Finite Group Databasebeta

← C2×#11724119 #13180557

Subgroups — C2×#11724119 #13180557

Order 1728.

1128 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 148 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×#88957C211yesmaximal, characteristic, complemented
2C2×D32×Dic3C211yesmaximal, characteristic
3C2×#88940C211yesmaximal, characteristic, complemented
4C2×#88934C211yesmaximal, characteristic, complemented
5C2×D32⋊Dic3C211yesmaximal, characteristic
6C2×C3×D32⋊C4C211yesmaximal, characteristic
7C2×Dic3⋊#157C211yesmaximal, characteristic, complemented
8#11724119C218yesmaximal, direct product factor
9C2×D32⋊Q8—31nomaximal
10C22×C3×D32C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C2×#6294C2211yescharacteristic
14C2×#6294C2211yescharacteristic
15C2×#6293—21no—
16C2×Dic3×C3⋊D3C2211yescharacteristic
17C2×D3×C3⋊Dic3—21no—
18C2×C3×#627—21no—
19C2×C3×#624C2211yescharacteristic
20C2×C3×D3×Dic3—21no—
21#88957—22no—
22#88957C2214yescomplemented
23D32×Dic3—22no—
24D32×Dic3C2214yes—
25#88940C2214yescomplemented
26#88934C2214yescomplemented
27D32⋊Dic3C2214yes—
28C3×D32⋊C4C2214yes—
29Dic3⋊#157C2214yescomplemented
30C22×D3×Dic3—61no—
31C2×C4×D32—31no—
32C2×#3797—31no—
33C2×D3×Dic6—61no—
34C2×C4⋊#157—31no—
35C2×#3773—31no—
36C2×#3773—31no—
37C2×D32⋊C4—31no—
38C2×D32⋊C4D311yescharacteristic
39D32⋊Q8—38no—
40C22×C3×C3⋊D3C2311yescharacteristic
41C22×C32×D3—21no—
42C2×C3×D32—21no—
43C2×C3×D32—22no—
44C2×C3×D32Q814yescomplemented
45C2×C3×D32C2312yes—
46C2×C3⋊#157—22no—
47C2×C3⋊#157C2312yes—
48C2×C3×#157—22no—
49C2×C3×#157C2312yes—
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55#6294—21no—
56#6294—21no—
57#6294C2312yes—
58#6294C2312yes—
59#6293—24no—
60Dic3×C3⋊D3D412yes—
61Dic3×C3⋊D3C2312yes—
62D3×C3⋊Dic3—24no—
63C3×#627—24no—
64C3×#624—21no—
65C3×#624C2312yes—
66C3×D3×Dic3—24no—
67C2×#878—91nomaximal
68C23×C3×D3—61no—
69C22×D32C2×D311yescharacteristic
70C22×#157—31no—
71C22×#157C2×D311yescharacteristic
72C22×C3⋊Dic3—61no—
73C2×C4×C3⋊D3—31no—
74C2×#634—61no—
75C22×C3×Dic3—61no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6—61no—
80C2×#627—21no—
81C2×#627—61no—
82C2×#627—61no—
83C2×#627—121no—
84C2×#624—31no—
85C2×#624—31no—
86C2×#624C2×D311yescharacteristic
87C2×D3×Dic3—21no—
88C2×D3×Dic3—64no—
89C2×D3×Dic3—64no—
90C2×D3×Dic3—61no—
91C2×D3×Dic3—61no—
92C2×D3×Dic3—61no—
93C2×D3×Dic3—61no—
94C2×D3×Dic3—61no—
95C2×D3×Dic3—61no—
96C2×D3×Dic3—121no—
97C4×D32—34no—
98C4×D32—62no—
99#3797—34no—
100#3797—62no—
101D3×Dic6—64no—
102D3×Dic6—64no—
103C4⋊#157—34no—
104#3773—34no—
105#3773—34no—
106D32⋊C4—34no—
107D32⋊C4C2×D314yes—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3C2×D411yescharacteristic
110C2×C3×C3⋊D3C2×Q812yes—
111C2×C3×C3⋊D3#4112yes—
112C2×C3×C3⋊D3C2411yescharacteristic
113C2×C32×D3—24no—
114C2×C32×D3—22no—
115C3×D32—24no—
116C3×D32—22no—
117C3×D32C2×Q814yescomplemented
118C3⋊#157—22no—
119C3×#157—22no—
120C3⋊(C3⋊Dic3)C2×D412yes—
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3—22no—
123C32×Dic3—22no—
124C32×Dic3C2×D412yes—
125C23×Dic3—91no—
126C2×D3×Q8—181no—
127C22×C4×D3—181no—
128C22×Dic6—91no—
129C2×C3×C22⋊C4—91no—
130C2×#217—91no—
131C2×C4⋊Dic3—91no—
132C2×Dic3⋊C4—91no—
133C2×Dic3⋊C4—91no—
134#878—98no—
135C23×C32—61no—
136C22×C3⋊D3C22×D311yescharacteristic
137C22×C3×D3—21no—
138C22×C3×D3—21no—
139C22×C3×D3—41no—
140C22×C3×D3—64no—
141C22×C3×D3—62no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—64no—
145C22×C3×D3—62no—
146C2×D32—21no—
147C2×D32—22no—
148C2×D32Dic614yescomplemented
149C2×D32C22×D312yes—
150C2×#157—22no—
151C2×#157—32no—
152C2×#157—62no—
153C2×#157C22×D312yes—
154C2×C32×C4—31no—
155C2×C32×C4—61no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—64no—
162C2×C3⋊Dic3—121no—
163C2×C3⋊Dic3—121no—
164C4×C3⋊D3—32no—
165C4×C3⋊D3—32no—
166#634—64no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C2×C3×Dic3—121no—
179C3×C4×D3—62no—
180C3×C4×D3—62no—
181C3×C4×D3—64no—
182C3×Dic6—64no—
183C3×Dic6—64no—
184#627—24no—
185#627—64no—
186#627—64no—
187#627—124no—
188#624—21no—
189#624—32no—
190#624—32no—
191#624—61no—
192#624—61no—
193#624C22×D312yes—
194D3×Dic3—24no—
195D3×Dic3—62no—
196D3×Dic3—62no—
197D3×Dic3—62no—
198D3×Dic3—62no—
199D3×Dic3—62no—
200D3×Dic3—62no—
201D3×Dic3—62no—
202D3×Dic3—62no—
203D3×Dic3—64no—
204D3×Dic3—64no—
205D3×Dic3—124no—
206C2×C22⋊Q8—271noSylow
207C2×C33C22⋊Q812yescomplemented
208C2×C33C22×D411yescharacteristic
209C3×C3⋊D3C22⋊Q812yescomplemented
210C3×C3⋊D3C22×Q811yescharacteristic
211C3×C3⋊D3C2×#4111yescharacteristic
212C32×D3—24no—
213C24×C3—91no—
214C23×D3—61no—
215C2×C3×Q8—181no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—181no—
219C22×Dic3—61no—
220C22×Dic3—94no—
221C22×Dic3—91no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C22×Dic3—91no—
225C22×Dic3—182no—
226C22×Dic3—181no—
227C22×Dic3—181no—
228C22×Dic3—361no—
229D3×Q8—184no—
230D3×Q8—184no—
231C2×C4×D3—61no—
232C2×C4×D3—61no—
233C2×C4×D3—184no—
234C2×C4×D3—181no—
235C2×C4×D3—181no—
236C2×C4×D3—181no—
237C2×C4×D3—181no—
238C2×C4×D3—184no—
239C2×C4×D3—181no—
240C2×Dic6—61no—
241C2×Dic6—61no—
242C2×Dic6—94no—
243C2×Dic6—182no—
244C2×Dic6—181no—
245C2×Dic6—181no—
246C2×Dic6—181no—
247C2×Dic6—361no—
248C3×C22⋊C4—94no—
249#217—94no—
250C4⋊Dic3—94no—
251Dic3⋊C4—94no—
252Dic3⋊C4—94no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—62no—
258C22×C32—64no—
259C22×C32D3×D411yescharacteristic
260C2×C3⋊D3C2×Dic612yes—
261C2×C3⋊D3D3×D411yescharacteristic
262C2×C3⋊D3#23712yes—
263C2×C3⋊D3C23×D311yescharacteristic
264C2×C3×D3—22no—
265C2×C3×D3—24no—
266C2×C3×D3—24no—
267C2×C3×D3—22no—
268C2×C3×D3—42no—
269C2×C3×D3—44no—
270C2×C3×D3—64no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—61no—
278C2×C3×D3—61no—
279C2×C3×D3—68no—
280C2×C3×D3—64no—
281D32—24no—
282D32—22no—
283D32C2×Dic614yescomplemented
284#157—22no—
285#157—62no—
286C32×C4—32no—
287C32×C4—62no—
288C3⋊Dic3—22no—
289C3⋊Dic3—22no—
290C3⋊Dic3—32no—
291C3⋊Dic3—62no—
292C3⋊Dic3—62no—
293C3⋊Dic3—122no—
294C3⋊Dic3—122no—
295C3×Dic3—22no—
296C3×Dic3—22no—
297C3×Dic3—22no—
298C3×Dic3—22no—
299C3×Dic3—42no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—122no—
305C3×Dic3—122no—
306C22×Q8—271no—
307C23×C4—271no—
308C22⋊Q8—278no—
309C2×C4⋊C4—271no—
310C2×C4⋊C4—271no—
311C2×C4⋊C4—271no—
312C2×C22⋊C4—91no—
313C2×C22⋊C4—271no—
314C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
315C23×C3—61no—
316C23×C3—61no—
317C23×C3—92no—
318C23×C3—94no—
319C23×C3—91no—
320C23×C3—121no—
321C23×C3—181no—
322C23×C3—182no—
323C22×D3—21no—
324C22×D3—64no—
325C22×D3—62no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—62no—
329C22×D3—64no—
330C3×Q8—184no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—121no—
338C2×C3×C4—181no—
339C2×C3×C4—184no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4—182no—
343C2×Dic3—21no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—64no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—91no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—92no—
355C2×Dic3—121no—
356C2×Dic3—121no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—184no—
362C2×Dic3—184no—
363C2×Dic3—182no—
364C2×Dic3—361no—
365C2×Dic3—361no—
366C2×Dic3—364no—
367C2×Dic3GO+(4,𝔽2)11yescharacteristic
368C4×D3—62no—
369C4×D3—62no—
370C4×D3—64no—
371C4×D3—182no—
372C4×D3—182no—
373C4×D3—182no—
374C4×D3—182no—
375C4×D3—182no—
376C4×D3—182no—
377C4×D3—182no—
378C4×D3—182no—
379C4×D3—184no—
380Dic6—64no—
381Dic6—64no—
382Dic6—184no—
383Dic6—184no—
384Dic6—184no—
385Dic6—364no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—22no—
389C2×C32—22no—
390C2×C32—42no—
391C2×C32—42no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32—64no—
395C2×C32#87812yescomplemented
396C2×C32C2×D3×D411yescharacteristic
397C3⋊D3#87812yescomplemented
398C3⋊D3C22×Dic611yescharacteristic
399C3⋊D3C2×#23711yescharacteristic
400C3×D3—24no—
401C3×D3—24no—
402C3×D3—44no—
403C3×D3—62no—
404C3×D3—62no—
405C3×D3—61no—
406C3×D3—61no—
407C3×D3—61no—
408C3×D3—61no—
409C24—91no—
410C2×Q8—181no—
411C2×Q8—274no—
412C2×Q8—541no—
413C2×Q8—542no—
414C22×C4—91no—
415C22×C4—91no—
416C22×C4—181no—
417C22×C4—271no—
418C22×C4—271no—
419C22×C4—271no—
420C22×C4—274no—
421C22×C4—271no—
422C22×C4—542no—
423C22×C4—541no—
424C4⋊C4—274no—
425C4⋊C4—274no—
426C4⋊C4—274no—
427C22⋊C4—94no—
428C22⋊C4—274no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—62no—
434C22×C3—64no—
435C22×C3—62no—
436C22×C3—64no—
437C22×C3—94no—
438C22×C3—91no—
439C22×C3—91no—
440C22×C3—92no—
441C22×C3—92no—
442C22×C3—122no—
443C22×C3—124no—
444C22×C3—182no—
445C22×C3—184no—
446C22×C3C2×GO+(4,𝔽2)11yescharacteristic
447C2×D3—22no—
448C2×D3—24no—
449C2×D3—64no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—68no—
459C2×D3—64no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—62no—
464C3×C4—122no—
465C3×C4—182no—
466C3×C4—182no—
467C3×C4—182no—
468Dic3—22no—
469Dic3—62no—
470Dic3—62no—
471Dic3—62no—
472Dic3—62no—
473Dic3—92no—
474Dic3—122no—
475Dic3—122no—
476Dic3—182no—
477Dic3—182no—
478Dic3—182no—
479Dic3—362no—
480Dic3—362no—
481Dic3C2×GO+(4,𝔽2)12yes—
482C32—21no—
483C32—21no—
484C32—41no—
485C32—41no—
486C32C2×#87811yescharacteristic, complemented
487C23—61no—
488C23—92no—
489C23—94no—
490C23—91no—
491C23—181no—
492C23—182no—
493Q8—184no—
494Q8—544no—
495C2×C4—31no—
496C2×C4—61no—
497C2×C4—92no—
498C2×C4—92no—
499C2×C4—184no—
500C2×C4—181no—
501C2×C4—182no—
502C2×C4—181no—
503C2×C4—181no—
504C2×C4—272no—
505C2×C4—272no—
506C2×C4—272no—
507C2×C4—272no—
508C2×C4—271no—
509C2×C4—272no—
510C2×C4—541no—
511C2×C4—541no—
512C2×C4—544no—
513C2×C4—542no—
514C2×C3—22no—
515C2×C3—22no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—42no—
519C2×C3—42no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—64no—
523C2×C3—64no—
524C2×C3—91no—
525C2×C3—91no—
526C2×C3—92no—
527C2×C3—124no—
528C2×C3D32⋊Q812yescomplemented
529C2×C3C22×GO+(4,𝔽2)11yescharacteristic
530D3—24no—
531D3—62no—
532D3—61no—
533D3—61no—
534D3—62no—
535D3—61no—
536D3—61no—
537C22—64no—
538C22—62no—
539C22—94no—
540C22—92no—
541C22—92no—
542C22—91no—
543C22—91no—
544C22—182no—
545C22—184no—
546C22D3×GO+(4,𝔽2)11yescharacteristic
547C4—32no—
548C4—62no—
549C4—182no—
550C4—182no—
551C4—182no—
552C4—272no—
553C4—542no—
554C3—21no—
555C3—21no—
556C3—41no—
557C3—41no—
558C3C2×D32⋊Q811yescharacteristic, complemented
559C2—64no—
560C2—91no—
561C2—91no—
562C2—92no—
563C2#1172411912yesdirect product factor
564C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
prime radicalC22×C3×D32C22
hypocenterC33C2×C22⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88957C2semidirect
3. C2×#88940C2semidirect
4. C2×#88934C2semidirect
7. C2×Dic3⋊#157C2semidirect
8. #11724119C2direct (N × H)
22. #88957C22semidirect
25. #88940C22semidirect
26. #88934C22semidirect
29. Dic3⋊#157C22semidirect
44. C2×C3×D32Q8semidirect
117. C3×D32C2×Q8semidirect
148. C2×D32Dic6semidirect
207. C2×C33C22⋊Q8semidirect
209. C3×C3⋊D3C22⋊Q8semidirect
283. D32C2×Dic6semidirect
314. C33C2×C22⋊Q8semidirect
395. C2×C32#878semidirect
397. C3⋊D3#878semidirect
486. C32C2×#878semidirect
528. C2×C3D32⋊Q8semidirect
558. C3C2×D32⋊Q8semidirect
563. C2#11724119direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889572
C2×D32×Dic32
C2×#889402
C2×#889342
C2×D32⋊Dic32
C2×C3×D32⋊C42
C2×Dic3⋊#1572
#117241192
C2×D32⋊Q83
C2×#8789