Finite Group Databasebeta

← C2×#11722780 #13167386

Subgroups — C2×#11722780 #13167386

Order 1728.

792 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 103 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×C4×#1113C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4#11722780C214yesmaximal, direct product factor
5C2×D32⋊C8—31nomaximal
6C22×#1113C411yescharacteristic
7C2×C3×C4×C3⋊D3—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6294—21no—
10C2×#6294C411yescharacteristic
11C4×#1113—21no—
12C4×#1113—21no—
13C4×#1113C412yes—
14C4×#1113C2212yes—
15C2×C3×#622—21no—
16C3×#3787—21no—
17C3×#3787C2212yescomplemented
18C2×C3×C3⋊(C3⋊C8)—21no—
19#88808—21no—
20#88808C2212yescomplemented
21C2×C4×D32—31no—
22C2×C4×D32—61no—
23C2×#3787D311yescharacteristic, complemented
24C2×#3710—31no—
25C2×D3×C3⋊C8—61no—
26D32⋊C8—34no—
27C22×C3×C3⋊D3—21no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C2×#1113—21no—
30C2×#1113—22no—
31C2×#1113C814yescomplemented
32C2×#1113C2×C412yes—
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3—22no—
38C3×C4×C3⋊D3C2×C411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3C2311yescharacteristic
42#6294—21no—
43#6294—22no—
44#6294—22no—
45#6294C2×C412yes—
46C3×#622—22no—
47C3×C3⋊(C3⋊C8)—22no—
48C2×(C2×D3)⋊C8—91nomaximal
49C22×D32—31no—
50C22×D32—61no—
51C2×C4×C3⋊D3—21no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—121no—
58C2×#624—31no—
59C2×#624—61no—
60C2×D3×Dic3—61no—
61C2×D3×Dic3—61no—
62C2×D3×Dic3—61no—
63C4×D32—32no—
64C4×D32—32no—
65C4×D32—62no—
66C4×D32—62no—
67C4×D32—62no—
68C4×D32—62no—
69C4×D32—61no—
70C4×D32—61no—
71C2×#622—21no—
72#3787—21no—
73#3787C2×D312yescomplemented
74C2×C3⋊(C3⋊C8)—21no—
75C2×C3×C3⋊C8—61no—
76C2×C3×C3⋊C8—61no—
77C2×C3×C3⋊C8—121no—
78#3710—32no—
79#3710—61no—
80D3×C3⋊C8—62no—
81D3×C3⋊C8—62no—
82C22×C33C22⋊C411yescharacteristic
83C2×C3×C3⋊D3—24no—
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3C22⋊C411yescharacteristic
86C2×C3×C3⋊D3C2×C812yes—
87C2×C3×C3⋊D3M1612yes—
88C2×C3×C3⋊D3C22×C411yescharacteristic
89#1113—24no—
90#1113—22no—
91#1113C2×C814yescomplemented
92C33×C4C22⋊C411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3C22⋊C411yescharacteristic
96C3×C3⋊Dic3C2×D411yescharacteristic
97C22×C4×D3—91no—
98C22×C4×D3—181no—
99C22×C3×C8—91no—
100C22×C3⋊C8—91no—
101C2×D3×C8—181no—
102(C2×D3)⋊C8—94no—
103C22×C3⋊D3—21no—
104C22×C3⋊D3C4×D311yescharacteristic
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C22×C3×D3—61no—
108C22×C3×D3—61no—
109C22×C3×D3—121no—
110C2×D32—34no—
111C2×D32—32no—
112C2×D32—64no—
113C2×D32—62no—
114C2×D32—64no—
115C2×D32—62no—
116C2×D32—62no—
117C2×D32—61no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C32×C4D1211yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3#6711yescharacteristic
125C4×C3⋊D3—22no—
126C4×C3⋊D3—22no—
127C4×C3⋊D3C4×D311yescharacteristic
128C4×C3⋊D3D1211yescharacteristic
129C4×C3⋊D3#6711yescharacteristic
130C4×C3⋊D3C22×D311yescharacteristic
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—62no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148C3×C4×D3—122no—
149C3×C4×D3—122no—
150#624—32no—
151#624—62no—
152#624—62no—
153#624—61no—
154D3×Dic3—62no—
155D3×Dic3—62no—
156D3×Dic3—62no—
157D3×Dic3—62no—
158D3×Dic3—62no—
159D3×Dic3—62no—
160#622—22no—
161C3⋊(C3⋊C8)—22no—
162C3×C3⋊C8—62no—
163C3×C3⋊C8—62no—
164C3×C3⋊C8—122no—
165C2×C22⋊C8—271noSylow
166C2×C33C22⋊C812yescomplemented
167C2×C33C2×C22⋊C411yescharacteristic
168C3×C3⋊D3—24no—
169C3×C3⋊D3C22⋊C812yescomplemented
170C3×C3⋊D3C22×C811yescharacteristic
171C3×C3⋊D3C2×M1611yescharacteristic
172C23×D3—91no—
173C23×D3—181no—
174C22×C3×C4—91no—
175C22×C3×C4—181no—
176C22×Dic3—91no—
177C22×Dic3—181no—
178C2×C4×D3—61no—
179C2×C4×D3—61no—
180C2×C4×D3—61no—
181C2×C4×D3—61no—
182C2×C4×D3—92no—
183C2×C4×D3—92no—
184C2×C4×D3—121no—
185C2×C4×D3—182no—
186C2×C4×D3—182no—
187C2×C4×D3—181no—
188C2×C4×D3—181no—
189C2×C4×D3—181no—
190C2×C4×D3—182no—
191C2×C4×D3—182no—
192C2×C4×D3—181no—
193C2×C4×D3—181no—
194C2×C3×C8—92no—
195C2×C3×C8—181no—
196C2×C3×C8—181no—
197C2×C3×C8—181no—
198C2×C3⋊C8—61no—
199C2×C3⋊C8—61no—
200C2×C3⋊C8—92no—
201C2×C3⋊C8—121no—
202C2×C3⋊C8—181no—
203D3×C8—182no—
204D3×C8—182no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32(C2×D3)⋊C411yescharacteristic
210C2×C3⋊D3—22no—
211C2×C3⋊D3—24no—
212C2×C3⋊D3D3×C812yes—
213C2×C3⋊D3ΓL(1,𝔽25)12yes—
214C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
215C2×C3⋊D3C2×C4×D311yescharacteristic
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—62no—
219C2×C3×D3—62no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—62no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C2×C3×D3—62no—
228C2×C3×D3—122no—
229C2×C3×D3—124no—
230D32—34no—
231D32—64no—
232D32—68no—
233D32—64no—
234D32—64no—
235D32—62no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4(C2×D3)⋊C411yescharacteristic
245C32×C4C2×D1211yescharacteristic
246C3⋊Dic3—22no—
247C3⋊Dic3(C2×D3)⋊C411yescharacteristic
248C3⋊Dic3C2×#6711yescharacteristic
249C3×Dic3—61no—
250C3×Dic3—61no—
251C3×Dic3—61no—
252C3×Dic3—61no—
253C3×Dic3—62no—
254C3×Dic3—62no—
255C3×Dic3—122no—
256C23×C4—271no—
257C22×C8—91no—
258C22×C8—271no—
259C22⋊C8—274no—
260C33C2×C22⋊C811yescharacteristic, complemented, Sylow
261C23×C3—91no—
262C23×C3—181no—
263C22×D3—61no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—92no—
268C22×D3—94no—
269C22×D3—121no—
270C22×D3—182no—
271C22×D3—184no—
272C22×D3—181no—
273C22×D3—182no—
274C22×D3—182no—
275C22×D3—184no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—91no—
281C2×C3×C4—91no—
282C2×C3×C4—91no—
283C2×C3×C4—91no—
284C2×C3×C4—91no—
285C2×C3×C4—182no—
286C2×C3×C4—182no—
287C2×C3×C4—181no—
288C2×C3×C4GO+(4,𝔽2)11yescharacteristic
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—92no—
294C2×Dic3—121no—
295C2×Dic3—181no—
296C2×Dic3—182no—
297C2×Dic3—182no—
298C4×D3—61no—
299C4×D3—61no—
300C4×D3—61no—
301C4×D3—61no—
302C4×D3—61no—
303C4×D3—61no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—62no—
307C4×D3—62no—
308C4×D3—62no—
309C4×D3—62no—
310C4×D3—122no—
311C4×D3—122no—
312C4×D3—182no—
313C4×D3—182no—
314C4×D3—182no—
315C4×D3—182no—
316C4×D3—182no—
317C4×D3—182no—
318C3×C8—182no—
319C3×C8—182no—
320C3⋊C8—62no—
321C3⋊C8—62no—
322C3⋊C8—122no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—22no—
326C2×C32—22no—
327C2×C32—42no—
328C2×C32—42no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32(C2×D3)⋊C812yescomplemented
332C2×C32C2×(C2×D3)⋊C411yescharacteristic
333C3⋊D3—24no—
334C3⋊D3(C2×D3)⋊C812yescomplemented
335C3⋊D3C2×D3×C811yescharacteristic
336C3⋊D3C2×ΓL(1,𝔽25)11yescharacteristic
337C3×D3—62no—
338C3×D3—62no—
339C3×D3—61no—
340C3×D3—61no—
341C3×D3—61no—
342C3×D3—61no—
343C3×D3—64no—
344C3×D3—64no—
345C3×D3—124no—
346C24—271no—
347C22×C4—91no—
348C22×C4—181no—
349C22×C4—272no—
350C22×C4—272no—
351C22×C4—271no—
352C22×C4—541no—
353C22×C4—541no—
354C22×C4—541no—
355C2×C8—92no—
356C2×C8—181no—
357C2×C8—181no—
358C2×C8—181no—
359C2×C8—272no—
360C2×C8—541no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—91no—
366C22×C3—91no—
367C22×C3—92no—
368C22×C3—92no—
369C22×C3—182no—
370C22×C3—184no—
371C22×C3D32⋊C411yescharacteristic
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—62no—
377C2×D3—62no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—64no—
382C2×D3—62no—
383C2×D3—64no—
384C2×D3—94no—
385C2×D3—122no—
386C2×D3—124no—
387C2×D3—184no—
388C2×D3—182no—
389C2×D3—184no—
390C2×D3—188no—
391C2×D3—184no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—41no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—91no—
401C3×C4—91no—
402C3×C4—182no—
403C3×C4D32⋊C411yescharacteristic
404C3×C4C2×GO+(4,𝔽2)11yescharacteristic
405Dic3—61no—
406Dic3—61no—
407Dic3—61no—
408Dic3—61no—
409Dic3—62no—
410Dic3—62no—
411Dic3—122no—
412C32—21no—
413C32—21no—
414C32—41no—
415C32—41no—
416C32C2×(C2×D3)⋊C811yescharacteristic, complemented
417C23—91no—
418C23—181no—
419C23—272no—
420C23—274no—
421C23—541no—
422C23—542no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—181no—
429C2×C4—182no—
430C2×C4—182no—
431C2×C4—272no—
432C2×C4—542no—
433C2×C4—542no—
434C2×C4—541no—
435C2×C4#157⋊D311yescharacteristic
436C8—182no—
437C8—182no—
438C2×C3—22no—
439C2×C3—22no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—42no—
443C2×C3—42no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—91no—
447C2×C3—91no—
448C2×C3—92no—
449C2×C3—184no—
450C2×C3D32⋊C812yescomplemented
451C2×C3C2×D32⋊C411yescharacteristic
452D3—62no—
453D3—61no—
454D3—61no—
455D3—62no—
456D3—61no—
457D3—61no—
458D3—64no—
459D3—64no—
460D3—124no—
461C22—92no—
462C22—92no—
463C22—91no—
464C22—91no—
465C22—184no—
466C22—182no—
467C22—274no—
468C22—542no—
469C22—544no—
470C22#8893911yescharacteristic
471C4—91no—
472C4—91no—
473C4—182no—
474C4#8893911yescharacteristic
475C4C2×#157⋊D311yescharacteristic
476C3—21no—
477C3—21no—
478C3—41no—
479C3—41no—
480C3C2×D32⋊C811yescharacteristic, complemented
481C2—91no—
482C2—92no—
483C2—91no—
484C2—184no—
485C2#1172278012yesdirect product factor
486C2C2×#8893911yescharacteristic
subgroupG/H
center, hypercenterC2×C4#157⋊D3
commutatorC3×C3⋊D3C22×C8
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC22×#1113C4
cosocleC3×C4×C3⋊D3C23
hypocenterC33C2×C22⋊C8
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3787C2semidirect
3. C2×#88808C2semidirect
4. #11722780C2direct (N × H)
17. C3×#3787C22semidirect
20. #88808C22semidirect
23. C2×#3787D3semidirect
31. C2×#1113C8semidirect
73. #3787C2×D3semidirect
91. #1113C2×C8semidirect
166. C2×C33C22⋊C8semidirect
169. C3×C3⋊D3C22⋊C8semidirect
260. C33C2×C22⋊C8semidirect
331. C2×C32(C2×D3)⋊C8semidirect
334. C3⋊D3(C2×D3)⋊C8semidirect
416. C32C2×(C2×D3)⋊C8semidirect
450. C2×C3D32⋊C8semidirect
480. C3C2×D32⋊C8semidirect
485. C2#11722780direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×C3×#37872
C2×#888082
#117227802
C2×D32⋊C83
C2×(C2×D3)⋊C89