Finite Group Databasebeta

← C2×#11721055 #13150586

Subgroups — C2×#11721055 #13150586

Order 1728.

1007 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C22×#6180C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4C2×#88430C212yesmaximal, complemented
5#11721055C218yesmaximal, direct product factor
6C2×((C3⋊D3)⋊C3)⋊C8C212yesmaximal, complemented
7C2×#63817—31nomaximal
8C2×#63814—31nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3C412yes—
11#3828⋊C3C414yes—
12#3828⋊C3C2214yes—
13C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×#634⋊C3C411yescharacteristic
16C2×#6180C2211yescharacteristic
17C2×#6180C2211yescharacteristic
18C2×#6180C2214yescomplemented
19#3748⋊C3C2214yescomplemented
20C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
21#88430C2218yescomplemented
22((C3⋊D3)⋊C3)⋊C8C2218yescomplemented
23C2×#3828D311yescharacteristic
24C2×C3×#235—31no—
25C2×#3748D311yescharacteristic, complemented
26C2×C3×#208—31no—
27C22×C3×C3⋊C8—31no—
28C22×C3×C3⋊C8—31no—
29C22×C3×C3⋊C8—61no—
30C2×#3712—32no—
31C2×#3711—32no—
32#63817—38no—
33C2×#3710—32no—
34#63814—38no—
35C2×D3×C3⋊C8—32no—
36C23×H(3,𝔽3)C2×C411yescharacteristic
37C22×(C3⋊D3)⋊C3C2×C412yes—
38C2×C4×H(3,𝔽3)C2×C411yescharacteristic
39C2×C4×H(3,𝔽3)C2×C412yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2312yes—
42#638⋊C3C2×C418yes—
43C2×(C3⋊Dic3)⋊C3C2×C412yes—
44C4⋊((C3⋊D3)⋊C3)C2×C414yes—
45C4×(C3⋊D3)⋊C3C2×C414yes—
46C4×(C3⋊D3)⋊C3C2314yes—
47#634⋊C3C2×C414yes—
48#6180C2312yes—
49#6180C2312yes—
50(C3⋊(C3⋊C8))⋊C3C2314yes—
51C2×#948—91no—
52C2×#907—91no—
53C2×#901—91no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C2×#638Dic312yes—
58#3828Dic314yes—
59#3828C2×D314yes—
60C2×C4⋊(C3⋊D3)Dic311yescharacteristic
61C2×C4×C3⋊D3C2×D312yes—
62C2×#634Dic311yescharacteristic
63C2×C3×#67—32no—
64C3×#235—34no—
65C3×#235—34no—
66C2×C3×D12—31no—
67C2×C3×C4×D3—32no—
68C2×C3×Dic6—31no—
69#3748C2×D314yescomplemented
70C2×C3⋊(C3⋊C8)C2×D312yes—
71C3×#208—34no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—34no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—61no—
80C2×C3×C3⋊C8—61no—
81C2×C3×C3⋊C8—64no—
82#3712—38no—
83#3711—38no—
84#3710—38no—
85D3×C3⋊C8—38no—
86C22×H(3,𝔽3)—22no—
87C22×H(3,𝔽3)C22×C412yes—
88C22×H(3,𝔽3)C22×C411yescharacteristic
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3C22×C414yes—
91C4×H(3,𝔽3)C22×C412yes—
92C4×H(3,𝔽3)C22×C411yescharacteristic
93C4×H(3,𝔽3)C2411yescharacteristic
94(C3⋊Dic3)⋊C3C22×C414yes—
95C2×C3×#41—91no—
96C2×#235—31no—
97C2×#235—91no—
98C2×C3×M16—91no—
99C22×C3×C8—91no—
100#948—98no—
101C2×#208—31no—
102C2×#208—91no—
103C2×#208—92no—
104C22×C3⋊C8—31no—
105C22×C3⋊C8—31no—
106C22×C3⋊C8—61no—
107C22×C3⋊C8—92no—
108#907—98no—
109#901—98no—
110C2×ΓL(1,𝔽25)—92no—
111C2×ΓL(1,𝔽25)—92no—
112C2×D3×C8—92no—
113C2×D3×C8—92no—
114C23×C32—21no—
115C23×C32C4×D311yescharacteristic
116C23×C32C2×Dic311yescharacteristic
117C22×C3⋊D3C2×Dic312yes—
118C22×C3×D3—32no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4C4×D311yescharacteristic
124C2×C32×C4C4×D312yes—
125C2×C32×C4C2×Dic311yescharacteristic
126C2×C32×C4C2×Dic312yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D312yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D312yes—
131#638C2×Dic318yes—
132C2×C3⋊Dic3C2×Dic312yes—
133C4⋊(C3⋊D3)C2×Dic314yes—
134C4×C3⋊D3C2×Dic314yes—
135C4×C3⋊D3C22×D314yes—
136#634C2×Dic314yes—
137C3×#67—38no—
138C2×C3×Dic3—32no—
139C3×D12—34no—
140C3×C4×D3—34no—
141C3×C4×D3—34no—
142C3×Dic6—34no—
143C3⋊(C3⋊C8)C22×D314yes—
144C3×C3⋊C8—34no—
145C3×C3⋊C8—32no—
146C3×C3⋊C8—32no—
147C3×C3⋊C8—32no—
148C3×C3⋊C8—32no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—62no—
151C2×#131—271noSylow
152C2×H(3,𝔽3)—22no—
153C2×H(3,𝔽3)#13112yescomplemented
154C2×H(3,𝔽3)C23×C411yescharacteristic
155(C3⋊D3)⋊C3—24no—
156C3×#41—94no—
157C3×#41—94no—
158C2×C3×Q8—91no—
159C2×C3×D4—91no—
160C2×C3×D4—92no—
161C22×C3×C4—31no—
162C22×C3×C4—31no—
163C22×C3×C4—61no—
164C22×C3×C4—92no—
165C22×C3×C4D3211yescharacteristic
166C2×#67—32no—
167C2×#67—92no—
168#235—34no—
169#235—34no—
170#235—94no—
171#235—94no—
172C2×D12—31no—
173C2×D12—91no—
174C2×C4×D3—32no—
175C2×C4×D3—92no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×M16—94no—
179C2×C3×C8—92no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C2×C3×C8—94no—
183#208—34no—
184#208—94no—
185#208—98no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—34no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—34no—
192C2×C3⋊C8—32no—
193C2×C3⋊C8—64no—
194C2×C3⋊C8—61no—
195C2×C3⋊C8—61no—
196C2×C3⋊C8—92no—
197C2×C3⋊C8—98no—
198ΓL(1,𝔽25)—98no—
199ΓL(1,𝔽25)—98no—
200D3×C8—98no—
201D3×C8—98no—
202C22×C32—24no—
203C22×C32—22no—
204C22×C32—22no—
205C22×C32—22no—
206C22×C32—21no—
207C22×C32C2×C4×D312yes—
208C22×C32C2×C4×D311yescharacteristic
209C22×C32C22×Dic312yes—
210C22×C32C22×Dic311yescharacteristic
211C2×C3⋊D3—24no—
212C2×C3⋊D3C22×Dic314yes—
213C2×C3×D3—34no—
214C2×C3×D3—64no—
215C32×C4—21no—
216C32×C4—22no—
217C32×C4—21no—
218C32×C4C2×C4×D312yes—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4C22×Dic312yes—
221C32×C4C22×Dic311yescharacteristic
222C32×C4C23×D311yescharacteristic
223C32×C4C23×D311yescharacteristic
224C3⋊Dic3C22×Dic314yes—
225C3×Dic3—34no—
226C2×#41—91no—
227#131—278no—
228C2×M16—91no—
229C2×M16—272no—
230C22×C8—91no—
231C22×C8—272no—
232H(3,𝔽3)C2×#13111yescharacteristic, complemented, Sylow
233C23×C3—31no—
234C23×C3—31no—
235C23×C3—61no—
236C23×C3—92no—
237C23×C3D3×Dic311yescharacteristic
238C22×D3—32no—
239C22×D3—92no—
240C3×Q8—94no—
241C3×D4—94no—
242C3×D4—98no—
243C2×C3×C4—31no—
244C2×C3×C4—32no—
245C2×C3×C4—31no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—32no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—61no—
252C2×C3×C4—62no—
253C2×C3×C4—62no—
254C2×C3×C4—61no—
255C2×C3×C4—94no—
256C2×C3×C4—94no—
257C2×C3×C4—92no—
258C2×C3×C4D3×Dic311yescharacteristic
259C2×C3×C4D3×Dic312yes—
260C2×C3×C4C2×D3212yes—
261C2×C3×C4C2×D3211yescharacteristic
262#67—38no—
263#67—98no—
264C2×Dic3—32no—
265C2×Dic3—92no—
266D12—34no—
267D12—94no—
268C4×D3—34no—
269C4×D3—34no—
270C4×D3—94no—
271C4×D3—94no—
272Dic6—34no—
273Dic6—94no—
274C3×C8—94no—
275C3×C8—92no—
276C3×C8—92no—
277C3⋊C8—32no—
278C3⋊C8—32no—
279C3⋊C8—32no—
280C3⋊C8—32no—
281C3⋊C8—34no—
282C3⋊C8—62no—
283C3⋊C8—62no—
284C3⋊C8—94no—
285C2×C32—24no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32#90712yescomplemented
291C2×C32#94812yescomplemented
292C2×C32C22×C4×D311yescharacteristic
293C2×C32C23×Dic311yescharacteristic
294C3⋊D3—24no—
295C3×D3—64no—
296#41—94no—
297#41—94no—
298C2×Q8—91no—
299C2×D4—92no—
300C2×D4—91no—
301C22×C4—92no—
302C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
303M16—94no—
304M16—278no—
305C2×C8—94no—
306C2×C8—91no—
307C2×C8—91no—
308C2×C8—92no—
309C2×C8—278no—
310C22×C3—22no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—32no—
314C22×C3—32no—
315C22×C3—62no—
316C22×C3—62no—
317C22×C3—64no—
318C22×C3—62no—
319C22×C3—61no—
320C22×C3—94no—
321C22×C3—184no—
322C22×C3C2×D3×Dic312yes—
323C22×C3C2×D3×Dic311yescharacteristic
324C2×D3—34no—
325C2×D3—64no—
326C2×D3—94no—
327C2×D3—184no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—32no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—32no—
334C3×C4—61no—
335C3×C4—61no—
336C3×C4—62no—
337C3×C4—94no—
338C3×C4C2×D3×Dic312yes—
339C3×C4C2×D3×Dic311yescharacteristic
340C3×C4C22×D3211yescharacteristic
341Dic3—34no—
342Dic3—94no—
343C32—21no—
344C32C2×#90711yescharacteristic, complemented
345C32C2×#94811yescharacteristic, complemented
346C23—92no—
347C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
348Q8—94no—
349D4—98no—
350D4—94no—
351C2×C4—92no—
352C2×C4—94no—
353C2×C4—94no—
354C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
355C2×C4((C3⋊D3)⋊C3)⋊C412yes—
356C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
357C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
358C8—92no—
359C8—92no—
360C8—94no—
361C2×C3—22no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—32no—
365C2×C3—32no—
366C2×C3—62no—
367C2×C3—62no—
368C2×C3—64no—
369C2×C3—61no—
370C2×C3—62no—
371C2×C3—184no—
372C2×C3#6381512yes—
373C2×C3C22×D3×Dic311yescharacteristic
374D3—64no—
375D3—184no—
376C22—22no—
377C22—94no—
378C22—184no—
379C22C2×((C3⋊D3)⋊C3)⋊C412yes—
380C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
381C4—94no—
382C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
383C4C2×((C3⋊D3)⋊C3)⋊C412yes—
384C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3C2×#6381511yescharacteristic
389C2—22no—
390C2—184no—
391C2#1172105512yesdirect product factor
392C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC2×#3828⋊C3C2
cosocleC4×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#131
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6180C2semidirect
3. C2×#3748⋊C3C2semidirect
4. C2×#88430C2semidirect
5. #11721055C2direct (N × H)
6. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
18. C2×#6180C22semidirect
19. #3748⋊C3C22semidirect
21. #88430C22semidirect
22. ((C3⋊D3)⋊C3)⋊C8C22semidirect
25. C2×#3748D3semidirect
69. #3748C2×D3semidirect
153. C2×H(3,𝔽3)#131semidirect
232. H(3,𝔽3)C2×#131semidirect
290. C2×C32#907semidirect
291. C2×C32#948semidirect
344. C32C2×#907semidirect
345. C32C2×#948semidirect
391. C2#11721055direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×#61802
C2×#3748⋊C32
C2×#884302
#117210552
C2×((C3⋊D3)⋊C3)⋊C82
C2×#638173
C2×#638143