Finite Group Databasebeta

← C2×#11721107 #13150627

Subgroups — C2×#11721107 #13150627

Order 1728.

1283 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6229C211yesmaximal, characteristic, complemented
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×#88450C212yesmaximal
5C2×#88448C212yesmaximal, complemented
6#11721107C218yesmaximal, direct product factor
7C2×#63871—31nomaximal
8C2×#63856—31nomaximal
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C2×#6229—24no—
13C2×#6229C2214yescomplemented
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3—24no—
17C2×C4×(C3⋊D3)⋊C3C2214yes—
18C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
19#88450C2218yes—
20#88448C2218yescomplemented
21C22×C4×C3⋊D3D311yescharacteristic
22C22×C3×C4×D3—31no—
23C22×C3×Dic6—31no—
24C22×C3×Dic6—31no—
25C22×C3×Dic6—61no—
26C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
27C2×C3×C4⋊Dic3—31no—
28C2×#3724—32no—
29C2×Dic3⋊Dic3—32no—
30C2×#3722—32no—
31C2×(C2×D3)⋊Dic3—32no—
32#63871—38no—
33#63856—38no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3—22no—
37C22×(C3⋊D3)⋊C3Q814yescomplemented
38C22×(C3⋊D3)⋊C3C2312yes—
39C2×C4×H(3,𝔽3)D414yes—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×#1084—24no—
42C2×#1084C2314yes—
43#6229—28no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C4×(C3⋊D3)⋊C3—28no—
49C2×#946—91no—
50C2×#924—91no—
51C2×#897—91no—
52C23×C3⋊D3C2×D311yescharacteristic
53C23×C3×D3—31no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C3⋊Dic3C2×D311yescharacteristic
58C22×C3⋊Dic3C2×D312yes—
59C2×C4×C3⋊D3—24no—
60C2×C4×C3⋊D3C2×D314yes—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—31no—
65C22×C3×Dic3—62no—
66C2×C3×C4×D3—34no—
67C2×C3×C4×D3—64no—
68C2×C3×Dic6—34no—
69C2×C3×Dic6—34no—
70C2×C3×Dic6—64no—
71C2×C3×Dic6—68no—
72C2×C3×Dic6—64no—
73C2×C3×Dic6—64no—
74C4⋊(C3⋊Dic3)C2×D314yescomplemented
75C3×C4⋊Dic3—34no—
76#3724—38no—
77Dic3⋊Dic3—38no—
78#3722—38no—
79(C2×D3)⋊Dic3—38no—
80C22×H(3,𝔽3)C2×D412yes—
81C22×H(3,𝔽3)C2×Q812yes—
82C22×H(3,𝔽3)#4112yes—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×(C3⋊D3)⋊C3—28no—
85C2×(C3⋊D3)⋊C3—22no—
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
88C4×H(3,𝔽3)C2×D414yes—
89#1084—24no—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—24no—
92C22×C3×Q8—91no—
93C23×C3×C4—91no—
94C22×C4×D3—31no—
95C22×C4×D3—91no—
96C22×Dic6—31no—
97C22×Dic6—31no—
98C22×Dic6—61no—
99C2×C3×C4⋊C4—91no—
100C2×#217—92no—
101#946—98no—
102C2×(C2×D3)⋊C4—92no—
103C2×(C2×D3)⋊C4—92no—
104C2×C4⋊Dic3—31no—
105C2×C4⋊Dic3—91no—
106C2×C4⋊Dic3—92no—
107#924—98no—
108C2×Dic3⋊C4—92no—
109C2×Dic3⋊C4—92no—
110#897—98no—
111C23×C32—21no—
112C23×C32C22×D311yescharacteristic
113C23×C32C22×D311yescharacteristic
114C22×C3⋊D3—22no—
115C22×C3⋊D3—22no—
116C22×C3⋊D3Dic614yescomplemented
117C22×C3⋊D3C22×D312yes—
118C22×C3×D3—34no—
119C22×C3×D3—32no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C2×C32×C4—24no—
123C2×C32×C4—22no—
124C2×C32×C4D1214yes—
125C2×C32×C4#6714yes—
126C2×C32×C4C22×D312yes—
127C2×C32×C4C22×D312yes—
128C2×C3⋊Dic3—24no—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3C22×D314yes—
131C2×C3⋊Dic3C22×D312yes—
132C4×C3⋊D3—28no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3—68no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—64no—
143C3×C4×D3—68no—
144C3×Dic6—616no—
145C3×Dic6—68no—
146C3×Dic6—68no—
147C2×C22⋊Q8—271noSylow
148C2×H(3,𝔽3)C22⋊Q814yescomplemented
149C2×H(3,𝔽3)C22×D411yescharacteristic
150C2×H(3,𝔽3)C22×Q811yescharacteristic
151C2×H(3,𝔽3)C2×#4111yescharacteristic
152(C3⋊D3)⋊C3—24no—
153C24×C3—91no—
154C23×D3—31no—
155C23×D3—91no—
156C2×C3×Q8—94no—
157C2×C3×Q8—184no—
158C22×C3×C4—31no—
159C22×C3×C4—31no—
160C22×C3×C4—61no—
161C22×C3×C4—94no—
162C22×C3×C4—91no—
163C22×C3×C4—92no—
164C22×C3×C4—92no—
165C22×C3×C4—184no—
166C22×C3×C4D3211yescharacteristic
167C22×Dic3—32no—
168C22×Dic3—32no—
169C22×Dic3—31no—
170C22×Dic3—32no—
171C22×Dic3—62no—
172C22×Dic3—92no—
173C22×Dic3—91no—
174C2×C4×D3—34no—
175C2×C4×D3—64no—
176C2×C4×D3—94no—
177C2×C4×D3—184no—
178C2×Dic6—34no—
179C2×Dic6—34no—
180C2×Dic6—64no—
181C2×Dic6—64no—
182C2×Dic6—64no—
183C2×Dic6—68no—
184C3×C4⋊C4—94no—
185#217—98no—
186(C2×D3)⋊C4—98no—
187(C2×D3)⋊C4—98no—
188C4⋊Dic3—34no—
189C4⋊Dic3—94no—
190C4⋊Dic3—98no—
191Dic3⋊C4—98no—
192Dic3⋊C4—98no—
193C22×C32—21no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32C2×Dic612yes—
198C22×C32C2×D1212yes—
199C22×C32#23512yes—
200C22×C32#23712yes—
201C22×C32D3×Q812yes—
202C22×C32C2×#6712yes—
203C22×C32C23×D311yescharacteristic
204C22×C32C23×D311yescharacteristic
205C2×C3⋊D3—28no—
206C2×C3⋊D3—22no—
207C2×C3⋊D3—22no—
208C2×C3⋊D3C2×Dic614yescomplemented
209C2×C3×D3—34no—
210C2×C3×D3—68no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C32×C4—24no—
214C32×C4C2×D1214yes—
215C32×C4C2×#6714yes—
216C3⋊Dic3—24no—
217C3⋊Dic3—22no—
218C3×Dic3—68no—
219C3×Dic3—62no—
220C3×Dic3—64no—
221C3×Dic3—64no—
222C3×Dic3—64no—
223C22×Q8—91no—
224C23×C4—91no—
225C22⋊Q8—278no—
226C2×C4⋊C4—91no—
227C2×C4⋊C4—272no—
228C2×C22⋊C4—272no—
229H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
230C23×C3—31no—
231C23×C3—31no—
232C23×C3—61no—
233C23×C3—92no—
234C23×C3—94no—
235C23×C3—182no—
236C23×C3—182no—
237C23×C3C2×D3211yescharacteristic
238C22×D3—32no—
239C22×D3—34no—
240C22×D3—62no—
241C22×D3—62no—
242C22×D3—94no—
243C22×D3—92no—
244C22×D3—182no—
245C22×D3—182no—
246C3×Q8—188no—
247C2×C3×C4—34no—
248C2×C3×C4—34no—
249C2×C3×C4—32no—
250C2×C3×C4—32no—
251C2×C3×C4—64no—
252C2×C3×C4—62no—
253C2×C3×C4—92no—
254C2×C3×C4—94no—
255C2×C3×C4—94no—
256C2×C3×C4—188no—
257C2×C3×C4—184no—
258C2×C3×C4—182no—
259C2×C3×C4—184no—
260C2×C3×C4#62614yes—
261C2×C3×C4C2×D3212yes—
262C2×Dic3—34no—
263C2×Dic3—34no—
264C2×Dic3—32no—
265C2×Dic3—34no—
266C2×Dic3—64no—
267C2×Dic3—64no—
268C2×Dic3—64no—
269C2×Dic3—68no—
270C2×Dic3—64no—
271C2×Dic3—62no—
272C2×Dic3—94no—
273C2×Dic3—92no—
274C2×Dic3—182no—
275C2×Dic3—184no—
276C4×D3—68no—
277C4×D3—188no—
278Dic6—68no—
279Dic6—68no—
280Dic6—616no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—24no—
285C2×C32#89714yescomplemented
286C2×C32#92414yescomplemented
287C2×C32C22×Dic611yescharacteristic
288C2×C32C22×D1211yescharacteristic
289C2×C32C2×#23511yescharacteristic
290C2×C32C2×#23711yescharacteristic
291C2×C32C2×D3×Q811yescharacteristic
292C2×C32C22×#6711yescharacteristic
293C3⋊D3—24no—
294C3×D3—64no—
295C24—91no—
296C2×Q8—94no—
297C2×Q8—184no—
298C22×C4—92no—
299C22×C4—92no—
300C22×C4—91no—
301C22×C4—94no—
302C22×C4—184no—
303C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
304C4⋊C4—94no—
305C4⋊C4—278no—
306C22⋊C4—278no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—32no—
312C22×C3—32no—
313C22×C3—32no—
314C22×C3—32no—
315C22×C3—61no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—62no—
319C22×C3—94no—
320C22×C3—182no—
321C22×C3—188no—
322C22×C3—182no—
323C22×C3D3×Dic612yes—
324C22×C3#379512yes—
325C22×C3C2×#62612yes—
326C22×C3C22×D3211yescharacteristic
327C2×D3—34no—
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—68no—
331C2×D3—94no—
332C2×D3—182no—
333C2×D3—188no—
334C2×D3—182no—
335C3×C4—34no—
336C3×C4—34no—
337C3×C4—64no—
338C3×C4—182no—
339C3×C4—184no—
340C3×C4—184no—
341C3×C4C2×#62614yes—
342Dic3—64no—
343Dic3—64no—
344Dic3—68no—
345Dic3—64no—
346Dic3—62no—
347Dic3—182no—
348Dic3—184no—
349C32—21no—
350C32C2×#89711yescharacteristic, complemented
351C32C2×#92411yescharacteristic, complemented
352C23—92no—
353C23—94no—
354C23—182no—
355C23—182no—
356C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
357Q8—188no—
358C2×C4—94no—
359C2×C4—94no—
360C2×C4—92no—
361C2×C4—184no—
362C2×C4—182no—
363C2×C4—184no—
364C2×C4—188no—
365C2×C4#620014yes—
366C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—34no—
370C2×C3—34no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—61no—
376C2×C3—64no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—184no—
380C2×C3#6385714yes—
381C2×C3C2×D3×Dic611yescharacteristic
382C2×C3C2×#379511yescharacteristic
383C2×C3C22×#62611yescharacteristic
384D3—64no—
385D3—184no—
386C22—94no—
387C22—182no—
388C22—188no—
389C22—182no—
390C22((C3⋊D3)⋊C3)⋊Q812yes—
391C22#8864812yes—
392C22C2×#620012yes—
393C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C4—182no—
395C4—184no—
396C4—184no—
397C4C2×#620014yes—
398C3—31no—
399C3—31no—
400C3—61no—
401C3C2×#6385711yescharacteristic
402C2—184no—
403C2#1172110714yesdirect product factor
404C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
405C2C2×#8864811yescharacteristic
406C2C22×#620011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6229C2semidirect
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88448C2semidirect
6. #11721107C2direct (N × H)
13. C2×#6229C22semidirect
18. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. #88448C22semidirect
26. C2×C4⋊(C3⋊Dic3)D3semidirect
37. C22×(C3⋊D3)⋊C3Q8semidirect
74. C4⋊(C3⋊Dic3)C2×D3semidirect
87. C2×(C3⋊D3)⋊C3C2×Q8semidirect
116. C22×C3⋊D3Dic6semidirect
148. C2×H(3,𝔽3)C22⋊Q8semidirect
208. C2×C3⋊D3C2×Dic6semidirect
229. H(3,𝔽3)C2×C22⋊Q8semidirect
285. C2×C32#897semidirect
286. C2×C32#924semidirect
350. C32C2×#897semidirect
351. C32C2×#924semidirect
403. C2#11721107direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C22×C4×(C3⋊D3)⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884502
C2×#884482
#117211072
C2×#638713
C2×#638563