2×D314yes—
58C22×C32×C4C2×D314yes—
59C22×C3×Dic3—34no—
60C22×C3×Dic3—34no—
61C22×C3×Dic3—34no—
62C22×C3×Dic3—34no—
63C32×C22⋊C4—28no—
64C3×#217—316no—
65C3×#217—68no—
66C3×Dic3⋊C4—68no—
67C3×C4×Dic3—68no—
68C22×H(3,𝔽3)—22no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)—22no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)—41no—
74C22×H(3,𝔽3)—41no—
75C22×H(3,𝔽3)C22×C411yescharacteristic
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4111yescharacteristic
80C22×H(3,𝔽3)#4111yescharacteristic
81C22×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)—41no—
84#1084—22no—
85#1084—22no—
86#1084—22no—
87#1084—41no—
88C2×C3×C4⋊C4—91no—
89C2×C3×C22⋊C4—34no—
90C2×C3×C22⋊C4—91no—
91C2×C3×C22⋊C4—91no—
92C2×C3×C22⋊C4C3⋊D311yescharacteristic
93C2×C3×C42—91no—
94C2×#217—34no—
95C2×#217—34no—
96C2×Dic3⋊C4—34no—
97C2×C4×Dic3—34no—
98C3×#95—91no—
99C3×#95—91no—
100C3⋊#95—34no—
101C3⋊#95—34no—
102C23×C32—24no—
103C23×C32—24no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32—24no—
107C23×C32—24no—
108C23×C32C4×D318yes—
109C23×C32C22×D314yes—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4#6718yes—
116C2×C3×Dic3—38no—
117C2×C3×Dic3—38no—
118C2×C3×Dic3—38no—
119C2×C3×Dic3—64no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128#388—91noSylow
129C2×H(3,𝔽3)—42no—
130C2×H(3,𝔽3)#11711yescharacteristic
131C2×H(3,𝔽3)C4×D411yescharacteristic
132C2×H(3,𝔽3)C4×D411yescharacteristic
133C2×H(3,𝔽3)#12111yescharacteristic
134C2×H(3,𝔽3)#12311yescharacteristic
135C2×H(3,𝔽3)#12411yescharacteristic
136C2×H(3,𝔽3)#12611yescharacteristic
137C24×C3—34no—
138C24×C3C2×C3⋊D311yescharacteristic
139C22×C3×C4—34no—
140C22×C3×C4—34no—
141C22×C3×C4—91no—
142C22×C3×C4—91no—
143C22×C3×C4—91no—
144C22×C3×C4—91no—
145C22×C3×C4C2×C3⋊D311yescharacteristic
146C22×C3×C4C2×C3⋊D311yescharacteristic
147C22×Dic3—34no—
148C22×Dic3—34no—
149C22×Dic3—34no—
150C22×Dic3—34no—
151C3×C4⋊C4—182no—
152C3×C22⋊C4—22no—
153C3×C22⋊C4—68no—
154C3×C22⋊C4—94no—
155C3×C22⋊C4—182no—
156C3×C42—182no—
157#217—316no—
158#217—68no—
159Dic3⋊C4—68no—
160C4×Dic3—68no—
161C22×C32—28no—
162C22×C32—28no—
163C22×C32—28no—
164C22×C32—44no—
165C22×C32—44no—
166C22×C32—44no—
167C22×C32—44no—
168C22×C32C2×C4×D314yes—
169C22×C32#23514yes—
170C22×C32D3×D414yes—
171C22×C32#23714yes—
172C22×C32#23714yes—
173C22×C32#23714yes—
174C22×C32C2×#6714yes—
175C32×C4—28no—
176C32×C4—44no—
177C3×Dic3—68no—
178C3×Dic3—68no—
179C3×Dic3—68no—
180C3×Dic3—124no—
181C2×C4⋊C4—91no—
182C2×C22⋊C4—91no—
183C2×C22⋊C4—91no—
184C2×C22⋊C4#27211yescharacteristic
185C2×C42—91no—
186#95—91no—
187#95—91no—
188H(3,𝔽3)#38811yescharacteristic, complemented, Sylow
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—38no—
196C23×C3—34no—
197C23×C3—64no—
198C23×C3—64no—
199C23×C3—64no—
200C23×C3—64no—
201C23×C3—64no—
202C23×C3—64no—
203C23×C3C4×C3⋊D312yes—
204C23×C3C22×C3⋊D311yescharacteristic
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—38no—
211C2×C3×C4—64no—
212C2×C3×C4—64no—
213C2×C3×C4—64no—
214C2×C3×C4—64no—
215C2×C3×C4—64no—
216C2×C3×C4—92no—
217C2×C3×C4—92no—
218C2×C3×C4—92no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4#63812yes—
229C2×Dic3—38no—
230C2×Dic3—38no—
231C2×Dic3—38no—
232C2×Dic3—64no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—64no—
236C2×Dic3—64no—
237C2×Dic3—64no—
238C2×Dic3—64no—
239C2×Dic3—64no—
240C2×Dic3—64no—
241C2×C32—48no—
242C2×C32#87714yes—
243C2×C32#87914yes—
244C2×C32#67⋊C414yes—
245C2×C32#88314yes—
246C2×C32C4×#6714yes—
247C2×C32#93614yes—
248C2×C32#93914yes—
249C24C2×#27211yescharacteristic
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4—91no—
254C22×C4C2×#27211yescharacteristic
255C22×C4C2×#27211yescharacteristic
256C4⋊C4—182no—
257C22⋊C4—22no—
258C22⋊C4—94no—
259C22⋊C4—182no—
260C42—182no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—34no—
265C22×C3—34no—
266C22×C3—34no—
267C22×C3—34no—
268C22×C3—34no—
269C22×C3—34no—
270C22×C3—34no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—68no—
276C22×C3—68no—
277C22×C3—68no—
278C22×C3—124no—
279C22×C3—124no—
280C22×C3—124no—
281C22×C3—124no—
282C22×C3C2×C4×C3⋊D311yescharacteristic
283C22×C3#382811yescharacteristic
284C22×C3D4×C3⋊D311yescharacteristic
285C22×C3#383011yescharacteristic
286C22×C3#383011yescharacteristic
287C22×C3#383011yescharacteristic
288C22×C3C2×#63811yescharacteristic
289C3×C4—22no—
290C3×C4—41no—
291C3×C4—68no—
292C3×C4—124no—
293C3×C4—182no—
294C3×C4—182no—
295C3×C4—182no—
296C3×C4—361no—
297Dic3—68no—
298Dic3—68no—
299Dic3—68no—
300Dic3—124no—
301C32#494514yescomplemented
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C4×#27212yes—
309C23C22×#27211yescharacteristic
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—92no—
316C2×C4—92no—
317C2×C4—92no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4#623512yes—
328C2×C3—34no—
329C2×C3—34no—
330C2×C3—34no—
331C2×C3—34no—
332C2×C3—34no—
333C2×C3—34no—
334C2×C3—34no—
335C2×C3—42no—
336C2×C3—128no—
337C2×C3#6408611yescharacteristic
338C2×C3#6408811yescharacteristic
339C2×C3#6409011yescharacteristic
340C2×C3#6409211yescharacteristic
341C2×C3C4×#63811yescharacteristic
342C2×C3#6414511yescharacteristic
343C2×C3#6414811yescharacteristic
344C22—22no—
345C22—22no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22C2×C4×#27211yescharacteristic
352C22#8874011yescharacteristic
353C22D4×#27211yescharacteristic
354C22#8874411yescharacteristic
355C22#8874411yescharacteristic
356C22#8874411yescharacteristic
357C22C2×#623511yescharacteristic
358C4—22no—
359C4—41no—
360C4—182no—
361C4—182no—
362C4—182no—
363C4—361no—
364C3—34no—
365C3#1059264411yescharacteristic
366C2—42no—
367C2#1172173311yescharacteristic
368C2#1172173511yescharacteristic
369C2#1172173711yescharacteristic
370C2#1172173911yescharacteristic
371C2C4×#623511yescharacteristic
372C2#1172184611yescharacteristic
373C2#1172184911yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
cosocleC23×H(3,𝔽3)C23
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#388
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
4. C2×#88540C2semidirect
5. C2×C4×#1084C2semidirect
6. #11720574C2semidirect
7. #11720574C2semidirect
18. #88552C4semidirect
188. H(3,𝔽3)#388semidirect
301. C32#4945semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×#885402
C2×C4×#10842
#117205742
C3×#49453