Finite Group Databasebeta

← #13144371

Subgroups — #13144371

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×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88542C211yesmaximal, characteristic
3C2×#88542C211yesmaximal, characteristic
4C2×#88542C211yesmaximal, characteristic
5C2×C4×#1084C211yesmaximal, characteristic, complemented
6#11720574C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4990—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
16C4⋊C4×H(3,𝔽3)—22no—
17#88542—22no—
18#88542—22no—
19#88542C414yescomplemented
20C4×#1084—22no—
21C2×C32×C4⋊C4D314yescomplemented
22C2×C3×(C2×D3)⋊C4—34no—
23C2×C3×(C2×D3)⋊C4—34no—
24C2×C3×(C2×D3)⋊C4—34no—
25C2×C3×C4×Dic3—34no—
26C3×C3⋊#95—34no—
27C3×C3⋊#95—34no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×#272—21no—
30C22×#272—21no—
31C22×#272—21no—
32C22×#272—21no—
33C22×#272—21no—
34C22×#272—21no—
35C22×#272C2×C412yes—
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)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2×C412yes—
44C2×C4×H(3,𝔽3)D412yes—
45C2×#1084—21no—
46C2×#1084—21no—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084C2×C412yes—
53C2×#1084D412yes—
54C3×#388—91no—
55#4990—34no—
56C23×C3×D3—34no—
57C22×C32×C4C2×D314yes—
58C22×C32×C4C2×D314yes—
59C22×C32×C4C2×D314yes—
60C22×C3×Dic3—34no—
61C22×C3×Dic3—34no—
62C22×C3×Dic3—34no—
63C32×C4⋊C4—28no—
64C3×(C2×D3)⋊C4—316no—
65C3×(C2×D3)⋊C4—68no—
66C3×(C2×D3)⋊C4—68no—
67C3×C4×Dic3—68no—
68C22×H(3,𝔽3)C22×C411yescharacteristic
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C22×H(3,𝔽3)#4111yescharacteristic
72C22×H(3,𝔽3)#4111yescharacteristic
73C22×H(3,𝔽3)#4111yescharacteristic
74C22×H(3,𝔽3)#4111yescharacteristic
75C2×#272—22no—
76C2×#272—22no—
77C2×#272—22no—
78C2×#272—41no—
79C2×#272—41no—
80C2×#272—41no—
81C2×#272—41no—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)—22no—
84C4×H(3,𝔽3)—41no—
85#1084—22no—
86#1084—22no—
87#1084—41no—
88C2×C3×C4⋊C4—34no—
89C2×C3×C4⋊C4C3⋊D311yescharacteristic
90C2×C3×C22⋊C4—91no—
91C2×C3×C22⋊C4—91no—
92C2×C3×C22⋊C4—91no—
93C2×C3×C42—91no—
94C2×(C2×D3)⋊C4—34no—
95C2×(C2×D3)⋊C4—34no—
96C2×(C2×D3)⋊C4—34no—
97C2×C4×Dic3—34no—
98C3×#95—91no—
99C3×#95—91no—
100C3⋊#95—34no—
101C3⋊#95—34no—
102C23×C32C22×D314yes—
103C22×C3×D3—38no—
104C22×C3×D3—64no—
105C22×C3×D3—64no—
106C22×C3×D3—64no—
107C22×C3×D3—64no—
108C22×C3×D3—64no—
109C22×C3×D3—64no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4C4×D318yes—
118C2×C32×C4#6718yes—
119C2×C3×Dic3—38no—
120C2×C3×Dic3—38no—
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)#11711yescharacteristic
130C2×H(3,𝔽3)C4×D411yescharacteristic
131C2×H(3,𝔽3)C4×D411yescharacteristic
132C2×H(3,𝔽3)#12111yescharacteristic
133C2×H(3,𝔽3)#12311yescharacteristic
134C2×H(3,𝔽3)#12411yescharacteristic
135C2×H(3,𝔽3)#12611yescharacteristic
136#272—42no—
137C24×C3—91no—
138C23×D3—34no—
139C22×C3×C4—34no—
140C22×C3×C4—34no—
141C22×C3×C4—34no—
142C22×C3×C4—91no—
143C22×C3×C4—91no—
144C22×C3×C4—91no—
145C22×C3×C4C2×C3⋊D311yescharacteristic
146C22×C3×C4C2×C3⋊D311yescharacteristic
147C22×C3×C4C2×C3⋊D311yescharacteristic
148C22×Dic3—34no—
149C22×Dic3—34no—
150C22×Dic3—34no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—68no—
153C3×C22⋊C4—94no—
154C3×C22⋊C4—182no—
155C3×C22⋊C4—182no—
156C3×C42—182no—
157(C2×D3)⋊C4—316no—
158(C2×D3)⋊C4—68no—
159(C2×D3)⋊C4—68no—
160C4×Dic3—68no—
161C22×C32C2×C4×D314yes—
162C22×C32#23514yes—
163C22×C32D3×D414yes—
164C22×C32#23714yes—
165C22×C32#23914yes—
166C22×C32#23914yes—
167C22×C32C2×#6714yes—
168C2×C3×D3—68no—
169C2×C3×D3—68no—
170C2×C3×D3—68no—
171C2×C3×D3—124no—
172C2×C3×D3—124no—
173C2×C3×D3—124no—
174C2×C3×D3—124no—
175C32×C4—28no—
176C32×C4—28no—
177C32×C4—44no—
178C3×Dic3—68no—
179C3×Dic3—68no—
180C3×Dic3—124no—
181C2×C4⋊C4#27211yescharacteristic, complemented
182C2×C22⋊C4—91no—
183C2×C22⋊C4—91no—
184C2×C22⋊C4—91no—
185C2×C42—91no—
186#95—91no—
187#95—91no—
188H(3,𝔽3)#38811yescharacteristic, complemented, Sylow
189C23×C3—34no—
190C23×C3—92no—
191C23×C3—181no—
192C23×C3—181no—
193C23×C3—181no—
194C23×C3—181no—
195C23×C3—181no—
196C23×C3—181no—
197C23×C3C22×C3⋊D311yescharacteristic
198C22×D3—38no—
199C22×D3—64no—
200C22×D3—64no—
201C22×D3—64no—
202C22×D3—64no—
203C22×D3—64no—
204C22×D3—64no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—38no—
213C2×C3×C4—38no—
214C2×C3×C4—64no—
215C2×C3×C4—64no—
216C2×C3×C4—64no—
217C2×C3×C4—64no—
218C2×C3×C4—64no—
219C2×C3×C4—64no—
220C2×C3×C4—64no—
221C2×C3×C4—92no—
222C2×C3×C4—92no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4C4×C3⋊D312yes—
231C2×C3×C4#63812yes—
232C2×Dic3—38no—
233C2×Dic3—38no—
234C2×Dic3—64no—
235C2×Dic3—64no—
236C2×Dic3—64no—
237C2×Dic3—64no—
238C2×Dic3—64no—
239C2×Dic3—64no—
240C2×Dic3—64no—
241C2×C32#89214yes—
242C2×C32#89314yes—
243C2×C32#89414yes—
244C2×C32#89814yes—
245C2×C32C4×#6714yes—
246C2×C32#93914yes—
247C2×C32#94714yes—
248C3×D3—128no—
249C24—91no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4C2×#27211yescharacteristic
254C22×C4C2×#27211yescharacteristic
255C22×C4C2×#27211yescharacteristic
256C4⋊C4—22no—
257C22⋊C4—94no—
258C22⋊C4—182no—
259C22⋊C4—182no—
260C42—182no—
261C22×C3—34no—
262C22×C3—34no—
263C22×C3—34no—
264C22×C3—34no—
265C22×C3—34no—
266C22×C3—34no—
267C22×C3—34no—
268C22×C3—182no—
269C22×C3—182no—
270C22×C3—182no—
271C22×C3—361no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—361no—
275C22×C3C2×C4×C3⋊D311yescharacteristic
276C22×C3#382811yescharacteristic
277C22×C3D4×C3⋊D311yescharacteristic
278C22×C3#383011yescharacteristic
279C22×C3#383211yescharacteristic
280C22×C3#383211yescharacteristic
281C22×C3C2×#63811yescharacteristic
282C2×D3—68no—
283C2×D3—68no—
284C2×D3—68no—
285C2×D3—124no—
286C2×D3—124no—
287C2×D3—124no—
288C2×D3—124no—
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—41no—
292C3×C4—68no—
293C3×C4—68no—
294C3×C4—124no—
295C3×C4—182no—
296C3×C4—182no—
297C3×C4—361no—
298Dic3—68no—
299Dic3—68no—
300Dic3—124no—
301C32#499014yescomplemented
302C23—92no—
303C23—181no—
304C23—181no—
305C23—181no—
306C23—181no—
307C23—181no—
308C23—181no—
309C23C22×#27211yescharacteristic
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—92no—
318C2×C4—92no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4C4×#27212yes—
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—362no—
336C2×C3#6410111yescharacteristic
337C2×C3#6410211yescharacteristic
338C2×C3#6410311yescharacteristic
339C2×C3#6410711yescharacteristic
340C2×C3C4×#63811yescharacteristic
341C2×C3#6414811yescharacteristic
342C2×C3#6415611yescharacteristic
343D3—128no—
344C22—182no—
345C22—182no—
346C22—182no—
347C22—361no—
348C22—361no—
349C22—361no—
350C22—361no—
351C22C2×C4×#27211yescharacteristic
352C22#8874011yescharacteristic
353C22D4×#27211yescharacteristic
354C22#8874411yescharacteristic
355C22#8874811yescharacteristic
356C22#8874811yescharacteristic
357C22C2×#623511yescharacteristic
358C4—22no—
359C4—22no—
360C4—41no—
361C4—182no—
362C4—182no—
363C4—361no—
364C3—34no—
365C3#1059268911yescharacteristic
366C2—362no—
367C2#1172176011yescharacteristic
368C2#1172176111yescharacteristic
369C2#1172176211yescharacteristic
370C2#1172176611yescharacteristic
371C2C4×#623511yescharacteristic
372C2#1172184911yescharacteristic
373C2#1172186411yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#388
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
5. C2×C4×#1084C2semidirect
6. #11720574C2semidirect
7. #11720574C2semidirect
19. #88542C4semidirect
21. C2×C32×C4⋊C4D3semidirect
181. C2×C4⋊C4#272semidirect
188. H(3,𝔽3)#388semidirect
301. C32#4990semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×#885422
C2×C4×#10842
#117205742
C3×#49903