Finite Group Databasebeta

← #13153860

Subgroups — #13153860

Order 1728.

1254 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
2C4×#6235C211yesmaximal, characteristic
3C4×#6235C212yesmaximal
4#11721824C211yesmaximal, characteristic, complemented
5#11721824C212yesmaximal, complemented
6#11721765C211yesmaximal, characteristic, complemented
7#11721764C212yesmaximal, complemented
8C4⋊C4×#272C211yesmaximal, characteristic, complemented
9#11721757C212yesmaximal, complemented
10#11721756C211yesmaximal, characteristic, complemented
11#11721755C211yesmaximal, characteristic, complemented
12C3×#5511—34nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2212yes—
15C2×#6235C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×C4×#272C2212yes—
18C2×#6229C2211yescharacteristic
19C2×#6229C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2212yes—
23#88552C2211yescharacteristic
24#88552C2212yes—
25#88542C2211yescharacteristic
26#88542C2212yes—
27C4⋊#1084C2211yescharacteristic
28C4⋊#1084C2212yes—
29#88540C2212yes—
30#88540C2212yes—
31#88540C2211yescharacteristic
32#88540C2212yes—
33#88540C2212yes—
34C4×#1084C2211yescharacteristic
35C4×#1084C2212yes—
36C2×C32×C4⋊C4D314yescomplemented
37C3×C4×#67—34no—
38C3×C4×#67—38no—
39C3×#924—38no—
40C3×#924—34no—
41C3×#897—34no—
42C3×#896—38no—
43C3×D3×C4⋊C4—34no—
44C3×#889—38no—
45C3×#888—34no—
46C3×#887—34no—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×#272C2311yescharacteristic
49C2×C4×H(3,𝔽3)—22no—
50C2×C4×H(3,𝔽3)—42no—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×C4×H(3,𝔽3)C2312yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55#6235Q814yescomplemented
56C2×#1084C2311yescharacteristic
57C2×#1084C2312yes—
58C2×#1084C2312yes—
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C4×#272—22no—
62C4×#272—42no—
63#6229—22no—
64#6229—42no—
65C3×#554—91no—
66#5511—34no—
67C22×C32×C4C2×D314yes—
68C22×C32×C4C2×D318yes—
69C2×C3×#67—34no—
70C2×C3×C4×D3—34no—
71C2×C3×C4×D3—38no—
72C2×C3×Dic6—34no—
73C2×C3×Dic6—38no—
74C32×C4⋊C4C2×D314yes—
75C32×C4⋊C4C2×D314yes—
76C32×C4⋊C4C2×D318yes—
77C3×#217—38no—
78C3×#217—34no—
79C3×(C2×D3)⋊C4—34no—
80C3×(C2×D3)⋊C4—38no—
81C3×C4⋊Dic3—34no—
82C3×C4⋊Dic3—38no—
83C3×Dic3⋊C4—38no—
84C3×Dic3⋊C4—38no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—38no—
87C3×Dic3⋊C4—38no—
88C3×C4×Dic3—34no—
89C3×C4×Dic3—38no—
90C22×H(3,𝔽3)—21no—
91C22×H(3,𝔽3)—21no—
92C22×H(3,𝔽3)C2×Q812yes—
93C22×H(3,𝔽3)C2411yescharacteristic
94C2×#272—21no—
95C2×#272—21no—
96C2×#272C2×Q812yes—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)—22no—
99C4×H(3,𝔽3)—22no—
100C4×H(3,𝔽3)#4112yes—
101#1084—22no—
102#1084—21no—
103#1084—21no—
104#1084—22no—
105#1084C2×Q812yes—
106C3×C4⋊Q8—91no—
107C3×#125—92no—
108C3×C22⋊Q8—92no—
109C3×C22⋊Q8—91no—
110C3×C22⋊Q8—92no—
111C3×C22⋊Q8—91no—
112C3×C4×Q8—91no—
113C3×C4×D4—92no—
114C3×C4×D4—91no—
115C2×C3×C4⋊C4—34no—
116C2×C3×C4⋊C4—91no—
117C2×C3×C4⋊C4C3⋊D311yescharacteristic
118C4×#67—34no—
119C4×#67—38no—
120#924—38no—
121#924—34no—
122#897—34no—
123#896—38no—
124D3×C4⋊C4—34no—
125#889—38no—
126#888—34no—
127#887—34no—
128C23×C32C22×D314yes—
129C22×C3×D3—34no—
130C2×C32×C4—28no—
131C2×C32×C4—48no—
132C2×C32×C4C22×D318yes—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D318yes—
135C2×C32×C4C22×D314yes—
136C3×#67—316no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—38no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—38no—
142C3×C4×D3—68no—
143C3×C4×D3—128no—
144C3×Dic6—68no—
145C3×Dic6—128no—
146#554—91noSylow
147C2×H(3,𝔽3)—22no—
148C2×H(3,𝔽3)C22×Q811yescharacteristic
149C2×H(3,𝔽3)C2×#4111yescharacteristic
150C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
151#272—22no—
152C2×C3×Q8—91no—
153C2×C3×Q8—92no—
154C2×C3×D4—91no—
155C22×C3×C4—34no—
156C22×C3×C4—38no—
157C22×C3×C4—91no—
158C22×C3×C4—92no—
159C22×C3×C4C2×C3⋊D311yescharacteristic
160C22×C3×C4C2×C3⋊D312yes—
161C2×#67—34no—
162C2×C4×D3—34no—
163C2×C4×D3—38no—
164C2×Dic6—34no—
165C2×Dic6—38no—
166C3×C4⋊C4—34no—
167C3×C4⋊C4—34no—
168C3×C4⋊C4—38no—
169C3×C4⋊C4—92no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4—92no—
172C3×C4⋊C4—92no—
173C3×C4⋊C4—92no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—92no—
176C3×C4⋊C4C2×C3⋊D311yescharacteristic
177C3×C4⋊C4C2×C3⋊D312yes—
178C3×C4⋊C4C2×C3⋊D311yescharacteristic
179C3×C22⋊C4—92no—
180C3×C22⋊C4—91no—
181C3×C22⋊C4—92no—
182C3×C22⋊C4—91no—
183C3×C42—92no—
184C3×C42—91no—
185#217—38no—
186#217—34no—
187(C2×D3)⋊C4—34no—
188(C2×D3)⋊C4—38no—
189C4⋊Dic3—34no—
190C4⋊Dic3—38no—
191Dic3⋊C4—38no—
192Dic3⋊C4—38no—
193Dic3⋊C4—34no—
194Dic3⋊C4—38no—
195Dic3⋊C4—38no—
196C4×Dic3—34no—
197C4×Dic3—38no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32D3×Q818yes—
201C22×C32C23×D314yes—
202C2×C3×D3—38no—
203C2×C3×D3—64no—
204C2×C3×D3—64no—
205C32×C4—24no—
206C32×C4—28no—
207C32×C4—28no—
208C32×C4#23518yes—
209C3×Dic3—38no—
210C3×Dic3—68no—
211C3×Dic3—64no—
212C3×Dic3—64no—
213C3×Dic3—68no—
214C4⋊Q8—91no—
215#125—92no—
216C22⋊Q8—92no—
217C22⋊Q8—91no—
218C22⋊Q8—92no—
219C22⋊Q8—91no—
220C4×Q8—91no—
221C4×D4—92no—
222C4×D4—91no—
223C2×C4⋊C4—91no—
224C2×C4⋊C4#27211yescharacteristic, complemented
225H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
226C23×C3—34no—
227C23×C3—91no—
228C23×C3C22×C3⋊D311yescharacteristic
229C22×D3—34no—
230C3×Q8—182no—
231C3×Q8—362no—
232C3×D4—94no—
233C2×C3×C4—22no—
234C2×C3×C4—34no—
235C2×C3×C4—38no—
236C2×C3×C4—34no—
237C2×C3×C4—38no—
238C2×C3×C4—42no—
239C2×C3×C4—68no—
240C2×C3×C4—91no—
241C2×C3×C4—92no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—92no—
245C2×C3×C4—128no—
246C2×C3×C4—182no—
247C2×C3×C4—362no—
248C2×C3×C4C22×C3⋊D312yes—
249C2×C3×C4C22×C3⋊D311yescharacteristic
250C2×C3×C4C22×C3⋊D312yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252#67—316no—
253C2×Dic3—34no—
254C2×Dic3—38no—
255C2×Dic3—34no—
256C2×Dic3—34no—
257C2×Dic3—38no—
258C4×D3—68no—
259C4×D3—128no—
260Dic6—68no—
261Dic6—128no—
262C2×C32—28no—
263C2×C32C2×#23514yes—
264C2×C32#100414yes—
265C2×C32C2×D3×Q814yes—
266C3×D3—68no—
267C2×Q8—91no—
268C2×Q8—92no—
269C2×D4—91no—
270C22×C4—91no—
271C22×C4—92no—
272C22×C4C2×#27211yescharacteristic
273C22×C4C2×#27212yes—
274C4⋊C4—92no—
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—92no—
278C4⋊C4—92no—
279C4⋊C4—92no—
280C4⋊C4—92no—
281C4⋊C4C2×#27211yescharacteristic
282C4⋊C4C2×#27212yes—
283C4⋊C4C2×#27211yescharacteristic
284C22⋊C4—92no—
285C22⋊C4—91no—
286C22⋊C4—92no—
287C22⋊C4—91no—
288C42—92no—
289C42—91no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—34no—
293C22×C3—38no—
294C22×C3—64no—
295C22×C3—64no—
296C22×C3—92no—
297C22×C3—181no—
298C22×C3—181no—
299C22×C3Q8×C3⋊D312yes—
300C22×C3C23×C3⋊D311yescharacteristic
301C2×D3—38no—
302C2×D3—64no—
303C2×D3—64no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—38no—
308C3×C4—68no—
309C3×C4—68no—
310C3×C4—64no—
311C3×C4—92no—
312C3×C4—181no—
313C3×C4—182no—
314C3×C4—182no—
315C3×C4—181no—
316C3×C4#382812yes—
317Dic3—38no—
318Dic3—68no—
319Dic3—64no—
320Dic3—68no—
321Dic3—64no—
322C32#551114yescomplemented
323C23—91no—
324C23C22×#27211yescharacteristic
325Q8—182no—
326Q8—362no—
327D4—94no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—91no—
333C2×C4—91no—
334C2×C4—92no—
335C2×C4—182no—
336C2×C4—362no—
337C2×C4C22×#27212yes—
338C2×C4C22×#27211yescharacteristic
339C2×C4C22×#27212yes—
340C2×C4C22×#27211yescharacteristic
341C2×C3—22no—
342C2×C3—34no—
343C2×C3—34no—
344C2×C3—34no—
345C2×C3—68no—
346C2×C3—182no—
347C2×C3C2×#382811yescharacteristic
348C2×C3#6436111yescharacteristic
349C2×C3C2×Q8×C3⋊D311yescharacteristic
350D3—68no—
351C22—21no—
352C22—21no—
353C22—92no—
354C22—181no—
355C22—181no—
356C22Q8×#27212yes—
357C22C23×#27211yescharacteristic
358C4—21no—
359C4—22no—
360C4—22no—
361C4—92no—
362C4—181no—
363C4—182no—
364C4—182no—
365C4—181no—
366C4#8874012yes—
367C3—34no—
368C3#1059598211yescharacteristic
369C2—22no—
370C2—182no—
371C2C2×#8874011yescharacteristic
372C2#1172266411yescharacteristic
373C2C2×Q8×#27211yescharacteristic
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)#554
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. #11721824C2semidirect
5. #11721824C2semidirect
6. #11721765C2semidirect
7. #11721764C2semidirect
8. C4⋊C4×#272C2semidirect
9. #11721757C2semidirect
10. #11721756C2semidirect
11. #11721755C2semidirect
36. C2×C32×C4⋊C4D3semidirect
55. #6235Q8semidirect
224. C2×C4⋊C4#272semidirect
225. H(3,𝔽3)#554semidirect
322. C32#5511semidirect

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
C4×#62352
#117218242
#117217652
#117217642
C4⋊C4×#2722
#117217572
#117217562
#117217552
C3×#55113