Finite Group Databasebeta

← C2×#11721054 #13150588

Subgroups — C2×#11721054 #13150588

Order 1728.

999 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C2×#88538C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4C2×#88430C212yesmaximal, complemented
5#11721054C218yesmaximal, direct product factor
6C2×((C3⋊D3)⋊C3)⋊C8C212yesmaximal, complemented
7C2×#63817—31nomaximal
8C2×#63815—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
16#88538C2214yescomplemented
17C2×#6180C2212yes—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2214yescomplemented
21#88430C2218yescomplemented
22((C3⋊D3)⋊C3)⋊C8C2218yescomplemented
23C2×#3828D311yescharacteristic
24C2×C3×#235—31no—
25C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
26C2×C3×#208—31no—
27C2×C3×#208—31no—
28C2×C3×#208—61no—
29C22×C3×C3⋊C8—31no—
30C2×#3712—32no—
31C2×#3711—32no—
32#63817—38no—
33C2×#3710—32no—
34#63815—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#6180C2314yes—
49(C3⋊(C3⋊C8))⋊C3C2312yes—
50(C3⋊(C3⋊C8))⋊C3C2312yes—
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—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D314yescomplemented
72C3×#208—34no—
73C3×#208—34no—
74C3×#208—64no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—62no—
81#3712—38no—
82#3711—38no—
83#3710—38no—
84D3×C3⋊C8—38no—
85C22×H(3,𝔽3)—22no—
86C22×H(3,𝔽3)C22×C412yes—
87C22×H(3,𝔽3)C22×C411yescharacteristic
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3C22×C414yes—
90C4×H(3,𝔽3)C22×C412yes—
91C4×H(3,𝔽3)C22×C411yescharacteristic
92C4×H(3,𝔽3)C2411yescharacteristic
93(C3⋊Dic3)⋊C3C22×C414yes—
94C2×C3×#41—91no—
95C2×#235—31no—
96C2×#235—91no—
97C2×C3×M16—91no—
98C22×C3×C8—91no—
99#948—98no—
100C2×#208—31no—
101C2×#208—31no—
102C2×#208—61no—
103C2×#208—92no—
104C22×C3⋊C8—31no—
105C22×C3⋊C8—91no—
106C22×C3⋊C8—92no—
107#907—98no—
108#901—98no—
109C2×ΓL(1,𝔽25)—92no—
110C2×ΓL(1,𝔽25)—92no—
111C2×D3×C8—92no—
112C2×D3×C8—92no—
113C23×C32—21no—
114C23×C32C4×D311yescharacteristic
115C23×C32C2×Dic311yescharacteristic
116C22×C3⋊D3C2×Dic312yes—
117C22×C3×D3—32no—
118C2×C32×C4—21no—
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4C4×D311yescharacteristic
123C2×C32×C4C4×D312yes—
124C2×C32×C4C2×Dic311yescharacteristic
125C2×C32×C4C2×Dic312yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D312yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D312yes—
130#638C2×Dic318yes—
131C2×C3⋊Dic3C2×Dic312yes—
132C4⋊(C3⋊D3)C2×Dic314yes—
133C4×C3⋊D3C2×Dic314yes—
134C4×C3⋊D3C22×D314yes—
135#634C2×Dic314yes—
136C3×#67—38no—
137C2×C3×Dic3—32no—
138C3×D12—34no—
139C3×C4×D3—34no—
140C3×C4×D3—34no—
141C3×Dic6—34no—
142C3⋊(C3⋊C8)C22×D312yes—
143C3⋊(C3⋊C8)C22×D312yes—
144C3×C3⋊C8—32no—
145C3×C3⋊C8—32no—
146C3×C3⋊C8—34no—
147C3×C3⋊C8—34no—
148C3×C3⋊C8—64no—
149C2×#131—271noSylow
150C2×H(3,𝔽3)—22no—
151C2×H(3,𝔽3)#13112yescomplemented
152C2×H(3,𝔽3)C23×C411yescharacteristic
153(C3⋊D3)⋊C3—24no—
154C3×#41—94no—
155C3×#41—94no—
156C2×C3×Q8—91no—
157C2×C3×D4—91no—
158C2×C3×D4—92no—
159C22×C3×C4—31no—
160C22×C3×C4—31no—
161C22×C3×C4—61no—
162C22×C3×C4—92no—
163C22×C3×C4D3211yescharacteristic
164C2×#67—32no—
165C2×#67—92no—
166#235—34no—
167#235—34no—
168#235—94no—
169#235—94no—
170C2×D12—31no—
171C2×D12—91no—
172C2×C4×D3—32no—
173C2×C4×D3—92no—
174C2×Dic6—31no—
175C2×Dic6—91no—
176C3×M16—94no—
177C2×C3×C8—91no—
178C2×C3×C8—91no—
179C2×C3×C8—94no—
180C2×C3×C8—92no—
181#208—34no—
182#208—34no—
183#208—64no—
184#208—98no—
185C2×C3⋊C8—32no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—34no—
190C2×C3⋊C8—62no—
191C2×C3⋊C8—91no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—94no—
194C2×C3⋊C8—98no—
195ΓL(1,𝔽25)—98no—
196ΓL(1,𝔽25)—98no—
197D3×C8—98no—
198D3×C8—98no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—22no—
203C22×C32—42no—
204C22×C32C2×C4×D312yes—
205C22×C32C2×C4×D311yescharacteristic
206C22×C32C22×Dic312yes—
207C22×C32C22×Dic311yescharacteristic
208C2×C3⋊D3—24no—
209C2×C3⋊D3C22×Dic314yes—
210C2×C3×D3—34no—
211C2×C3×D3—64no—
212C32×C4—21no—
213C32×C4—22no—
214C32×C4—21no—
215C32×C4C2×C4×D312yes—
216C32×C4C2×C4×D311yescharacteristic
217C32×C4C22×Dic312yes—
218C32×C4C22×Dic311yescharacteristic
219C32×C4C23×D311yescharacteristic
220C32×C4C23×D311yescharacteristic
221C3⋊Dic3C22×Dic314yes—
222C3×Dic3—34no—
223C2×#41—91no—
224#131—278no—
225C2×M16—91no—
226C2×M16—272no—
227C22×C8—91no—
228C22×C8—272no—
229H(3,𝔽3)C2×#13111yescharacteristic, complemented, Sylow
230C23×C3—31no—
231C23×C3—31no—
232C23×C3—61no—
233C23×C3—92no—
234C23×C3D3×Dic311yescharacteristic
235C22×D3—32no—
236C22×D3—92no—
237C3×Q8—94no—
238C3×D4—94no—
239C3×D4—98no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—31no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—61no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4—61no—
252C2×C3×C4—94no—
253C2×C3×C4—94no—
254C2×C3×C4—92no—
255C2×C3×C4D3×Dic311yescharacteristic
256C2×C3×C4D3×Dic312yes—
257C2×C3×C4C2×D3212yes—
258C2×C3×C4C2×D3211yescharacteristic
259#67—38no—
260#67—98no—
261C2×Dic3—32no—
262C2×Dic3—92no—
263D12—34no—
264D12—94no—
265C4×D3—34no—
266C4×D3—34no—
267C4×D3—94no—
268C4×D3—94no—
269Dic6—34no—
270Dic6—94no—
271C3×C8—92no—
272C3×C8—92no—
273C3×C8—94no—
274C3⋊C8—34no—
275C3⋊C8—34no—
276C3⋊C8—32no—
277C3⋊C8—32no—
278C3⋊C8—64no—
279C3⋊C8—92no—
280C3⋊C8—92no—
281C2×C32—22no—
282C2×C32—22no—
283C2×C32—21no—
284C2×C32—22no—
285C2×C32—42no—
286C2×C32#90112yescomplemented
287C2×C32#94812yescomplemented
288C2×C32C22×C4×D311yescharacteristic
289C2×C32C23×Dic311yescharacteristic
290C3⋊D3—24no—
291C3×D3—64no—
292#41—94no—
293#41—94no—
294C2×Q8—91no—
295C2×D4—92no—
296C2×D4—91no—
297C22×C4—92no—
298C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
299M16—94no—
300M16—278no—
301C2×C8—94no—
302C2×C8—91no—
303C2×C8—91no—
304C2×C8—92no—
305C2×C8—278no—
306C22×C3—22no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—62no—
312C22×C3—62no—
313C22×C3—61no—
314C22×C3—62no—
315C22×C3—94no—
316C22×C3—122no—
317C22×C3—184no—
318C22×C3C2×D3×Dic312yes—
319C22×C3C2×D3×Dic311yescharacteristic
320C2×D3—34no—
321C2×D3—64no—
322C2×D3—94no—
323C2×D3—184no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—32no—
327C3×C4—31no—
328C3×C4—31no—
329C3×C4—32no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—62no—
333C3×C4—94no—
334C3×C4C2×D3×Dic312yes—
335C3×C4C2×D3×Dic311yescharacteristic
336C3×C4C22×D3211yescharacteristic
337Dic3—34no—
338Dic3—94no—
339C32—21no—
340C32C2×#90111yescharacteristic, complemented
341C32C2×#94811yescharacteristic, complemented
342C23—92no—
343C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
344Q8—94no—
345D4—98no—
346D4—94no—
347C2×C4—92no—
348C2×C4—94no—
349C2×C4—94no—
350C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
351C2×C4((C3⋊D3)⋊C3)⋊C412yes—
352C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
353C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
354C8—92no—
355C8—92no—
356C8—94no—
357C2×C3—22no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—32no—
361C2×C3—32no—
362C2×C3—62no—
363C2×C3—62no—
364C2×C3—61no—
365C2×C3—62no—
366C2×C3—122no—
367C2×C3—184no—
368C2×C3#6381412yes—
369C2×C3C22×D3×Dic311yescharacteristic
370D3—64no—
371D3—184no—
372C22—22no—
373C22—94no—
374C22—184no—
375C22C2×((C3⋊D3)⋊C3)⋊C412yes—
376C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
377C4—94no—
378C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
379C4C2×((C3⋊D3)⋊C3)⋊C412yes—
380C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C3—31no—
382C3—31no—
383C3—61no—
384C3C2×#6381411yescharacteristic
385C2—22no—
386C2—184no—
387C2#1172105412yesdirect product factor
388C2C22×((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. C2×#88538C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. C2×#88430C2semidirect
5. #11721054C2direct (N × H)
6. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
16. #88538C22semidirect
20. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
21. #88430C22semidirect
22. ((C3⋊D3)⋊C3)⋊C8C22semidirect
25. C22×C3⋊(C3⋊C8)D3semidirect
71. C2×C3⋊(C3⋊C8)C2×D3semidirect
151. C2×H(3,𝔽3)#131semidirect
229. H(3,𝔽3)C2×#131semidirect
286. C2×C32#901semidirect
287. C2×C32#948semidirect
340. C32C2×#901semidirect
341. C32C2×#948semidirect
387. C2#11721054direct (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
C2×#885382
C22×(C3⋊(C3⋊C8))⋊C32
C2×#884302
#117210542
C2×((C3⋊D3)⋊C3)⋊C82
C2×#638173
C2×#638153