Finite Group Databasebeta

← C2×#11721324 #13152092

Subgroups — C2×#11721324 #13152092

Order 1728.

2553 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 2553 conjugacy classes of subgroups — too many to draw. Only its 219 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
2C22×#6235C211yesmaximal, characteristic, complemented
3C22×#638⋊C3C212yesmaximal, complemented
4#11721324C218yesmaximal, direct product factor
5C2×#88451C211yesmaximal, characteristic, complemented
6C2×#88450C212yesmaximal, complemented
7C2×#63983—32nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic, complemented
9C23×#272C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×((C3⋊D3)⋊C3)⋊C2—24no—
12C22×((C3⋊D3)⋊C3)⋊C2—28no—
13C22×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
14C2×#6235—24no—
15C2×#6235C2214yescomplemented
16C22×#1084C2211yescharacteristic, complemented
17C2×#638⋊C3—28no—
18C2×#638⋊C3C2218yescomplemented
19C22×(C3⋊Dic3)⋊C3C2212yescomplemented
20#88451C2214yescomplemented
21#88450C2218yescomplemented
22C23×D32—32no—
23C22×#638D312yescomplemented
24C22×C3×#67—32no—
25C22×C3×#67—32no—
26C22×C3×#67—61no—
27#63983—316no—
28C2×#3722—32no—
29C2×(C2×D3)⋊Dic3—32no—
30C2×(C2×D3)⋊Dic3—32no—
31C23×H(3,𝔽3)—22no—
32C23×H(3,𝔽3)—24no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C23×H(3,𝔽3)C2312yescomplemented
35C22×#272—24no—
36C22×#272D414yescomplemented
37C22×#272C2312yes—
38C22×(C3⋊D3)⋊C3—24no—
39C22×(C3⋊D3)⋊C3—24no—
40C22×(C3⋊D3)⋊C3D418yescomplemented
41C22×(C3⋊D3)⋊C3C2314yes—
42C2×((C3⋊D3)⋊C3)⋊C2—232no—
43C2×((C3⋊D3)⋊C3)⋊C2—28no—
44C2×((C3⋊D3)⋊C3)⋊C2—216no—
45#6235—28no—
46C2×#1084—22no—
47C2×#1084C2312yescomplemented
48#638⋊C3—216no—
49C2×(C3⋊Dic3)⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3C2314yescomplemented
51C2×#937—91no—
52C2×#937—92no—
53C24×C32—21no—
54C24×C32C2×D312yescomplemented
55C23×C3⋊D3C2×D312yes—
56C23×C3×D3—32no—
57C23×C3×D3—32no—
58C23×C3×D3—61no—
59C22×D32—38no—
60C22×D32—68no—
61C22×D32—68no—
62C22×D32—68no—
63C2×#638—28no—
64C2×#638C2×D318yescomplemented
65C22×C3⋊Dic3C2×D312yescomplemented
66C2×C3×#67—38no—
67C2×C3×#67—38no—
68C2×C3×#67—68no—
69C2×C3×#67—68no—
70C2×C3×#67—64no—
71C2×C3×#67—68no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—32no—
74C22×C3×Dic3—61no—
75#3722—38no—
76(C2×D3)⋊Dic3—38no—
77(C2×D3)⋊Dic3—38no—
78C22×H(3,𝔽3)—22no—
79C22×H(3,𝔽3)—24no—
80C22×H(3,𝔽3)—44no—
81C22×H(3,𝔽3)C2×D412yes—
82C22×H(3,𝔽3)C2×D414yes—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×#272—28no—
85C2×#272—24no—
86C2×#272C2×D414yescomplemented
87C2×(C3⋊D3)⋊C3—216no—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3C2×D418yescomplemented
91((C3⋊D3)⋊C3)⋊C2—232no—
92#1084—22no—
93(C3⋊Dic3)⋊C3—24no—
94C24×D3—91no—
95C24×D3—92no—
96C22×C3×D4—92no—
97C22×C3×D4—91no—
98C22×#67—32no—
99C22×#67—32no—
100C22×#67—61no—
101C22×#67—92no—
102C2×#217—91no—
103C2×#217—92no—
104#937—98no—
105#937—916no—
106C2×(C2×D3)⋊C4—92no—
107C2×(C2×D3)⋊C4—92no—
108C2×(C2×D3)⋊C4—92no—
109C23×C32—24no—
110C23×C32—24no—
111C23×C32—24no—
112C23×C32—24no—
113C23×C32—21no—
114C23×C32—22no—
115C23×C32—44no—
116C23×C32C22×D312yes—
117C23×C32C22×D314yescomplemented
118C22×C3⋊D3—24no—
119C22×C3⋊D3—24no—
120C22×C3⋊D3#6718yescomplemented
121C22×C3⋊D3C22×D314yes—
122C22×C3×D3—38no—
123C22×C3×D3—38no—
124C22×C3×D3—34no—
125C22×C3×D3—34no—
126C22×C3×D3—64no—
127C22×C3×D3—62no—
128C22×C3×D3—64no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—64no—
132C22×C3×D3—124no—
133C2×D32—616no—
134C2×D32—616no—
135C2×D32—664no—
136C2×D32—616no—
137#638—216no—
138C2×C3⋊Dic3—24no—
139C2×C3⋊Dic3C22×D314yescomplemented
140C3×#67—616no—
141C3×#67—616no—
142C3×#67—616no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—64no—
149C2×#120—271noSylow
150C2×H(3,𝔽3)—42no—
151C2×H(3,𝔽3)#12014yescomplemented
152C2×H(3,𝔽3)C22×D411yescharacteristic
153C2×H(3,𝔽3)C22×D412yes—
154#272—24no—
155(C3⋊D3)⋊C3—28no—
156C24×C3—32no—
157C24×C3—61no—
158C24×C3—92no—
159C24×C3—91no—
160C24×C3D3211yescharacteristic
161C23×D3—32no—
162C23×D3—32no—
163C23×D3—61no—
164C23×D3—92no—
165C23×D3—94no—
166C23×D3—98no—
167C23×D3—188no—
168C23×D3—188no—
169C23×D3—184no—
170C23×D3—188no—
171C23×D3—188no—
172C2×C3×D4—98no—
173C2×C3×D4—94no—
174C2×C3×D4—184no—
175C2×C3×D4—188no—
176C22×C3×C4—92no—
177C22×C3×C4—91no—
178C2×#67—38no—
179C2×#67—38no—
180C2×#67—64no—
181C2×#67—68no—
182C2×#67—68no—
183C2×#67—68no—
184C2×#67—98no—
185C2×#67—188no—
186C22×Dic3—32no—
187C22×Dic3—32no—
188C22×Dic3—61no—
189C22×Dic3—92no—
190#217—94no—
191#217—98no—
192(C2×D3)⋊C4—98no—
193(C2×D3)⋊C4—98no—
194(C2×D3)⋊C4—98no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—21no—
200C22×C32—24no—
201C22×C32—22no—
202C22×C32—44no—
203C22×C32—48no—
204C22×C32—48no—
205C22×C32D3×D414yes—
206C22×C32D3×D414yes—
207C22×C32C2×#6714yes—
208C22×C32C23×D312yes—
209C2×C3⋊D3—216no—
210C2×C3⋊D3—24no—
211C2×C3⋊D3—24no—
212C2×C3⋊D3C2×#6718yescomplemented
213C2×C3×D3—38no—
214C2×C3×D3—38no—
215C2×C3×D3—64no—
216C2×C3×D3—616no—
217C2×C3×D3—616no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—124no—
223C2×C3×D3—128no—
224D32—664no—
225C3⋊Dic3—24no—
226C3×Dic3—64no—
227C3×Dic3—64no—
228C3×Dic3—64no—
229C25—271no—
230C22×D4—91no—
231C22×D4—92no—
232#120—278no—
233C2×C22⋊C4—272no—
234C2×C22⋊C4—271no—
235H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
236C23×C3—24no—
237C23×C3—22no—
238C23×C3—32no—
239C23×C3—34no—
240C23×C3—64no—
241C23×C3—64no—
242C23×C3—64no—
243C23×C3—64no—
244C23×C3—61no—
245C23×C3—62no—
246C23×C3—94no—
247C23×C3—98no—
248C23×C3—92no—
249C23×C3—94no—
250C23×C3—124no—
251C23×C3—184no—
252C23×C3—184no—
253C23×C3—184no—
254C23×C3C2×D3211yescharacteristic
255C23×C3C2×D3212yes—
256C22×D3—34no—
257C22×D3—38no—
258C22×D3—34no—
259C22×D3—38no—
260C22×D3—62no—
261C22×D3—64no—
262C22×D3—64no—
263C22×D3—64no—
264C22×D3—64no—
265C22×D3—64no—
266C22×D3—98no—
267C22×D3—94no—
268C22×D3—124no—
269C22×D3—184no—
270C22×D3—184no—
271C22×D3—1816no—
272C22×D3—1816no—
273C22×D3—188no—
274C22×D3—1832no—
275C22×D3—1816no—
276C22×D3—1816no—
277C22×D3—1864no—
278C3×D4—188no—
279C3×D4—1816no—
280C2×C3×C4—94no—
281C2×C3×C4—92no—
282C2×C3×C4—184no—
283C2×C3×C4—182no—
284#67—616no—
285#67—616no—
286#67—616no—
287#67—1816no—
288C2×Dic3—34no—
289C2×Dic3—34no—
290C2×Dic3—62no—
291C2×Dic3—64no—
292C2×Dic3—64no—
293C2×Dic3—64no—
294C2×Dic3—94no—
295C2×Dic3—184no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32—24no—
299C2×C32—44no—
300C2×C32—44no—
301C2×C32#93718yescomplemented
302C2×C32C2×D3×D412yes—
303C2×C32C2×D3×D412yes—
304C2×C32C22×#6712yes—
305C3⋊D3—28no—
306C3×D3—68no—
307C3×D3—68no—
308C3×D3—124no—
309C24—91no—
310C24—92no—
311C24—274no—
312C24—548no—
313C24—544no—
314C24((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
315C2×D4—94no—
316C2×D4—98no—
317C2×D4—184no—
318C2×D4—188no—
319C22×C4—91no—
320C22×C4—92no—
321C22⋊C4—278no—
322C22⋊C4—274no—
323C22×C3—24no—
324C22×C3—22no—
325C22×C3—32no—
326C22×C3—34no—
327C22×C3—34no—
328C22×C3—34no—
329C22×C3—44no—
330C22×C3—64no—
331C22×C3—64no—
332C22×C3—64no—
333C22×C3—64no—
334C22×C3—61no—
335C22×C3—64no—
336C22×C3—62no—
337C22×C3—98no—
338C22×C3—94no—
339C22×C3—124no—
340C22×C3—128no—
341C22×C3—128no—
342C22×C3—184no—
343C22×C3—184no—
344C22×C3—1816no—
345C22×C3—184no—
346C22×C3—188no—
347C22×C3D3×#6714yes—
348C22×C3#381112yes—
349C22×C3C22×D3211yescharacteristic
350C2×D3—38no—
351C2×D3—38no—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—64no—
355C2×D3—616no—
356C2×D3—64no—
357C2×D3—64no—
358C2×D3—616no—
359C2×D3—98no—
360C2×D3—124no—
361C2×D3—128no—
362C2×D3—184no—
363C2×D3—184no—
364C2×D3—1816no—
365C2×D3—1832no—
366C2×D3—1864no—
367C3×C4—184no—
368C3×C4—182no—
369Dic3—64no—
370Dic3—64no—
371Dic3—64no—
372Dic3—184no—
373C32—21no—
374C32C2×#93712yescomplemented
375C23—24no—
376C23—22no—
377C23—92no—
378C23—94no—
379C23—98no—
380C23—94no—
381C23—184no—
382C23—184no—
383C23—184no—
384C23—5416no—
385C23—548no—
386C23—5432no—
387C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
388C23C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
389D4—188no—
390D4—1816no—
391C2×C4—92no—
392C2×C4—94no—
393C2×C4—184no—
394C2×C4—182no—
395C2×C3—32no—
396C2×C3—32no—
397C2×C3—32no—
398C2×C3—38no—
399C2×C3—42no—
400C2×C3—62no—
401C2×C3—61no—
402C2×C3—64no—
403C2×C3—124no—
404C2×C3—124no—
405C2×C3—188no—
406C2×C3—184no—
407C2×C3#6398314yes—
408C2×C3C2×D3×#6712yes—
409C2×C3C2×#381111yescharacteristic
410D3—68no—
411D3—68no—
412D3—124no—
413D3—188no—
414C22—24no—
415C22—22no—
416C22—44no—
417C22—98no—
418C22—94no—
419C22—184no—
420C22—184no—
421C22—184no—
422C22—188no—
423C22—1816no—
424C22—5432no—
425C22#8867614yes—
426C22#8867712yes—
427C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C4—184no—
429C4—182no—
430C3—32no—
431C3—61no—
432C3C2×#6398311yescharacteristic
433C2—42no—
434C2—184no—
435C2—188no—
436C2#1172132414yesdirect product factor
437C2C2×#8867612yes—
438C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#120
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×((C3⋊D3)⋊C3)⋊C2C2semidirect
2. C22×#6235C2semidirect
3. C22×#638⋊C3C2semidirect
4. #11721324C2direct (N × H)
5. C2×#88451C2semidirect
6. C2×#88450C2semidirect
8. C24×H(3,𝔽3)C22semidirect
13. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
15. C2×#6235C22semidirect
16. C22×#1084C22semidirect
18. C2×#638⋊C3C22semidirect
19. C22×(C3⋊Dic3)⋊C3C22semidirect
20. #88451C22semidirect
21. #88450C22semidirect
23. C22×#638D3semidirect
34. C23×H(3,𝔽3)C23semidirect
36. C22×#272D4semidirect
40. C22×(C3⋊D3)⋊C3D4semidirect
47. C2×#1084C23semidirect
50. C2×(C3⋊Dic3)⋊C3C23semidirect
54. C24×C32C2×D3semidirect
64. C2×#638C2×D3semidirect
65. C22×C3⋊Dic3C2×D3semidirect
86. C2×#272C2×D4semidirect
90. C2×(C3⋊D3)⋊C3C2×D4semidirect
117. C23×C32C22×D3semidirect
120. C22×C3⋊D3#67semidirect
139. C2×C3⋊Dic3C22×D3semidirect
151. C2×H(3,𝔽3)#120semidirect
212. C2×C3⋊D3C2×#67semidirect
235. H(3,𝔽3)C2×#120semidirect
301. C2×C32#937semidirect
314. C24((C3⋊D3)⋊C3)⋊C2semidirect
374. C32C2×#937semidirect
388. C23C2×((C3⋊D3)⋊C3)⋊C2semidirect
436. C2#11721324direct (N × H)

Maximal subgroups

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

subgroupindex
C23×((C3⋊D3)⋊C3)⋊C22
C22×#62352
C22×#638⋊C32
#117213242
C2×#884512
C2×#884502
C2×#639833