Finite Group Databasebeta

← #13154302

Subgroups — #13154302

Order 1728.

1188 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#11721864C212yesmaximal, complemented
3#11721863C212yesmaximal, complemented
4#11721766C212yesmaximal, complemented
5#11721765C211yesmaximal, characteristic, complemented
6#11721763C211yesmaximal, characteristic
7#11721762C212yesmaximal, complemented
8#11721761C211yesmaximal, characteristic, complemented
9#11721755C211yesmaximal, characteristic, complemented
10#11721721C211yesmaximal, characteristic, complemented
11#11721718C211yesmaximal, characteristic, complemented
12C3×#5733—34nomaximal
13C2×Q8×H(3,𝔽3)C2212yes—
14C2×C4⋊#272C2211yescharacteristic
15C2×C4⋊#272C2212yes—
16C2×C4×#272C2211yescharacteristic
17C2×C4×#272C2212yes—
18C2×#6229C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23#88542C2212yes—
24#88542C2211yescharacteristic
25#88542C2212yes—
26#88542C2212yes—
27#88542C2212yes—
28#88542C2212yes—
29#88542C2211yescharacteristic
30C4⋊#1084C2212yes—
31#88540C2212yes—
32#88540C2211yescharacteristic
33#88540C2211yescharacteristic
34C4×#1084C2211yescharacteristic
35C4×#1084C2212yes—
36C32×C4⋊Q8D314yescomplemented
37C3×#947—38no—
38C3×#946—38no—
39C3×#898—38no—
40C3×#897—34no—
41C3×#895—34no—
42C3×#894—38no—
43C3×#893—34no—
44C3×#887—34no—
45C3×#875—34no—
46C3×#872—34no—
47C22×#272C2311yescharacteristic
48C22×#272C2312yes—
49Q8×H(3,𝔽3)—42no—
50C2×C4×H(3,𝔽3)C2312yes—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×#1084C2312yes—
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C4⋊#272—22no—
60C4⋊#272—42no—
61C4×#272—22no—
62C4×#272—42no—
63#6229—22no—
64C3×#542—91no—
65#5733—34no—
66C2×C32×Q8C2×D318yes—
67C2×C3×D12—34no—
68C2×C3×D12—38no—
69C2×C3×C4×D3—34no—
70C2×C3×C4×D3—38no—
71C2×C3×Dic6—34no—
72C32×C4⋊C4C2×D314yes—
73C32×C4⋊C4C2×D314yes—
74C32×C4⋊C4C2×D318yes—
75C32×C42C2×D314yes—
76C3×(C2×D3)⋊C4—38no—
77C3×(C2×D3)⋊C4—34no—
78C3×(C2×D3)⋊C4—38no—
79C3×(C2×D3)⋊C4—38no—
80C3×(C2×D3)⋊C4—38no—
81C3×(C2×D3)⋊C4—38no—
82C3×(C2×D3)⋊C4—34no—
83C3×C4⋊Dic3—38no—
84C3×Dic3⋊C4—38no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—34no—
87C3×C4×Dic3—34no—
88C3×C4×Dic3—38no—
89C22×H(3,𝔽3)C2411yescharacteristic
90C2×#272—21no—
91C2×#272—22no—
92C2×#272—22no—
93C2×#272—21no—
94C2×#272—21no—
95C2×#272—22no—
96C4×H(3,𝔽3)—22no—
97C4×H(3,𝔽3)—22no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)#4112yes—
101#1084—22no—
102#1084—21no—
103#1084—21no—
104#1084—21no—
105C3×C4⋊Q8—34no—
106C3×C4⋊Q8C3⋊D311yescharacteristic
107C3×#126—92no—
108C3×#124—92no—
109C3×#124—91no—
110C3×#123—92no—
111C3×C22⋊Q8—92no—
112C3×C22⋊Q8—91no—
113C3×#121—91no—
114C3×C4×Q8—91no—
115C3×C4×D4—91no—
116C3×#117—91no—
117#947—38no—
118#946—38no—
119#898—38no—
120#897—34no—
121#895—34no—
122#894—38no—
123#893—34no—
124#887—34no—
125#875—34no—
126#872—34no—
127C22×C3×D3—38no—
128C22×C3×D3—34no—
129C32×Q8—48no—
130C2×C32×C4C22×D318yes—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D318yes—
134C2×C32×C4C22×D314yes—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—38no—
138C2×C3×Dic3—34no—
139C3×D12—68no—
140C3×D12—128no—
141C3×C4×D3—68no—
142C3×C4×D3—128no—
143C3×Dic6—68no—
144#542—91noSylow
145C2×H(3,𝔽3)C2×#4111yescharacteristic
146C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
147C2×H(3,𝔽3)2−1+411yescharacteristic
148#272—41no—
149#272—42no—
150C2×C3×Q8—38no—
151C2×C3×Q8—91no—
152C2×C3×Q8C2×C3⋊D312yes—
153C2×C3×D4—91no—
154C2×C3×D4—92no—
155C22×C3×C4—91no—
156C22×C3×C4—92no—
157C2×D12—34no—
158C2×D12—38no—
159C2×C4×D3—34no—
160C2×C4×D3—38no—
161C2×Dic6—34no—
162C3×C4⋊C4—34no—
163C3×C4⋊C4—34no—
164C3×C4⋊C4—38no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—92no—
168C3×C4⋊C4—92no—
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170C3×C4⋊C4C2×C3⋊D311yescharacteristic
171C3×C4⋊C4C2×C3⋊D312yes—
172C3×C22⋊C4—91no—
173C3×C22⋊C4—91no—
174C3×C22⋊C4—92no—
175C3×C22⋊C4—92no—
176C3×C22⋊C4—92no—
177C3×C22⋊C4—92no—
178C3×C22⋊C4—92no—
179C3×C42—34no—
180C3×C42—91no—
181C3×C42—92no—
182C3×C42C2×C3⋊D311yescharacteristic
183(C2×D3)⋊C4—34no—
184(C2×D3)⋊C4—34no—
185(C2×D3)⋊C4—38no—
186(C2×D3)⋊C4—38no—
187(C2×D3)⋊C4—38no—
188(C2×D3)⋊C4—38no—
189(C2×D3)⋊C4—38no—
190C4⋊Dic3—38no—
191Dic3⋊C4—34no—
192Dic3⋊C4—34no—
193Dic3⋊C4—38no—
194C4×Dic3—34no—
195C4×Dic3—38no—
196C22×C32C23×D314yes—
197C2×C3×D3—68no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—68no—
201C2×C3×D3—68no—
202C2×C3×D3—64no—
203C32×C4—28no—
204C32×C4—28no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4#23918yes—
208C3×Dic3—64no—
209C3×Dic3—68no—
210C3×Dic3—64no—
211C3×Dic3—64no—
212C4⋊Q8#27211yescharacteristic, complemented
213#126—92no—
214#124—92no—
215#124—91no—
216#123—92no—
217C22⋊Q8—92no—
218C22⋊Q8—91no—
219#121—91no—
220C4×Q8—91no—
221C4×D4—91no—
222#117—91no—
223H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
224C23×C3—92no—
225C23×C3—91no—
226C22×D3—34no—
227C22×D3—38no—
228C3×Q8—42no—
229C3×Q8—128no—
230C3×Q8—182no—
231C3×D4—182no—
232C3×D4—362no—
233C2×C3×C4—34no—
234C2×C3×C4—38no—
235C2×C3×C4—34no—
236C2×C3×C4—34no—
237C2×C3×C4—38no—
238C2×C3×C4—91no—
239C2×C3×C4—92no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—182no—
243C2×C3×C4—362no—
244C2×C3×C4C22×C3⋊D312yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246C2×C3×C4C22×C3⋊D311yescharacteristic
247C2×C3×C4C22×C3⋊D312yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249C2×Dic3—34no—
250C2×Dic3—34no—
251C2×Dic3—38no—
252C2×Dic3—34no—
253D12—68no—
254D12—128no—
255C4×D3—68no—
256C4×D3—128no—
257Dic6—68no—
258C2×C32#100414yes—
259C2×C32C2×#23914yes—
260C2×C32#100714yes—
261C3×D3—128no—
262C3×D3—124no—
263C2×Q8—91no—
264C2×Q8C2×#27212yes—
265C2×D4—91no—
266C2×D4—92no—
267C22×C4—91no—
268C22×C4—92no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—92no—
272C4⋊C4—92no—
273C4⋊C4C2×#27211yescharacteristic
274C4⋊C4C2×#27211yescharacteristic
275C4⋊C4C2×#27212yes—
276C22⋊C4—91no—
277C22⋊C4—91no—
278C22⋊C4—92no—
279C22⋊C4—92no—
280C22⋊C4—92no—
281C22⋊C4—92no—
282C22⋊C4—92no—
283C42—91no—
284C42—92no—
285C42C2×#27211yescharacteristic
286C22×C3—34no—
287C22×C3—182no—
288C22×C3—181no—
289C22×C3—181no—
290C22×C3—182no—
291C22×C3—182no—
292C22×C3—181no—
293C22×C3C23×C3⋊D311yescharacteristic
294C2×D3—64no—
295C2×D3—68no—
296C2×D3—68no—
297C2×D3—64no—
298C2×D3—64no—
299C2×D3—68no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—38no—
305C3×C4—68no—
306C3×C4—68no—
307C3×C4—64no—
308C3×C4—64no—
309C3×C4—181no—
310C3×C4—181no—
311C3×C4—182no—
312C3×C4—181no—
313C3×C4#383212yes—
314Dic3—68no—
315Dic3—64no—
316Dic3—64no—
317Dic3—64no—
318C32#573314yescomplemented
319C23—92no—
320C23—91no—
321Q8—42no—
322Q8—182no—
323D4—182no—
324D4—362no—
325C2×C4—91no—
326C2×C4—92no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—182no—
330C2×C4—362no—
331C2×C4C22×#27212yes—
332C2×C4C22×#27211yescharacteristic
333C2×C4C22×#27211yescharacteristic
334C2×C4C22×#27212yes—
335C2×C4C22×#27211yescharacteristic
336C2×C3—34no—
337C2×C3—34no—
338C2×C3—34no—
339C2×C3—362no—
340C2×C3—361no—
341C2×C3#6436111yescharacteristic
342C2×C3C2×#383211yescharacteristic
343C2×C3#6436411yescharacteristic
344D3—124no—
345D3—128no—
346C22—182no—
347C22—181no—
348C22—181no—
349C22—182no—
350C22—182no—
351C22—181no—
352C22C23×#27211yescharacteristic
353C4—22no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—181no—
358C4—181no—
359C4—182no—
360C4—181no—
361C4#8874812yes—
362C3—34no—
363C3#1059620411yescharacteristic
364C2—362no—
365C2—361no—
366C2#1172266411yescharacteristic
367C2C2×#8874811yescharacteristic
368C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
prime radical#11721763C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#542
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
2. #11721864C2semidirect
3. #11721863C2semidirect
4. #11721766C2semidirect
5. #11721765C2semidirect
7. #11721762C2semidirect
8. #11721761C2semidirect
9. #11721755C2semidirect
10. #11721721C2semidirect
11. #11721718C2semidirect
36. C32×C4⋊Q8D3semidirect
212. C4⋊Q8#272semidirect
223. H(3,𝔽3)#542semidirect
318. C32#5733semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#117218642
#117218632
#117217662
#117217652
#117217632
#117217622
#117217612
#117217552
#117217212
#117217182
C3×#57333