Finite Group Databasebeta

← #13153862

Subgroups — #13153862

Order 1728.

1177 conjugacy classes of subgroups.

The subgroup lattice has 365 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
2#11721829C212yesmaximal
3C4×#6235C211yesmaximal, characteristic
4#11721826C212yesmaximal, complemented
5#11721824C211yesmaximal, characteristic, complemented
6#11721766C212yesmaximal, complemented
7#11721765C211yesmaximal, characteristic, complemented
8#11721760C211yesmaximal, characteristic, complemented
9#11721758C211yesmaximal, characteristic, complemented
10#11721757C212yesmaximal, complemented
11#11721755C211yesmaximal, characteristic, complemented
12C3×#5513—34nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2212yes—
15C2×#6235C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#6229C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21#88552C2212yes—
22#88552C2211yescharacteristic
23#88542C2212yes—
24#88542C2212yes—
25#88542C2211yescharacteristic
26C4⋊#1084C2212yes—
27C4⋊#1084C2211yescharacteristic
28C4⋊#1084C2212yes—
29#88540C2212yes—
30#88540C2211yescharacteristic
31#88540C2212yes—
32#88540C2212yes—
33C4×#1084C2211yescharacteristic
34C4×#1084C2212yes—
35C4×#1084C2212yes—
36C2×C32×C4⋊C4D314yescomplemented
37C3×#929—38no—
38C3×C4×#67—34no—
39C3×#926—38no—
40C3×#924—34no—
41C3×#898—38no—
42C3×#897—34no—
43C3×#892—34no—
44C3×#890—34no—
45C3×#889—38no—
46C3×#887—34no—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×#272C2311yescharacteristic
49C2×C4×H(3,𝔽3)—24no—
50C2×C4×H(3,𝔽3)—22no—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2312yes—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55#6235—22no—
56C2×#1084C2312yes—
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×#1084C2312yes—
61C4×#272—22no—
62#6229—22no—
63C3×#552—91no—
64#5513—34no—
65C22×C32×C4C2×D314yes—
66C22×C32×C4C2×D318yes—
67C2×C3×#67—34no—
68C2×C3×C4×D3—34no—
69C2×C3×Dic6—34no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C2×D318yes—
72C32×C4⋊C4C2×D314yes—
73C3×#217—38no—
74C3×#217—34no—
75C3×(C2×D3)⋊C4—38no—
76C3×(C2×D3)⋊C4—38no—
77C3×(C2×D3)⋊C4—34no—
78C3×C4⋊Dic3—38no—
79C3×C4⋊Dic3—34no—
80C3×C4⋊Dic3—38no—
81C3×Dic3⋊C4—38no—
82C3×Dic3⋊C4—34no—
83C3×Dic3⋊C4—38no—
84C3×Dic3⋊C4—38no—
85C3×C4×Dic3—34no—
86C3×C4×Dic3—38no—
87C3×C4×Dic3—38no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)#4112yes—
91C22×H(3,𝔽3)C2411yescharacteristic
92C2×#272—21no—
93C2×#272—21no—
94C2×#272—21no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)—22no—
97C4×H(3,𝔽3)—22no—
98C4×H(3,𝔽3)#4112yes—
99#1084—22no—
100#1084—21no—
101#1084—21no—
102#1084—21no—
103#1084—22no—
104C3×#126—92no—
105C3×#125—92no—
106C3×#125—91no—
107C3×#123—92no—
108C3×C22⋊Q8—91no—
109C3×C22⋊Q8—91no—
110C3×C4×Q8—91no—
111C3×C4×D4—91no—
112C3×#117—92no—
113C3×#117—91no—
114C2×C3×C4⋊C4—34no—
115C2×C3×C4⋊C4C3⋊D311yescharacteristic
116#929—38no—
117C4×#67—34no—
118#926—38no—
119#924—34no—
120#898—38no—
121#897—34no—
122#892—34no—
123#890—34no—
124#889—38no—
125#887—34no—
126C23×C32C22×D314yes—
127C22×C3×D3—34no—
128C2×C32×C4—216no—
129C2×C32×C4—28no—
130C2×C32×C4C22×D318yes—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D318yes—
133C2×C32×C4C22×D314yes—
134C3×#67—68no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—38no—
139C2×C3×Dic3—38no—
140C3×C4×D3—68no—
141C3×Dic6—68no—
142#552—91noSylow
143C2×H(3,𝔽3)—22no—
144C2×H(3,𝔽3)C2×#4111yescharacteristic
145C2×H(3,𝔽3)C2×#4111yescharacteristic
146C2×H(3,𝔽3)2−1+411yescharacteristic
147#272—41no—
148C2×C3×Q8—91no—
149C2×C3×D4—91no—
150C22×C3×C4—34no—
151C22×C3×C4—38no—
152C22×C3×C4—91no—
153C22×C3×C4C2×C3⋊D312yes—
154C22×C3×C4C2×C3⋊D311yescharacteristic
155C2×#67—34no—
156C2×C4×D3—34no—
157C2×Dic6—34no—
158C3×C4⋊C4—34no—
159C3×C4⋊C4—38no—
160C3×C4⋊C4—34no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—92no—
163C3×C4⋊C4—92no—
164C3×C4⋊C4—92no—
165C3×C4⋊C4—92no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—92no—
168C3×C4⋊C4C2×C3⋊D311yescharacteristic
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170C3×C4⋊C4C2×C3⋊D312yes—
171C3×C22⋊C4—91no—
172C3×C22⋊C4—92no—
173C3×C22⋊C4—92no—
174C3×C22⋊C4—91no—
175C3×C22⋊C4—92no—
176C3×C42—92no—
177C3×C42—92no—
178C3×C42—91no—
179#217—38no—
180#217—34no—
181(C2×D3)⋊C4—38no—
182(C2×D3)⋊C4—38no—
183(C2×D3)⋊C4—34no—
184C4⋊Dic3—38no—
185C4⋊Dic3—34no—
186C4⋊Dic3—38no—
187Dic3⋊C4—38no—
188Dic3⋊C4—34no—
189Dic3⋊C4—38no—
190Dic3⋊C4—38no—
191C4×Dic3—34no—
192C4×Dic3—38no—
193C4×Dic3—38no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32#23718yes—
197C22×C32C23×D314yes—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—64no—
201C32×C4—24no—
202C32×C4—28no—
203C32×C4—28no—
204C32×C4#23518yes—
205C3×Dic3—68no—
206C3×Dic3—64no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—68no—
210#126—92no—
211#125—92no—
212#125—91no—
213#123—92no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C4×Q8—91no—
217C4×D4—91no—
218#117—92no—
219#117—91no—
220C2×C4⋊C4#27211yescharacteristic, complemented
221H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
222C23×C3—34no—
223C23×C3—91no—
224C23×C3C22×C3⋊D311yescharacteristic
225C22×D3—34no—
226C3×Q8—182no—
227C3×D4—182no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—34no—
231C2×C3×C4—38no—
232C2×C3×C4—34no—
233C2×C3×C4—38no—
234C2×C3×C4—68no—
235C2×C3×C4—616no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—92no—
239C2×C3×C4—92no—
240C2×C3×C4—91no—
241C2×C3×C4—182no—
242C2×C3×C4C22×C3⋊D312yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244C2×C3×C4C22×C3⋊D312yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246#67—68no—
247C2×Dic3—34no—
248C2×Dic3—34no—
249C2×Dic3—34no—
250C2×Dic3—38no—
251C2×Dic3—38no—
252C4×D3—68no—
253Dic6—68no—
254C2×C32—28no—
255C2×C32C2×#23514yes—
256C2×C32C2×#23714yes—
257C2×C32#100714yes—
258C3×D3—124no—
259C2×Q8—91no—
260C2×D4—91no—
261C22×C4—91no—
262C22×C4C2×#27212yes—
263C22×C4C2×#27211yescharacteristic
264C4⋊C4—92no—
265C4⋊C4—91no—
266C4⋊C4—92no—
267C4⋊C4—91no—
268C4⋊C4—92no—
269C4⋊C4—92no—
270C4⋊C4—92no—
271C4⋊C4C2×#27211yescharacteristic
272C4⋊C4C2×#27211yescharacteristic
273C4⋊C4C2×#27212yes—
274C22⋊C4—91no—
275C22⋊C4—92no—
276C22⋊C4—92no—
277C22⋊C4—91no—
278C22⋊C4—92no—
279C42—92no—
280C42—92no—
281C42—91no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—34no—
285C22×C3—38no—
286C22×C3—64no—
287C22×C3—64no—
288C22×C3—181no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3#383012yes—
292C22×C3C23×C3⋊D311yescharacteristic
293C2×D3—64no—
294C2×D3—64no—
295C2×D3—64no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—38no—
300C3×C4—64no—
301C3×C4—68no—
302C3×C4—68no—
303C3×C4—181no—
304C3×C4—182no—
305C3×C4—181no—
306C3×C4—181no—
307C3×C4—182no—
308C3×C4#382812yes—
309Dic3—68no—
310Dic3—64no—
311Dic3—64no—
312Dic3—64no—
313Dic3—68no—
314C32#551314yescomplemented
315C23—91no—
316C23C22×#27211yescharacteristic
317Q8—182no—
318D4—182no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—92no—
324C2×C4—92no—
325C2×C4—91no—
326C2×C4—182no—
327C2×C4C22×#27212yes—
328C2×C4C22×#27211yescharacteristic
329C2×C4C22×#27212yes—
330C2×C4C22×#27211yescharacteristic
331C2×C3—22no—
332C2×C3—34no—
333C2×C3—34no—
334C2×C3—34no—
335C2×C3—68no—
336C2×C3—361no—
337C2×C3C2×#382811yescharacteristic
338C2×C3C2×#383011yescharacteristic
339C2×C3#6436411yescharacteristic
340D3—124no—
341C22—21no—
342C22—21no—
343C22—181no—
344C22—181no—
345C22—181no—
346C22#8874412yes—
347C22C23×#27211yescharacteristic
348C4—21no—
349C4—22no—
350C4—22no—
351C4—181no—
352C4—182no—
353C4—181no—
354C4—181no—
355C4—182no—
356C4#8874012yes—
357C3—34no—
358C3#1059598411yescharacteristic
359C2—22no—
360C2—361no—
361C2C2×#8874011yescharacteristic
362C2C2×#8874411yescharacteristic
363C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC2×#6235C22
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#552
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×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
4. #11721826C2semidirect
5. #11721824C2semidirect
6. #11721766C2semidirect
7. #11721765C2semidirect
8. #11721760C2semidirect
9. #11721758C2semidirect
10. #11721757C2semidirect
11. #11721755C2semidirect
36. C2×C32×C4⋊C4D3semidirect
220. C2×C4⋊C4#272semidirect
221. H(3,𝔽3)#552semidirect
314. C32#5513semidirect

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
#117218292
C4×#62352
#117218262
#117218242
#117217662
#117217652
#117217602
#117217582
#117217572
#117217552
C3×#55133