Finite Group Databasebeta

← #13154224

Subgroups — #13154224

Order 1728.

1133 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#11721765C212yesmaximal
3#11721764C212yesmaximal
4#11721760C211yesmaximal, characteristic
5C4⋊C4×#272C211yesmaximal, characteristic
6#11721758C211yesmaximal, characteristic, complemented
7#11721757C212yesmaximal, complemented
8#11721756C211yesmaximal, characteristic, complemented
9#11721756C212yesmaximal, complemented
10#11721718C211yesmaximal, characteristic
11#11721715C211yesmaximal, characteristic, complemented
12C3×#5689—34nomaximal
13C2×C4×#272C2212yes—
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2212yes—
16C2×#6229C2212yes—
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2212yes—
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C42×H(3,𝔽3)C2211yescharacteristic
22#88542C2212yes—
23#88542C2211yescharacteristic
24#88542C2211yescharacteristic
25C4⋊#1084C2212yes—
26C4⋊#1084C2211yescharacteristic
27C4⋊#1084C2212yes—
28C4⋊#1084C2211yescharacteristic
29#88540C2211yescharacteristic
30#88540C2212yes—
31#88540C2212yes—
32#88540C2211yescharacteristic
33#88540C2212yes—
34C4×#1084C2212yes—
35C4×#1084C2211yescharacteristic
36C32×#125D314yescomplemented
37C3×#897—38no—
38C3×#896—38no—
39C3×#892—34no—
40C3×D3×C4⋊C4—34no—
41C3×#890—34no—
42C3×#889—38no—
43C3×#888—38no—
44C3×#888—34no—
45C3×#872—34no—
46C3×#869—34no—
47C22×#272C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2312yes—
50C2×C4×H(3,𝔽3)C2312yes—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×#1084C2311yescharacteristic
54C2×#1084C2312yes—
55C2×#1084C2312yes—
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C4×#272—42no—
59C4×#272Q814yescomplemented
60#6229—42no—
61#6229—42no—
62C3×#544—91no—
63#5689—34no—
64C2×C3×C4×D3—34no—
65C2×C3×C4×D3—38no—
66C2×C3×Dic6—38no—
67C2×C3×Dic6—38no—
68C32×C4⋊C4C2×D314yes—
69C32×C4⋊C4C2×D318yes—
70C32×C4⋊C4C2×D318yes—
71C32×C4⋊C4C2×D314yes—
72C32×C42C2×D314yes—
73C3×(C2×D3)⋊C4—38no—
74C3×(C2×D3)⋊C4—34no—
75C3×(C2×D3)⋊C4—34no—
76C3×C4⋊Dic3—38no—
77C3×C4⋊Dic3—34no—
78C3×C4⋊Dic3—38no—
79C3×C4⋊Dic3—34no—
80C3×Dic3⋊C4—34no—
81C3×Dic3⋊C4—38no—
82C3×Dic3⋊C4—38no—
83C3×Dic3⋊C4—34no—
84C3×Dic3⋊C4—38no—
85C3×C4×Dic3—38no—
86C3×C4×Dic3—34no—
87C22×H(3,𝔽3)C2411yescharacteristic
88C2×#272—21no—
89C2×#272—21no—
90C2×#272C2×Q812yes—
91C4×H(3,𝔽3)—21no—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)—22no—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×Q812yes—
96#1084—21no—
97#1084—22no—
98#1084—22no—
99#1084—21no—
100#1084C2×Q812yes—
101C3×C4⋊Q8—92no—
102C3×C4⋊Q8—91no—
103C3×C4⋊Q8—91no—
104C3×#125—34no—
105C3×#125—92no—
106C3×#125—91no—
107C3×#125C3⋊D311yescharacteristic
108C3×C22⋊Q8—92no—
109C3×C22⋊Q8—92no—
110C3×#117—91no—
111C3×#117—91no—
112C2×C3×C4⋊C4—91no—
113#897—38no—
114#896—38no—
115#892—34no—
116D3×C4⋊C4—34no—
117#890—34no—
118#889—38no—
119#888—34no—
120#888—38no—
121#872—34no—
122#869—34no—
123C22×C3×D3—34no—
124C2×C32×C4C22×D318yes—
125C2×C32×C4C22×D318yes—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D314yes—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—38no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—38no—
134C3×C4×D3—316no—
135C3×C4×D3—128no—
136C3×Dic6—128no—
137C3×Dic6—128no—
138#544—91noSylow
139C2×H(3,𝔽3)C22×Q811yescharacteristic
140C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
141C2×H(3,𝔽3)2−1+411yescharacteristic
142#272—22no—
143C2×C3×Q8—92no—
144C2×C3×Q8—92no—
145C22×C3×C4—91no—
146C22×C3×C4—92no—
147C2×C4×D3—34no—
148C2×C4×D3—38no—
149C2×Dic6—38no—
150C2×Dic6—38no—
151C3×C4⋊C4—34no—
152C3×C4⋊C4—38no—
153C3×C4⋊C4—38no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—92no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—92no—
159C3×C4⋊C4—92no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—92no—
163C3×C4⋊C4—92no—
164C3×C4⋊C4C2×C3⋊D311yescharacteristic
165C3×C4⋊C4C2×C3⋊D311yescharacteristic
166C3×C4⋊C4C2×C3⋊D312yes—
167C3×C4⋊C4C2×C3⋊D312yes—
168C3×C22⋊C4—91no—
169C3×C22⋊C4—92no—
170C3×C22⋊C4—91no—
171C3×C42—34no—
172C3×C42—92no—
173C3×C42—91no—
174C3×C42C2×C3⋊D311yescharacteristic
175(C2×D3)⋊C4—34no—
176(C2×D3)⋊C4—34no—
177(C2×D3)⋊C4—38no—
178C4⋊Dic3—34no—
179C4⋊Dic3—38no—
180C4⋊Dic3—34no—
181C4⋊Dic3—38no—
182Dic3⋊C4—34no—
183Dic3⋊C4—34no—
184Dic3⋊C4—38no—
185Dic3⋊C4—38no—
186Dic3⋊C4—38no—
187C4×Dic3—34no—
188C4×Dic3—38no—
189C22×C32C23×D314yes—
190C2×C3×D3—38no—
191C2×C3×D3—64no—
192C2×C3×D3—64no—
193C32×C4—24no—
194C32×C4—28no—
195C32×C4—28no—
196C32×C4—24no—
197C32×C4D3×Q818yes—
198C3×Dic3—38no—
199C3×Dic3—64no—
200C3×Dic3—68no—
201C3×Dic3—68no—
202C3×Dic3—64no—
203C4⋊Q8—92no—
204C4⋊Q8—91no—
205C4⋊Q8—91no—
206#125—92no—
207#125—91no—
208#125#27211yescharacteristic, complemented
209C22⋊Q8—92no—
210C22⋊Q8—92no—
211#117—91no—
212#117—91no—
213C2×C4⋊C4—91no—
214H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—34no—
217C3×Q8—362no—
218C3×Q8—362no—
219C2×C3×C4—34no—
220C2×C3×C4—38no—
221C2×C3×C4—38no—
222C2×C3×C4—34no—
223C2×C3×C4—34no—
224C2×C3×C4—91no—
225C2×C3×C4—94no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4—362no—
231C2×C3×C4C22×C3⋊D312yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×C3×C4C22×C3⋊D312yes—
235C2×C3×C4C22×C3⋊D311yescharacteristic
236C2×Dic3—34no—
237C2×Dic3—34no—
238C2×Dic3—34no—
239C2×Dic3—38no—
240C2×Dic3—38no—
241C4×D3—316no—
242C4×D3—128no—
243Dic6—128no—
244Dic6—128no—
245C2×C32C2×D3×Q814yes—
246C2×C32#100914yes—
247C2×C32#101014yes—
248C3×D3—68no—
249C2×Q8—92no—
250C2×Q8—92no—
251C22×C4—91no—
252C22×C4—92no—
253C4⋊C4—92no—
254C4⋊C4—91no—
255C4⋊C4—91no—
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—92no—
259C4⋊C4—92no—
260C4⋊C4—92no—
261C4⋊C4—92no—
262C4⋊C4C2×#27211yescharacteristic
263C4⋊C4C2×#27211yescharacteristic
264C4⋊C4C2×#27212yes—
265C4⋊C4C2×#27212yes—
266C22⋊C4—91no—
267C22⋊C4—92no—
268C22⋊C4—91no—
269C42—92no—
270C42—91no—
271C42C2×#27211yescharacteristic
272C22×C3—34no—
273C22×C3—92no—
274C22×C3—181no—
275C22×C3—181no—
276C22×C3C23×C3⋊D311yescharacteristic
277C2×D3—38no—
278C2×D3—64no—
279C2×D3—64no—
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—38no—
285C3×C4—64no—
286C3×C4—68no—
287C3×C4—68no—
288C3×C4—64no—
289C3×C4—92no—
290C3×C4—182no—
291C3×C4—181no—
292C3×C4—182no—
293C3×C4—181no—
294C3×C4Q8×C3⋊D312yes—
295Dic3—38no—
296Dic3—64no—
297Dic3—68no—
298Dic3—68no—
299Dic3—64no—
300C32#568914yescomplemented
301C23—91no—
302Q8—362no—
303Q8—362no—
304C2×C4—91no—
305C2×C4—94no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—92no—
309C2×C4—92no—
310C2×C4—362no—
311C2×C4C22×#27212yes—
312C2×C4C22×#27211yescharacteristic
313C2×C4C22×#27211yescharacteristic
314C2×C4C22×#27212yes—
315C2×C4C22×#27211yescharacteristic
316C2×C3—34no—
317C2×C3—34no—
318C2×C3—34no—
319C2×C3—182no—
320C2×C3C2×Q8×C3⋊D311yescharacteristic
321C2×C3#6436611yescharacteristic
322C2×C3#6436711yescharacteristic
323D3—68no—
324C22—92no—
325C22—181no—
326C22—181no—
327C22C23×#27211yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—21no—
332C4—92no—
333C4—182no—
334C4—181no—
335C4—182no—
336C4—181no—
337C4Q8×#27212yes—
338C3—34no—
339C3#1059616011yescharacteristic
340C2—182no—
341C2C2×Q8×#27211yescharacteristic
342C2#1172267511yescharacteristic
343C2#1172267611yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radicalC22×#272C23
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#544
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
6. #11721758C2semidirect
7. #11721757C2semidirect
8. #11721756C2semidirect
9. #11721756C2semidirect
11. #11721715C2semidirect
36. C32×#125D3semidirect
59. C4×#272Q8semidirect
208. #125#272semidirect
214. H(3,𝔽3)#544semidirect
300. C32#5689semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217652
#117217642
#117217602
C4⋊C4×#2722
#117217582
#117217572
#117217562
#117217182
#117217152
C3×#56893