Finite Group Databasebeta

← C2×#64151⋊C3 #13153655

Subgroups — C2×#64151⋊C3 #13153655

Order 1728.

1059 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic
3#64151⋊C3C218yesmaximal, direct product factor
4C2×#3755⋊C3C212yesmaximal, complemented
5C2×#3756⋊C3C212yesmaximal, complemented
6C2×#3748⋊C3C211yesmaximal, characteristic, complemented
7C2×#64151C311yesmaximal, characteristic, complemented
8C2×C3×#942—31nomaximal
9C2×Q8×H(3,𝔽3)—21no—
10C2×Q8×H(3,𝔽3)—22no—
11C2×Q8×H(3,𝔽3)C2212yes—
12C2×Q8×H(3,𝔽3)C2214yescomplemented
13C22×C4×H(3,𝔽3)—21no—
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3—21no—
16#3828⋊C3—22no—
17#3828⋊C3C2214yes—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×#634⋊C3C2211yescharacteristic
21#3755⋊C3C2218yescomplemented
22#3756⋊C3C2218yescomplemented
23#3748⋊C3C2214yescomplemented
24C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
25C22×C32×Q8—21no—
26C22×C32×Q8D311yescharacteristic, complemented
27C22×C32×Q8C2×C311yescharacteristic, complemented
28C2×#3828C2×C311yescharacteristic
29C2×C3×#235—31no—
30C2×#3756C2×C312yescomplemented
31#64151C2×C318yescomplemented
32C2×#3755C2×C312yescomplemented
33C2×#3748C2×C311yescharacteristic, complemented
34C2×C3×#216—32no—
35C3×#942—38no—
36C2×C3×#215—32no—
37C2×C3×#208—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—22no—
41Q8×H(3,𝔽3)—24no—
42Q8×H(3,𝔽3)C2314yes—
43C2×C4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)—24no—
45C2×C4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)D412yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2312yes—
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)C2312yes—
53C4×(C3⋊D3)⋊C3—24no—
54#634⋊C3—21no—
55#634⋊C3C2312yes—
56(C3⋊(C3⋊C8))⋊C3C2314yes—
57C2×C3×#137—91no—
58C2×#942—31no—
59C2×#942—91no—
60C2×C32×Q8—22no—
61C2×C32×Q8—24no—
62C2×C32×Q8—22no—
63C2×C32×Q8—24no—
64C2×C32×Q8—21no—
65C2×C32×Q8—22no—
66C2×C32×Q8—21no—
67C2×C32×Q8—22no—
68C2×C32×Q8C2×D312yes—
69C2×C32×Q8C2×D314yescomplemented
70C2×C32×Q8C22×C312yes—
71C2×C32×Q8C22×C314yescomplemented
72C22×C32×C4—22no—
73C22×C32×C4—21no—
74C22×C32×C4—21no—
75C22×C32×C4—21no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C22×C311yescharacteristic
78C2×#638—21no—
79#3828—22no—
80#3828C22×C314yes—
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4×C3⋊D3—21no—
83C2×#634C22×C311yescharacteristic
84C2×C3×#67—61no—
85C3×#235—34no—
86C3×#235—62no—
87C2×C3×D12—31no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—31no—
90#3756C22×C318yescomplemented
91#3755C22×C318yescomplemented
92#3748C22×C314yescomplemented
93C2×C3⋊(C3⋊C8)C22×C312yes—
94C3×#216—38no—
95C3×#215—38no—
96C3×#208—34no—
97C2×C3×C3⋊C8—32no—
98C22×H(3,𝔽3)—22no—
99C22×H(3,𝔽3)C2×D412yes—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—42no—
103C4×H(3,𝔽3)—24no—
104C4×H(3,𝔽3)C2×D411yescharacteristic
105C4×H(3,𝔽3)C2×D412yes—
106C4×H(3,𝔽3)C2411yescharacteristic
107(C3⋊Dic3)⋊C3—22no—
108C2×C3×#41—91no—
109C22×C3×Q8—31no—
110C22×C3×Q8—31no—
111C22×C3×Q8—61no—
112C22×C3×Q8C3×D311yescharacteristic
113C2×#235—31no—
114C2×#235—91no—
115C3×#137—98no—
116C2×C3×Q16—92no—
117C2×C3×SD16—92no—
118C2×C3×M16—91no—
119C2×#216—32no—
120C2×#216—92no—
121#942—38no—
122#942—98no—
123C2×#215—32no—
124C2×#215—92no—
125C2×#208—31no—
126C2×#208—91no—
127C23×C32—21no—
128C23×C32#6711yescharacteristic
129C23×C32C3×D411yescharacteristic
130C22×C3⋊D3—21no—
131C22×C3×D3—61no—
132C32×Q8—24no—
133C32×Q8—28no—
134C32×Q8—22no—
135C32×Q8—24no—
136C32×Q8—22no—
137C32×Q8—24no—
138C32×Q8—24no—
139C32×Q8C22×D314yes—
140C32×Q8C23×C314yes—
141C2×C32×C4—24no—
142C2×C32×C4—28no—
143C2×C32×C4—22no—
144C2×C32×C4—24no—
145C2×C32×C4—22no—
146C2×C32×C4—24no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4#6711yescharacteristic
152C2×C32×C4#6712yes—
153C2×C32×C4C3×D411yescharacteristic
154C2×C32×C4C3×D412yes—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D312yes—
157C2×C32×C4C23×C311yescharacteristic
158C2×C32×C4C23×C312yes—
159#638—24no—
160C2×C3⋊Dic3—21no—
161C4⋊(C3⋊D3)—21no—
162C4⋊(C3⋊D3)C23×C312yes—
163C4×C3⋊D3—24no—
164#634—21no—
165#634C23×C312yes—
166C3×#67—64no—
167C2×C3×Dic3—61no—
168C3×D12—32no—
169C3×D12—61no—
170C3×C4×D3—64no—
171C3×Dic6—32no—
172C3×Dic6—61no—
173C3⋊(C3⋊C8)C23×C314yes—
174C3×C3⋊C8—34no—
175C2×#137—91noSylow
176C2×H(3,𝔽3)—22no—
177C2×H(3,𝔽3)#13712yescomplemented
178C2×H(3,𝔽3)C22×D411yescharacteristic
179(C3⋊D3)⋊C3—42no—
180C3×#41—94no—
181C3×#41—182no—
182C2×C3×Q8—21no—
183C2×C3×Q8—22no—
184C2×C3×Q8—32no—
185C2×C3×Q8—32no—
186C2×C3×Q8—34no—
187C2×C3×Q8—34no—
188C2×C3×Q8—61no—
189C2×C3×Q8—62no—
190C2×C3×Q8—61no—
191C2×C3×Q8—62no—
192C2×C3×Q8—64no—
193C2×C3×Q8—62no—
194C2×C3×Q8—62no—
195C2×C3×Q8—64no—
196C2×C3×Q8—91no—
197C2×C3×Q8C2×C3×D312yes—
198C2×C3×Q8C2×C3×D314yes—
199C2×C3×D4—91no—
200C2×C3×D4—181no—
201C22×C3×C4—21no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C22×C3×C4—62no—
207C22×C3×C4—61no—
208C22×C3×C4—181no—
209C22×C3×C4C2×C3×D311yescharacteristic
210C2×#67—61no—
211C2×#67—181no—
212#235—34no—
213#235—62no—
214#235—94no—
215#235—182no—
216C2×D12—31no—
217C2×D12—91no—
218C2×C4×D3—61no—
219C2×C4×D3—181no—
220C2×Dic6—31no—
221C2×Dic6—91no—
222C3×Q16—98no—
223C3×SD16—98no—
224C3×M16—94no—
225C2×C3×C8—92no—
226#216—38no—
227#216—98no—
228#215—38no—
229#215—98no—
230#208—34no—
231#208—94no—
232C2×C3⋊C8—32no—
233C2×C3⋊C8—92no—
234C22×C32—24no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32C2×#6712yes—
240C22×C32C2×#6711yescharacteristic
241C22×C32C2×C3×D412yes—
242C22×C32C2×C3×D411yescharacteristic
243C2×C3⋊D3—22no—
244C2×C3⋊D3—42no—
245C2×C3×D3—62no—
246C2×C3×D3—122no—
247C32×C4—28no—
248C32×C4—24no—
249C32×C4—24no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—22no—
253C32×C4C2×#6711yescharacteristic
254C32×C4C2×#6712yes—
255C32×C4C2×C3×D411yescharacteristic
256C32×C4C2×C3×D412yes—
257C32×C4C23×D311yescharacteristic
258C32×C4C24×C311yescharacteristic
259C3⋊Dic3—22no—
260C3×Dic3—62no—
261C2×#41—91no—
262C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
263#137—98no—
264C2×Q16—92no—
265C2×SD16—92no—
266C2×M16—91no—
267H(3,𝔽3)C2×#13711yescharacteristic, complemented, Sylow
268C23×C3—31no—
269C23×C3—31no—
270C23×C3—61no—
271C23×C3—181no—
272C23×C3C3×#6711yescharacteristic
273C22×D3—61no—
274C22×D3—181no—
275C3×Q8—22no—
276C3×Q8—24no—
277C3×Q8—34no—
278C3×Q8—34no—
279C3×Q8—62no—
280C3×Q8—62no—
281C3×Q8—64no—
282C3×Q8—64no—
283C3×Q8—64no—
284C3×Q8—68no—
285C3×Q8—64no—
286C3×Q8—92no—
287C3×Q8—181no—
288C3×Q8C22×C3×D314yes—
289C3×D4—92no—
290C3×D4—181no—
291C3×D4—184no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—31no—
295C2×C3×C4—32no—
296C2×C3×C4—31no—
297C2×C3×C4—32no—
298C2×C3×C4—31no—
299C2×C3×C4—32no—
300C2×C3×C4—31no—
301C2×C3×C4—32no—
302C2×C3×C4—64no—
303C2×C3×C4—64no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—68no—
307C2×C3×C4—64no—
308C2×C3×C4—62no—
309C2×C3×C4—61no—
310C2×C3×C4—62no—
311C2×C3×C4—61no—
312C2×C3×C4—184no—
313C2×C3×C4—181no—
314C2×C3×C4C3×#6711yescharacteristic
315C2×C3×C4C3×#6712yes—
316C2×C3×C4C22×C3×D311yescharacteristic
317C2×C3×C4C22×C3×D312yes—
318#67—64no—
319#67—184no—
320C2×Dic3—61no—
321C2×Dic3—181no—
322D12—32no—
323D12—61no—
324D12—92no—
325D12—181no—
326C4×D3—64no—
327C4×D3—184no—
328Dic6—32no—
329Dic6—61no—
330Dic6—92no—
331Dic6—181no—
332C3×C8—94no—
333C3⋊C8—34no—
334C3⋊C8—94no—
335C2×C32—24no—
336C2×C32—22no—
337C2×C32—22no—
338C2×C32—22no—
339C2×C32—21no—
340C2×C32#94212yescomplemented
341C2×C32C3×#13712yescomplemented
342C2×C32C22×#6711yescharacteristic
343C2×C32C22×C3×D411yescharacteristic
344C3⋊D3—42no—
345C3×D3—122no—
346#41—94no—
347#41—182no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—91no—
351C2×Q8C2×(C3⋊D3)⋊C312yes—
352C2×Q8C2×(C3⋊D3)⋊C314yescomplemented
353C2×D4—91no—
354C2×D4—181no—
355C22×C4—21no—
356C22×C4—181no—
357C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
358Q16—98no—
359SD16—98no—
360M16—94no—
361C2×C8—92no—
362C22×C3—22no—
363C22×C3—32no—
364C22×C3—32no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—62no—
368C22×C3—62no—
369C22×C3—64no—
370C22×C3—62no—
371C22×C3—61no—
372C22×C3—182no—
373C22×C3—362no—
374C22×C3C2×C3×#6712yes—
375C22×C3C2×C3×#6711yescharacteristic
376C2×D3—62no—
377C2×D3—122no—
378C2×D3—182no—
379C2×D3—362no—
380C3×C4—24no—
381C3×C4—31no—
382C3×C4—31no—
383C3×C4—31no—
384C3×C4—32no—
385C3×C4—32no—
386C3×C4—31no—
387C3×C4—64no—
388C3×C4—64no—
389C3×C4—68no—
390C3×C4—61no—
391C3×C4—62no—
392C3×C4—61no—
393C3×C4—182no—
394C3×C4C2×C3×#6711yescharacteristic
395C3×C4C2×C3×#6712yes—
396C3×C4C23×C3×D311yescharacteristic
397Dic3—62no—
398Dic3—182no—
399C32—21no—
400C32C2×#94211yescharacteristic, complemented
401C32C2×C3×#13711yescharacteristic, complemented
402C23—181no—
403C23#638⋊C311yescharacteristic
404Q8—22no—
405Q8—24no—
406Q8—92no—
407Q8—181no—
408Q8C22×(C3⋊D3)⋊C314yes—
409D4—92no—
410D4—184no—
411D4—181no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—181no—
415C2×C4—184no—
416C2×C4#638⋊C312yes—
417C2×C4#638⋊C311yescharacteristic
418C2×C4C22×(C3⋊D3)⋊C312yes—
419C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
420C8—94no—
421C2×C3—22no—
422C2×C3—31no—
423C2×C3—31no—
424C2×C3—32no—
425C2×C3—32no—
426C2×C3—62no—
427C2×C3—62no—
428C2×C3—64no—
429C2×C3—61no—
430C2×C3—62no—
431C2×C3—362no—
432C2×C3C3×#94212yes—
433C2×C3C22×C3×#6711yescharacteristic
434D3—122no—
435D3—362no—
436C22—22no—
437C22—182no—
438C22—362no—
439C22C2×#638⋊C312yes—
440C22C2×#638⋊C311yescharacteristic
441C4—24no—
442C4—182no—
443C4C2×#638⋊C312yes—
444C4C2×#638⋊C311yescharacteristic
445C4C23×(C3⋊D3)⋊C311yescharacteristic
446C3—31no—
447C3—31no—
448C3—61no—
449C3C2×C3×#94211yescharacteristic
450C2—22no—
451C2—362no—
452C2#64151⋊C312yesdirect product factor
453C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×#3828⋊C3C2
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C2×C3×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
3. #64151⋊C3C2direct (N × H)
4. C2×#3755⋊C3C2semidirect
5. C2×#3756⋊C3C2semidirect
6. C2×#3748⋊C3C2semidirect
7. C2×#64151C3semidirect
12. C2×Q8×H(3,𝔽3)C22semidirect
21. #3755⋊C3C22semidirect
22. #3756⋊C3C22semidirect
23. #3748⋊C3C22semidirect
26. C22×C32×Q8D3semidirect
27. C22×C32×Q8C2×C3semidirect
30. C2×#3756C2×C3semidirect
31. #64151C2×C3semidirect
32. C2×#3755C2×C3semidirect
33. C2×#3748C2×C3semidirect
69. C2×C32×Q8C2×D3semidirect
71. C2×C32×Q8C22×C3semidirect
90. #3756C22×C3semidirect
91. #3755C22×C3semidirect
92. #3748C22×C3semidirect
177. C2×H(3,𝔽3)#137semidirect
262. C22×Q8(C3⋊D3)⋊C3semidirect
267. H(3,𝔽3)C2×#137semidirect
340. C2×C32#942semidirect
341. C2×C32C3×#137semidirect
352. C2×Q8C2×(C3⋊D3)⋊C3semidirect
400. C32C2×#942semidirect
401. C32C2×C3×#137semidirect
452. C2#64151⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×#3828⋊C32
#64151⋊C32
C2×#3755⋊C32
C2×#3756⋊C32
C2×#3748⋊C32
C2×#641513
C2×C3×#9423