Finite Group Databasebeta

← C2×#11722839 #13170572

Subgroups — C2×#11722839 #13170572

Order 1728.

1426 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 212 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
1C2×#89040C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic
3C2×#88792C212yesmaximal, complemented
4C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×#3716C212yesmaximal, complemented
6#11722839C218yesmaximal, direct product factor
7C2×D3×#214—32nomaximal
8C2×#63796—31nomaximal
9C22×C3×D32—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634C2212yescomplemented
12C2×C32×D12C2212yes—
13C2×#6296—21no—
14C2×#6294—21no—
15#89040—22no—
16#89040C2214yescomplemented
17C2×C3×#625—21no—
18C3×#3802—22no—
19C3×#3802C2214yes—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88792C2218yescomplemented
23C3⋊D3×C3⋊C8C2214yescomplemented
24C3×#3716C2218yescomplemented
25C2×C3×D3×D4—32no—
26C2×#3802D311yescharacteristic
27C2×#3797—31no—
28C2×D3×Dic6—32no—
29C2×#3754—32no—
30C2×C8×C3⋊D3—31no—
31C2×C3×#214—32no—
32C2×C3×#204—32no—
33D3×#214—316no—
34C2×#3716D312yescomplemented
35C2×#3715—32no—
36C2×#3715—32no—
37C2×#3715—62no—
38C2×D3×C3⋊C8—32no—
39C2×D3×C3⋊C8—61no—
40#63796—38no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—22no—
43C2×C3×D32—24no—
44C2×C3×D32—44no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3D412yes—
49C3×C4×C3⋊D3C2312yes—
50C3×#634—22no—
51C3×#634C2314yescomplemented
52C32×D12—22no—
53C32×D12C2314yes—
54#6296—24no—
55#6294—24no—
56C3×#625—24no—
57C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
58C32×C3⋊C8C2312yescomplemented
59C22×#214—91no—
60C2×D3×SD16—92no—
61C23×C3×D3—62no—
62C22×D32—21no—
63C2×C32×D4—32no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×#634C2×D312yescomplemented
66C2×C3×#67—62no—
67C3×D3×D4—38no—
68C3×D3×D4—64no—
69C2×C3×D12—22no—
70C2×C3×D12C2×D312yes—
71C2×C3×D12C2×D312yes—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—61no—
74C2×C3×Dic6—32no—
75C2×C3×Dic6—32no—
76C2×C3×Dic6—62no—
77C2×#627—62no—
78C2×#627—61no—
79C2×#625—21no—
80C2×#624—61no—
81C2×D3×Dic3—62no—
82#3802—22no—
83#3802C2×D314yes—
84#3797—34no—
85#3797—62no—
86D3×Dic6—38no—
87D3×Dic6—64no—
88C2×C32×C8—31no—
89#3754—38no—
90C2×C3⋊(C3⋊C8)—32no—
91C2×C3⋊(C3⋊C8)—31no—
92C2×C3⋊(C3⋊C8)—61no—
93C2×C3⋊(C3⋊C8)—62no—
94C2×C3⋊(C3⋊C8)—121no—
95C8×C3⋊D3—34no—
96C3×#214—38no—
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8C2×D312yescomplemented
99C3×#204—38no—
100#3716C2×D318yescomplemented
101#3715—38no—
102#3715—38no—
103#3715—68no—
104D3×C3⋊C8—38no—
105D3×C3⋊C8—64no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3SD1614yescomplemented
108C2×C3×C3⋊D3C2×D412yes—
109C2×C32×D3—24no—
110C2×C32×D3—44no—
111C3×D32—48no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2411yescharacteristic
114C3×C3⋊Dic3—24no—
115C3×C3⋊Dic3C2×D412yes—
116C22×C3×D4—91no—
117C2×D3×Q8—92no—
118C2×D3×D4—32no—
119C22×Dic6—91no—
120C2×C3×SD16—92no—
121C2×#215—92no—
122C2×#214—32no—
123C2×#214—98no—
124C2×#214—92no—
125C2×#214—92no—
126C2×#214—182no—
127C22×C3⋊C8—91no—
128D3×SD16—916no—
129C2×#204—32no—
130C2×D3×C8—92no—
131C2×D3×C8—181no—
132C23×C32—62no—
133C22×C3⋊D3#6711yescharacteristic
134C22×C3×D3—22no—
135C22×C3×D3—22no—
136C22×C3×D3—32no—
137C22×C3×D3—42no—
138C22×C3×D3—61no—
139C22×C3×D3—68no—
140C22×C3×D3—128no—
141C2×D32—24no—
142C2×D32—44no—
143C32×D4—34no—
144C32×D4—62no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—41no—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3#6711yescharacteristic
152C4×C3⋊D3#6712yes—
153C4×C3⋊D3C22×D312yes—
154#634—22no—
155#634C22×D314yescomplemented
156C3×#67—68no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—122no—
162C3×D12—22no—
163C3×D12—22no—
164C3×D12—24no—
165C3×D12—42no—
166C3×D12C22×D314yes—
167C3×D12C22×D314yes—
168C3×C4×D3—34no—
169C3×C4×D3—34no—
170C3×C4×D3—62no—
171C3×C4×D3—62no—
172C3×Dic6—34no—
173C3×Dic6—34no—
174C3×Dic6—64no—
175C3×Dic6—62no—
176C3×Dic6—62no—
177C3×Dic6—122no—
178#627—68no—
179#627—64no—
180#625—24no—
181#624—64no—
182D3×Dic3—68no—
183C32×C8—32no—
184C3⋊(C3⋊C8)—34no—
185C3⋊(C3⋊C8)—32no—
186C3⋊(C3⋊C8)—62no—
187C3⋊(C3⋊C8)—64no—
188C3⋊(C3⋊C8)—122no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8C22×D314yescomplemented
191C22×SD16—271noSylow
192C2×C33C2×SD1612yescomplemented
193C2×C33C22×D411yescharacteristic
194C3×C3⋊D3C2×SD1614yescomplemented
195C32×D3—44no—
196C24×C3—181no—
197C23×D3—62no—
198C2×C3×Q8—92no—
199C2×C3×D4—32no—
200C2×C3×D4—32no—
201C2×C3×D4—62no—
202C2×C3×D4—94no—
203C2×C3×D4—182no—
204C2×C3×D4—181no—
205C22×C3×C4—91no—
206C2×#67—62no—
207C22×Dic3—181no—
208D3×Q8—98no—
209D3×Q8—184no—
210D3×D4—38no—
211D3×D4—64no—
212C2×D12D3212yes—
213C2×C4×D3—32no—
214C2×C4×D3—61no—
215C2×C4×D3—182no—
216C2×Dic6—32no—
217C2×Dic6—32no—
218C2×Dic6—62no—
219C2×Dic6—94no—
220C2×Dic6—182no—
221C2×Dic6—182no—
222C2×Dic6—181no—
223C3×SD16—98no—
224C2×C3×C8—32no—
225C2×C3×C8—61no—
226#215—98no—
227#214—38no—
228#214—98no—
229#214—98no—
230#214—188no—
231C2×C3⋊C8—92no—
232C2×C3⋊C8—91no—
233C2×C3⋊C8—94no—
234C2×C3⋊C8—181no—
235C2×C3⋊C8—182no—
236C2×C3⋊C8—361no—
237C2×C3⋊C8D3211yescharacteristic, complemented
238#204—38no—
239D3×C8—98no—
240D3×C8—184no—
241C22×C32—21no—
242C22×C32—22no—
243C22×C32—41no—
244C22×C32—64no—
245C22×C32—124no—
246C22×C32D3×D412yes—
247C22×C32C2×#6711yescharacteristic
248C2×C3⋊D3#21414yescomplemented
249C2×C3⋊D3C2×#6712yes—
250C2×C3×D3—24no—
251C2×C3×D3—24no—
252C2×C3×D3—38no—
253C2×C3×D3—34no—
254C2×C3×D3—44no—
255C2×C3×D3—44no—
256C2×C3×D3—44no—
257C2×C3×D3—64no—
258C2×C3×D3—62no—
259C2×C3×D3—84no—
260C2×C3×D3—1216no—
261D32—48no—
262C32×C4—21no—
263C32×C4—22no—
264C32×C4—21no—
265C32×C4—22no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4D3×D412yes—
269C32×C4C2×#6711yescharacteristic
270C32×C4C23×D312yes—
271C32×C4C23×D311yescharacteristic
272C3⋊Dic3—24no—
273C3⋊Dic3C2×#6712yes—
274C3×Dic3—34no—
275C3×Dic3—62no—
276C3×Dic3—64no—
277C3×Dic3—64no—
278C3×Dic3—124no—
279C22×Q8—271no—
280C22×D4—91no—
281C2×SD16—92no—
282C2×SD16—278no—
283C2×SD16—272no—
284C22×C8—271no—
285C33C22×SD1611yescharacteristic, complemented, Sylow
286C23×C3—62no—
287C23×C3—62no—
288C23×C3—91no—
289C23×C3—122no—
290C23×C3—184no—
291C23×C3—364no—
292C22×D3—22no—
293C22×D3—32no—
294C22×D3—61no—
295C22×D3—68no—
296C22×D3—128no—
297C3×Q8—94no—
298C3×Q8—182no—
299C3×D4—34no—
300C3×D4—34no—
301C3×D4—64no—
302C3×D4—62no—
303C3×D4—62no—
304C3×D4—122no—
305C3×D4—184no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—91no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—182no—
313C2×C3×C4C2×D3211yescharacteristic
314C2×C3×C4C2×D3212yes—
315#67—68no—
316C2×Dic3—32no—
317C2×Dic3—61no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—122no—
321C2×Dic3—184no—
322D12—22no—
323D12C2×D3214yes—
324C4×D3—34no—
325C4×D3—34no—
326C4×D3—62no—
327C4×D3—62no—
328C4×D3—188no—
329Dic6—34no—
330Dic6—34no—
331Dic6—64no—
332Dic6—62no—
333Dic6—62no—
334Dic6—122no—
335Dic6—188no—
336Dic6—184no—
337C3×C8—34no—
338C3×C8—62no—
339C3⋊C8—94no—
340C3⋊C8—92no—
341C3⋊C8—182no—
342C3⋊C8—184no—
343C3⋊C8—362no—
344C3⋊C8C2×D3212yescomplemented
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—24no—
349C2×C32—42no—
350C2×C32—41no—
351C2×C32—124no—
352C2×C32D3×SD1614yescomplemented
353C2×C32C2×#21412yescomplemented
354C2×C32C2×D3×D412yes—
355C2×C32C22×#6711yescharacteristic
356C3⋊D3C2×#21414yescomplemented
357C3×D3—38no—
358C3×D3—44no—
359C3×D3—44no—
360C3×D3—64no—
361C3×D3—84no—
362C24—181no—
363C2×Q8—92no—
364C2×Q8—274no—
365C2×Q8—541no—
366C2×Q8—542no—
367C2×D4—32no—
368C2×D4—94no—
369C2×D4—182no—
370C2×D4—181no—
371C22×C4—91no—
372C22×C4—541no—
373SD16—98no—
374SD16—278no—
375C2×C8—31no—
376C2×C8—274no—
377C2×C8—271no—
378C22×C3—21no—
379C22×C3—22no—
380C22×C3—41no—
381C22×C3—64no—
382C22×C3—64no—
383C22×C3—92no—
384C22×C3—94no—
385C22×C3—124no—
386C22×C3—124no—
387C22×C3—124no—
388C22×C3—244no—
389C22×C3—368no—
390C22×C3#380211yescharacteristic
391C22×C3D3×#6712yes—
392C2×D3—24no—
393C2×D3—34no—
394C2×D3—38no—
395C2×D3—44no—
396C2×D3—62no—
397C2×D3—64no—
398C2×D3—1216no—
399C3×C4—21no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—21no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—92no—
406C3×C4—184no—
407C3×C4#380211yescharacteristic
408C3×C4D3×#6712yes—
409C3×C4C22×D3211yescharacteristic
410C3×C4C22×D3212yes—
411Dic3—34no—
412Dic3—62no—
413Dic3—64no—
414Dic3—64no—
415Dic3—124no—
416C32—22no—
417C32—21no—
418C32—41no—
419C32C2×D3×SD1612yescomplemented
420C32C22×#21411yescharacteristic, complemented
421C23—62no—
422C23—91no—
423C23—184no—
424C23—364no—
425Q8—94no—
426Q8—182no—
427Q8—544no—
428D4—34no—
429D4—62no—
430D4—184no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—182no—
435C2×C4—544no—
436C2×C4D3311yescharacteristic
437C8—32no—
438C8—272no—
439C2×C3—22no—
440C2×C3—24no—
441C2×C3—21no—
442C2×C3—22no—
443C2×C3—42no—
444C2×C3—41no—
445C2×C3—94no—
446C2×C3—124no—
447C2×C3—124no—
448C2×C3—244no—
449C2×C3#6379612yescomplemented
450C2×C3D3×#21414yescomplemented
451C2×C3C2×#380211yescharacteristic
452C2×C3C2×D3×#6712yes—
453D3—38no—
454D3—44no—
455D3—64no—
456C22—64no—
457C22—94no—
458C22—92no—
459C22—124no—
460C22—368no—
461C22#8895311yescharacteristic
462C4—92no—
463C4—184no—
464C4#8895311yescharacteristic
465C4C2×D3311yescharacteristic
466C3—21no—
467C3—22no—
468C3—41no—
469C3C2×#6379611yescharacteristic, complemented
470C3C2×D3×#21412yescomplemented
471C2—94no—
472C2—124no—
473C2#1172283912yesdirect product factor
474C2C2×#8895311yescharacteristic
subgroupG/H
centerC22#88953
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
hypercenterC2×C4D33
hypocenterC33C22×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
3. C2×#88792C2semidirect
4. C2×C3⋊D3×C3⋊C8C2semidirect
5. C2×C3×#3716C2semidirect
6. #11722839C2direct (N × H)
11. C2×C3×#634C22semidirect
16. #89040C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C2×C32×C3⋊C8C22semidirect
22. #88792C22semidirect
23. C3⋊D3×C3⋊C8C22semidirect
24. C3×#3716C22semidirect
34. C2×#3716D3semidirect
51. C3×#634C23semidirect
57. C3⋊(C3⋊(C3⋊C8))C23semidirect
58. C32×C3⋊C8C23semidirect
65. C2×#634C2×D3semidirect
98. C2×C3×C3⋊C8C2×D3semidirect
100. #3716C2×D3semidirect
107. C2×C3×C3⋊D3SD16semidirect
155. #634C22×D3semidirect
190. C3×C3⋊C8C22×D3semidirect
192. C2×C33C2×SD16semidirect
194. C3×C3⋊D3C2×SD16semidirect
237. C2×C3⋊C8D32semidirect
248. C2×C3⋊D3#214semidirect
285. C33C22×SD16semidirect
344. C3⋊C8C2×D32semidirect
352. C2×C32D3×SD16semidirect
353. C2×C32C2×#214semidirect
356. C3⋊D3C2×#214semidirect
419. C32C2×D3×SD16semidirect
420. C32C22×#214semidirect
449. C2×C3#63796semidirect
450. C2×C3D3×#214semidirect
469. C3C2×#63796semidirect
470. C3C2×D3×#214semidirect
473. C2#11722839direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#38022
C2×#887922
C2×C3⋊D3×C3⋊C82
C2×C3×#37162
#117228392
C2×D3×#2143
C2×#637963