Finite Group Databasebeta

← #13144298

Subgroups — #13144298

Order 1728.

1023 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88552C211yesmaximal, characteristic
3C2×#88552C211yesmaximal, characteristic
4C2×C4⋊#1084C211yesmaximal, characteristic, complemented
5C2×#88540C211yesmaximal, characteristic, complemented
6C2×#88540C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8C3×#4948—34nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22⋊C4×H(3,𝔽3)—22no—
17#88552—22no—
18#88552—22no—
19C4⋊#1084—22no—
20#88540—22no—
21#88540—22no—
22C2×C32×C22⋊C4D314yes—
23C2×C3×#217—34no—
24C2×C3×#217—34no—
25C2×C3×C4⋊Dic3—34no—
26C2×C3×Dic3⋊C4—34no—
27C2×C3×Dic3⋊C4—34no—
28C3×C3⋊#95—34no—
29C23×H(3,𝔽3)—21no—
30C23×H(3,𝔽3)—21no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)—21no—
33C23×H(3,𝔽3)—21no—
34C23×H(3,𝔽3)—21no—
35C23×H(3,𝔽3)Q812yes—
36C23×H(3,𝔽3)C2311yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D412yes—
43C2×#1084—21no—
44C2×#1084—21no—
45C2×#1084—21no—
46C2×#1084—21no—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084—21no—
53C2×#1084D412yes—
54C2×#1084D412yes—
55C3×#396—91no—
56#4948—34no—
57C24×C32C2×D314yes—
58C22×C32×C4C2×D314yes—
59C22×C32×C4C2×D314yes—
60C22×C3×Dic3—34no—
61C22×C3×Dic3—34no—
62C22×C3×Dic3—34no—
63C22×C3×Dic3—34no—
64C32×C22⋊C4—28no—
65C3×#217—68no—
66C3×#217—68no—
67C3×C4⋊Dic3—68no—
68C3×Dic3⋊C4—68no—
69C3×Dic3⋊C4—68no—
70C22×H(3,𝔽3)—22no—
71C22×H(3,𝔽3)—22no—
72C22×H(3,𝔽3)—22no—
73C22×H(3,𝔽3)—41no—
74C22×H(3,𝔽3)—41no—
75C22×H(3,𝔽3)—41no—
76C22×H(3,𝔽3)—41no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C22×H(3,𝔽3)C2×Q811yescharacteristic
81C22×H(3,𝔽3)#4111yescharacteristic
82C22×H(3,𝔽3)#4111yescharacteristic
83C22×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)—22no—
85C4×H(3,𝔽3)—41no—
86#1084—22no—
87#1084—22no—
88#1084—41no—
89#1084—41no—
90C2×C3×C4⋊C4—91no—
91C2×C3×C4⋊C4—91no—
92C2×C3×C4⋊C4—91no—
93C2×C3×C22⋊C4—34no—
94C2×C3×C22⋊C4—91no—
95C2×C3×C22⋊C4—91no—
96C2×C3×C22⋊C4C3⋊D311yescharacteristic
97C2×#217—34no—
98C2×#217—34no—
99C2×C4⋊Dic3—34no—
100C2×Dic3⋊C4—34no—
101C2×Dic3⋊C4—34no—
102C3×#95—91no—
103C3⋊#95—34no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32—24no—
107C23×C32—24no—
108C23×C32—24no—
109C23×C32—24no—
110C23×C32Dic618yes—
111C23×C32C22×D314yes—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4#6718yes—
118C2×C3×Dic3—38no—
119C2×C3×Dic3—38no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129C2×C3×Dic3—64no—
130#396—91noSylow
131C2×H(3,𝔽3)—42no—
132C2×H(3,𝔽3)#12111yescharacteristic
133C2×H(3,𝔽3)#12111yescharacteristic
134C2×H(3,𝔽3)#12111yescharacteristic
135C2×H(3,𝔽3)C22⋊Q811yescharacteristic
136C2×H(3,𝔽3)C22⋊Q811yescharacteristic
137C2×H(3,𝔽3)C22⋊Q811yescharacteristic
138C2×H(3,𝔽3)#12611yescharacteristic
139C24×C3—34no—
140C24×C3C2×C3⋊D311yescharacteristic
141C22×C3×C4—34no—
142C22×C3×C4—34no—
143C22×C3×C4—91no—
144C22×C3×C4—91no—
145C22×C3×C4—91no—
146C22×C3×C4—91no—
147C22×C3×C4C2×C3⋊D311yescharacteristic
148C22×C3×C4C2×C3⋊D311yescharacteristic
149C22×Dic3—34no—
150C22×Dic3—34no—
151C22×Dic3—34no—
152C22×Dic3—34no—
153C3×C4⋊C4—182no—
154C3×C4⋊C4—182no—
155C3×C4⋊C4—182no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—68no—
158C3×C22⋊C4—182no—
159C3×C22⋊C4—182no—
160#217—68no—
161#217—68no—
162C4⋊Dic3—68no—
163Dic3⋊C4—68no—
164Dic3⋊C4—68no—
165C22×C32—28no—
166C22×C32—28no—
167C22×C32—28no—
168C22×C32—44no—
169C22×C32—44no—
170C22×C32—44no—
171C22×C32—44no—
172C22×C32C2×Dic614yes—
173C22×C32#23514yes—
174C22×C32D3×D414yes—
175C22×C32D3×D414yes—
176C22×C32#23714yes—
177C22×C32#23714yes—
178C22×C32C2×#6714yes—
179C32×C4—28no—
180C32×C4—44no—
181C3×Dic3—68no—
182C3×Dic3—68no—
183C3×Dic3—124no—
184C3×Dic3—124no—
185C2×C4⋊C4—91no—
186C2×C4⋊C4—91no—
187C2×C4⋊C4—91no—
188C2×C22⋊C4—91no—
189C2×C22⋊C4—91no—
190C2×C22⋊C4#27211yescharacteristic
191#95—91no—
192H(3,𝔽3)#39611yescharacteristic, complemented, Sylow
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—38no—
200C23×C3—34no—
201C23×C3—64no—
202C23×C3—64no—
203C23×C3—64no—
204C23×C3—64no—
205C23×C3—64no—
206C23×C3—64no—
207C23×C3#63412yes—
208C23×C3C22×C3⋊D311yescharacteristic
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—38no—
215C2×C3×C4—64no—
216C2×C3×C4—64no—
217C2×C3×C4—64no—
218C2×C3×C4—64no—
219C2×C3×C4—64no—
220C2×C3×C4—92no—
221C2×C3×C4—92no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4#63812yes—
233C2×Dic3—38no—
234C2×Dic3—38no—
235C2×Dic3—64no—
236C2×Dic3—64no—
237C2×Dic3—64no—
238C2×Dic3—64no—
239C2×Dic3—64no—
240C2×Dic3—64no—
241C2×Dic3—64no—
242C2×Dic3—64no—
243C2×Dic3—64no—
244C2×Dic3—64no—
245C2×C32—48no—
246C2×C32#87814yes—
247C2×C32#87814yes—
248C2×C32#87914yes—
249C2×C32#88414yes—
250C2×C32#92414yes—
251C2×C32#93814yes—
252C2×C32#93914yes—
253C24C2×#27211yescharacteristic
254C22×C4—91no—
255C22×C4—91no—
256C22×C4—91no—
257C22×C4—91no—
258C22×C4C2×#27211yescharacteristic
259C22×C4C2×#27211yescharacteristic
260C4⋊C4—182no—
261C4⋊C4—182no—
262C4⋊C4—182no—
263C22⋊C4—22no—
264C22⋊C4—182no—
265C22⋊C4—182no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—34no—
270C22×C3—34no—
271C22×C3—34no—
272C22×C3—34no—
273C22×C3—34no—
274C22×C3—34no—
275C22×C3—34no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—68no—
281C22×C3—68no—
282C22×C3—68no—
283C22×C3—124no—
284C22×C3—124no—
285C22×C3—124no—
286C22×C3—124no—
287C22×C3C2×#63411yescharacteristic
288C22×C3#382811yescharacteristic
289C22×C3D4×C3⋊D311yescharacteristic
290C22×C3D4×C3⋊D311yescharacteristic
291C22×C3#383011yescharacteristic
292C22×C3#383011yescharacteristic
293C22×C3C2×#63811yescharacteristic
294C3×C4—22no—
295C3×C4—41no—
296C3×C4—68no—
297C3×C4—124no—
298C3×C4—182no—
299C3×C4—182no—
300C3×C4—361no—
301C3×C4—361no—
302Dic3—68no—
303Dic3—68no—
304Dic3—124no—
305Dic3—124no—
306C32#494814yescomplemented
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23#622912yes—
314C23C22×#27211yescharacteristic
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—92no—
321C2×C4—92no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4#623512yes—
333C2×C3—34no—
334C2×C3—34no—
335C2×C3—34no—
336C2×C3—34no—
337C2×C3—34no—
338C2×C3—34no—
339C2×C3—34no—
340C2×C3—42no—
341C2×C3—128no—
342C2×C3#6408711yescharacteristic
343C2×C3#6408711yescharacteristic
344C2×C3#6408811yescharacteristic
345C2×C3#6409311yescharacteristic
346C2×C3#6413311yescharacteristic
347C2×C3#6414711yescharacteristic
348C2×C3#6414811yescharacteristic
349C22—22no—
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22C2×#622911yescharacteristic
357C22#8874011yescharacteristic
358C22D4×#27211yescharacteristic
359C22D4×#27211yescharacteristic
360C22#8874411yescharacteristic
361C22#8874411yescharacteristic
362C22C2×#623511yescharacteristic
363C4—22no—
364C4—41no—
365C4—182no—
366C4—182no—
367C4—361no—
368C4—361no—
369C3—34no—
370C3#1059264711yescharacteristic
371C2—42no—
372C2#1172173411yescharacteristic
373C2#1172173411yescharacteristic
374C2#1172173511yescharacteristic
375C2#1172174011yescharacteristic
376C2#1172182411yescharacteristic
377C2#1172184811yescharacteristic
378C2#1172184911yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, cosocleC23×H(3,𝔽3)C23
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)#396
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊#1084C2semidirect
5. C2×#88540C2semidirect
6. C2×#88540C2semidirect
7. #11720574C2semidirect
192. H(3,𝔽3)#396semidirect
306. C32#4948semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#885522
C2×C4⋊#10842
C2×#885402
#117205742
C3×#49483