Finite Group Databasebeta

← C2×#11724346 #13181162

Subgroups — C2×#11724346 #13181162

Order 1728.

1264 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1264 conjugacy classes of subgroups — too many to draw. Only its 143 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×#89049C211yesmaximal, characteristic
2C22×#6294C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
4C2×#88940C212yesmaximal, complemented
5#11724346C218yesmaximal, direct product factor
6C2×#88939C212yesmaximal, complemented
7C2×#64233—31nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×#1113C2211yescharacteristic
10C22×C3×#157C2212yes—
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13#89049—22no—
14#89049C2214yes—
15C2×#6295—21no—
16C2×#6294—22no—
17C2×#6294—21no—
18C2×#6294C2212yes—
19C2×#6294C2211yescharacteristic
20C2×#6294C2214yescomplemented
21C3×C22⋊#157C2214yescomplemented
22#88940C2218yescomplemented
23#88939C2218yescomplemented
24C2×#3811—31no—
25C2×D3×#67—61no—
26C22×#624—31no—
27C22×D3×Dic3—61no—
28C2×C22⋊#157D311yescharacteristic, complemented
29C2×#3773—32no—
30#64233—38no—
31C2×D32⋊C4—32no—
32C23×C33D411yescharacteristic
33C22×C3×C3⋊D3—24no—
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3D411yescharacteristic
36C22×C3×C3⋊D3D412yes—
37C22×C3×C3⋊D3C2312yes—
38C22×C3×C3⋊D3C2311yescharacteristic
39C2×#1113—24no—
40C2×#1113—21no—
41C2×#1113C2312yes—
42C2×C3×#157—24no—
43C2×C3×#157C2314yes—
44C3×#638—24no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—24no—
47C2×C3×C3⋊Dic3—21no—
48#6295—24no—
49#6294—22no—
50#6294—24no—
51#6294—21no—
52#6294C2314yes—
53#6294C2312yes—
54C2×#886—91nomaximal
55C23×C3⋊D3C2×D311yescharacteristic
56C23×C3×D3—61no—
57C23×C3×D3—61no—
58C22×D32—31no—
59C22×D32—61no—
60C22×#157C2×D312yes—
61C2×#638—21no—
62C22×C3⋊Dic3—21no—
63C2×C3×#67—61no—
64C2×C3×#67—61no—
65C2×C3×#67—121no—
66C22×C3×Dic3—61no—
67C22×C3×Dic3—61no—
68C22×C3×Dic3—121no—
69#3811—34no—
70#3811—62no—
71D3×#67—64no—
72D3×#67—64no—
73C2×#626—61no—
74C2×#626—61no—
75C2×#625—61no—
76C2×#624—32no—
77C2×#624—34no—
78C2×#624—31no—
79C2×#624—61no—
80C2×#624—62no—
81C2×D3×Dic3—62no—
82C2×D3×Dic3—64no—
83C2×D3×Dic3—61no—
84C2×D3×Dic3—64no—
85C2×D3×Dic3—62no—
86C22⋊#157C2×D314yescomplemented
87#3773—38no—
88D32⋊C4—38no—
89C22×C33—22no—
90C22×C33C2×D412yes—
91C22×C33C2×D411yescharacteristic
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—24no—
94C2×C3×C3⋊D3—24no—
95C2×C3×C3⋊D3—22no—
96C2×C3×C3⋊D3—42no—
97C2×C3×C3⋊D3C2×D412yes—
98C2×C3×C3⋊D3C2×D411yescharacteristic
99C2×C3×C3⋊D3#4114yes—
100C2×C3×C3⋊D3C2411yescharacteristic
101#1113—24no—
102#1113—42no—
103C3×#157—24no—
104C3×C3⋊Dic3—24no—
105C3×C3⋊Dic3—22no—
106C22×#67—91no—
107C23×Dic3—91no—
108C2×D3×D4—181no—
109C22×C4×D3—181no—
110C2×C3×C22⋊C4—91no—
111C2×(C2×D3)⋊C4—92no—
112C2×C4⋊Dic3—92no—
113#886—98no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C23×C32—41no—
118C23×C32D1211yescharacteristic
119C22×C3⋊D3—24no—
120C22×C3⋊D3—21no—
121C22×C3⋊D3D1211yescharacteristic
122C22×C3⋊D3D1212yes—
123C22×C3⋊D3C22×D312yes—
124C22×C3⋊D3C22×D311yescharacteristic
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—62no—
128C22×C3×D3—61no—
129C22×C3×D3—62no—
130C22×C3×D3—61no—
131C22×C3×D3—62no—
132C22×C3×D3—62no—
133C22×C3×D3—68no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—124no—
137C22×C3×D3—121no—
138C2×D32—32no—
139C2×D32—62no—
140C2×D32—62no—
141C2×D32—61no—
142C2×D32—64no—
143C2×D32—124no—
144C2×#157—24no—
145C2×#157C22×D314yes—
146#638—24no—
147C2×C3⋊Dic3—22no—
148C2×C3⋊Dic3—24no—
149C2×C3⋊Dic3—21no—
150C3×#67—64no—
151C3×#67—64no—
152C3×#67—124no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—122no—
160C2×C3×Dic3—124no—
161C2×C3×Dic3—121no—
162#626—64no—
163#626—64no—
164#625—64no—
165#624—34no—
166#624—32no—
167#624—64no—
168#624—62no—
169#624—61no—
170D3×Dic3—64no—
171D3×Dic3—62no—
172D3×Dic3—64no—
173D3×Dic3—68no—
174D3×Dic3—62no—
175C2×#123—271noSylow
176C2×C33—22no—
177C2×C33#12312yescomplemented
178C2×C33C22×D411yescharacteristic
179C3×C3⋊D3—22no—
180C3×C3⋊D3—42no—
181C3×C3⋊D3#12312yescomplemented
182C3×C3⋊D3C2×#4112yes—
183C24×C3—91no—
184C23×D3—61no—
185C23×D3—61no—
186C23×D3—91no—
187C23×D3—181no—
188C2×C3×D4—181no—
189C22×C3×C4—92no—
190C22×C3×C4—181no—
191C2×#67—61no—
192C2×#67—61no—
193C2×#67—94no—
194C2×#67—121no—
195C2×#67—181no—
196C2×#67—181no—
197C2×#67—182no—
198C22×Dic3—61no—
199C22×Dic3—61no—
200C22×Dic3—94no—
201C22×Dic3—91no—
202C22×Dic3—92no—
203C22×Dic3—121no—
204C22×Dic3—182no—
205C22×Dic3—181no—
206D3×D4—184no—
207D3×D4—184no—
208C2×D12—181no—
209C2×C4×D3—182no—
210C2×C4×D3—184no—
211C2×C4×D3—181no—
212C2×C4×D3—184no—
213C2×C4×D3—182no—
214C3×C22⋊C4—94no—
215(C2×D3)⋊C4—98no—
216C4⋊Dic3—98no—
217C22×C32—24no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—42no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—44no—
229C22×C32—82no—
230C22×C32C2×D1212yes—
231C22×C32C2×D1211yescharacteristic
232C2×C3⋊D3—22no—
233C2×C3⋊D3—24no—
234C2×C3⋊D3—24no—
235C2×C3⋊D3—22no—
236C2×C3⋊D3—42no—
237C2×C3⋊D3C2×D1212yes—
238C2×C3⋊D3C2×D1211yescharacteristic
239C2×C3⋊D3#23714yes—
240C2×C3⋊D3C23×D311yescharacteristic
241C2×C3×D3—61no—
242C2×C3×D3—62no—
243C2×C3×D3—61no—
244C2×C3×D3—64no—
245C2×C3×D3—64no—
246C2×C3×D3—61no—
247C2×C3×D3—61no—
248C2×C3×D3—62no—
249C2×C3×D3—68no—
250C2×C3×D3—68no—
251C2×C3×D3—64no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—124no—
255C2×C3×D3—124no—
256C2×C3×D3—122no—
257C2×C3×D3—122no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—242no—
261D32—64no—
262D32—124no—
263D32—124no—
264D32—122no—
265#157—24no—
266C3⋊Dic3—24no—
267C3⋊Dic3—22no—
268C3×Dic3—64no—
269C3×Dic3—62no—
270C3×Dic3—64no—
271C3×Dic3—62no—
272C3×Dic3—124no—
273C3×Dic3—122no—
274C22×D4—271no—
275C23×C4—271no—
276#123—278no—
277C2×C4⋊C4—272no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—272no—
280C33C2×#12311yescharacteristic, complemented, Sylow
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—91no—
286C23×C3—92no—
287C23×C3—92no—
288C23×C3—91no—
289C23×C3—184no—
290C23×C3—181no—
291C23×C3GO+(4,𝔽2)11yescharacteristic
292C22×D3—62no—
293C22×D3—61no—
294C22×D3—61no—
295C22×D3—62no—
296C22×D3—61no—
297C22×D3—61no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3—68no—
301C22×D3—61no—
302C22×D3—61no—
303C22×D3—92no—
304C22×D3—124no—
305C22×D3—121no—
306C22×D3—182no—
307C22×D3—181no—
308C22×D3—182no—
309C22×D3—184no—
310C22×D3—364no—
311C3×D4—184no—
312C2×C3×C4—94no—
313C2×C3×C4—184no—
314C2×C3×C4—182no—
315C2×C3×C4—181no—
316C2×C3×C4—184no—
317#67—64no—
318#67—64no—
319#67—124no—
320#67—184no—
321#67—184no—
322C2×Dic3—64no—
323C2×Dic3—62no—
324C2×Dic3—61no—
325C2×Dic3—64no—
326C2×Dic3—62no—
327C2×Dic3—61no—
328C2×Dic3—92no—
329C2×Dic3—94no—
330C2×Dic3—124no—
331C2×Dic3—122no—
332C2×Dic3—121no—
333C2×Dic3—182no—
334C2×Dic3—184no—
335C2×Dic3—181no—
336D12—184no—
337C4×D3—184no—
338C4×D3—182no—
339C4×D3—184no—
340C4×D3—188no—
341C4×D3—182no—
342C2×C32—24no—
343C2×C32—22no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—42no—
349C2×C32—42no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—42no—
353C2×C32—44no—
354C2×C32—82no—
355C2×C32#88612yescomplemented
356C2×C32C22×D1211yescharacteristic
357C3⋊D3—22no—
358C3⋊D3—42no—
359C3⋊D3#88612yescomplemented
360C3⋊D3C2×#23712yes—
361C3×D3—62no—
362C3×D3—62no—
363C3×D3—62no—
364C3×D3—62no—
365C3×D3—64no—
366C3×D3—122no—
367C3×D3—122no—
368C3×D3—122no—
369C3×D3—242no—
370C24—91no—
371C24—271no—
372C2×D4—181no—
373C2×D4—274no—
374C2×D4—541no—
375C2×D4—542no—
376C22×C4—92no—
377C22×C4—181no—
378C22×C4—272no—
379C22×C4—271no—
380C22×C4—274no—
381C22×C4—542no—
382C22×C4—541no—
383C4⋊C4—278no—
384C22⋊C4—94no—
385C22⋊C4—278no—
386C22×C3—24no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—42no—
393C22×C3—44no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—82no—
399C22×C3—91no—
400C22×C3—91no—
401C22×C3—94no—
402C22×C3—92no—
403C22×C3—184no—
404C22×C3—184no—
405C22×C3—182no—
406C22×C3—182no—
407C22×C3—362no—
408C22×C3C2×GO+(4,𝔽2)12yes—
409C22×C3C2×GO+(4,𝔽2)11yescharacteristic
410C2×D3—61no—
411C2×D3—62no—
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—61no—
415C2×D3—64no—
416C2×D3—64no—
417C2×D3—62no—
418C2×D3—68no—
419C2×D3—68no—
420C2×D3—64no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—124no—
424C2×D3—124no—
425C2×D3—122no—
426C2×D3—122no—
427C2×D3—122no—
428C2×D3—122no—
429C2×D3—184no—
430C2×D3—242no—
431C2×D3—364no—
432C2×D3—364no—
433C2×D3—362no—
434C3×C4—184no—
435C3×C4—182no—
436C3×C4—184no—
437Dic3—64no—
438Dic3—62no—
439Dic3—64no—
440Dic3—62no—
441Dic3—124no—
442Dic3—122no—
443C32—21no—
444C32—21no—
445C32—41no—
446C32—41no—
447C32C2×#88611yescharacteristic, complemented
448C23—91no—
449C23—92no—
450C23—92no—
451C23—91no—
452C23—184no—
453C23—181no—
454C23—272no—
455C23—541no—
456C23—544no—
457C23#157⋊D311yescharacteristic
458D4—184no—
459D4—544no—
460C2×C4—94no—
461C2×C4—184no—
462C2×C4—184no—
463C2×C4—182no—
464C2×C4—181no—
465C2×C4—274no—
466C2×C4—272no—
467C2×C4—542no—
468C2×C4—544no—
469C2×C4—541no—
470C2×C3—24no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—22no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—42no—
477C2×C3—44no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—82no—
483C2×C3—92no—
484C2×C3—92no—
485C2×C3—182no—
486C2×C3—362no—
487C2×C3#6423312yescomplemented
488C2×C3C22×GO+(4,𝔽2)11yescharacteristic
489D3—62no—
490D3—62no—
491D3—62no—
492D3—62no—
493D3—64no—
494D3—122no—
495D3—122no—
496D3—122no—
497D3—242no—
498C22—22no—
499C22—94no—
500C22—91no—
501C22—91no—
502C22—92no—
503C22—184no—
504C22—184no—
505C22—182no—
506C22—182no—
507C22—362no—
508C22—544no—
509C22—1082no—
510C22C2×#157⋊D312yes—
511C22C2×#157⋊D311yescharacteristic
512C4—184no—
513C4—184no—
514C4—182no—
515C3—21no—
516C3—21no—
517C3—41no—
518C3—41no—
519C3C2×#6423311yescharacteristic, complemented
520C2—22no—
521C2—92no—
522C2—92no—
523C2—182no—
524C2—362no—
525C2#1172434612yesdirect product factor
526C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×#89049C2
hypercenterC23#157⋊D3
hypocenterC33C2×#123
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×#6294C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. C2×#88940C2semidirect
5. #11724346C2direct (N × H)
6. C2×#88939C2semidirect
20. C2×#6294C22semidirect
21. C3×C22⋊#157C22semidirect
22. #88940C22semidirect
23. #88939C22semidirect
28. C2×C22⋊#157D3semidirect
86. C22⋊#157C2×D3semidirect
177. C2×C33#123semidirect
181. C3×C3⋊D3#123semidirect
280. C33C2×#123semidirect
355. C2×C32#886semidirect
359. C3⋊D3#886semidirect
447. C32C2×#886semidirect
487. C2×C3#64233semidirect
519. C3C2×#64233semidirect
525. C2#11724346direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890492
C22×#62942
C2×C3×C22⋊#1572
C2×#889402
#117243462
C2×#889392
C2×#642333
C2×#8869