Finite Group Databasebeta

← #13154229

Subgroups — #13154229

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
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#11721766C212yesmaximal, complemented
3#11721765C211yesmaximal, characteristic, complemented
4#11721765C211yesmaximal, characteristic, complemented
5#11721762C212yesmaximal
6#11721760C212yesmaximal, complemented
7#11721760C211yesmaximal, characteristic, complemented
8C4⋊C4×#272C211yesmaximal, characteristic, complemented
9#11721758C212yesmaximal, complemented
10C4×C4⋊#272C211yesmaximal, characteristic
11C4×#6229C211yesmaximal, characteristic, complemented
12C3×#5694—34nomaximal
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272C2212yes—
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#6229C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23#88542C2211yescharacteristic
24#88542C2212yes—
25#88542C2212yes—
26#88542C2212yes—
27#88542C2211yescharacteristic
28C4⋊#1084C2211yescharacteristic
29C4⋊#1084C2212yes—
30C4⋊#1084C2212yes—
31C4⋊#1084C2211yescharacteristic
32#88540C2212yes—
33#88540C2212yes—
34C4×#1084C2212yes—
35C4×#1084C2212yes—
36C32×#125D314yescomplemented
37C3×#898—38no—
38C3×#897—34no—
39C3×#897—34no—
40C3×#894—38no—
41C3×#892—34no—
42C3×#892—38no—
43C3×D3×C4⋊C4—34no—
44C3×#890—38no—
45C3×C4×D12—34no—
46C3×C4×Dic6—34no—
47C22×#272C2311yescharacteristic
48C22×#272C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2312yes—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×C4×H(3,𝔽3)C2312yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×#1084C2312yes—
55C2×#1084C2312yes—
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C4⋊#272—22no—
59C4×#272—24no—
60C4×#272—22no—
61C4×#272—22no—
62#6229—22no—
63C3×#552—91no—
64#5694—34no—
65C2×C3×D12—34no—
66C2×C3×C4×D3—38no—
67C2×C3×C4×D3—34no—
68C2×C3×C4×D3—34no—
69C2×C3×Dic6—34no—
70C32×C4⋊C4C2×D318yes—
71C32×C4⋊C4C2×D314yes—
72C32×C4⋊C4C2×D314yes—
73C32×C4⋊C4C2×D318yes—
74C32×C42C2×D314yes—
75C3×(C2×D3)⋊C4—34no—
76C3×(C2×D3)⋊C4—38no—
77C3×(C2×D3)⋊C4—38no—
78C3×(C2×D3)⋊C4—38no—
79C3×(C2×D3)⋊C4—34no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3—38no—
82C3×C4⋊Dic3—38no—
83C3×C4⋊Dic3—34no—
84C3×Dic3⋊C4—38no—
85C3×Dic3⋊C4—38no—
86C3×C4×Dic3—38no—
87C3×C4×Dic3—38no—
88C22×H(3,𝔽3)C2411yescharacteristic
89C2×#272—21no—
90C2×#272—21no—
91C2×#272—21no—
92C2×#272—21no—
93C2×#272—21no—
94C2×#272#4112yes—
95C4×H(3,𝔽3)—22no—
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)—22no—
99C4×H(3,𝔽3)#4112yes—
100#1084—22no—
101#1084—21no—
102#1084—21no—
103#1084—22no—
104C3×#126—92no—
105C3×#125—34no—
106C3×#125—92no—
107C3×#125C3⋊D311yescharacteristic
108C3×#123—92no—
109C3×C22⋊Q8—91no—
110C3×C22⋊Q8—91no—
111C3×C4×Q8—91no—
112C3×C4×D4—91no—
113C3×#117—91no—
114C3×#117—92no—
115C2×C3×C4⋊C4—91no—
116#898—38no—
117#897—34no—
118#897—34no—
119#894—38no—
120#892—38no—
121#892—34no—
122D3×C4⋊C4—34no—
123#890—38no—
124C4×D12—34no—
125C4×Dic6—34no—
126C22×C3×D3—34no—
127C22×C3×D3—34no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D318yes—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D318yes—
132C2×C32×C4C22×D314yes—
133C2×C3×Dic3—38no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—38no—
136C2×C3×Dic3—34no—
137C3×D12—68no—
138C3×C4×D3—616no—
139C3×C4×D3—68no—
140C3×C4×D3—68no—
141C3×Dic6—68no—
142#552—91noSylow
143C2×H(3,𝔽3)C2×#4111yescharacteristic
144C2×H(3,𝔽3)C2×#4111yescharacteristic
145C2×H(3,𝔽3)2−1+411yescharacteristic
146#272—22no—
147#272—41no—
148C2×C3×Q8—91no—
149C2×C3×D4—91no—
150C22×C3×C4—92no—
151C22×C3×C4—91no—
152C22×C3×C4—91no—
153C2×D12—34no—
154C2×C4×D3—34no—
155C2×C4×D3—34no—
156C2×C4×D3—38no—
157C2×Dic6—34no—
158C3×C4⋊C4—38no—
159C3×C4⋊C4—34no—
160C3×C4⋊C4—34no—
161C3×C4⋊C4—38no—
162C3×C4⋊C4—92no—
163C3×C4⋊C4—92no—
164C3×C4⋊C4—91no—
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×C4⋊C4C2×C3⋊D312yes—
172C3×C22⋊C4—91no—
173C3×C22⋊C4—92no—
174C3×C22⋊C4—92no—
175C3×C22⋊C4—91no—
176C3×C22⋊C4—92no—
177C3×C42—34no—
178C3×C42—92no—
179C3×C42—92no—
180C3×C42C2×C3⋊D311yescharacteristic
181(C2×D3)⋊C4—34no—
182(C2×D3)⋊C4—34no—
183(C2×D3)⋊C4—38no—
184(C2×D3)⋊C4—38no—
185(C2×D3)⋊C4—38no—
186C4⋊Dic3—34no—
187C4⋊Dic3—34no—
188C4⋊Dic3—38no—
189C4⋊Dic3—38no—
190Dic3⋊C4—38no—
191Dic3⋊C4—38no—
192C4×Dic3—38no—
193C4×Dic3—38no—
194C22×C32C23×D314yes—
195C2×C3×D3—38no—
196C2×C3×D3—64no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—64no—
201C32×C4—28no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—28no—
205C32×C4#23918yes—
206C3×Dic3—68no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—68no—
210#126—92no—
211#125—92no—
212#125#27211yescharacteristic, complemented
213#123—92no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C4×Q8—91no—
217C4×D4—91no—
218#117—91no—
219#117—92no—
220C2×C4⋊C4—91no—
221H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
222C23×C3—91no—
223C23×C3—91no—
224C22×D3—34no—
225C22×D3—34no—
226C3×Q8—182no—
227C3×D4—182no—
228C2×C3×C4—34no—
229C2×C3×C4—34no—
230C2×C3×C4—38no—
231C2×C3×C4—34no—
232C2×C3×C4—38no—
233C2×C3×C4—91no—
234C2×C3×C4—92no—
235C2×C3×C4—92no—
236C2×C3×C4—91no—
237C2×C3×C4—184no—
238C2×C3×C4—182no—
239C2×C3×C4—182no—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241C2×C3×C4C22×C3⋊D312yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×C3×C4C22×C3⋊D312yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—38no—
246C2×Dic3—34no—
247C2×Dic3—38no—
248C2×Dic3—34no—
249D12—68no—
250C4×D3—68no—
251C4×D3—68no—
252C4×D3—616no—
253Dic6—68no—
254C2×C32C2×#23914yes—
255C2×C32D3×#4114yes—
256C2×C32#101014yes—
257C3×D3—68no—
258C3×D3—124no—
259C2×Q8—91no—
260C2×D4—91no—
261C22×C4—92no—
262C22×C4—91no—
263C22×C4—91no—
264C4⋊C4—91no—
265C4⋊C4—92no—
266C4⋊C4—92no—
267C4⋊C4—91no—
268C4⋊C4—92no—
269C4⋊C4—92no—
270C4⋊C4C2×#27211yescharacteristic
271C4⋊C4C2×#27211yescharacteristic
272C4⋊C4C2×#27212yes—
273C4⋊C4C2×#27212yes—
274C22⋊C4—91no—
275C22⋊C4—92no—
276C22⋊C4—92no—
277C22⋊C4—91no—
278C22⋊C4—92no—
279C42—92no—
280C42—92no—
281C42C2×#27211yescharacteristic
282C22×C3—34no—
283C22×C3—92no—
284C22×C3—181no—
285C22×C3—181no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3—181no—
289C22×C3C23×C3⋊D311yescharacteristic
290C2×D3—38no—
291C2×D3—64no—
292C2×D3—64no—
293C2×D3—64no—
294C2×D3—64no—
295C2×D3—64no—
296C3×C4—22no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—22no—
300C3×C4—38no—
301C3×C4—68no—
302C3×C4—64no—
303C3×C4—64no—
304C3×C4—68no—
305C3×C4—181no—
306C3×C4—181no—
307C3×C4—182no—
308C3×C4—182no—
309C3×C4#383212yes—
310Dic3—68no—
311Dic3—64no—
312Dic3—64no—
313Dic3—68no—
314C32#569414yescomplemented
315C23—91no—
316C23—91no—
317Q8—182no—
318D4—182no—
319C2×C4—91no—
320C2×C4—92no—
321C2×C4—92no—
322C2×C4—91no—
323C2×C4—184no—
324C2×C4—182no—
325C2×C4—182no—
326C2×C4C22×#27211yescharacteristic
327C2×C4C22×#27212yes—
328C2×C4C22×#27211yescharacteristic
329C2×C4C22×#27212yes—
330C2×C4C22×#27211yescharacteristic
331C2×C3—34no—
332C2×C3—34no—
333C2×C3—34no—
334C2×C3—182no—
335C2×C3—361no—
336C2×C3C2×#383211yescharacteristic
337C2×C3#41×C3⋊D311yescharacteristic
338C2×C3#6436711yescharacteristic
339D3—68no—
340D3—124no—
341C22—92no—
342C22—181no—
343C22—181no—
344C22—181no—
345C22—181no—
346C22—181no—
347C22C23×#27211yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—22no—
352C4—181no—
353C4—181no—
354C4—182no—
355C4—182no—
356C4#8874812yes—
357C3—34no—
358C3#1059616511yescharacteristic
359C2—182no—
360C2—361no—
361C2C2×#8874811yescharacteristic
362C2#41×#27211yescharacteristic
363C2#1172267611yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#552
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #11721766C2semidirect
3. #11721765C2semidirect
4. #11721765C2semidirect
6. #11721760C2semidirect
7. #11721760C2semidirect
8. C4⋊C4×#272C2semidirect
9. #11721758C2semidirect
11. C4×#6229C2semidirect
36. C32×#125D3semidirect
212. #125#272semidirect
221. H(3,𝔽3)#552semidirect
314. C32#5694semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217662
#117217652
#117217622
#117217602
C4⋊C4×#2722
#117217582
C4×C4⋊#2722
C4×#62292
C3×#56943