Finite Group Databasebeta

← C2×#64098⋊C3 #13152430

Subgroups — C2×#64098⋊C3 #13152430

Order 1728.

883 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
3C2×#3750⋊C3C211yesmaximal, characteristic
4C2×#3750⋊C3C211yesmaximal, characteristic
5C2×#3750⋊C3C211yesmaximal, characteristic
6C2×#3750⋊C3C211yesmaximal, characteristic
7C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
8#64098⋊C3C218yesmaximal, direct product factor
9C2×#64098C311yesmaximal, characteristic, complemented
10C2×C3×#889—31nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2214yes—
20#3750⋊C3C2214yes—
21#3750⋊C3C2214yes—
22#3750⋊C3C2214yes—
23#3750⋊C3C2214yes—
24C4×(C3⋊Dic3)⋊C3C2214yes—
25C2×C32×C4⋊C4—21no—
26C2×C32×C4⋊C4D311yescharacteristic
27C2×C32×C4⋊C4C2×C311yescharacteristic
28C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
29C2×#3750C2×C311yescharacteristic
30C2×#3750C2×C311yescharacteristic
31C2×#3750C2×C311yescharacteristic
32C2×#3750C2×C311yescharacteristic
33C2×C4×C3⋊Dic3C2×C311yescharacteristic
34#64098C2×C318yescomplemented
35C2×C3×C4⋊Dic3—31no—
36C2×C3×Dic3⋊C4—31no—
37C2×C3×Dic3⋊C4—31no—
38C2×C3×Dic3⋊C4—31no—
39C2×C3×Dic3⋊C4—31no—
40C2×C3×C4×Dic3—31no—
41C3×#889—38no—
42C23×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2312yes—
47C2×C4×H(3,𝔽3)C2312yes—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×(C3⋊Dic3)⋊C3—22no—
50C2×(C3⋊Dic3)⋊C3—22no—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3Q814yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×C3×#125—91no—
58C2×#889—31no—
59C2×#889—91no—
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C22×C32×C4C22×C311yescharacteristic
69C22×C3⋊Dic3C22×C311yescharacteristic
70C22×C3⋊Dic3C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C32×C4⋊C4—24no—
78C32×C4⋊C4C2×D314yes—
79C32×C4⋊C4C22×C314yes—
80C4⋊(C3⋊Dic3)C22×C314yes—
81#3750C22×C314yes—
82#3750C22×C314yes—
83#3750C22×C314yes—
84#3750C22×C314yes—
85C4×C3⋊Dic3C22×C314yes—
86C3×C4⋊Dic3—34no—
87C3×Dic3⋊C4—34no—
88C3×Dic3⋊C4—34no—
89C3×Dic3⋊C4—34no—
90C3×Dic3⋊C4—34no—
91C3×C4×Dic3—34no—
92C22×H(3,𝔽3)C2×Q812yes—
93C22×H(3,𝔽3)#4112yes—
94C22×H(3,𝔽3)#4112yes—
95C22×H(3,𝔽3)C2411yescharacteristic
96C4×H(3,𝔽3)—22no—
97C4×H(3,𝔽3)—22no—
98C4×H(3,𝔽3)—22no—
99(C3⋊Dic3)⋊C3—22no—
100(C3⋊Dic3)⋊C3—22no—
101(C3⋊Dic3)⋊C3—22no—
102(C3⋊Dic3)⋊C3C2×Q814yes—
103C3×#125—98no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—31no—
106C2×C3×C4⋊C4—61no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C4⋊C4—91no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C4⋊C4—91no—
112C2×C3×C4⋊C4C3×D311yescharacteristic
113C2×C3×C42—91no—
114C2×C4⋊Dic3—31no—
115C2×C4⋊Dic3—91no—
116C2×Dic3⋊C4—31no—
117C2×Dic3⋊C4—31no—
118C2×Dic3⋊C4—31no—
119C2×Dic3⋊C4—31no—
120C2×Dic3⋊C4—91no—
121C2×Dic3⋊C4—91no—
122C2×Dic3⋊C4—91no—
123C2×Dic3⋊C4—91no—
124C2×C4×Dic3—31no—
125C2×C4×Dic3—91no—
126#889—38no—
127#889—98no—
128C23×C32—21no—
129C23×C32C22×D311yescharacteristic
130C23×C32C23×C311yescharacteristic
131C2×C32×C4—24no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—42no—
142C2×C32×C4—42no—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D312yes—
145C2×C32×C4C22×D312yes—
146C2×C32×C4C23×C312yes—
147C2×C32×C4C23×C312yes—
148C2×C32×C4C23×C312yes—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3C3×Q814yes—
153C2×C3⋊Dic3C23×C312yes—
154C2×C3⋊Dic3C23×C312yes—
155C2×C3⋊Dic3C23×C312yes—
156C2×C3⋊Dic3C23×C312yes—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×#125—91noSylow
166C2×H(3,𝔽3)#12514yescomplemented
167C2×H(3,𝔽3)C22×Q811yescharacteristic
168C2×H(3,𝔽3)C2×#4111yescharacteristic
169C2×H(3,𝔽3)C2×#4111yescharacteristic
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—61no—
177C22×C3×C4—61no—
178C22×C3×C4—61no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4—91no—
183C22×C3×C4C2×C3×D311yescharacteristic
184C22×C3×C4C2×C3×D311yescharacteristic
185C22×C3×C4C2×C3×D311yescharacteristic
186C22×Dic3—31no—
187C22×Dic3—31no—
188C22×Dic3—31no—
189C22×Dic3—31no—
190C22×Dic3—91no—
191C22×Dic3—91no—
192C22×Dic3—91no—
193C22×Dic3—91no—
194C3×C4⋊C4—34no—
195C3×C4⋊C4—34no—
196C3×C4⋊C4—64no—
197C3×C4⋊C4—94no—
198C3×C4⋊C4—94no—
199C3×C4⋊C4—94no—
200C3×C4⋊C4—94no—
201C3×C4⋊C4—94no—
202C3×C4⋊C4C2×C3×D314yes—
203C3×C42—94no—
204C4⋊Dic3—34no—
205C4⋊Dic3—94no—
206Dic3⋊C4—34no—
207Dic3⋊C4—34no—
208Dic3⋊C4—34no—
209Dic3⋊C4—34no—
210Dic3⋊C4—94no—
211Dic3⋊C4—94no—
212Dic3⋊C4—94no—
213Dic3⋊C4—94no—
214C4×Dic3—34no—
215C4×Dic3—94no—
216C22×C32—21no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32#23512yes—
221C22×C32#23712yes—
222C22×C32D3×Q812yes—
223C22×C32C2×C3×Q812yes—
224C22×C32C3×#4112yes—
225C22×C32C3×#4112yes—
226C22×C32C23×D311yescharacteristic
227C22×C32C24×C311yescharacteristic
228C32×C4—24no—
229C32×C4—22no—
230C32×C4—22no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—42no—
236C32×C4—42no—
237C3⋊Dic3—22no—
238C3⋊Dic3—22no—
239C3⋊Dic3—22no—
240C3⋊Dic3C2×C3×Q814yes—
241C3×Dic3—34no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C3×Dic3—62no—
245#125—98no—
246C2×C4⋊C4—91no—
247C2×C4⋊C4—91no—
248C2×C4⋊C4—91no—
249C2×C4⋊C4—91no—
250C2×C4⋊C4—91no—
251C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
252C2×C42—91no—
253H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
254C23×C3—31no—
255C23×C3—31no—
256C23×C3—61no—
257C23×C3C22×C3×D311yescharacteristic
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—32no—
262C2×C3×C4—32no—
263C2×C3×C4—32no—
264C2×C3×C4—32no—
265C2×C3×C4—32no—
266C2×C3×C4—32no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—64no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—92no—
278C2×C3×C4—94no—
279C2×C3×C4—92no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4—122no—
283C2×C3×C4—122no—
284C2×C3×C4—182no—
285C2×C3×C4—182no—
286C2×C3×C4—182no—
287C2×C3×C4C22×C3×D312yes—
288C2×C3×C4C22×C3×D312yes—
289C2×C3×C4C22×C3×D312yes—
290C2×Dic3—32no—
291C2×Dic3—34no—
292C2×Dic3—32no—
293C2×Dic3—32no—
294C2×Dic3—32no—
295C2×Dic3—62no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—94no—
299C2×Dic3—92no—
300C2×Dic3—92no—
301C2×Dic3—92no—
302C2×Dic3—92no—
303C2×Dic3—182no—
304C2×Dic3—182no—
305C2×Dic3—182no—
306C2×C32—24no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32#88914yescomplemented
311C2×C32C3×#12514yescomplemented
312C2×C32C2×#23511yescharacteristic
313C2×C32C2×#23711yescharacteristic
314C2×C32C2×D3×Q811yescharacteristic
315C2×C32C22×C3×Q811yescharacteristic
316C2×C32C2×C3×#4111yescharacteristic
317C2×C32C2×C3×#4111yescharacteristic
318C22×C4—91no—
319C22×C4—91no—
320C22×C4—91no—
321C22×C4—91no—
322C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
323C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
324C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
325C4⋊C4—94no—
326C4⋊C4—94no—
327C4⋊C4—94no—
328C4⋊C4—94no—
329C4⋊C4—94no—
330C4⋊C4C2×(C3⋊D3)⋊C314yes—
331C42—94no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—32no—
335C22×C3—32no—
336C22×C3—32no—
337C22×C3—32no—
338C22×C3—32no—
339C22×C3—32no—
340C22×C3—61no—
341C22×C3—62no—
342C22×C3—62no—
343C22×C3—62no—
344C22×C3C3×#23512yes—
345C22×C3C3×#23712yes—
346C22×C3C3×D3×Q812yes—
347C22×C3C23×C3×D311yescharacteristic
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—62no—
352C3×C4—62no—
353C3×C4—62no—
354C3×C4—62no—
355C3×C4—62no—
356C3×C4—62no—
357C3×C4—64no—
358C3×C4—94no—
359C3×C4—122no—
360C3×C4—122no—
361C3×C4—182no—
362C3×C4—182no—
363C3×C4—182no—
364Dic3—34no—
365Dic3—62no—
366Dic3—62no—
367Dic3—62no—
368Dic3—94no—
369Dic3—182no—
370Dic3—182no—
371Dic3—182no—
372C32—21no—
373C32C2×#88911yescharacteristic, complemented
374C32C2×C3×#12511yescharacteristic, complemented
375C23C22×(C3⋊D3)⋊C311yescharacteristic
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—92no—
380C2×C4—94no—
381C2×C4—92no—
382C2×C4—92no—
383C2×C4—92no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4C22×(C3⋊D3)⋊C312yes—
388C2×C4C22×(C3⋊D3)⋊C312yes—
389C2×C4C22×(C3⋊D3)⋊C312yes—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—34no—
397C2×C3—34no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—64no—
402C2×C3C3×#88914yes—
403C2×C3C2×C3×#23511yescharacteristic
404C2×C3C2×C3×#23711yescharacteristic
405C2×C3C2×C3×D3×Q811yescharacteristic
406C22#3828⋊C312yes—
407C22#3830⋊C312yes—
408C22Q8×(C3⋊D3)⋊C312yes—
409C22C23×(C3⋊D3)⋊C311yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—94no—
414C4—182no—
415C4—182no—
416C4—182no—
417C3—31no—
418C3—31no—
419C3—61no—
420C3C2×C3×#88911yescharacteristic
421C2#64098⋊C314yesdirect product factor
422C2C2×#3828⋊C311yescharacteristic
423C2C2×#3830⋊C311yescharacteristic
424C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC23×H(3,𝔽3)C23
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#125
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #64098⋊C3C2direct (N × H)
9. C2×#64098C3semidirect
34. #64098C2×C3semidirect
166. C2×H(3,𝔽3)#125semidirect
253. H(3,𝔽3)C2×#125semidirect
310. C2×C32#889semidirect
311. C2×C32C3×#125semidirect
373. C32C2×#889semidirect
374. C32C2×C3×#125semidirect
421. C2#64098⋊C3direct (N × H)

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
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#64098⋊C32
C2×#640983
C2×C3×#8893